1-manifolds

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1 Introduction

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$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, 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: let $\chi$ denote the Euler characteristic, $\pi_0(M)$ the number of connected components of $M$ and $\pi_0(\partial M)$ the connected components of the boundary of $M$. * $(\chi(S^1), \pi_0(S^1), \pi_0(\partial S^1)) = (0, 1, 0)$, * $(\chi(I), \pi_0(I), \pi_0(\partial I)) = (1, 1, 2)$, * $(\chi(\Rr), \pi_0(\Rr), \pi_0(\partial\Rr)) = (1, 1, 0)$, * $(\chi(\Rr_+), \pi_0(\Rr_+), \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\mathbb R\mid x\ge0\}$$\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$$\mathbb R^n$. To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.

2 Construction and examples

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

3 Invariants

Let $M$$M$ be a 1-manifold: let $\chi$$\chi$ denote the Euler characteristic, $\pi_0(M)$$\pi_0(M)$ the number of connected components of $M$$M$ and $\pi_0(\partial M)$$\pi_0(\partial M)$ the connected components of the boundary of $M$$M$.

• $(\chi(S^1), \pi_0(S^1), \pi_0(\partial S^1)) = (0, 1, 0)$$(\chi(S^1), \pi_0(S^1), \pi_0(\partial S^1)) = (0, 1, 0)$,
• $(\chi(I), \pi_0(I), \pi_0(\partial I)) = (1, 1, 2)$$(\chi(I), \pi_0(I), \pi_0(\partial I)) = (1, 1, 2)$,
• $(\chi(\Rr), \pi_0(\Rr), \pi_0(\partial\Rr)) = (1, 1, 0)$$(\chi(\Rr), \pi_0(\Rr), \pi_0(\partial\Rr)) = (1, 1, 0)$,
• $(\chi(\Rr_+), \pi_0(\Rr_+), \pi_0(\partial \Rr_+)) = (1, 1, 1)$$(\chi(\Rr_+), \pi_0(\Rr_+), \pi_0(\partial \Rr_+)) = (1, 1, 1)$,

4 Classification and characterization

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

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

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

Any connected non-compact 1-manifold without boundary is homeomorphic to the line $\mathbb R$$\mathbb R$.

Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to the half-line $\mathbb R_+$$\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$$\mathbb R$ is homeomorphic either to $\mathbb R$$\mathbb R$ or $S^1$$S^1$.

5 Further discussion

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.

6 References

This page has not been refereed. The information given here might be incomplete or provisional.