1-manifolds

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1 Introduction

According to the general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., its topological structure has a countable base), satisfies the Hausdorff axiom (any two different points have disjoint neighborhoods) and each point of which has a neighbourhood homeomorphic either to the real line $\Rr$${{Stub}} == Introduction == <wikitex>; According to the general definition of manifold, a manifold of dimension 1 is a topological space which is [[Wikipedia:Second_countable|second countable]] (i.e., its topological structure has a countable base), satisfies the [[Wikipedia:Hausdorff_space|Hausdorff axiom]] (any two different points have disjoint neighborhoods) and each point of which has a neighbourhood homeomorphic either to the [[Wikipedia:Real_line|real line]] $\Rr $ or to the half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$. Manifolds of dimension 1 are called [[Wikipedia:Curve|''curves'']], but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term ''curve'' may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$. To be on the safe side, we use an unambiguous term ''manifold of dimension 1'' or ''1-manifold''. For other expositions about \Rr$ or to the half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$$\Rr_+=\{x\in\Rr\mid x\ge0\}$.

Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$$\Rr^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

For other expositions about $1$$1$-manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].

2 Examples

• The real line: $\mathbb R$$\mathbb R$

• The half-line: $\mathbb R_+$$\mathbb R_+$

• The circle: $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$

• The closed interval: $I=[0,1]$$I=[0,1]$

3 Topological classification

3.1 Reduction to classification of connected manifolds

The following elementary facts hold for $n$$n$-manifolds of any dimension $n$$n$.

Any manifold is homeomorphic to the disjoint sum of its connected components.

A connected component of an $n$$n$-manifold is a $n$$n$-manifold.

Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.

3.2 Topological classification of connected 1-manifolds

Theorem 3.1. Any connected 1-manifold is homeomorphic to one of the following 4 manifolds:

1. real line $\mathbb R$$\mathbb R$
2. half-line $\mathbb R_+$$\mathbb R_+$
3. circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$
4. closed interval $I=[0,1]$$I=[0,1]$.

No two of these manifolds are homeomorphic to each other.

3.3 Characterizing the topological type of a connected 1-manifold

Theorem 3.2.

1. Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$$\Rr$.
2. Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$$\Rr_+$.
3. Any connected closed 1-manifold is homeomorphic to $S^1$$S^1$.
4. Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$$I$.

Thus, for connected 1-manifolds, two invariants, compactness and presence of boundary, form a complete system of topological invariants. Each of the invariants takes two values.

Theorems 3.1 and 3.2 above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result.

3.4 About proofs of the classification theorems

The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:

Lemma 3.3. Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$$\Rr$ is homeomorphic either to $\Rr$$\Rr$ or $S^1$$S^1$.

Lemma 3.4. If a topological space $X$$X$ can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to $\Rr$$\Rr$, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.

3.5 Corollaries: homotopy classification

Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.

It follows immediately from Theorem 3.1.

3.6 Corollaries: cobordisms of 0-manifolds

Theorem 3.6. A compact 0-manifold $X$$X$ bounds a compact 1-manifold iff the number of points in $X$$X$ is even.

Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.

4 Orders and orientations

4.1 Interval topology

Most of properties specific for 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.

Open intervals $(a,b)=\{x\in\Rr\mid a<x<b\}$$(a,b)=\{x\in\Rr\mid a<x<b\}$ and open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$ form the base of the (standard) topology on $\Rr$$\Rr$. On $\Rr_+$$\Rr_+$ and $I$$I$, the (standard) topology is induced from the standard topology on $\Rr$$\Rr$, and can be described in terms of the induced order.

This way of introducing a topological structure can be applied in any (linearly) ordered set $X$$X$.

On $S^1$$S^1$ the topology is defined by the standard cyclic order. A cyclic order on a set $X$$X$ is usually defined as a ternary relation on $X$$X$. It can be described also by fixing a linear order $x<_ay$$x<_ay$ on $X\smallsetminus a$$X\smallsetminus a$ for each $a\in X$$a\in X$. The linear orders on $X\smallsetminus a$$X\smallsetminus a$ and $X\smallsetminus b$$X\smallsetminus b$ describe the same cyclic order on $X$$X$ if $b<_ax\Longleftrightarrow x<_ba$$b<_ax\Longleftrightarrow x<_ba$ for each $x\in X\smallsetminus\{a,b\}$$x\in X\smallsetminus\{a,b\}$. Thus all the orders $<_a$$<_a$ can be recovered form any one of them, and hence the cyclic order is determined by any one of the orders $<_a$$<_a$.

Open rays in $S^1\smallsetminus a$$S^1\smallsetminus a$ for all $a\in S^1$$a\in S^1$ form the base of the standard topology on $S^1$$S^1$. The same way of introducing a topological structure can be applied in any cyclically ordered set.

Topological structure of either kind, on a linearly ordered or cyclically ordered set, desrcibed above will be called interval topology.

4.2 Orientations

An orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.

An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.

Theorem 4.1. Every connected non-closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to $\Rr$$\Rr$, or $\Rr_+$$\Rr_+$, or $I$$I$. On each of these 1-manifolds there are two linear orders, $<$$<$ and $>$$>$, defining the topology. We have to prove that there is no other linear order defining the same topology.

A linear order $<$$<$ on a set $X$$X$ is encoded in the system of sets $X_a=\{x\in X\mid a<x\}$$X_a=\{x\in X\mid a<x\}$ for $a\in X$$a\in X$. A system of susbsets $X_a\subset X\smallsetminus a$$X_a\subset X\smallsetminus a$ with $a\in X$$a\in X$ comes from a linear order on $X$$X$ iff either $X_a\subset X_b$$X_a\subset X_b$, or $X_a\supset X_b$$X_a\supset X_b$, or $X_a=X_b$$X_a=X_b$ for any $a,b\in X$$a,b\in X$ and $X_a=X_b$$X_a=X_b$ only if $a=b$$a=b$. In the interval topology on $X$$X$, each $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ is open.

Each interior point $a$$a$ of a connected non-closed 1-manifold $X$$X$ splits $X$$X$ to two open connected sets. Any linear order on $X$$X$ defining the topology of $X$$X$ would give rise to splitting of each of these two sets into its intesections with $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ which are disjoint and open. Due to connectedness of the set, it coincides with one of these intersections. Hence the order coincides with one of two standard orders.

$\square$$\square$

An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.

Theorem 4.2. Every connected closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected closed 1-manifold is homeomorphic either to $S^1$$S^1$. On $S^1$$S^1$ there are cyclic orders opposite to each other that define the topology. We have to prove that there is no other cyclic order defining the same topology.

As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.

$\square$$\square$

An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).

Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of $n$$n$ connected components, then it admits $2^n$$2^n$ orientations.

4.3 Self-homeomorphisms

Theorem 4.4. A map $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a homeomorphism iff $h$$h$ is a monotone bijection.

Proof. Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a homeomorphism. First, observe that $h$$h$ maps every ray to a ray. Indeed, for any $x\in\Rr$$x\in\Rr$, the map $h$$h$ induces a homeomorphism $\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$$\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$. The rays $(-\infty,x)$$(-\infty,x)$ and $(x,\infty)$$(x,\infty)$ are connected components of $\Rr\smallsetminus x$$\Rr\smallsetminus x$. Therefore their images are connected components $(-\infty,h(x))$$(-\infty,h(x))$ and $(h(x),\infty)$$(h(x),\infty)$ of $\Rr\smallsetminus h(x)$$\Rr\smallsetminus h(x)$.

Observe that rays have the same direction iff one of them is contained in the other one, that is their intersection is a ray. Therefore two rays of the same dirction are mapped by $h$$h$ to rays with the same direction. Thus rays $(x,+\infty)$$(x,+\infty)$ are mapped either all to rays $(h(x),+\infty)$$(h(x),+\infty)$ or all to $(-\infty,h(x))$$(-\infty,h(x))$. Thus $h$$h$ is monotone.

Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a monotone bijection. Then the image and preimage under $h$$h$ of any open interval is an open interval. Therefore, both $h$$h$ and $h^{-1}$$h^{-1}$ are continuous, and hence $h$$h$ is a homeomorphism.

$\square$$\square$

The following theorem can be proved similarly or can be deduced from Theorem 4.4

Theorem 4.5.

1. A map $h:I\to I$$h:I\to I$ is a homeomorphism iff $h$$h$ is a monotone bijection.
2. A map $h:\Rr_+\to\Rr_+$$h:\Rr_+\to\Rr_+$ is a homeomorphism iff $h$$h$ is a monotone increasing bijection.
3. A map $h:S^1\to S^1$$h:S^1\to S^1$ is a homeomorphism iff $h$$h$ is a bijection that either preserves or reverses the cyclic order of points on $S^1$$S^1$.

Any self-homeomorphism of a connected 1-manifold either preserve orientation, or reverses it. The half-line $\Rr_+$$\Rr_+$ does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\Rr_+$ admits an orientation reversing map.

Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of $\Rr_+$$\Rr_+$ and $\Rr_-$$\Rr_-$ with the orientations induced by the standard order.

4.4 Characterizations of connected compact 1-manifolds in terms of separating points

A subset $A$$A$ of a topological space $X$$X$ is said to separate $X$$X$ if $X\smallsetminus A$$X\smallsetminus A$ can be presented as a union of two disjoint open sets.

Theorem 4.6. (See [Moor1920].) Let $X$$X$ be a connected compact Hausdorff second countable topological space.

1. If every two points separate $X$$X$, then $X$$X$ is homeomorphic to the circle.
2. If each point, with two exceptions, separates $X$$X$, then $X$$X$ is homeomorphic to $I$$I$.

Any point $a\in\Rr$$a\in\Rr$ splits $\Rr$$\Rr$ to two disjoint open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$.

Theorem 4.7. (See [Ward1936].) Let $X$$X$ be a connected locally compact Hausdorff second countable topological space.

1. If the complement of each point in $X$$X$ consists of two connected components, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.
2. If $X$$X$ contains a point

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   $b$ such that $X\smallsetminus b$$X\smallsetminus b$ is connected and $X\smallsetminus a$$X\smallsetminus a$ consists of two connected components for each $a\in X$$a\in X$, $a\ne b$$a\ne b$, then $X$$X$ is homeomorphic to $\Rr_+$$\Rr_+$.

5 Invariants

5.1 Basic invariants

As follows from the Theorems 3.1 and 3.2 above, the following invariants

• the number of connected components,
• the compactness of each connected component,
• and the number of boundary points of each connected component

determine the topological type of a 1-manifold.

5.2 Homotopy invariants

The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial. The fundamental group $\pi_1(X,x_0)$$\pi_1(X,x_0)$ is infinite cyclic group, if the connected component of $X$$X$ containing $x_0$$x_0$ is homeomorphic to circle, and trivial otherwise.

5.3 Homology invariants

Let $X$$X$ be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union

of some numbers of copies of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, $S^1$$S^1$ and $I$$I$:

$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$ $$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$$

Then $H_0(X)$$H_0(X)$ is a free abelian group of rank equal to the number $a+b+c+d$$a+b+c+d$ of all connected components of $X$$X$ and $H_1(X)$$H_1(X)$ is a free abelian group of rank equal to the number $c$$c$ of closed (compact without boundary) components of $X$$X$.

Relative homology groups: $H_0(X,\partial X)$$H_0(X,\partial X)$ is a free abelian group of rank equal to the number $a+c$$a+c$ of connected components of $X$$X$ without boundary; $H_1(X,\partial X)$$H_1(X,\partial X)$ is a free abelian group of rank equal to the number $c+d$$c+d$ of compact components of $X$$X$. So,

$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$ $$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$$

Numbers $a$$a$,

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$b$, $c$$c$, and $d$$d$ and the topological type of $X$$X$ can be recovered from the ranks of these groups.

Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):

$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$ $$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$$

The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So

$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$ $$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$$

A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over $S^1$$S^1$ has non-trivial monodromy, then all the homology groups are trivial.

5.4 Tangent bundle invariants

The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.

6 Additional structures

6.1 Triangulations

Any 1-manifold admits a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism.

A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$$\ge3$. Similarly, the topological type of a triangulation of $I$$I$ is defined by the number of 1-simplices, which can take any positive integral value.

6.2 Smooth structures

Any 1-manifold admits a smooth structure.

If smooth 1-manifolds $X$$X$ and $Y$$Y$ are homeomorphic, then they are also diffeomorphic. Moreover,

Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.

$\square$$\square$

6.3 Inner metrics

Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$$s:I\to X$ in a metric space $X$$X$ with metric $d:X\times X\to \Rr_+$$d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$$\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$.

Any connected 1-manifold admits an inner metric. A connected 1-manifold with an inner metric is defined up to isometry by the diameter of the space. Recall that the diameter of a metric space $X$$X$ with metric $d:X\times X\to\Rr$$d:X\times X\to\Rr$ is $\sup\{ d(x,y)\mid x,y\in X\}$$\sup\{ d(x,y)\mid x,y\in X\}$.

For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.

1. For $\Rr$$\Rr$ with diameter $D\in (0,\infty]$$D\in (0,\infty]$ this is $(-D/2,D/2)$$(-D/2,D/2)$.
2. For $\Rr_+$$\Rr_+$ with diameter $D\in(0,\infty]$$D\in(0,\infty]$ this is $[0,D)$$[0,D)$.
3. For a circle with inner metric of diameter $D\in (0,\infty)$$D\in (0,\infty)$ this is the circle $\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$$\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$ of radius $D/\pi$$D/\pi$ on the plane with the inner metric.
4. For $I$$I$ with diameter $D\in(0,\infty)$$D\in(0,\infty)$ this is $[0,D]$$[0,D]$.

An inner metric on a connected 1-manifold defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above.

7 Constructions

7.1 Surgery

Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.

If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).

An index 1 surgery preserving orientation on closed 1-manifold changes the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components.

7.2 Connected sums

The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$$\Rr$ is a disjoint sum of two copies of $\Rr$$\Rr$.

Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example

$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$ $$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$$

8 Groups of self-homeomorphisms

8.1 Mapping class groups

Recall that the mapping class group of a manifold $X$$X$ is the quotient group of the group $\operatorname{Homeo} (X)$$\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$$X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$$X$ is $\pi_0(\operatorname{Homeo}(X))$$\pi_0(\operatorname{Homeo}(X))$.

An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:

Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.

This is a corollary of the following two obvious lemmas.

Lemma 8.2. On rectilinear isotopy. Let $X$$X$ be one of the following 1-manifolds: $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, or $I$$I$. Let $f,g:X\to X$$f,g:X\to X$ be two monotone bijections that are either both increasing or both decreasing. Then the family $h_t=(1-t)f+tg:X\to X$$h_t=(1-t)f+tg:X\to X$ with $t\in[0,1]$$t\in[0,1]$ consists of monotone bijections (and hence is an isotopy between $f$$f$ and $g$$g$).

Lemma 8.3. Let $f,g:S^1\to S^1$$f,g:S^1\to S^1$ be two bijections that either both preserve or both reverse the standard cyclic order of points on $S^1$$S^1$. Let $f$$f$ and $g$$g$ coincide at $x\in S^1$$x\in S^1$. Then $f$$f$ and $g$$g$ are isotopic via the canonical isotopy which is stationary at $x$$x$ and is provided on the complement of $x$$x$ by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of $\Rr$$\Rr$.

