1-manifolds

Revision as of 18:06, 1 October 2009

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 neighborhood homeomorphic either to line $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; 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 neighborhood homeomorphic either to line $\Rr $ or half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$. 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 $\mathbb R^n$. To be on the safe side, we use an unambiguous term ''manifold of dimension 1'' or ''1-manifold''. </wikitex> == Examples == <wikitex>; * Real line $\mathbb R$ * Half-line $\mathbb R_+$ * Circle $S^1=\{(x,y)\in\Rr^2\mid x^2+y^2=1\}$ * Closed interval $I=[0,1]$ </wikitex> <!-- COMMENT O.V.: About usage of Euler characteristic. A non-compact space has many Euler characteristics. The notion of Euler characteristic depends on the properties that one wants to preserve. If one wants to have invariance under homotopy equivalences, then the additivity would be lost. If one wants to keep additivity, then it is not the alternating sum of Betti numbers. If one wants to keep topological invariance and additivity, then $\chi(\Rr)=\chi(I)-2\chi(pt)=1-2=-1$, similarly $\chi(\Rr_+)=0$. In all the cases under consideration the latter Euler characteristic is the alternating sum of the ranks of homology groups with closed support (Borel-Moore homology). Pretty legitimate! I would not recommend to use the Euler characteristic in a non-compact situation without a detailed specification of the definition. The problem of recognizing of the topological type is solved by the two invariants that I mentioned below: compactness and presence of boundary. Other invariants can be used and deserve consideration. However, I suggest to place the section Invariants after the section Classification. END OF COMMENT --> == Classification == ==== Reduction to classification of connected manifolds ==== <wikitex>; Any manifold is homeomorphic to disjoint sum of its connected components. A connected component of a 1-manifold is a 1-manifolds. Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic. </wikitex> ==== Topological classification of connected 1-manifolds ==== <wikitex>; {{beginthm|Theorem}} 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}} </wikitex> ==== Characterizing the topological type of a connected 1-manifold ==== <wikitex>; * Any connected closed 1-manifold is homeomorphic to $S^1$. * Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$. * 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 $\Rr_+$. 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. </wikitex> ==== Remarks ==== <wikitex>; Proofs of the results above are elementary. The core of them is the following simple {{beginthm|Lemma}} Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$ is homeomorphic either to $\Rr$ or $S^1$.{{endthm}} The theorems 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. </wikitex> === Corollaries === ==== Homotopy classification of 1-manifolds ==== <wikitex>; Each connected 1-manifold is either contractible, or homotopy equivalent to circle. </wikitex> ==== 0-manifolds cobordant to zero ==== <wikitex>; A compact 0-manifold $X$ bounds a compact 1-manifold iff the number of points in $X$ is even. </wikitex> ==== Smooth structures ==== <wikitex>; ''Any 1-manifold admits a smooth structure.'' If smooth 1-manifolds $X$ and $Y$ are homeomorphic, then they are also diffeomorphic. Moreover, any homeomorphism $X\to Y$ can be approximated in the $C^0$-topology by a diffeomorphism. Technically this can be considered as a corollary of the following simple theorem: a map $\Rr\to\Rr$ is a homeomorphism iff it is a continuous monotone bijection. </wikitex> == Invariants == <wikitex>; As follows from the classification theorems, ''connectedness'', ''compactness'' and presence of ''boundary'' play fundamental role in topology of 1-manifolds. Homotopy invariants are extremely simple. </wikitex> == Further discussion == ==== Connected sums ==== <wikitex>; The very term ''connected sum'' is compromised in dimension 1. Indeed, 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 $\Rr$. Moreover, connected sum, as an operation on topological types of 1-manifolds, is not well-defined. Indeed, both $I\amalg\Rr$ and $\Rr_+\amalg\Rr_+$ can be presented as a connected sum of two copies of $\Rr_+$. </wikitex> ==== Orientations ==== <wikitex>; Orientation of a 1-manifold can be interpreted as 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. Half-line $\Rr_+$ does not admit a homeomorphism reversing orientation. Any connected 1-manifold non-homeomorphic to $\Rr_+$ admits an orientation reversing map. </wikitex> ==== Mapping class groups ==== <wikitex>; </wikitex> ==== Finite group actions ==== <wikitex>; </wikitex> ==== Surgery ==== <wikitex>; </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}\Rr$ or half-line $\Rr_+=\{x\in\Rr\mid x\ge0\}$.

