1-manifolds

Revision as of 05:25, 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 general definition of manifold, a manifold of dimension 1 is a topological space which is second countable (i.e., has a countable base), satisfies the Hausdorff axiom (any two different point in it have disjoint neighborhoods) and each point of which has a neighborhood homeomorphic either to line $\Rr $ or half-line $\Rr_+=\{x\in\mathbb R\mid x\ge0\}$. 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> == Construction and examples == <wikitex>; Real line $\mathbb R$, half-line $\mathbb R_+$, circle $S^1=\{(x,y)\in\mathbb R^2\mid x^2+y^2=1\}$, closed interval $I=[0,1]$. </wikitex> == Invariants == <wikitex>; Let $M$ be a 1-manifold and let $(\chi, \pi_0, \pi_0 \partial)(M)$ be the triple consisting of the Euler characteristic of $M$, the number of connected components of $M$ and number of connected components of the boundary of $M$. * $(\chi, \pi_0, \pi_0 \partial )(S^1) = (0, 1, 0)$, * $(\chi, \pi_0, \pi_0 \partial )(I) = (1, 1, 2)$, * $(\chi, \pi_0, \pi_0 \partial )(\Rr) = (1, 1, 0)$, * $(\chi, \pi_0, \pi_0 \partial )(\Rr_+) = (1, 1, 1)$. </wikitex> == Classification and characterization == <wikitex>; Any 1-manifold is homeomorphic to disjoint sum of its connected components. The connected components of a 1-manifold are 1-manifolds. Two manifolds are homeomorphic iff there exists a one-to-one correspondence between their components such that the corresponding components are homeomorphic. Any connected 1-manifold is homeomorphic either to real line $\mathbb R$, half-line $\mathbb R_+$, circle $S^1=\{(x,y)\in\mathbb R^2\mid x^2+y^2=1\}$, closed interval $I=[0,1]$. Any connected closed 1-manifold is homeomorphic to the circle $S^1$. Any connected compact 1-manifold with non-empty boundary is homeomorphic to the interval $I$. Any connected non-compact 1-manifold without boundary is homeomorphic to the line $\mathbb R$. Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to the half-line $\mathbb R_+$. Thus we see that for connected 1-manifolds two invariants, compactness and presence of boundary, taking each two values form a complete system of topological invariants. The classification theorems stated above can be deduced from the following simple Lemma: a Hausdorff topological space which can be covered by two open subsets which are homeomorphic to $\mathbb R$ is homeomorphic either to $\mathbb R$ or $S^1$. </wikitex> == Further discussion == <wikitex>; The results above solve topological classification problem for 1-manifolds in the most effective way that one can wish. Surprisingly, many textbooks manage not to mention it. </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 $\mathbb R^n$. 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 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.

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.

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, connectedness, compactness and presence of boundary play fundamental role in topology of 1-manifolds. Homotopy invariants are extremely simple.

5 Further discussion

5.1 Connected sums

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 $\Rr_+$.

5.2 Orientations

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.

5.3 Mapping class groups

5.4 Finite group actions

5.5 Surgery

6 References