Corollary 8.4. $\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$$\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$ and $\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$$\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$

Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and $\operatorname{Homeo}$$\operatorname{Homeo}$ is replaced by $\operatorname{Diffeo}$$\operatorname{Diffeo}$.

8.2 Homotopy types of groups of self-homeomorphisms

The group $\operatorname{Homeo}(S^1)$$\operatorname{Homeo}(S^1)$ contains $O(2)$$O(2)$ as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point $x\in S^1$$x\in S^1$, Lemma 8.3 provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$$\operatorname{Homeo}(S^1)\to O(2)$.

Similarly, the group of self-homeomorphisms of $S^1$$S^1$ isotopic to identity contains $SO(2)=S^1$$SO(2)=S^1$ as a subgroup, which is its deformation retract.

The groups of self-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to $f=\operatorname{id}$$f=\operatorname{id}$ and an arbitrary $g$$g$.

Thus for each connected 1-manifold $X$$X$ the group of homeomorphisms $X \to X$$X \to X$ isotopic to identity is homotopy equivalent to $X$$X$.

9 Finite group actions

Consider an action of a finite group $G$$G$ on a 1-manifold $X$$X$.

9.1 Free actions

For any point $a\in X$$a\in X$, its orbit $Ga\subset X$$Ga\subset X$ is a finite set and has an invariant neighborhood $U$$U$ whose connected components are disjoint open sets, each of them contains exactly one point of $Ga$$Ga$, and either all the components are homeomorphic to $\Rr$$\Rr$, or all homeomorphic to $\Rr_+$$\Rr_+$.

If the action is free, then the orbit space $X/G$$X/G$ is a 1-manifold and the natural projection $X\to X/G$$X\to X/G$ is a covering.

Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.

A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold $X$$X$ has a contractible orbit space $Y=X/G$$Y=X/G$, then $X$$X$ is a disjoint union of copies of $Y$$Y$ and $G$$G$ permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.

Coverings $X\to S^1$$X\to S^1$ with connected $X$$X$ are in one-to-one correspondence with subgroups of finite indices of $\pi_1(S^1)=\mathbb Z$$\pi_1(S^1)=\mathbb Z$. For each index $m\in\mathbb Z$$m\in\mathbb Z$ there is one subgroup, and hence one covering. The total space is homeomorphic to $S^1$$S^1$, and the covering is equivalent to $S^1\to S^1:z\mapsto z^m$$S^1\to S^1:z\mapsto z^m$. In the corresponding action, the group is cyclic of order $m$$m$, it acts on $S^1$$S^1$ by rotations.

In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.

Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.

Theorem 9.2. If a finite group $G$$G$ acts freely on the circle than $G$$G$ is cyclic. Any finite cyclic group has a linear free action on $S^1$$S^1$. Any free action of a finite cyclic group on $S^1$$S^1$ is conjugate to a linear action.

9.2 Asymmetry of a half-line

Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.

Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).

Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.

$\square$$\square$

</wikitex>

9.3 Actions on line and segment

Theorem 9.4. The only orientation preserving homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is the identity.

Proof. Fix a homeomorphism $f:(0,\infty)\to\Rr$$f:(0,\infty)\to\Rr$ (say, define it by formula $f:x\mapsto (x^2- 1)/x$$f:x\mapsto (x^2- 1)/x$). Consider a homeomorphism $(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$$(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$. It preserves orientation (since $h$$h$ preserves orientation). So, it is a monotone increasing bijection $(0,\infty)\to(0,\infty)$$(0,\infty)\to(0,\infty)$ of finite order. It can be extended to $\Rr_+$$\Rr_+$ by letting $0\mapsto0$$0\mapsto0$. The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.

$\square$$\square$

Theorem 9.5. Any orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is of order two. It is conjugate to the symmetry against a point.

Proof. An orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a monotone decreasing bijection. Consider the function $x\mapsto h(x)-x$$x\mapsto h(x)-x$. It is also monotone decreasing bijection $\Rr\to\Rr$$\Rr\to\Rr$ and hence there exists a unique $a\in\Rr$$a\in\Rr$ such that $h(a)-a=0$$h(a)-a=0$, that is $h(a)=a$$h(a)=a$.

The homeomorphism $h$$h$ maps each connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$ to a connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$. The connected components are open rays $(-\infty,a)$$(-\infty,a)$ and $(a,\infty)$$(a,\infty)$. If each of them is mapped to itself, then $h$$h$ defines a homeomorphism of a finite order of the closed rays $(-\infty,a]$$(-\infty,a]$ and $[a,\infty)$$[a,\infty)$. Then by Theorem 9.3, $h$$h$ is identity, which contradicts to our assumption. Thus, $h([a,\infty))=(-\infty,a]$$h([a,\infty))=(-\infty,a]$ and $h((-\infty,a])=[a,\infty)$$h((-\infty,a])=[a,\infty)$. Then $h^2$$h^2$ preserves the rays, and, by Theorem 9.3, is identity. Thus $h$$h$ has order two.

Choose a homeomorphism $f:[0,\infty)\to[a,\infty)$$f:[0,\infty)\to[a,\infty)$. Define function $g:(-infty,0]\to(-\infty,a]$$g:(-infty,0]\to(-\infty,a]$ by formula $g(x)=hf(-x)$$g(x)=hf(-x)$. It's a homeomorphism. Together, $f$$f$ and $g$$g$ form a homeomorphism $\phi:\Rr\to\Rr$$\phi:\Rr\to\Rr$. As easy to check, $\phi^{-1}h\phi(x)=-x$$\phi^{-1}h\phi(x)=-x$.

$\square$$\square$

Theorem 9.6. A non-trivial finite group acting effectively on $\Rr$$\Rr$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, $f$$f$ and $g$$g$ of the line $\Rr$$\Rr$. Their composition $f\circ g$$f\circ g$ preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, $fg=1$$fg=1$ and hence $f=g^{-1}$$f=g^{-1}$. But $g^2=1$$g^2=1$. Therefore $g^{-1}=g$$g^{-1}=g$ and $f=g^{-1}=g$$f=g^{-1}=g$.

$\square$$\square$

Corollary 9.7. A non-trivial finite group acting effectively on $I$$I$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. Any auto-homeomorphism of $I$$I$ preserves the boundary and the interior of $I$$I$. Hence an effective finite group action on $I$$I$ induces an action of finite group on the interior of $I$$I$. An auto-homeomorphism of $I$$I$ is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of $\operatorname{Int}I$$\operatorname{Int}I$ has a unique extension to auto-homeomorphism of $I$$I$. The interior $\operatorname{Int}I$$\operatorname{Int}I$ is homeomorphic to $\Rr$$\Rr$.

$\square$$\square$

9.4 Actions on circle

{{beginthm|Theorem} Any periodic orientation reversing homeomorphism $S^1\to S^1$$S^1\to S^1$ is an involution (i.e., has period 2). It is conjugate to a symmetry of $S^1$$S^1$ against its diameter. </div>

Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.

Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.

$\square$$\square$

Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

Theorem 9.8. A periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ has no fixed point. It is conjugate to a rotation.

Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.

For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ have no fixed points. Therefore, these powers form a cyclic group freely acting on $S^1$$S^1$. See Theorem 9.2.

$\square$$\square$

Theorem 9.9. A finite group acting effectively on $S^1$$S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons.

Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.

Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group $D_2$$D_2$. If the number of orientation preserving homeomorphisms is $n>2$$n>2$, then the whole group is called the dihedral group $D_n$$D_n$. It is the symmetry group of an $n$$n$-sided regular polygon.

$\square$$\square$

10 Relatives of 1-manifolds

10.1 Non-Hausdorff 1-manifolds

If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable.

Indeed, one can take the disjoint union of two copies of the line $\Rr$$\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.

11 References

• [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
• [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
• [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
• [Moor1920] R.L.Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), 333—347.
• [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02

12 External links

• The Encylopedia of Mathematics article on one-dimensional manifolds.
• The Encylopedia of Mathematics article on lines.
• The Wikipedia page about curves.

$-manifolds, see \cite{Ghys2001}, \cite{Gale1987} and also \cite{Fuks&Rokhlin1984|Sections 3.1.1.16-19}. == Examples == ; * The real line: $\mathbb R$ * The half-line: $\mathbb R_+$ * The circle: $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$ * The closed interval: $I=[0,1]$ == Topological classification == ==== Reduction to classification of connected manifolds ==== ; The following elementary facts hold for $n$-manifolds of any dimension $n$. Any manifold is homeomorphic to the disjoint sum of its connected components. A connected component of an $n$-manifold is a $n$-manifold. Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic. ==== Topological classification of connected 1-manifolds ==== ; {{beginthm|Theorem}} \label{thm:classification} Any connected 1-manifold is homeomorphic to one of the following 4 manifolds: # real line $\mathbb R$ # half-line $\mathbb R_+$ # circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$ # closed interval $I=[0,1]$. No two of these manifolds are homeomorphic to each other. {{endthm}} ==== Characterizing the topological type of a connected 1-manifold ==== ; {{beginthm|Theorem}} \label{thm:characterisation} # Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$. # Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$. # Any connected closed 1-manifold is homeomorphic to $S^1$. # Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$. {{endthm}} Thus, for connected 1-manifolds, two invariants, compactness and presence of boundary, form a complete system of topological invariants. Each of the invariants takes two values. Theorems \ref{thm:classification} and \ref{thm:characterisation} above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result. ==== About proofs of the classification theorems ==== ; The proofs of Theorems \ref{thm:classification} and \ref{thm:characterisation} above are elementary. They can be found, e.g., in \cite{Fuks&Rokhlin1984| Sections 3.1.1.16-19}. The core of them are the following simple lemmas: {{beginthm|Lemma}} Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$ is homeomorphic either to $\Rr$ or $S^1$.{{endthm}} {{beginthm|Lemma}} If a topological space $X$ can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to $\Rr$, then $X$ is homeomorphic to $\Rr$. {{endthm}} ==== Corollaries: homotopy classification ==== ; {{beginthm|Theorem}} Each connected 1-manifold is either contractible, or homotopy equivalent to circle. {{endthm}} It follows immediately from Theorem \ref{thm:classification}. ==== Corollaries: cobordisms of 0-manifolds ==== ; {{beginthm|Theorem}} A compact 0-manifold $X$ bounds a compact 1-manifold iff the number of points in $X$ is even. {{endthm}} {{beginthm|Corollary}} Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2. {{endthm}} == Orders and orientations == ==== Interval topology ==== ; Most of properties specific for 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points. Open intervals $(a,b)=\{x\in\Rr\mid a ==== Orientations ==== ; An [[Orientation_of_manifolds|orientation]] of a 1-manifold can be interpreted via linear or cyclic orderings of their points. An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold. {{beginthm|Theorem}}\label{thm:orientations-on-non-closed} Every connected non-closed 1-manifold admits exactly two orientations. {{endthm}} {{beginproof}} By Theorem \ref{thm:characterisation}, a coonected non-closed 1-manifold is homeomorphic either to $\Rr$, or $\Rr_+$, or $I$. On each of these 1-manifolds there are two linear orders, $<$ and $>$, defining the topology. We have to prove that there is no other linear order defining the same topology. A linear order $<$ on a set $X$ is encoded in the system of sets $X_a=\{x\in X\mid a ==== Self-homeomorphisms ==== ; {{beginthm|Theorem}}\label{thm:homeomorphisms-of-line} A map $h:\Rr\to\Rr$ is a homeomorphism iff $h$ is a monotone bijection. {{endthm}} {{beginproof}} Let $h:\Rr\to\Rr$ be a homeomorphism. First, observe that $h$ maps every ray to a ray. Indeed, for any $x\in\Rr$, the map $h$ induces a homeomorphism $\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$. The rays $(-\infty,x)$ and $(x,\infty)$ are connected components of $\Rr\smallsetminus x$. Therefore their images are connected components $(-\infty,h(x))$ and $(h(x),\infty)$ of $\Rr\smallsetminus h(x)$. Observe that rays have the same direction iff one of them is contained in the other one, that is their intersection is a ray. Therefore two rays of the same dirction are mapped by $h$ to rays with the same direction. Thus rays $(x,+\infty)$ are mapped either all to rays $(h(x),+\infty)$ or all to $(-\infty,h(x))$. Thus $h$ is monotone. Let $h:\Rr\to\Rr$ be a monotone bijection. Then the image and preimage under $h$ of any open interval is an open interval. Therefore, both $h$ and $h^{-1}$ are continuous, and hence $h$ is a homeomorphism. {{endproof}} The following theorem can be proved similarly or can be deduced from Theorem \ref{thm:homeomorphisms-of-line} {{beginthm|Theorem}}\label{thm:homeomorphisms-of-others} # A map $h:I\to I$ is a homeomorphism iff $h$ is a monotone bijection. # A map $h:\Rr_+\to\Rr_+$ is a homeomorphism iff $h$ is a monotone increasing bijection. # A map $h:S^1\to S^1$ is a homeomorphism iff $h$ is a bijection that either preserves or reverses the cyclic order of points on $S^1$. {{endthm}} Any self-homeomorphism of a connected 1-manifold either preserve orientation, or reverses it. The half-line $\Rr_+$ does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$ admits an orientation reversing map. Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of $\Rr_+$ and $\Rr_-$ with the orientations induced by the standard order. ==== Characterizations of connected compact 1-manifolds in terms of separating points ==== ; A subset $A$ of a topological space $X$ is said to ''separate'' $X$ if $X\smallsetminus A$ can be presented as a union of two disjoint open sets. {{beginthm|Theorem}}\label{thm:compact-characterisation} (See \cite{Moor1920}.) Let $X$ be a connected compact Hausdorff second countable topological space. # If every two points separate $X$, then $X$ is homeomorphic to the circle. # If each point, with two exceptions, separates $X$, then $X$ is homeomorphic to $I$. {{endthm}} Any point $a\in\Rr$ splits $\Rr$ to two disjoint open rays $(-\infty,a)=\{x\in\Rr\mid x == Invariants == ==== Basic invariants ==== ; As follows from the Theorems \ref{thm:classification} and \ref{thm:characterisation} above, the following invariants * the number of connected components, * the compactness of each connected component, * and the number of boundary points of each connected component determine the topological type of a 1-manifold. ==== Homotopy invariants ==== ; The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$ are trivial. The fundamental group $\pi_1(X,x_0)$ is infinite cyclic group, if the connected component of $X$ containing $x_0$ is homeomorphic to circle, and trivial otherwise. ==== Homology invariants ==== ; Let $X$ be a 1-manifold with a finite number of connected components. By Theorem \ref{thm:classification}, it is homeomorphic to a disjoint union of some numbers of copies of $\Rr$, $\Rr_+$, $S^1$ and $I$: $$X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$$ Then $H_0(X)$ is a free abelian group of rank equal to the number $a+b+c+d$ of all connected components of $X$ and $H_1(X)$ is a free abelian group of rank equal to the number $c$ of closed (compact without boundary) components of $X$. Relative homology groups: $H_0(X,\partial X)$ is a free abelian group of rank equal to the number $a+c$ of connected components of $X$ without boundary; $H_1(X,\partial X)$ is a free abelian group of rank equal to the number $c+d$ of compact components of $X$. So, $$H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$$ Numbers $a$, $b$, $c$, and $d$ and the topological type of $X$ can be recovered from the ranks of these groups. Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology): $$H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$$ The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So $$H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$$ A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over $S^1$ has non-trivial monodromy, then all the homology groups are trivial. ==== Tangent bundle invariants ==== ; The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial. == Additional structures == ==== Triangulations ==== ; Any 1-manifold admits a [[Wikipedia:Triangulation_(topology)|triangulation]]. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism. A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$. Similarly, the topological type of a triangulation of $I$ is defined by the number of 1-simplices, which can take any positive integral value. ==== Smooth structures ==== ; ''Any 1-manifold admits a [[Wikipedia:Smooth_structure|smooth structure]].'' If smooth 1-manifolds $X$ and $Y$ are homeomorphic, then they are also diffeomorphic. Moreover,
{{beginthm|Theorem}} Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$-topology by a diffeomorphism. {{endthm}} {{beginproof}} By Theorems \ref{thm:homeomorphisms-of-line} and \ref{thm:homeomorphisms-of-others}, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points. {{endproof}} ==== Inner metrics ==== ; Recall that a [[Wikipedia:Metric_space|metric]] on a path-connected space is said to be ''inner'' if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$ in a metric space $X$ with metric $d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0 == Constructions == ==== Surgery ==== ; Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary. If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold). An index 1 surgery preserving orientation on closed 1-manifold changes the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components. ==== Connected sums ==== ; The notion of [[Parametric_connected_sum#Connected_sum|connected sum]] is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term ''connected sum'' can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$ is a disjoint sum of two copies of $\Rr$. Note that connected sum is only a well defined operation on ''oriented manifolds'' and one has to be careful with the orientations. For example $$ \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$$ == Groups of self-homeomorphisms == ==== Mapping class groups ==== ; Recall that the ''mapping class group'' of a manifold $X$ is the quotient group of the group $\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$ is $\pi_0(\operatorname{Homeo}(X))$. An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic: {{beginthm|Theorem}}\label{thm:isotopy} Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic. {{endthm}} This is a corollary of the following two obvious lemmas. {{beginthm|Lemma}}\label{thm:rectilinear-isotopy} '''On rectilinear isotopy.''' Let $X$ be one of the following 1-manifolds: $\Rr$, $\Rr_+$, or $I$. Let $f,g:X\to X$ be two monotone bijections that are either both increasing or both decreasing. Then the family $h_t=(1-t)f+tg:X\to X$ with $t\in[0,1]$ consists of monotone bijections (and hence is an isotopy between $f$ and $g$). {{endthm}} {{beginthm|Lemma}}\label{thm:circle-isotopy} Let $f,g:S^1\to S^1$ be two bijections that either both preserve or both reverse the standard cyclic order of points on $S^1$. Let $f$ and $g$ coincide at $x\in S^1$. Then $f$ and $g$ are isotopic via the canonical isotopy which is stationary at $x$ and is provided on the complement of $x$ by stereographic projections and the rectilinear isotopy from Lemma \ref{thm:rectilinear-isotopy} of the corresponding self-homeomorphisms of $\Rr$. {{endthm}} {{beginthm|Corollary}} $\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$ and $\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$ {{endthm}} '''Remark.''' All the statements in this section remains true, if everywhere the word ''homeomorphism'' is replaced by the word ''diffeomorphism'' and $\operatorname{Homeo}$ is replaced by $\operatorname{Diffeo}$. ==== Homotopy types of groups of self-homeomorphisms ==== ; The group $\operatorname{Homeo}(S^1)$ contains $O(2)$ as a subgroup, which is its deformation retract. It follows from Lemma \ref{thm:circle-isotopy}. More precisely, for each point $x\in S^1$, Lemma \ref{thm:circle-isotopy} provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$. Similarly, the group of self-homeomorphisms of $S^1$ isotopic to identity contains $SO(2)=S^1$ as a subgroup, which is its deformation retract. The groups of self-homeomorphisms of $\Rr$, $\Rr_+$ and $I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma \ref{thm:rectilinear-isotopy} applied to $f=\operatorname{id}$ and an arbitrary $g$. Thus for each connected 1-manifold $X$ the group of homeomorphisms $X \to X$ isotopic to identity is homotopy equivalent to $X$. == Finite group actions == ; Consider an action of a finite group $G$ on a 1-manifold $X$. ==== Free actions ==== ; For any point $a\in X$, its orbit $Ga\subset X$ is a finite set and has an invariant neighborhood $U$ whose connected components are disjoint open sets, each of them contains exactly one point of $Ga$, and either all the components are homeomorphic to $\Rr$, or all homeomorphic to $\Rr_+$. If the action is free, then the orbit space $X/G$ is a 1-manifold and the natural projection $X\to X/G$ is a covering. Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds. A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold $X$ has a contractible orbit space $Y=X/G$, then $X$ is a disjoint union of copies of $Y$ and $G$ permutes these copies. In particular, there is no non-trivial free group action on a ''connected'' 1-manifold having contractible orbit space. Coverings $X\to S^1$ with connected $X$ are in one-to-one correspondence with subgroups of finite indices of $\pi_1(S^1)=\mathbb Z$. For each index $m\in\mathbb Z$ there is one subgroup, and hence one covering. The total space is homeomorphic to $S^1$, and the covering is equivalent to $S^1\to S^1:z\mapsto z^m$. In the corresponding action, the group is cyclic of order $m$, it acts on $S^1$ by rotations. In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems. {{beginthm|Theorem}}\label{thm:free-action-on-contractible} There is no non-trivial free finite group action on a contractible 1-manifold. {{endthm}} {{beginthm|Theorem}}\label{thm:free-action-on-circle} If a finite group $G$ acts freely on the circle than $G$ is cyclic. Any finite cyclic group has a linear free action on $S^1$. Any free action of a finite cyclic group on $S^1$ is conjugate to a linear action. {{endthm}} ==== Asymmetry of a half-line ==== {{beginthm|Theorem}}\label{thm:no-action-on-ray} There is no non-trivial action of a finite group in $\Rr_+$. {{endthm}} {{beginproof}} We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $\Rr$ or to the half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$$\Rr_+=\{x\in\Rr\mid x\ge0\}$.

Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$$\Rr^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

For other expositions about $1$$1$-manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].

2 Examples

• The real line: $\mathbb R$$\mathbb R$

• The half-line: $\mathbb R_+$$\mathbb R_+$

• The circle: $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$

• The closed interval: $I=[0,1]$$I=[0,1]$

3 Topological classification

3.1 Reduction to classification of connected manifolds

The following elementary facts hold for $n$$n$-manifolds of any dimension $n$$n$.

Any manifold is homeomorphic to the disjoint sum of its connected components.

A connected component of an $n$$n$-manifold is a $n$$n$-manifold.

Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.

3.2 Topological classification of connected 1-manifolds

Theorem 3.1. Any connected 1-manifold is homeomorphic to one of the following 4 manifolds:

1. real line $\mathbb R$$\mathbb R$
2. half-line $\mathbb R_+$$\mathbb R_+$
3. circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$
4. closed interval $I=[0,1]$$I=[0,1]$.

No two of these manifolds are homeomorphic to each other.

3.3 Characterizing the topological type of a connected 1-manifold

Theorem 3.2.

1. Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$$\Rr$.
2. Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$$\Rr_+$.
3. Any connected closed 1-manifold is homeomorphic to $S^1$$S^1$.
4. Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$$I$.

Thus, for connected 1-manifolds, two invariants, compactness and presence of boundary, form a complete system of topological invariants. Each of the invariants takes two values.

Theorems 3.1 and 3.2 above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result.

3.4 About proofs of the classification theorems

The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:

Lemma 3.3. Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$$\Rr$ is homeomorphic either to $\Rr$$\Rr$ or $S^1$$S^1$.

Lemma 3.4. If a topological space $X$$X$ can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to $\Rr$$\Rr$, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.

3.5 Corollaries: homotopy classification

Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.

It follows immediately from Theorem 3.1.

3.6 Corollaries: cobordisms of 0-manifolds

Theorem 3.6. A compact 0-manifold $X$$X$ bounds a compact 1-manifold iff the number of points in $X$$X$ is even.

Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.

4 Orders and orientations

4.1 Interval topology

Most of properties specific for 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.

Open intervals $(a,b)=\{x\in\Rr\mid a<x<b\}$$(a,b)=\{x\in\Rr\mid a<x<b\}$ and open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$ form the base of the (standard) topology on $\Rr$$\Rr$. On $\Rr_+$$\Rr_+$ and $I$$I$, the (standard) topology is induced from the standard topology on $\Rr$$\Rr$, and can be described in terms of the induced order.

This way of introducing a topological structure can be applied in any (linearly) ordered set $X$$X$.

On $S^1$$S^1$ the topology is defined by the standard cyclic order. A cyclic order on a set $X$$X$ is usually defined as a ternary relation on $X$$X$. It can be described also by fixing a linear order $x<_ay$$x<_ay$ on $X\smallsetminus a$$X\smallsetminus a$ for each $a\in X$$a\in X$. The linear orders on $X\smallsetminus a$$X\smallsetminus a$ and $X\smallsetminus b$$X\smallsetminus b$ describe the same cyclic order on $X$$X$ if $b<_ax\Longleftrightarrow x<_ba$$b<_ax\Longleftrightarrow x<_ba$ for each $x\in X\smallsetminus\{a,b\}$$x\in X\smallsetminus\{a,b\}$. Thus all the orders $<_a$$<_a$ can be recovered form any one of them, and hence the cyclic order is determined by any one of the orders $<_a$$<_a$.

Open rays in $S^1\smallsetminus a$$S^1\smallsetminus a$ for all $a\in S^1$$a\in S^1$ form the base of the standard topology on $S^1$$S^1$. The same way of introducing a topological structure can be applied in any cyclically ordered set.

Topological structure of either kind, on a linearly ordered or cyclically ordered set, desrcibed above will be called interval topology.

4.2 Orientations

An orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.

An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.

Theorem 4.1. Every connected non-closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to $\Rr$$\Rr$, or $\Rr_+$$\Rr_+$, or $I$$I$. On each of these 1-manifolds there are two linear orders, $<$$<$ and $>$$>$, defining the topology. We have to prove that there is no other linear order defining the same topology.

A linear order $<$$<$ on a set $X$$X$ is encoded in the system of sets $X_a=\{x\in X\mid a<x\}$$X_a=\{x\in X\mid a<x\}$ for $a\in X$$a\in X$. A system of susbsets $X_a\subset X\smallsetminus a$$X_a\subset X\smallsetminus a$ with $a\in X$$a\in X$ comes from a linear order on $X$$X$ iff either $X_a\subset X_b$$X_a\subset X_b$, or $X_a\supset X_b$$X_a\supset X_b$, or $X_a=X_b$$X_a=X_b$ for any $a,b\in X$$a,b\in X$ and $X_a=X_b$$X_a=X_b$ only if $a=b$$a=b$. In the interval topology on $X$$X$, each $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ is open.

Each interior point $a$$a$ of a connected non-closed 1-manifold $X$$X$ splits $X$$X$ to two open connected sets. Any linear order on $X$$X$ defining the topology of $X$$X$ would give rise to splitting of each of these two sets into its intesections with $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ which are disjoint and open. Due to connectedness of the set, it coincides with one of these intersections. Hence the order coincides with one of two standard orders.

$\square$$\square$

An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.

Theorem 4.2. Every connected closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected closed 1-manifold is homeomorphic either to $S^1$$S^1$. On $S^1$$S^1$ there are cyclic orders opposite to each other that define the topology. We have to prove that there is no other cyclic order defining the same topology.

As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.

$\square$$\square$

An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).

Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of $n$$n$ connected components, then it admits $2^n$$2^n$ orientations.

4.3 Self-homeomorphisms

Theorem 4.4. A map $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a homeomorphism iff $h$$h$ is a monotone bijection.

Proof. Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a homeomorphism. First, observe that $h$$h$ maps every ray to a ray. Indeed, for any $x\in\Rr$$x\in\Rr$, the map $h$$h$ induces a homeomorphism $\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$$\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$. The rays $(-\infty,x)$$(-\infty,x)$ and $(x,\infty)$$(x,\infty)$ are connected components of $\Rr\smallsetminus x$$\Rr\smallsetminus x$. Therefore their images are connected components $(-\infty,h(x))$$(-\infty,h(x))$ and $(h(x),\infty)$$(h(x),\infty)$ of $\Rr\smallsetminus h(x)$$\Rr\smallsetminus h(x)$.

Observe that rays have the same direction iff one of them is contained in the other one, that is their intersection is a ray. Therefore two rays of the same dirction are mapped by $h$$h$ to rays with the same direction. Thus rays $(x,+\infty)$$(x,+\infty)$ are mapped either all to rays $(h(x),+\infty)$$(h(x),+\infty)$ or all to $(-\infty,h(x))$$(-\infty,h(x))$. Thus $h$$h$ is monotone.

Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a monotone bijection. Then the image and preimage under $h$$h$ of any open interval is an open interval. Therefore, both $h$$h$ and $h^{-1}$$h^{-1}$ are continuous, and hence $h$$h$ is a homeomorphism.

$\square$$\square$

The following theorem can be proved similarly or can be deduced from Theorem 4.4

Theorem 4.5.

1. A map $h:I\to I$$h:I\to I$ is a homeomorphism iff $h$$h$ is a monotone bijection.
2. A map $h:\Rr_+\to\Rr_+$$h:\Rr_+\to\Rr_+$ is a homeomorphism iff $h$$h$ is a monotone increasing bijection.
3. A map $h:S^1\to S^1$$h:S^1\to S^1$ is a homeomorphism iff $h$$h$ is a bijection that either preserves or reverses the cyclic order of points on $S^1$$S^1$.

Any self-homeomorphism of a connected 1-manifold either preserve orientation, or reverses it. The half-line $\Rr_+$$\Rr_+$ does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\Rr_+$ admits an orientation reversing map.

Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of $\Rr_+$$\Rr_+$ and $\Rr_-$$\Rr_-$ with the orientations induced by the standard order.

4.4 Characterizations of connected compact 1-manifolds in terms of separating points

A subset $A$$A$ of a topological space $X$$X$ is said to separate $X$$X$ if $X\smallsetminus A$$X\smallsetminus A$ can be presented as a union of two disjoint open sets.

Theorem 4.6. (See [Moor1920].) Let $X$$X$ be a connected compact Hausdorff second countable topological space.

1. If every two points separate $X$$X$, then $X$$X$ is homeomorphic to the circle.
2. If each point, with two exceptions, separates $X$$X$, then $X$$X$ is homeomorphic to $I$$I$.

Any point $a\in\Rr$$a\in\Rr$ splits $\Rr$$\Rr$ to two disjoint open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$.

Theorem 4.7. (See [Ward1936].) Let $X$$X$ be a connected locally compact Hausdorff second countable topological space.

