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

Specific properties of 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.

For other expositions about $1$$1$-manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Chapter 3, Section 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 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

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

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

3.6 Characterizations of connected 1-manifolds in terms of set-theoretic topology

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 3.7. (See [Moor,1920].) 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$.

Theorem 3.8. (See [Ward,1936].) 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 $b$$b$ such that $X\smallsetminus b$$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_+$.

3.7 Non-Hausdorff complications

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.

4 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.

The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial.

The tangent bundles of 1-manifolds are trivial.

5 Further discussion

5.1 Self-homeomorphisms

Theorem 5.1. 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 5.1

Theorem 5.2.

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$.

5.2 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.

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.

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

Any 1-manifold admits an orientation.

The half-line $\Rr_+$$\Rr_+$ does not admit a homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\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.

5.3 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 5.3. Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Proof. By Theorems 5.1 and 5.2, 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$

5.4 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.

5.5 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.

5.6 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 5.4. 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 5.5. 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 5.6. 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$$x$ by stereographic projections and the rectilinear isotopy from Lemma 5.5 of the corresponding self-homeomorphisms of $\Rr$$\Rr$.

Corollary 5.7. $\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}$.

5.7 Homotopy types of groups of auto-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 5.6. More precisely, for each point $x\in S^1$$x\in S^1$, Lemma 5.6 provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$$\operatorname{Homeo}(S^1)\to O(2)$.

Similarly, the group of auto-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 auto-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 5.5 applied to $f=\operatorname{id}$$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$.

5.8 Finite group actions

There are no non-trivial free finite group actions on contractible 1-manifolds. 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.

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.

Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

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.

A non-trivial finite group acting effectively on $\Rr^1$$\Rr^1$ or $[0,1]$$[0,1]$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

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

5.9 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.

5.10 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

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

6 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
• [Moor,1920] R.L.Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), 333—347.
• [Ward,1936] A.J.Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198.

7 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|Chapter 3, Section 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: $I=[0,1]$ == 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 ==== ; {{beginthm|Theorem}} '''Homotopy classification of 1-manifolds.''' Each connected 1-manifold is either contractible, or homotopy equivalent to circle. {{endthm}} {{beginthm|Theorem}} '''0-manifolds cobordant to zero.''' A compact 0-manifold $X$ bounds a compact 1-manifold iff the number of points in $X$ is even. {{endthm}} ==== Characterizations of connected 1-manifolds in terms of set-theoretic topology ==== ; 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{Moor,1920}.) 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}} {{beginthm|Theorem}}\label{thm:non-compact-characterisation} (See \cite{Ward,1936}.) Let $X$ be a connected locally compact Hausdorff second countable topological space. # If the complement of each point in $X$ consists of two connected components, then $X$ is homeomorphic to $\Rr$. # If $X$ contains a point $b$ such that $X\smallsetminus b$ is connected and $X\smallsetminus a$ consists of two connected components for each $a\in X$, $a\ne b$, then $X$ is homeomorphic to $\Rr_+$. {{endthm}} ==== Non-Hausdorff complications ==== ; 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. == 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. The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$ are trivial. The tangent bundles of 1-manifolds are trivial. == Further discussion == ==== 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}} ==== 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. 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. An orientation of an arbitrary 1-manifold is a collection of orientations of its connected components (each component is equipped with an orientation). Any 1-manifold admits an orientation. The half-line $\Rr_+$ does not admit a 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. ==== 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}} ==== 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. ==== 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 ==== 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 auto-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 auto-homeomorphisms of $S^1$ isotopic to identity contains $SO(2)=S^1$ as a subgroup, which is its deformation retract. The groups of auto-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 ==== ; There are no non-trivial free finite group actions on contractible 1-manifolds. 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. 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. Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation. 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. A non-trivial finite group acting effectively on $\Rr^1$ or $[0,1]$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point. There is no non-trivial action of a finite group in $\Rr_+$. ==== 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 $$ \Rr_+ \sharp \Rr_+ \cong \Rr_+ \amalg \Rr_+ \quad \text{but} \quad \Rr_+ \sharp (-\Rr_+) \cong I \amalg \Rr_+ .$$ == 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.

Specific properties of 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.

For other expositions about $1$$1$-manifolds, see [Ghys2001], [Gale1987] and also [Fuks&Rokhlin1984, Chapter 3, Section 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 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

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

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

3.6 Characterizations of connected 1-manifolds in terms of set-theoretic topology

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 3.7. (See [Moor,1920].) 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$.

Theorem 3.8. (See [Ward,1936].) 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 $b$$b$ such that $X\smallsetminus b$$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_+$.

3.7 Non-Hausdorff complications

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.

4 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.

The homotopy invariants of 1-manifolds are extremely simple. All homology and homotopy groups of dimensions $>1$$>1$ are trivial.

The tangent bundles of 1-manifolds are trivial.

5 Further discussion

5.1 Self-homeomorphisms

Theorem 5.1. 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 5.1

Theorem 5.2.

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$.

5.2 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.

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.

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

Any 1-manifold admits an orientation.

The half-line $\Rr_+$$\Rr_+$ does not admit a homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$$\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.

5.3 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 5.3. Any homeomorphism between two smooth 1-manifolds can be approximated in the $C^0$$C^0$-topology by a diffeomorphism.

Proof. By Theorems 5.1 and 5.2, 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$

5.4 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.

5.5 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.

5.6 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 5.4. 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 5.5. 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 5.6. 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$$x$ by stereographic projections and the rectilinear isotopy from Lemma 5.5 of the corresponding self-homeomorphisms of $\Rr$$\Rr$.

Corollary 5.7. $\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}$.

5.7 Homotopy types of groups of auto-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 5.6. More precisely, for each point $x\in S^1$$x\in S^1$, Lemma 5.6 provides a deformation retraction $\operatorname{Homeo}(S^1)\to O(2)$$\operatorname{Homeo}(S^1)\to O(2)$.

Similarly, the group of auto-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 auto-homeomorphisms of $\Rr$$\Rr$, $\Rr_+$$\Rr_+$ and $I$$I$ which are isotopic to identity are contractible. The contraction is provided by the rectilinear isotopy from Lemma 5.5 applied to $f=\operatorname{id}$$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$.

5.8 Finite group actions

There are no non-trivial free finite group actions on contractible 1-manifolds. 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.

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.

Any non-trivial periodic homeomorphism of a connected 1-manifold with a fixed point is an involution reversing orientation.

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.

A non-trivial finite group acting effectively on $\Rr^1$$\Rr^1$ or $[0,1]$$[0,1]$ is a cyclic group of order 2. The action of the non-unit element is conjugate to the symmetry against a point.

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

5.9 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.

5.10 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

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

6 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
• [Moor,1920] R.L.Moore, Concerning simple continuous curves, Trans. Amer. Math. Soc. 21 (1920), 333—347.
• [Ward,1936] A.J.Ward, The topological characterization of an open linear interval, Proc. London Math. Soc.(2) 41 (1936), 191-198.

7 External links

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