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 $\Rr^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

2 Examples

• Real line $\mathbb R$

• Half-line $\mathbb R_+$

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

• Closed interval $I=[0,1]$

3 Classification

3.1 Reduction to classification of connected manifolds

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

A connected component of a 1-manifold is a 1-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

3.3 Characterizing the topological type of a connected 1-manifold

• Any connected closed 1-manifold is homeomorphic to $S^1$.

• Any connected compact 1-manifold with non-empty boundary is homeomorphic to $I$.

• 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 $\Rr_+$.

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.

3.4 Remarks

Proofs of the results above are elementary. The core of them is the following simple

Lemma 3.2. Any connected 1-manifold covered by two open subsets homeomorphic to $\Rr$ is homeomorphic either to $\Rr$ or $S^1$.

The theorems 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.

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 two copies of line $\Rr$ and identify an open set in one of them with its copy in the other one by the identity map. The result satisfies all the requirements from the definition of 1-manifold except the Hausdorff axiom.

3.5 Corollaries

3.5.1 Homotopy classification of 1-manifolds

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

3.5.2 0-manifolds cobordant to zero

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

3.5.3 Smooth structures

Any 1-manifold admits a smooth structure.

If smooth 1-manifolds $X$ and $Y$ are homeomorphic, then they are also diffeomorphic. Moreover,
any homeomorphism $X\to Y$ can be approximated in the $C^0$-topology by a diffeomorphism.

Technically this can be considered as a corollary of the following simple theorem:
a map $\Rr\to\Rr$ is a homeomorphism iff it is a continuous monotone bijection.

4 Invariants

As follows from the classification theorems,

• the number of connected components,
• compactness of a connected component,
• and boundary of a connected component
  

play fundamental role in topology of 1-manifolds.

Homotopy invariants are extremely simple.

5 Further discussion

5.1 Connected sums

The notion of connected sum is defined for 1-manifolds, but it does not work in the same way as for manifolds of higher dimensions.

Recall that a manifold $X$ of dimension $n$ is a connected sum of manifolds $A$ and $B$ if there are $n$-disks $D_A\subset A$ and $D_B\subset B$ and a homeomorphism $h:D_A\to D_B$ such that $X$ is homeomorphic to $(A\cup_h B)\smallsetminus\operatorname{Int}D_A$ (that is to the result of attaching of $A$ to $B$ by $h$ with the image of interior of the identified disks $D_A$ and $D_B$ removed). One writes $X=A\sharp B$, however the topology of $X$ may depend not only on $A$ and $B$, but also on $D_A$, $D_B$ and $h$. The dependence disappears if all connected components of $A$ are homeomorphic to each other, all connected components of $B$ are homeomorphic to each other, and a homeomorphism $D_A\to D_A$ reversing orientation can be extended to a homeomorphism $A\to A$, or a homeomorphism $D_B\to D_B$ reversing orientation can be extended to a homeomorphism $B\to B$. For a collection of manifolds satisfying these conditions, a connected sum can be considered as operation on topological types.

The very term connected sum is compromised in dimension 1. Indeed, 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 $\Rr$.

Moreover, connected sum, as an operation on topological types of 1-manifolds, is not well-defined. Indeed, both disjoint sums $I\amalg\Rr$ and $\Rr_+\amalg\Rr_+$ can be presented as a connected sum of two copies of $\Rr_+$.

5.2 Orientations

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.

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.

5.3 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 $\pi_0(\operatorname{Homeo}(X))$.

A reversing orientation homeomorphism cannot be isotopic to the identity. For an auto-homeomorphism of a connected 1-manifold this is the only obstruction to being isotopic to the identity. Therefore $\pi_0(\operatorname{Homeo}(S^1))=\pi_0(\operatorname{Homeo}(\Rr))=\pi_0(\operatorname{Homeo}(I))=\mathbb{Z}/_2$, while $\pi_0(\operatorname{Homeo} \Rr_+)=0$.

5.4 Finite group actions

5.5 Surgery

6 References