1. If the complement of each point in $X$$X$ consists of two connected components, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.
2. If $X$$X$ contains a point

   Tex syntax error

   $b$ such that $X\smallsetminus b$$X\smallsetminus b$ is connected and $X\smallsetminus a$$X\smallsetminus a$ consists of two connected components for each $a\in X$$a\in X$, $a\ne b$$a\ne b$, then $X$$X$ is homeomorphic to $\Rr_+$$\Rr_+$.

5 Invariants

5.1 Basic invariants

As follows from the Theorems 3.1 and 3.2 above, the following invariants

• the number of connected components,
• the compactness of each connected component,
• and the number of boundary points of each connected component

determine the topological type of a 1-manifold.

5.2 Homotopy invariants

The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial. The fundamental group $\pi_1(X,x_0)$$\pi_1(X,x_0)$ is infinite cyclic group, if the connected component of $X$$X$ containing $x_0$$x_0$ is homeomorphic to circle, and trivial otherwise.

5.3 Homology invariants

Let $X$$X$ be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union

of some numbers of copies of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, $S^1$$S^1$ and $I$$I$:

$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$ $$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$$

Then $H_0(X)$$H_0(X)$ is a free abelian group of rank equal to the number $a+b+c+d$$a+b+c+d$ of all connected components of $X$$X$ and $H_1(X)$$H_1(X)$ is a free abelian group of rank equal to the number $c$$c$ of closed (compact without boundary) components of $X$$X$.

Relative homology groups: $H_0(X,\partial X)$$H_0(X,\partial X)$ is a free abelian group of rank equal to the number $a+c$$a+c$ of connected components of $X$$X$ without boundary; $H_1(X,\partial X)$$H_1(X,\partial X)$ is a free abelian group of rank equal to the number $c+d$$c+d$ of compact components of $X$$X$. So,

$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$ $$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$$

Numbers $a$$a$,

Tex syntax error

$b$, $c$$c$, and $d$$d$ and the topological type of $X$$X$ can be recovered from the ranks of these groups.

Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):

$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$ $$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$$

The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So

$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$ $$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$$

A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over $S^1$$S^1$ has non-trivial monodromy, then all the homology groups are trivial.

5.4 Tangent bundle invariants

The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.

6 Additional structures

6.1 Triangulations

Any 1-manifold admits a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism.

A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$$\ge3$. Similarly, the topological type of a triangulation of $I$$I$ is defined by the number of 1-simplices, which can take any positive integral value.

6.2 Smooth structures

Any 1-manifold admits a smooth structure.

If smooth 1-manifolds $X$$X$ and $Y$$Y$ are homeomorphic, then they are also diffeomorphic. Moreover,

Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.

$\square$$\square$

6.3 Inner metrics

Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$$s:I\to X$ in a metric space $X$$X$ with metric $d:X\times X\to \Rr_+$$d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$$\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$.

Any connected 1-manifold admits an inner metric. A connected 1-manifold with an inner metric is defined up to isometry by the diameter of the space. Recall that the diameter of a metric space $X$$X$ with metric $d:X\times X\to\Rr$$d:X\times X\to\Rr$ is $\sup\{ d(x,y)\mid x,y\in X\}$$\sup\{ d(x,y)\mid x,y\in X\}$.

For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.

1. For $\Rr$$\Rr$ with diameter $D\in (0,\infty]$$D\in (0,\infty]$ this is $(-D/2,D/2)$$(-D/2,D/2)$.
2. For $\Rr_+$$\Rr_+$ with diameter $D\in(0,\infty]$$D\in(0,\infty]$ this is $[0,D)$$[0,D)$.
3. For a circle with inner metric of diameter $D\in (0,\infty)$$D\in (0,\infty)$ this is the circle $\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$$\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$ of radius $D/\pi$$D/\pi$ on the plane with the inner metric.
4. For $I$$I$ with diameter $D\in(0,\infty)$$D\in(0,\infty)$ this is $[0,D]$$[0,D]$.

An inner metric on a connected 1-manifold defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above.

7 Constructions

7.1 Surgery

Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.

If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).

An index 1 surgery preserving orientation on closed 1-manifold changes the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components.

7.2 Connected sums

The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$$\Rr$ is a disjoint sum of two copies of $\Rr$$\Rr$.

Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example

$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$ $$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$$

8 Groups of self-homeomorphisms

8.1 Mapping class groups

Recall that the mapping class group of a manifold $X$$X$ is the quotient group of the group $\operatorname{Homeo} (X)$$\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$$X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$$X$ is $\pi_0(\operatorname{Homeo}(X))$$\pi_0(\operatorname{Homeo}(X))$.

An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:

Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.

This is a corollary of the following two obvious lemmas.

Lemma 8.2. On rectilinear isotopy. Let $X$$X$ be one of the following 1-manifolds: $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, or $I$$I$. Let $f,g:X\to X$$f,g:X\to X$ be two monotone bijections that are either both increasing or both decreasing. Then the family $h_t=(1-t)f+tg:X\to X$$h_t=(1-t)f+tg:X\to X$ with $t\in[0,1]$$t\in[0,1]$ consists of monotone bijections (and hence is an isotopy between $f$$f$ and $g$$g$).

Lemma 8.3. Let $f,g:S^1\to S^1$$f,g:S^1\to S^1$ be two bijections that either both preserve or both reverse the standard cyclic order of points on $S^1$$S^1$. Let $f$$f$ and $g$$g$ coincide at $x\in S^1$$x\in S^1$. Then $f$$f$ and $g$$g$ are isotopic via the canonical isotopy which is stationary at $x$$x$ and is provided on the complement of $x$$x$ by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of $\Rr$$\Rr$.

Corollary 8.4. $\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$$\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$ and $\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$$\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$

Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and $\operatorname{Homeo}$$\operatorname{Homeo}$ is replaced by $\operatorname{Diffeo}$$\operatorname{Diffeo}$.

8.2 Homotopy types of groups of self-homeomorphisms

The group $\operatorname{Homeo}(S^1)$$\operatorname{Homeo}(S^1)$ contains $O(2)$$O(2)$ as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point $x\in S^1$$x\in S^1$, Lemma 8.3 provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$$\operatorname{Homeo}(S^1)\to O(2)$.

Similarly, the group of self-homeomorphisms of $S^1$$S^1$ isotopic to identity contains $SO(2)=S^1$$SO(2)=S^1$ as a subgroup, which is its deformation retract.

The groups of self-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to $f=\operatorname{id}$$f=\operatorname{id}$ and an arbitrary $g$$g$.

Thus for each connected 1-manifold $X$$X$ the group of homeomorphisms $X \to X$$X \to X$ isotopic to identity is homotopy equivalent to $X$$X$.

9 Finite group actions

Consider an action of a finite group $G$$G$ on a 1-manifold $X$$X$.

9.1 Free actions

For any point $a\in X$$a\in X$, its orbit $Ga\subset X$$Ga\subset X$ is a finite set and has an invariant neighborhood $U$$U$ whose connected components are disjoint open sets, each of them contains exactly one point of $Ga$$Ga$, and either all the components are homeomorphic to $\Rr$$\Rr$, or all homeomorphic to $\Rr_+$$\Rr_+$.

If the action is free, then the orbit space $X/G$$X/G$ is a 1-manifold and the natural projection $X\to X/G$$X\to X/G$ is a covering.

Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.

A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold $X$$X$ has a contractible orbit space $Y=X/G$$Y=X/G$, then $X$$X$ is a disjoint union of copies of $Y$$Y$ and $G$$G$ permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.

Coverings $X\to S^1$$X\to S^1$ with connected $X$$X$ are in one-to-one correspondence with subgroups of finite indices of $\pi_1(S^1)=\mathbb Z$$\pi_1(S^1)=\mathbb Z$. For each index $m\in\mathbb Z$$m\in\mathbb Z$ there is one subgroup, and hence one covering. The total space is homeomorphic to $S^1$$S^1$, and the covering is equivalent to $S^1\to S^1:z\mapsto z^m$$S^1\to S^1:z\mapsto z^m$. In the corresponding action, the group is cyclic of order $m$$m$, it acts on $S^1$$S^1$ by rotations.

In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.

Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.

Theorem 9.2. If a finite group $G$$G$ acts freely on the circle than $G$$G$ is cyclic. Any finite cyclic group has a linear free action on $S^1$$S^1$. Any free action of a finite cyclic group on $S^1$$S^1$ is conjugate to a linear action.

9.2 Asymmetry of a half-line

Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.

Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).

Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.

$\square$$\square$

</wikitex>

9.3 Actions on line and segment

Theorem 9.4. The only orientation preserving homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is the identity.

Proof. Fix a homeomorphism $f:(0,\infty)\to\Rr$$f:(0,\infty)\to\Rr$ (say, define it by formula $f:x\mapsto (x^2- 1)/x$$f:x\mapsto (x^2- 1)/x$). Consider a homeomorphism $(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$$(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$. It preserves orientation (since $h$$h$ preserves orientation). So, it is a monotone increasing bijection $(0,\infty)\to(0,\infty)$$(0,\infty)\to(0,\infty)$ of finite order. It can be extended to $\Rr_+$$\Rr_+$ by letting $0\mapsto0$$0\mapsto0$. The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.

$\square$$\square$

Theorem 9.5. Any orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is of order two. It is conjugate to the symmetry against a point.

Proof. An orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a monotone decreasing bijection. Consider the function $x\mapsto h(x)-x$$x\mapsto h(x)-x$. It is also monotone decreasing bijection $\Rr\to\Rr$$\Rr\to\Rr$ and hence there exists a unique $a\in\Rr$$a\in\Rr$ such that $h(a)-a=0$$h(a)-a=0$, that is $h(a)=a$$h(a)=a$.

The homeomorphism $h$$h$ maps each connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$ to a connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$. The connected components are open rays $(-\infty,a)$$(-\infty,a)$ and $(a,\infty)$$(a,\infty)$. If each of them is mapped to itself, then $h$$h$ defines a homeomorphism of a finite order of the closed rays $(-\infty,a]$$(-\infty,a]$ and $[a,\infty)$$[a,\infty)$. Then by Theorem 9.3, $h$$h$ is identity, which contradicts to our assumption. Thus, $h([a,\infty))=(-\infty,a]$$h([a,\infty))=(-\infty,a]$ and $h((-\infty,a])=[a,\infty)$$h((-\infty,a])=[a,\infty)$. Then $h^2$$h^2$ preserves the rays, and, by Theorem 9.3, is identity. Thus $h$$h$ has order two.

Choose a homeomorphism $f:[0,\infty)\to[a,\infty)$$f:[0,\infty)\to[a,\infty)$. Define function $g:(-infty,0]\to(-\infty,a]$$g:(-infty,0]\to(-\infty,a]$ by formula $g(x)=hf(-x)$$g(x)=hf(-x)$. It's a homeomorphism. Together, $f$$f$ and $g$$g$ form a homeomorphism $\phi:\Rr\to\Rr$$\phi:\Rr\to\Rr$. As easy to check, $\phi^{-1}h\phi(x)=-x$$\phi^{-1}h\phi(x)=-x$.

$\square$$\square$

Theorem 9.6. A non-trivial finite group acting effectively on $\Rr$$\Rr$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, $f$$f$ and $g$$g$ of the line $\Rr$$\Rr$. Their composition $f\circ g$$f\circ g$ preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, $fg=1$$fg=1$ and hence $f=g^{-1}$$f=g^{-1}$. But $g^2=1$$g^2=1$. Therefore $g^{-1}=g$$g^{-1}=g$ and $f=g^{-1}=g$$f=g^{-1}=g$.

$\square$$\square$

Corollary 9.7. A non-trivial finite group acting effectively on $I$$I$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. Any auto-homeomorphism of $I$$I$ preserves the boundary and the interior of $I$$I$. Hence an effective finite group action on $I$$I$ induces an action of finite group on the interior of $I$$I$. An auto-homeomorphism of $I$$I$ is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of $\operatorname{Int}I$$\operatorname{Int}I$ has a unique extension to auto-homeomorphism of $I$$I$. The interior $\operatorname{Int}I$$\operatorname{Int}I$ is homeomorphic to $\Rr$$\Rr$.

$\square$$\square$

9.4 Actions on circle

{{beginthm|Theorem} Any periodic orientation reversing homeomorphism $S^1\to S^1$$S^1\to S^1$ is an involution (i.e., has period 2). It is conjugate to a symmetry of $S^1$$S^1$ against its diameter. </div>

Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.

Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.

$\square$$\square$

Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

Theorem 9.8. A periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ has no fixed point. It is conjugate to a rotation.

Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.

For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ have no fixed points. Therefore, these powers form a cyclic group freely acting on $S^1$$S^1$. See Theorem 9.2.

$\square$$\square$

Theorem 9.9. A finite group acting effectively on $S^1$$S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons.

Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.

Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group $D_2$$D_2$. If the number of orientation preserving homeomorphisms is $n>2$$n>2$, then the whole group is called the dihedral group $D_n$$D_n$. It is the symmetry group of an $n$$n$-sided regular polygon.

$\square$$\square$

10 Relatives of 1-manifolds

10.1 Non-Hausdorff 1-manifolds

If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable.

Indeed, one can take the disjoint union of two copies of the line $\Rr$$\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.

11 References

• [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
• [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
• [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
• [Moor1920] R.L.Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), 333—347.
• [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02

12 External links

• The Encylopedia of Mathematics article on one-dimensional manifolds.
• The Encylopedia of Mathematics article on lines.
• The Wikipedia page about curves.

\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b ==== Actions on line and segment ==== ; {{beginthm|Theorem}}\label{thm:orientation-preserving-on-line} The only orientation preserving homeomorphism $h:\Rr\to\Rr$ of finite order is the identity. {{endthm}} {{beginproof}} Fix a homeomorphism $f:(0,\infty)\to\Rr$ (say, define it by formula $f:x\mapsto (x^2- 1)/x$). Consider a homeomorphism $ (0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$. It preserves orientation (since $h$ preserves orientation). So, it is a monotone increasing bijection $(0,\infty)\to(0,\infty)$ of finite order. It can be extended to $\Rr_+$ by letting $\Rr$ or to the half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$$\Rr_+=\{x\in\Rr\mid x\ge0\}$.

Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$$\Rr^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

For other expositions about $1$$1$-manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].

2 Examples

• The real line: $\mathbb R$$\mathbb R$

• The half-line: $\mathbb R_+$$\mathbb R_+$

• The circle: $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$

• The closed interval: $I=[0,1]$$I=[0,1]$

3 Topological classification

3.1 Reduction to classification of connected manifolds

The following elementary facts hold for $n$$n$-manifolds of any dimension $n$$n$.

Any manifold is homeomorphic to the disjoint sum of its connected components.

A connected component of an $n$$n$-manifold is a $n$$n$-manifold.

Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.

3.2 Topological classification of connected 1-manifolds

Theorem 3.1. Any connected 1-manifold is homeomorphic to one of the following 4 manifolds:

1. real line $\mathbb R$$\mathbb R$
2. half-line $\mathbb R_+$$\mathbb R_+$
3. circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$
4. closed interval $I=[0,1]$$I=[0,1]$.

No two of these manifolds are homeomorphic to each other.

3.3 Characterizing the topological type of a connected 1-manifold

Theorem 3.2.

1. Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$$\Rr$.
2. Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$$\Rr_+$.
3. Any connected closed 1-manifold is homeomorphic to $S^1$$S^1$.
4. Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$$I$.

Thus, for connected 1-manifolds, two invariants, compactness and presence of boundary, form a complete system of topological invariants. Each of the invariants takes two values.

Theorems 3.1 and 3.2 above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result.

3.4 About proofs of the classification theorems

The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:

Lemma 3.3. Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$$\Rr$ is homeomorphic either to $\Rr$$\Rr$ or $S^1$$S^1$.

Lemma 3.4. If a topological space $X$$X$ can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to $\Rr$$\Rr$, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.

3.5 Corollaries: homotopy classification

Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.

It follows immediately from Theorem 3.1.

3.6 Corollaries: cobordisms of 0-manifolds

Theorem 3.6. A compact 0-manifold $X$$X$ bounds a compact 1-manifold iff the number of points in $X$$X$ is even.

Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.

4 Orders and orientations

4.1 Interval topology

Most of properties specific for 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.

Open intervals $(a,b)=\{x\in\Rr\mid a<x<b\}$$(a,b)=\{x\in\Rr\mid a<x<b\}$ and open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$ form the base of the (standard) topology on $\Rr$$\Rr$. On $\Rr_+$$\Rr_+$ and $I$$I$, the (standard) topology is induced from the standard topology on $\Rr$$\Rr$, and can be described in terms of the induced order.

This way of introducing a topological structure can be applied in any (linearly) ordered set $X$$X$.

On $S^1$$S^1$ the topology is defined by the standard cyclic order. A cyclic order on a set $X$$X$ is usually defined as a ternary relation on $X$$X$. It can be described also by fixing a linear order $x<_ay$$x<_ay$ on $X\smallsetminus a$$X\smallsetminus a$ for each $a\in X$$a\in X$. The linear orders on $X\smallsetminus a$$X\smallsetminus a$ and $X\smallsetminus b$$X\smallsetminus b$ describe the same cyclic order on $X$$X$ if $b<_ax\Longleftrightarrow x<_ba$$b<_ax\Longleftrightarrow x<_ba$ for each $x\in X\smallsetminus\{a,b\}$$x\in X\smallsetminus\{a,b\}$. Thus all the orders $<_a$$<_a$ can be recovered form any one of them, and hence the cyclic order is determined by any one of the orders $<_a$$<_a$.

Open rays in $S^1\smallsetminus a$$S^1\smallsetminus a$ for all $a\in S^1$$a\in S^1$ form the base of the standard topology on $S^1$$S^1$. The same way of introducing a topological structure can be applied in any cyclically ordered set.

Topological structure of either kind, on a linearly ordered or cyclically ordered set, desrcibed above will be called interval topology.

4.2 Orientations

An orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.

An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.

Theorem 4.1. Every connected non-closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to $\Rr$$\Rr$, or $\Rr_+$$\Rr_+$, or $I$$I$. On each of these 1-manifolds there are two linear orders, $<$$<$ and $>$$>$, defining the topology. We have to prove that there is no other linear order defining the same topology.

A linear order $<$$<$ on a set $X$$X$ is encoded in the system of sets $X_a=\{x\in X\mid a<x\}$$X_a=\{x\in X\mid a<x\}$ for $a\in X$$a\in X$. A system of susbsets $X_a\subset X\smallsetminus a$$X_a\subset X\smallsetminus a$ with $a\in X$$a\in X$ comes from a linear order on $X$$X$ iff either $X_a\subset X_b$$X_a\subset X_b$, or $X_a\supset X_b$$X_a\supset X_b$, or $X_a=X_b$$X_a=X_b$ for any $a,b\in X$$a,b\in X$ and $X_a=X_b$$X_a=X_b$ only if $a=b$$a=b$. In the interval topology on $X$$X$, each $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ is open.

Each interior point $a$$a$ of a connected non-closed 1-manifold $X$$X$ splits $X$$X$ to two open connected sets. Any linear order on $X$$X$ defining the topology of $X$$X$ would give rise to splitting of each of these two sets into its intesections with $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ which are disjoint and open. Due to connectedness of the set, it coincides with one of these intersections. Hence the order coincides with one of two standard orders.

$\square$$\square$

An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.

Theorem 4.2. Every connected closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected closed 1-manifold is homeomorphic either to $S^1$$S^1$. On $S^1$$S^1$ there are cyclic orders opposite to each other that define the topology. We have to prove that there is no other cyclic order defining the same topology.

As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.

$\square$$\square$

An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).

Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of $n$$n$ connected components, then it admits $2^n$$2^n$ orientations.

4.3 Self-homeomorphisms

Theorem 4.4. A map $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a homeomorphism iff $h$$h$ is a monotone bijection.

Proof. Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a homeomorphism. First, observe that $h$$h$ maps every ray to a ray. Indeed, for any $x\in\Rr$$x\in\Rr$, the map $h$$h$ induces a homeomorphism $\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$$\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$. The rays $(-\infty,x)$$(-\infty,x)$ and $(x,\infty)$$(x,\infty)$ are connected components of $\Rr\smallsetminus x$$\Rr\smallsetminus x$. Therefore their images are connected components $(-\infty,h(x))$$(-\infty,h(x))$ and $(h(x),\infty)$$(h(x),\infty)$ of $\Rr\smallsetminus h(x)$$\Rr\smallsetminus h(x)$.

Observe that rays have the same direction iff one of them is contained in the other one, that is their intersection is a ray. Therefore two rays of the same dirction are mapped by $h$$h$ to rays with the same direction. Thus rays $(x,+\infty)$$(x,+\infty)$ are mapped either all to rays $(h(x),+\infty)$$(h(x),+\infty)$ or all to $(-\infty,h(x))$$(-\infty,h(x))$. Thus $h$$h$ is monotone.

Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a monotone bijection. Then the image and preimage under $h$$h$ of any open interval is an open interval. Therefore, both $h$$h$ and $h^{-1}$$h^{-1}$ are continuous, and hence $h$$h$ is a homeomorphism.

$\square$$\square$

The following theorem can be proved similarly or can be deduced from Theorem 4.4

Theorem 4.5.

1. A map $h:I\to I$$h:I\to I$ is a homeomorphism iff $h$$h$ is a monotone bijection.
2. A map $h:\Rr_+\to\Rr_+$$h:\Rr_+\to\Rr_+$ is a homeomorphism iff $h$$h$ is a monotone increasing bijection.
3. A map $h:S^1\to S^1$$h:S^1\to S^1$ is a homeomorphism iff $h$$h$ is a bijection that either preserves or reverses the cyclic order of points on $S^1$$S^1$.

Any self-homeomorphism of a connected 1-manifold either preserve orientation, or reverses it. The half-line $\Rr_+$$\Rr_+$ does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\Rr_+$ admits an orientation reversing map.

Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of $\Rr_+$$\Rr_+$ and $\Rr_-$$\Rr_-$ with the orientations induced by the standard order.

4.4 Characterizations of connected compact 1-manifolds in terms of separating points

A subset $A$$A$ of a topological space $X$$X$ is said to separate $X$$X$ if $X\smallsetminus A$$X\smallsetminus A$ can be presented as a union of two disjoint open sets.

Theorem 4.6. (See [Moor1920].) Let $X$$X$ be a connected compact Hausdorff second countable topological space.

1. If every two points separate $X$$X$, then $X$$X$ is homeomorphic to the circle.
2. If each point, with two exceptions, separates $X$$X$, then $X$$X$ is homeomorphic to $I$$I$.

Any point $a\in\Rr$$a\in\Rr$ splits $\Rr$$\Rr$ to two disjoint open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$.

Theorem 4.7. (See [Ward1936].) Let $X$$X$ be a connected locally compact Hausdorff second countable topological space.

1. If the complement of each point in $X$$X$ consists of two connected components, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.
2. If $X$$X$ contains a point

   Tex syntax error

   $b$ such that $X\smallsetminus b$$X\smallsetminus b$ is connected and $X\smallsetminus a$$X\smallsetminus a$ consists of two connected components for each $a\in X$$a\in X$, $a\ne b$$a\ne b$, then $X$$X$ is homeomorphic to $\Rr_+$$\Rr_+$.

5 Invariants

5.1 Basic invariants

As follows from the Theorems 3.1 and 3.2 above, the following invariants

• the number of connected components,
• the compactness of each connected component,
• and the number of boundary points of each connected component

determine the topological type of a 1-manifold.

5.2 Homotopy invariants

The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial. The fundamental group $\pi_1(X,x_0)$$\pi_1(X,x_0)$ is infinite cyclic group, if the connected component of $X$$X$ containing $x_0$$x_0$ is homeomorphic to circle, and trivial otherwise.

5.3 Homology invariants

Let $X$$X$ be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union

of some numbers of copies of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, $S^1$$S^1$ and $I$$I$:

$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$ $$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$$

Then $H_0(X)$$H_0(X)$ is a free abelian group of rank equal to the number $a+b+c+d$$a+b+c+d$ of all connected components of $X$$X$ and $H_1(X)$$H_1(X)$ is a free abelian group of rank equal to the number $c$$c$ of closed (compact without boundary) components of $X$$X$.

Relative homology groups: $H_0(X,\partial X)$$H_0(X,\partial X)$ is a free abelian group of rank equal to the number $a+c$$a+c$ of connected components of $X$$X$ without boundary; $H_1(X,\partial X)$$H_1(X,\partial X)$ is a free abelian group of rank equal to the number $c+d$$c+d$ of compact components of $X$$X$. So,

$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$ $$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$$

Numbers $a$$a$,

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$b$, $c$$c$, and $d$$d$ and the topological type of $X$$X$ can be recovered from the ranks of these groups.

Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):

$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$ $$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$$

The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So

$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$ $$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$$

A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over $S^1$$S^1$ has non-trivial monodromy, then all the homology groups are trivial.

5.4 Tangent bundle invariants

The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.

6 Additional structures

6.1 Triangulations

Any 1-manifold admits a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism.

A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$$\ge3$. Similarly, the topological type of a triangulation of $I$$I$ is defined by the number of 1-simplices, which can take any positive integral value.

6.2 Smooth structures

Any 1-manifold admits a smooth structure.

If smooth 1-manifolds $X$$X$ and $Y$$Y$ are homeomorphic, then they are also diffeomorphic. Moreover,

Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.

$\square$$\square$

6.3 Inner metrics

Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$$s:I\to X$ in a metric space $X$$X$ with metric $d:X\times X\to \Rr_+$$d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$$\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$.

Any connected 1-manifold admits an inner metric. A connected 1-manifold with an inner metric is defined up to isometry by the diameter of the space. Recall that the diameter of a metric space $X$$X$ with metric $d:X\times X\to\Rr$$d:X\times X\to\Rr$ is $\sup\{ d(x,y)\mid x,y\in X\}$$\sup\{ d(x,y)\mid x,y\in X\}$.

For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.

1. For $\Rr$$\Rr$ with diameter $D\in (0,\infty]$$D\in (0,\infty]$ this is $(-D/2,D/2)$$(-D/2,D/2)$.
2. For $\Rr_+$$\Rr_+$ with diameter $D\in(0,\infty]$$D\in(0,\infty]$ this is $[0,D)$$[0,D)$.
3. For a circle with inner metric of diameter $D\in (0,\infty)$$D\in (0,\infty)$ this is the circle $\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$$\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$ of radius $D/\pi$$D/\pi$ on the plane with the inner metric.
4. For $I$$I$ with diameter $D\in(0,\infty)$$D\in(0,\infty)$ this is $[0,D]$$[0,D]$.

An inner metric on a connected 1-manifold defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above.

7 Constructions

7.1 Surgery

Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.

If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).

An index 1 surgery preserving orientation on closed 1-manifold changes the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components.

7.2 Connected sums

The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$$\Rr$ is a disjoint sum of two copies of $\Rr$$\Rr$.

Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example

$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$ $$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$$

8 Groups of self-homeomorphisms

8.1 Mapping class groups

Recall that the mapping class group of a manifold $X$$X$ is the quotient group of the group $\operatorname{Homeo} (X)$$\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$$X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$$X$ is $\pi_0(\operatorname{Homeo}(X))$$\pi_0(\operatorname{Homeo}(X))$.

An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:

Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.

This is a corollary of the following two obvious lemmas.

Lemma 8.2. On rectilinear isotopy. Let $X$$X$ be one of the following 1-manifolds: $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, or $I$$I$. Let $f,g:X\to X$$f,g:X\to X$ be two monotone bijections that are either both increasing or both decreasing. Then the family $h_t=(1-t)f+tg:X\to X$$h_t=(1-t)f+tg:X\to X$ with $t\in[0,1]$$t\in[0,1]$ consists of monotone bijections (and hence is an isotopy between $f$$f$ and $g$$g$).

Lemma 8.3. Let $f,g:S^1\to S^1$$f,g:S^1\to S^1$ be two bijections that either both preserve or both reverse the standard cyclic order of points on $S^1$$S^1$. Let $f$$f$ and $g$$g$ coincide at $x\in S^1$$x\in S^1$. Then $f$$f$ and $g$$g$ are isotopic via the canonical isotopy which is stationary at $x$$x$ and is provided on the complement of $x$$x$ by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of $\Rr$$\Rr$.

Corollary 8.4. $\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$$\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$ and $\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$$\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$

Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and $\operatorname{Homeo}$$\operatorname{Homeo}$ is replaced by $\operatorname{Diffeo}$$\operatorname{Diffeo}$.

8.2 Homotopy types of groups of self-homeomorphisms

The group $\operatorname{Homeo}(S^1)$$\operatorname{Homeo}(S^1)$ contains $O(2)$$O(2)$ as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point $x\in S^1$$x\in S^1$, Lemma 8.3 provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$$\operatorname{Homeo}(S^1)\to O(2)$.

Similarly, the group of self-homeomorphisms of $S^1$$S^1$ isotopic to identity contains $SO(2)=S^1$$SO(2)=S^1$ as a subgroup, which is its deformation retract.

The groups of self-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to $f=\operatorname{id}$$f=\operatorname{id}$ and an arbitrary $g$$g$.

Thus for each connected 1-manifold $X$$X$ the group of homeomorphisms $X \to X$$X \to X$ isotopic to identity is homotopy equivalent to $X$$X$.

9 Finite group actions

Consider an action of a finite group $G$$G$ on a 1-manifold $X$$X$.

9.1 Free actions

For any point $a\in X$$a\in X$, its orbit $Ga\subset X$$Ga\subset X$ is a finite set and has an invariant neighborhood $U$$U$ whose connected components are disjoint open sets, each of them contains exactly one point of $Ga$$Ga$, and either all the components are homeomorphic to $\Rr$$\Rr$, or all homeomorphic to $\Rr_+$$\Rr_+$.

If the action is free, then the orbit space $X/G$$X/G$ is a 1-manifold and the natural projection $X\to X/G$$X\to X/G$ is a covering.

Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.

A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold $X$$X$ has a contractible orbit space $Y=X/G$$Y=X/G$, then $X$$X$ is a disjoint union of copies of $Y$$Y$ and $G$$G$ permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.

Coverings $X\to S^1$$X\to S^1$ with connected $X$$X$ are in one-to-one correspondence with subgroups of finite indices of $\pi_1(S^1)=\mathbb Z$$\pi_1(S^1)=\mathbb Z$. For each index $m\in\mathbb Z$$m\in\mathbb Z$ there is one subgroup, and hence one covering. The total space is homeomorphic to $S^1$$S^1$, and the covering is equivalent to $S^1\to S^1:z\mapsto z^m$$S^1\to S^1:z\mapsto z^m$. In the corresponding action, the group is cyclic of order $m$$m$, it acts on $S^1$$S^1$ by rotations.

In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.

Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.

Theorem 9.2. If a finite group $G$$G$ acts freely on the circle than $G$$G$ is cyclic. Any finite cyclic group has a linear free action on $S^1$$S^1$. Any free action of a finite cyclic group on $S^1$$S^1$ is conjugate to a linear action.

9.2 Asymmetry of a half-line

Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.

Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).

Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.

$\square$$\square$

</wikitex>

9.3 Actions on line and segment

Theorem 9.4. The only orientation preserving homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is the identity.

Proof. Fix a homeomorphism $f:(0,\infty)\to\Rr$$f:(0,\infty)\to\Rr$ (say, define it by formula $f:x\mapsto (x^2- 1)/x$$f:x\mapsto (x^2- 1)/x$). Consider a homeomorphism $(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$$(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$. It preserves orientation (since $h$$h$ preserves orientation). So, it is a monotone increasing bijection $(0,\infty)\to(0,\infty)$$(0,\infty)\to(0,\infty)$ of finite order. It can be extended to $\Rr_+$$\Rr_+$ by letting $0\mapsto0$$0\mapsto0$. The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.

$\square$$\square$

Theorem 9.5. Any orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is of order two. It is conjugate to the symmetry against a point.

Proof. An orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a monotone decreasing bijection. Consider the function $x\mapsto h(x)-x$$x\mapsto h(x)-x$. It is also monotone decreasing bijection $\Rr\to\Rr$$\Rr\to\Rr$ and hence there exists a unique $a\in\Rr$$a\in\Rr$ such that $h(a)-a=0$$h(a)-a=0$, that is $h(a)=a$$h(a)=a$.

The homeomorphism $h$$h$ maps each connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$ to a connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$. The connected components are open rays $(-\infty,a)$$(-\infty,a)$ and $(a,\infty)$$(a,\infty)$. If each of them is mapped to itself, then $h$$h$ defines a homeomorphism of a finite order of the closed rays $(-\infty,a]$$(-\infty,a]$ and $[a,\infty)$$[a,\infty)$. Then by Theorem 9.3, $h$$h$ is identity, which contradicts to our assumption. Thus, $h([a,\infty))=(-\infty,a]$$h([a,\infty))=(-\infty,a]$ and $h((-\infty,a])=[a,\infty)$$h((-\infty,a])=[a,\infty)$. Then $h^2$$h^2$ preserves the rays, and, by Theorem 9.3, is identity. Thus $h$$h$ has order two.

Choose a homeomorphism $f:[0,\infty)\to[a,\infty)$$f:[0,\infty)\to[a,\infty)$. Define function $g:(-infty,0]\to(-\infty,a]$$g:(-infty,0]\to(-\infty,a]$ by formula $g(x)=hf(-x)$$g(x)=hf(-x)$. It's a homeomorphism. Together, $f$$f$ and $g$$g$ form a homeomorphism $\phi:\Rr\to\Rr$$\phi:\Rr\to\Rr$. As easy to check, $\phi^{-1}h\phi(x)=-x$$\phi^{-1}h\phi(x)=-x$.

$\square$$\square$

Theorem 9.6. A non-trivial finite group acting effectively on $\Rr$$\Rr$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, $f$$f$ and $g$$g$ of the line $\Rr$$\Rr$. Their composition $f\circ g$$f\circ g$ preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, $fg=1$$fg=1$ and hence $f=g^{-1}$$f=g^{-1}$. But $g^2=1$$g^2=1$. Therefore $g^{-1}=g$$g^{-1}=g$ and $f=g^{-1}=g$$f=g^{-1}=g$.

$\square$$\square$

Corollary 9.7. A non-trivial finite group acting effectively on $I$$I$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. Any auto-homeomorphism of $I$$I$ preserves the boundary and the interior of $I$$I$. Hence an effective finite group action on $I$$I$ induces an action of finite group on the interior of $I$$I$. An auto-homeomorphism of $I$$I$ is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of $\operatorname{Int}I$$\operatorname{Int}I$ has a unique extension to auto-homeomorphism of $I$$I$. The interior $\operatorname{Int}I$$\operatorname{Int}I$ is homeomorphic to $\Rr$$\Rr$.

$\square$$\square$

9.4 Actions on circle

{{beginthm|Theorem} Any periodic orientation reversing homeomorphism $S^1\to S^1$$S^1\to S^1$ is an involution (i.e., has period 2). It is conjugate to a symmetry of $S^1$$S^1$ against its diameter. </div>

Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.

Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.

$\square$$\square$

Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

Theorem 9.8. A periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ has no fixed point. It is conjugate to a rotation.

Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.

For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ have no fixed points. Therefore, these powers form a cyclic group freely acting on $S^1$$S^1$. See Theorem 9.2.

$\square$$\square$

Theorem 9.9. A finite group acting effectively on $S^1$$S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons.

Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.

Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group $D_2$$D_2$. If the number of orientation preserving homeomorphisms is $n>2$$n>2$, then the whole group is called the dihedral group $D_n$$D_n$. It is the symmetry group of an $n$$n$-sided regular polygon.

$\square$$\square$

10 Relatives of 1-manifolds

10.1 Non-Hausdorff 1-manifolds

If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable.

Indeed, one can take the disjoint union of two copies of the line $\Rr$$\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.

11 References

• [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
• [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
• [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
• [Moor1920] R.L.Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), 333—347.
• [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02

12 External links

• The Encylopedia of Mathematics article on one-dimensional manifolds.
• The Encylopedia of Mathematics article on lines.
• The Wikipedia page about curves.

\mapsto0$. The extended homeomorphism has the same finite order. But by Theorem \ref{thm:no-action-on-ray} any such homeomorphism is the identity. {{endproof}} {{beginthm|Theorem}}\label{thm:orientation-reversing-on-line} Any orientation reversing homeomorphism $h:\Rr\to\Rr$ of finite order is of order two. It is conjugate to the symmetry against a point. {{endthm}} {{beginproof}} An orientation reversing homeomorphism $h:\Rr\to\Rr$ is a monotone decreasing bijection. Consider the function $x\mapsto h(x)-x$. It is also monotone decreasing bijection $\Rr\to\Rr$ and hence there exists a unique $a\in\Rr$ such that $h(a)-a=0$, that is $h(a)=a$. The homeomorphism $h$ maps each connected component of $\Rr\smallsetminus a$ to a connected component of $\Rr\smallsetminus a$. The connected components are open rays $(-\infty,a)$ and $(a,\infty)$. If each of them is mapped to itself, then $h$ defines a homeomorphism of a finite order of the closed rays $(-\infty,a]$ and $[a,\infty)$. Then by Theorem \ref{thm:no-action-on-ray}, $h$ is identity, which contradicts to our assumption. Thus, $h([a,\infty))=(-\infty,a]$ and $h((-\infty,a])=[a,\infty)$. Then $h^2$ preserves the rays, and, by Theorem \ref{thm:no-action-on-ray}, is identity. Thus $h$ has order two. Choose a homeomorphism $f:[0,\infty)\to[a,\infty)$. Define function $g:(-infty,0]\to(-\infty,a]$ by formula $g(x)=hf(-x)$. It's a homeomorphism. Together, $f$ and $g$ form a homeomorphism $\phi:\Rr\to\Rr$. As easy to check, $\phi^{-1}h\phi(x)=-x$. {{endproof}} {{beginthm|Theorem}}\label{thm:actions-on-line} A non-trivial finite group acting effectively on $\Rr$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point. {{endthm}} {{beginproof}} As follows from Corollaries \ref{thm:orientation-preserving-on-line} and \ref{thm:orientation-reversing-on-line}, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, $f$ and $g$ of the line $\Rr$. Their composition $f\circ g$ preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem \ref{thm:orientation-preserving-on-line}, it is identity. So, $fg=1$ and hence $f=g^{-1}$. But $g^2=1$. Therefore $g^{-1}=g$ and $f=g^{-1}=g$. {{endproof}} {{beginthm|Corollary}}\label{thm:actions-on-segment} A non-trivial finite group acting effectively on $I$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point. {{endthm}} {{beginproof}} Any auto-homeomorphism of $I$ preserves the boundary and the interior of $I$. Hence an effective finite group action on $I$ induces an action of finite group on the interior of $I$. An auto-homeomorphism of $I$ is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of $\operatorname{Int}I$ has a unique extension to auto-homeomorphism of $I$. The interior $\operatorname{Int}I$ is homeomorphic to $\Rr$. {{endproof}} ==== Actions on circle ==== ; {{beginthm|Theorem}\label{thm:orientation-reversing-on-circle} Any periodic orientation reversing homeomorphism $S^1\to S^1$ is an involution (i.e., has period 2). It is conjugate to a symmetry of $S^1$ against its diameter. {{endthm}} {{beginproof}} Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2. Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary \ref{thm:orientation-reversing-on-line}, which gives the required result. {{endproof}} Observe that by theorems \ref{thm:orientation-reversing-on-line}, \ref{thm:actions-on-segment}, and \ref{thm:orientation-reversing-on-circle} any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation. {{beginthm|Theorem}}\label{thm:orientation-preserving-on-circle} A periodic non-identity orientation preserving homeomorphism $S^1\to S^1$ has no fixed point. It is conjugate to a rotation. {{endthm}} {{beginproof}} If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem \ref{thm:orientation-preserving-on-line} would conclude that it is identity and hence the whole homeomorphism is identity. For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism $S^1\to S^1$ have no fixed points. Therefore, these powers form a cyclic group freely acting on $S^1$. See Theorem \ref{thm:free-action-on-circle}. {{endproof}} {{beginthm|Theorem}} A finite group acting effectively on $S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons. {{endthm}} {{beginproof}} If all the homeomorphisms in the action preserve orientation, then by Theorem \ref{thm:orientation-preserving-on-circle} the action is free, and the result follows from Theorem \ref{thm:free-action-on-circle}. Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's ''Vierergruppe'' or ''dihedral'' group $D_2$. If the number of orientation preserving homeomorphisms is $n>2$, then the whole group is called the dihedral group $D_n$. It is the symmetry group of an $n$-sided regular polygon. {{endproof}} == Relatives of 1-manifolds == ==== Non-Hausdorff 1-manifolds ==== ; If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable. Indeed, one can take the disjoint union of two copies of the line $\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space. == References == {{#RefList:}} == External links == * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/One-dimensional_manifold one-dimensional manifolds]. * The Encylopedia of Mathematics article on [http://www.encyclopediaofmath.org/index.php/Line_(curve) lines]. * The Wikipedia page about [[Wikipedia:Curve|curves]]. [[Category:Manifolds]]\Rr or to the half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$$\Rr_+=\{x\in\Rr\mid x\ge0\}$.

Manifolds of dimension 1 are called curves, but this name may lead to a confusion, because many mathematical objects share it. Even in the context of topology, the term curve may mean not only a manifold of dimension 1 with an additional structure, but, for instance, an immersion of a smooth manifold of dimension 1 to Euclidean space $\Rr^n$$\Rr^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

For other expositions about $1$$1$-manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Sections 3.1.1.16-19].

2 Examples

• The real line: $\mathbb R$$\mathbb R$

• The half-line: $\mathbb R_+$$\mathbb R_+$

• The circle: $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$

• The closed interval: $I=[0,1]$$I=[0,1]$

3 Topological classification

3.1 Reduction to classification of connected manifolds

The following elementary facts hold for $n$$n$-manifolds of any dimension $n$$n$.

Any manifold is homeomorphic to the disjoint sum of its connected components.

A connected component of an $n$$n$-manifold is a $n$$n$-manifold.

Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic.

3.2 Topological classification of connected 1-manifolds

Theorem 3.1. Any connected 1-manifold is homeomorphic to one of the following 4 manifolds:

1. real line $\mathbb R$$\mathbb R$
2. half-line $\mathbb R_+$$\mathbb R_+$
3. circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$$S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$
4. closed interval $I=[0,1]$$I=[0,1]$.

No two of these manifolds are homeomorphic to each other.

3.3 Characterizing the topological type of a connected 1-manifold

Theorem 3.2.

1. Any connected non-compact 1-manifold without boundary is homeomorphic to $\Rr$$\Rr$.
2. Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to $\Rr_+$$\Rr_+$.
3. Any connected closed 1-manifold is homeomorphic to $S^1$$S^1$.
4. Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$$I$.

Thus, for connected 1-manifolds, two invariants, compactness and presence of boundary, form a complete system of topological invariants. Each of the invariants takes two values.

Theorems 3.1 and 3.2 above solve the topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many Topology textbooks manage not to mention this fundamental result.

3.4 About proofs of the classification theorems

The proofs of Theorems 3.1 and 3.2 above are elementary. They can be found, e.g., in [Fuks&Rokhlin1984, Sections 3.1.1.16-19]. The core of them are the following simple lemmas:

Lemma 3.3. Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$$\Rr$ is homeomorphic either to $\Rr$$\Rr$ or $S^1$$S^1$.

Lemma 3.4. If a topological space $X$$X$ can be represented as the union of a nondecreasing sequence of open subsets, all homeomorphic to $\Rr$$\Rr$, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.

3.5 Corollaries: homotopy classification

Theorem 3.5. Each connected 1-manifold is either contractible, or homotopy equivalent to circle.

It follows immediately from Theorem 3.1.

3.6 Corollaries: cobordisms of 0-manifolds

Theorem 3.6. A compact 0-manifold $X$$X$ bounds a compact 1-manifold iff the number of points in $X$$X$ is even.

Corollary 3.7. Two compact 0-manifolds are cordant iff their numbers of points are congruent modulo 2.

4 Orders and orientations

4.1 Interval topology

Most of properties specific for 1-manifolds can be related to the fact that the topological structure in a 1-manifold is defined by linear or cyclic ordering of points.

Open intervals $(a,b)=\{x\in\Rr\mid a<x<b\}$$(a,b)=\{x\in\Rr\mid a<x<b\}$ and open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$ form the base of the (standard) topology on $\Rr$$\Rr$. On $\Rr_+$$\Rr_+$ and $I$$I$, the (standard) topology is induced from the standard topology on $\Rr$$\Rr$, and can be described in terms of the induced order.

This way of introducing a topological structure can be applied in any (linearly) ordered set $X$$X$.

On $S^1$$S^1$ the topology is defined by the standard cyclic order. A cyclic order on a set $X$$X$ is usually defined as a ternary relation on $X$$X$. It can be described also by fixing a linear order $x<_ay$$x<_ay$ on $X\smallsetminus a$$X\smallsetminus a$ for each $a\in X$$a\in X$. The linear orders on $X\smallsetminus a$$X\smallsetminus a$ and $X\smallsetminus b$$X\smallsetminus b$ describe the same cyclic order on $X$$X$ if $b<_ax\Longleftrightarrow x<_ba$$b<_ax\Longleftrightarrow x<_ba$ for each $x\in X\smallsetminus\{a,b\}$$x\in X\smallsetminus\{a,b\}$. Thus all the orders $<_a$$<_a$ can be recovered form any one of them, and hence the cyclic order is determined by any one of the orders $<_a$$<_a$.

Open rays in $S^1\smallsetminus a$$S^1\smallsetminus a$ for all $a\in S^1$$a\in S^1$ form the base of the standard topology on $S^1$$S^1$. The same way of introducing a topological structure can be applied in any cyclically ordered set.

Topological structure of either kind, on a linearly ordered or cyclically ordered set, desrcibed above will be called interval topology.

4.2 Orientations

An orientation of a 1-manifold can be interpreted via linear or cyclic orderings of their points.

An orientation of a connected non-closed 1-manifold is a linear order on the set of its points such that the corresponding interval topology coincides with the topology of this manifold.

Theorem 4.1. Every connected non-closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected non-closed 1-manifold is homeomorphic either to $\Rr$$\Rr$, or $\Rr_+$$\Rr_+$, or $I$$I$. On each of these 1-manifolds there are two linear orders, $<$$<$ and $>$$>$, defining the topology. We have to prove that there is no other linear order defining the same topology.

A linear order $<$$<$ on a set $X$$X$ is encoded in the system of sets $X_a=\{x\in X\mid a<x\}$$X_a=\{x\in X\mid a<x\}$ for $a\in X$$a\in X$. A system of susbsets $X_a\subset X\smallsetminus a$$X_a\subset X\smallsetminus a$ with $a\in X$$a\in X$ comes from a linear order on $X$$X$ iff either $X_a\subset X_b$$X_a\subset X_b$, or $X_a\supset X_b$$X_a\supset X_b$, or $X_a=X_b$$X_a=X_b$ for any $a,b\in X$$a,b\in X$ and $X_a=X_b$$X_a=X_b$ only if $a=b$$a=b$. In the interval topology on $X$$X$, each $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ is open.

Each interior point $a$$a$ of a connected non-closed 1-manifold $X$$X$ splits $X$$X$ to two open connected sets. Any linear order on $X$$X$ defining the topology of $X$$X$ would give rise to splitting of each of these two sets into its intesections with $X_a$$X_a$ and $X\smallsetminus(X_a\cup a)$$X\smallsetminus(X_a\cup a)$ which are disjoint and open. Due to connectedness of the set, it coincides with one of these intersections. Hence the order coincides with one of two standard orders.

$\square$$\square$

An orientation of a connected closed 1-manifold is a cyclic order on the set of its points such that the topology of this cyclic order coincides with the topology of the 1-manifold.

Theorem 4.2. Every connected closed 1-manifold admits exactly two orientations.

Proof. By Theorem 3.2, a coonected closed 1-manifold is homeomorphic either to $S^1$$S^1$. On $S^1$$S^1$ there are cyclic orders opposite to each other that define the topology. We have to prove that there is no other cyclic order defining the same topology.

As a cyclic order is determined by a linear order on the complement of any point, orientations of a connected closed 1-manifold are defined by orientations of the complement of any point. By Theorem 4.1, the complement has exactly two orientations, so the circle cannot have more.

$\square$$\square$

An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation).

Corollary 4.3. Any 1-manifold admits an orientation. If the 1-manifold consists of $n$$n$ connected components, then it admits $2^n$$2^n$ orientations.

4.3 Self-homeomorphisms

Theorem 4.4. A map $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a homeomorphism iff $h$$h$ is a monotone bijection.

Proof. Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a homeomorphism. First, observe that $h$$h$ maps every ray to a ray. Indeed, for any $x\in\Rr$$x\in\Rr$, the map $h$$h$ induces a homeomorphism $\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$$\Rr\smallsetminus x\to\Rr\smallsetminus h(x)$. The rays $(-\infty,x)$$(-\infty,x)$ and $(x,\infty)$$(x,\infty)$ are connected components of $\Rr\smallsetminus x$$\Rr\smallsetminus x$. Therefore their images are connected components $(-\infty,h(x))$$(-\infty,h(x))$ and $(h(x),\infty)$$(h(x),\infty)$ of $\Rr\smallsetminus h(x)$$\Rr\smallsetminus h(x)$.

Observe that rays have the same direction iff one of them is contained in the other one, that is their intersection is a ray. Therefore two rays of the same dirction are mapped by $h$$h$ to rays with the same direction. Thus rays $(x,+\infty)$$(x,+\infty)$ are mapped either all to rays $(h(x),+\infty)$$(h(x),+\infty)$ or all to $(-\infty,h(x))$$(-\infty,h(x))$. Thus $h$$h$ is monotone.

Let $h:\Rr\to\Rr$$h:\Rr\to\Rr$ be a monotone bijection. Then the image and preimage under $h$$h$ of any open interval is an open interval. Therefore, both $h$$h$ and $h^{-1}$$h^{-1}$ are continuous, and hence $h$$h$ is a homeomorphism.

$\square$$\square$

The following theorem can be proved similarly or can be deduced from Theorem 4.4

Theorem 4.5.

1. A map $h:I\to I$$h:I\to I$ is a homeomorphism iff $h$$h$ is a monotone bijection.
2. A map $h:\Rr_+\to\Rr_+$$h:\Rr_+\to\Rr_+$ is a homeomorphism iff $h$$h$ is a monotone increasing bijection.
3. A map $h:S^1\to S^1$$h:S^1\to S^1$ is a homeomorphism iff $h$$h$ is a bijection that either preserves or reverses the cyclic order of points on $S^1$$S^1$.

Any self-homeomorphism of a connected 1-manifold either preserve orientation, or reverses it. The half-line $\Rr_+$$\Rr_+$ does not admit a self-homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\Rr_+$ admits an orientation reversing map.

Thus, there are 5 topological types of oriented connected 1-manifolds: the topological type of the non-oriented half-line splits into the oriented topological types of $\Rr_+$$\Rr_+$ and $\Rr_-$$\Rr_-$ with the orientations induced by the standard order.

4.4 Characterizations of connected compact 1-manifolds in terms of separating points

A subset $A$$A$ of a topological space $X$$X$ is said to separate $X$$X$ if $X\smallsetminus A$$X\smallsetminus A$ can be presented as a union of two disjoint open sets.

Theorem 4.6. (See [Moor1920].) Let $X$$X$ be a connected compact Hausdorff second countable topological space.

1. If every two points separate $X$$X$, then $X$$X$ is homeomorphic to the circle.
2. If each point, with two exceptions, separates $X$$X$, then $X$$X$ is homeomorphic to $I$$I$.

Any point $a\in\Rr$$a\in\Rr$ splits $\Rr$$\Rr$ to two disjoint open rays $(-\infty,a)=\{x\in\Rr\mid x<a\}$$(-\infty,a)=\{x\in\Rr\mid x<a\}$ and $(a,\infty)=\{x\in\Rr\mid a<x\}$$(a,\infty)=\{x\in\Rr\mid a<x\}$.

Theorem 4.7. (See [Ward1936].) Let $X$$X$ be a connected locally compact Hausdorff second countable topological space.

1. If the complement of each point in $X$$X$ consists of two connected components, then $X$$X$ is homeomorphic to $\Rr$$\Rr$.
2. If $X$$X$ contains a point

   Tex syntax error

   $b$ such that $X\smallsetminus b$$X\smallsetminus b$ is connected and $X\smallsetminus a$$X\smallsetminus a$ consists of two connected components for each $a\in X$$a\in X$, $a\ne b$$a\ne b$, then $X$$X$ is homeomorphic to $\Rr_+$$\Rr_+$.

5 Invariants

5.1 Basic invariants

As follows from the Theorems 3.1 and 3.2 above, the following invariants

• the number of connected components,
• the compactness of each connected component,
• and the number of boundary points of each connected component

determine the topological type of a 1-manifold.

5.2 Homotopy invariants

The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial. The fundamental group $\pi_1(X,x_0)$$\pi_1(X,x_0)$ is infinite cyclic group, if the connected component of $X$$X$ containing $x_0$$x_0$ is homeomorphic to circle, and trivial otherwise.

5.3 Homology invariants

Let $X$$X$ be a 1-manifold with a finite number of connected components. By Theorem 3.1, it is homeomorphic to a disjoint union

of some numbers of copies of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, $S^1$$S^1$ and $I$$I$:

$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$ $$\displaystyle X=a \Rr\amalg b\Rr_+ \amalg c S^1\amalg d I.$$

Then $H_0(X)$$H_0(X)$ is a free abelian group of rank equal to the number $a+b+c+d$$a+b+c+d$ of all connected components of $X$$X$ and $H_1(X)$$H_1(X)$ is a free abelian group of rank equal to the number $c$$c$ of closed (compact without boundary) components of $X$$X$.

Relative homology groups: $H_0(X,\partial X)$$H_0(X,\partial X)$ is a free abelian group of rank equal to the number $a+c$$a+c$ of connected components of $X$$X$ without boundary; $H_1(X,\partial X)$$H_1(X,\partial X)$ is a free abelian group of rank equal to the number $c+d$$c+d$ of compact components of $X$$X$. So,

$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$ $$\displaystyle H_0(X)=\mathbb Z^{a+b+c+d},\ \ \ H_1(X)=\mathbb Z^c, \ \ \ H_0(X,\partial X)=\mathbb Z^{a+c},\ \ \ H_1(X,\partial X)=\mathbb Z^{c+d}.$$

Numbers $a$$a$,

Tex syntax error

$b$, $c$$c$, and $d$$d$ and the topological type of $X$$X$ can be recovered from the ranks of these groups.

Above by homology we mean homology with compact support. The homology with closed support (Borel-Moore homology):

$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$ $$\displaystyle H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^{fr}_1(X)=\mathbb Z^{a+c}, \ \ \ H^{fr}_0(X,\partial X)=\mathbb Z^{c},\ \ \ H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}.$$

The Poincare duality is an isomorphism between usual cohomogy (recall that the usual cohomology has closed support) and the relative Borel-Moore homology of the complementary dimension. So

$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$ $$\displaystyle H^1(X;\mathbb Z)\to H^{fr}_0(X,\partial X)=\mathbb Z^{c}, \ \ \ H^0(X;\mathbb Z)\to H^{fr}_1(X,\partial X)=\mathbb Z^{a+b+c+d}, \ \ \ H^1(X,\partial X;\mathbb Z)\to H^{fr}_0(X)=\mathbb Z^{c+d}, \ \ \ H^0(X,\partial X;\mathbb Z)\to H^{fr}_1(X)=\mathbb Z^{a+c}.$$

A local coefficient system on a 1-manifold homeomorphic to the circle, may be non-trivial. E.g., if the local coefficient system over $S^1$$S^1$ has non-trivial monodromy, then all the homology groups are trivial.

5.4 Tangent bundle invariants

The tangent bundles of 1-manifolds are trivial. Thus all the characteristic classes are trivial.

6 Additional structures

6.1 Triangulations

Any 1-manifold admits a triangulation. A triangulation of a non-compact connected 1-manifold is unique up to homeomorphism.

A compact 1-manifold has non-homeomorphic triangulations, but they are easy to classify up to homeomorphism. On circle the topological type of a triangulation is defined by the number of 1-simplices. This number can take any integral value $\ge3$$\ge3$. Similarly, the topological type of a triangulation of $I$$I$ is defined by the number of 1-simplices, which can take any positive integral value.

6.2 Smooth structures

Any 1-manifold admits a smooth structure.

If smooth 1-manifolds $X$$X$ and $Y$$Y$ are homeomorphic, then they are also diffeomorphic. Moreover,

Theorem 6.1. Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Proof. By Theorems 4.4 and 4.5, a homeomorphism is monotone in the appropriate sense. Choose a net of points in the source such that the image of each of them is sufficiently close to the images of its neighbors. Take a smooth monotone bijection coinciding with the homeomorphism at the chosen points.

$\square$$\square$

6.3 Inner metrics

Recall that a metric on a path-connected space is said to be inner if the distance between any two points is equal to the infimum of lengths of paths connecting the points, and that the length of a path $s:I\to X$$s:I\to X$ in a metric space $X$$X$ with metric $d:X\times X\to \Rr_+$$d:X\times X\to \Rr_+$ is $\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$$\inf\{\sum_{i=1}^n d(s(t_{i-1}),s(t_{i})\mid \text{ all sequences } 0=t_0<t_1<\dots<t_n=1\}$.

Any connected 1-manifold admits an inner metric. A connected 1-manifold with an inner metric is defined up to isometry by the diameter of the space. Recall that the diameter of a metric space $X$$X$ with metric $d:X\times X\to\Rr$$d:X\times X\to\Rr$ is $\sup\{ d(x,y)\mid x,y\in X\}$$\sup\{ d(x,y)\mid x,y\in X\}$.

For each value of the diameter there is a standard model for the inner metric space. For the four homeomorphism types of connected 1-manifolds these standard models are as follows.

1. For $\Rr$$\Rr$ with diameter $D\in (0,\infty]$$D\in (0,\infty]$ this is $(-D/2,D/2)$$(-D/2,D/2)$.
2. For $\Rr_+$$\Rr_+$ with diameter $D\in(0,\infty]$$D\in(0,\infty]$ this is $[0,D)$$[0,D)$.
3. For a circle with inner metric of diameter $D\in (0,\infty)$$D\in (0,\infty)$ this is the circle $\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$$\{(x,y)\in\Rr^2\mid x^2+y^2=D^2/\pi^2\}$ of radius $D/\pi$$D/\pi$ on the plane with the inner metric.
4. For $I$$I$ with diameter $D\in(0,\infty)$$D\in(0,\infty)$ this is $[0,D]$$[0,D]$.

An inner metric on a connected 1-manifold defines a unique smooth structure on the manifold, namely, the smooth structure induced by the isometry to the corresponding standard model from the list above.

7 Constructions

7.1 Surgery

Any 1-manifold can be transformed by surgeries to any other 1-manifold with the same boundary.

If given two 1-manifolds with the same boundary are oriented and the induced orientations on the boundary coincide, then the surgery can be chosen preserving the orientation (this means that the corresponding cobordism is an oriented 2-manifold and its orientation induces on the boundary the given orientation on one of the 1-manifolds and the orientation opposite to the given one on the other 1-manifold).

An index 1 surgery preserving orientation on closed 1-manifold changes the number of connected components by 1. An index 1 surgery on a closed 1-manifold, which does not preserve any orientation, preserves the number of connected components.

7.2 Connected sums

The notion of connected sum is defined for 1-manifolds, but the connectivity of the outcome is different in dimension 1 compared to other dimensions. Indeed term connected sum can be misleading in dimension 1 since a connected sum of connected 1-manifolds may be not connected. For example a connected sum of two copies of $\Rr$$\Rr$ is a disjoint sum of two copies of $\Rr$$\Rr$.

Note that connected sum is only a well defined operation on oriented manifolds and one has to be careful with the orientations. For example

$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$ $$\displaystyle \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$$

8 Groups of self-homeomorphisms

8.1 Mapping class groups

Recall that the mapping class group of a manifold $X$$X$ is the quotient group of the group $\operatorname{Homeo} (X)$$\operatorname{Homeo} (X)$ of all homeomorphisms $X\to X$$X\to X$ by the normal subgroup of homeomorphisms isotopic to the identity. In other words, the mapping class group of $X$$X$ is $\pi_0(\operatorname{Homeo}(X))$$\pi_0(\operatorname{Homeo}(X))$.

An orientation reversing homeomorphism cannot be isotopic to an orientation preserving homeomorphism. For auto-homeomorphisms of a connected 1-manifold this is the only obstruction to being isotopic:

Theorem 8.1. Any two auto-homeomorphisms of a connected 1-manifold that are either both orientation preserving, or both orientation reversing are isotopic.

This is a corollary of the following two obvious lemmas.

Lemma 8.2. On rectilinear isotopy. Let $X$$X$ be one of the following 1-manifolds: $\Rr$$\Rr$, $\Rr_+$$\Rr_+$, or $I$$I$. Let $f,g:X\to X$$f,g:X\to X$ be two monotone bijections that are either both increasing or both decreasing. Then the family $h_t=(1-t)f+tg:X\to X$$h_t=(1-t)f+tg:X\to X$ with $t\in[0,1]$$t\in[0,1]$ consists of monotone bijections (and hence is an isotopy between $f$$f$ and $g$$g$).

Lemma 8.3. Let $f,g:S^1\to S^1$$f,g:S^1\to S^1$ be two bijections that either both preserve or both reverse the standard cyclic order of points on $S^1$$S^1$. Let $f$$f$ and $g$$g$ coincide at $x\in S^1$$x\in S^1$. Then $f$$f$ and $g$$g$ are isotopic via the canonical isotopy which is stationary at $x$$x$ and is provided on the complement of $x$$x$ by stereographic projections and the rectilinear isotopy from Lemma 8.2 of the corresponding self-homeomorphisms of $\Rr$$\Rr$.

Corollary 8.4. $\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$$\pi_0(\operatorname{Homeo}(S^1)) \cong \pi_0(\operatorname{Homeo}(\Rr)) \cong \pi_0(\operatorname{Homeo}(I)) \cong \mathbb{Z}/2$ and $\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$$\pi_0(\operatorname{Homeo} \Rr_+) \cong 0.$

Remark. All the statements in this section remains true, if everywhere the word homeomorphism is replaced by the word diffeomorphism and $\operatorname{Homeo}$$\operatorname{Homeo}$ is replaced by $\operatorname{Diffeo}$$\operatorname{Diffeo}$.

8.2 Homotopy types of groups of self-homeomorphisms

The group $\operatorname{Homeo}(S^1)$$\operatorname{Homeo}(S^1)$ contains $O(2)$$O(2)$ as a subgroup, which is its deformation retract. It follows from Lemma 8.3. More precisely, for each point $x\in S^1$$x\in S^1$, Lemma 8.3 provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$$\operatorname{Homeo}(S^1)\to O(2)$.

Similarly, the group of self-homeomorphisms of $S^1$$S^1$ isotopic to identity contains $SO(2)=S^1$$SO(2)=S^1$ as a subgroup, which is its deformation retract.

The groups of self-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 8.2 applied to $f=\operatorname{id}$$f=\operatorname{id}$ and an arbitrary $g$$g$.

Thus for each connected 1-manifold $X$$X$ the group of homeomorphisms $X \to X$$X \to X$ isotopic to identity is homotopy equivalent to $X$$X$.

9 Finite group actions

Consider an action of a finite group $G$$G$ on a 1-manifold $X$$X$.

9.1 Free actions

For any point $a\in X$$a\in X$, its orbit $Ga\subset X$$Ga\subset X$ is a finite set and has an invariant neighborhood $U$$U$ whose connected components are disjoint open sets, each of them contains exactly one point of $Ga$$Ga$, and either all the components are homeomorphic to $\Rr$$\Rr$, or all homeomorphic to $\Rr_+$$\Rr_+$.

If the action is free, then the orbit space $X/G$$X/G$ is a 1-manifold and the natural projection $X\to X/G$$X\to X/G$ is a covering.

Therefore the theory of coverings gives a simple classification of free finite group actions on 1-manifolds.

A contractible 1-manifold has no non-trivial covering. Thus, if a free finite group action on 1-manifold $X$$X$ has a contractible orbit space $Y=X/G$$Y=X/G$, then $X$$X$ is a disjoint union of copies of $Y$$Y$ and $G$$G$ permutes these copies. In particular, there is no non-trivial free group action on a connected 1-manifold having contractible orbit space.

Coverings $X\to S^1$$X\to S^1$ with connected $X$$X$ are in one-to-one correspondence with subgroups of finite indices of $\pi_1(S^1)=\mathbb Z$$\pi_1(S^1)=\mathbb Z$. For each index $m\in\mathbb Z$$m\in\mathbb Z$ there is one subgroup, and hence one covering. The total space is homeomorphic to $S^1$$S^1$, and the covering is equivalent to $S^1\to S^1:z\mapsto z^m$$S^1\to S^1:z\mapsto z^m$. In the corresponding action, the group is cyclic of order $m$$m$, it acts on $S^1$$S^1$ by rotations.

In this classification of free finite group actions on connected 1-manifolds, the orbit space plays the main role. However, it is easy to reformulate it with emphasis the 1-manifold on which the group acts. This is done in the next two theorems.

Theorem 9.1. There is no non-trivial free finite group action on a contractible 1-manifold.

Theorem 9.2. If a finite group $G$$G$ acts freely on the circle than $G$$G$ is cyclic. Any finite cyclic group has a linear free action on $S^1$$S^1$. Any free action of a finite cyclic group on $S^1$$S^1$ is conjugate to a linear action.

9.2 Asymmetry of a half-line

Theorem 9.3. There is no non-trivial action of a finite group in $\Rr_+$.

Proof. We will prove that the only homeomorphism $h:\Rr_+\to\Rr_+$ of finite order is the identity. Obsreve, first that any homeomorphism $\Rr_+\to\Rr_+$ preserves the only boundary point $0\in\Rr_+$. Assume that $h$ is a homeomorphism $\Rr_+\to\Rr_+$ of finite order $m$, and there exists $a\in\Rr_+$ such that $h(a)=b\ne a$. Then $h([0,a])=[0,b]$. Without loss of generality, we may assume that $b<a$ (otherwise just replace $h$ by $h^{-1}$).

Then $h([0,a]=[0,b]\subset[0,a]$ and $a\not\in h([0,a])$. By the assumption, $h^m(a)=a$. On the other hand, $h^m(a)\in h^m([0,a])\subset h^{m-1}([0,a])\subset\dots \subset h([0,a])\not\ni a$.

$\square$$\square$

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9.3 Actions on line and segment

Theorem 9.4. The only orientation preserving homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is the identity.

Proof. Fix a homeomorphism $f:(0,\infty)\to\Rr$$f:(0,\infty)\to\Rr$ (say, define it by formula $f:x\mapsto (x^2- 1)/x$$f:x\mapsto (x^2- 1)/x$). Consider a homeomorphism $(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$$(0,\infty)\to(0,\infty): x\mapsto f^{-1}hf(x)$. It preserves orientation (since $h$$h$ preserves orientation). So, it is a monotone increasing bijection $(0,\infty)\to(0,\infty)$$(0,\infty)\to(0,\infty)$ of finite order. It can be extended to $\Rr_+$$\Rr_+$ by letting $0\mapsto0$$0\mapsto0$. The extended homeomorphism has the same finite order. But by Theorem 9.3 any such homeomorphism is the identity.

$\square$$\square$

Theorem 9.5. Any orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ of finite order is of order two. It is conjugate to the symmetry against a point.

Proof. An orientation reversing homeomorphism $h:\Rr\to\Rr$$h:\Rr\to\Rr$ is a monotone decreasing bijection. Consider the function $x\mapsto h(x)-x$$x\mapsto h(x)-x$. It is also monotone decreasing bijection $\Rr\to\Rr$$\Rr\to\Rr$ and hence there exists a unique $a\in\Rr$$a\in\Rr$ such that $h(a)-a=0$$h(a)-a=0$, that is $h(a)=a$$h(a)=a$.

The homeomorphism $h$$h$ maps each connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$ to a connected component of $\Rr\smallsetminus a$$\Rr\smallsetminus a$. The connected components are open rays $(-\infty,a)$$(-\infty,a)$ and $(a,\infty)$$(a,\infty)$. If each of them is mapped to itself, then $h$$h$ defines a homeomorphism of a finite order of the closed rays $(-\infty,a]$$(-\infty,a]$ and $[a,\infty)$$[a,\infty)$. Then by Theorem 9.3, $h$$h$ is identity, which contradicts to our assumption. Thus, $h([a,\infty))=(-\infty,a]$$h([a,\infty))=(-\infty,a]$ and $h((-\infty,a])=[a,\infty)$$h((-\infty,a])=[a,\infty)$. Then $h^2$$h^2$ preserves the rays, and, by Theorem 9.3, is identity. Thus $h$$h$ has order two.

Choose a homeomorphism $f:[0,\infty)\to[a,\infty)$$f:[0,\infty)\to[a,\infty)$. Define function $g:(-infty,0]\to(-\infty,a]$$g:(-infty,0]\to(-\infty,a]$ by formula $g(x)=hf(-x)$$g(x)=hf(-x)$. It's a homeomorphism. Together, $f$$f$ and $g$$g$ form a homeomorphism $\phi:\Rr\to\Rr$$\phi:\Rr\to\Rr$. As easy to check, $\phi^{-1}h\phi(x)=-x$$\phi^{-1}h\phi(x)=-x$.

$\square$$\square$

Theorem 9.6. A non-trivial finite group acting effectively on $\Rr$$\Rr$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. As follows from Corollaries 9.4 and 9.5, any non-trivial element of the group is an orientation reversing involution. We have to prove that the group contains at most one such element. Assume that there are two orientation reversing homeomorphisms, $f$$f$ and $g$$g$ of the line $\Rr$$\Rr$. Their composition $f\circ g$$f\circ g$ preserves orientation. Since it belongs to a finite group, it has finite order. By Theorem 9.4, it is identity. So, $fg=1$$fg=1$ and hence $f=g^{-1}$$f=g^{-1}$. But $g^2=1$$g^2=1$. Therefore $g^{-1}=g$$g^{-1}=g$ and $f=g^{-1}=g$$f=g^{-1}=g$.

$\square$$\square$

Corollary 9.7. A non-trivial finite group acting effectively on $I$$I$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

Proof. Any auto-homeomorphism of $I$$I$ preserves the boundary and the interior of $I$$I$. Hence an effective finite group action on $I$$I$ induces an action of finite group on the interior of $I$$I$. An auto-homeomorphism of $I$$I$ is recovered from its restriction to the interior. Moreover, any auto-homeomorphism of $\operatorname{Int}I$$\operatorname{Int}I$ has a unique extension to auto-homeomorphism of $I$$I$. The interior $\operatorname{Int}I$$\operatorname{Int}I$ is homeomorphic to $\Rr$$\Rr$.

$\square$$\square$

9.4 Actions on circle

{{beginthm|Theorem} Any periodic orientation reversing homeomorphism $S^1\to S^1$$S^1\to S^1$ is an involution (i.e., has period 2). It is conjugate to a symmetry of $S^1$$S^1$ against its diameter. </div>

Proof. Observe first that any orientation reversing auto-homeomorphism of the circle has a fixed point. One can prove this by elementary arguments, but we just refer to the Lefschetz Fixed Point Theorem: the Lefschetz number of such homeomorphism is 2.

Consider the complement of a fixed point. The homeomorphism resticted to it satisfies the conditions of Corollary 9.5, which gives the required result.

$\square$$\square$

Observe that by theorems 9.5, 9.7, and 9.7 any non-identity periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

Theorem 9.8. A periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ has no fixed point. It is conjugate to a rotation.

Proof. If if had a fixed point, then we could consider its restriction to the complement of this point, and by Theorem 9.4 would conclude that it is identity and hence the whole homeomorphism is identity.

For the same reasons, the non-identity powers of our periodic non-identity orientation preserving homeomorphism $S^1\to S^1$$S^1\to S^1$ have no fixed points. Therefore, these powers form a cyclic group freely acting on $S^1$$S^1$. See Theorem 9.2.

$\square$$\square$

Theorem 9.9. A finite group acting effectively on $S^1$$S^1$ is either cyclic or dihedral, and the action is conjugate to a linear one. The standard actions of cyclic and dihedral groups on circle are provided by the symmetry groups of regular polygons.

Proof. If all the homeomorphisms in the action preserve orientation, then by Theorem 9.8 the action is free, and the result follows from Theorem 9.2.

Assume that the action contains an orientation reversing homeomorphism. The orientation preserving homeomorphisms from the action form a cyclic subgroup as above. It is of index 2. Its complement consists of orientation reversing involutions. If the subgroup of orientation preserving homeomorphisms is trivial, then the whole group is of order 2 and the only non-trivial element is an orientation resersing involution. When the group contains two orientation preserving homeomorphisms, the whole group is the cartesian product of two cyclic groups of order 2. It is called Klein's Vierergruppe or dihedral group $D_2$$D_2$. If the number of orientation preserving homeomorphisms is $n>2$$n>2$, then the whole group is called the dihedral group $D_n$$D_n$. It is the symmetry group of an $n$$n$-sided regular polygon.

$\square$$\square$

10 Relatives of 1-manifolds

10.1 Non-Hausdorff 1-manifolds

If we enlarge the collection of spaces by getting rid of the Hausdorff property, then the number of topological types of connected spaces becomes uncountable.

Indeed, one can take the disjoint union of two copies of the line $\Rr$$\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The quotient space is connected and satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom. In this way one can construct uncountably many pairwise non-homeomorphic spaces. To prove that they are not homeomorphic, one can use, for example, the topological type of the subset formed by those points that do not separate the space.

11 References

• [Fuks&Rokhlin1984] D. B. Fuks and V. A. Rokhlin, Beginner's course in topology. Geometric chapters. Translated from the Russian by A. Iacob. Universitext. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1984. MR759162 (86a:57001) Zbl 0562.54003
• [Gale1987] D. Gale, The Teaching of Mathematics: The Classification of 1-Manifolds: A Take-Home Exam, Amer. Math. Monthly 94 (1987), no.2, 170–175. MR1541035 Zbl 0621.57001
• [Ghys2001] E. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no.3-4, 329–407. MR1876932 (2003a:37032) Zbl 1044.37033
• [Moor1920] R.L.Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), 333—347.
• [Ward1936] A. J. Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198. MR1577110 Zbl 62.0693.02

12 External links

• The Encylopedia of Mathematics article on one-dimensional manifolds.
• The Encylopedia of Mathematics article on lines.
• The Wikipedia page about curves.