1-manifolds
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− | Let $M$ be a 1-manifold | + | Let $M$ be a 1-manifold and let $(\chi, \pi_0, \pi_0 \partial)(M)$ be the triple consisting of |
− | * $(\chi | + | the Euler characteristic of $M$, the number of connected components of $M$ and number of connected components of the boundary of $M$. |
− | * $(\chi | + | * $(\chi, \pi_0, \pi_0 \partial )(S^1) = (0, 1, 0)$, |
− | * $(\chi | + | * $(\chi, \pi_0, \pi_0 \partial )(I) = (1, 1, 2)$, |
− | * $(\chi | + | * $(\chi, \pi_0, \pi_0 \partial )(\Rr) = (1, 1, 0)$, |
+ | * $(\chi, \pi_0, \pi_0 \partial )(\Rr_+) = (1, 1, 1)$. | ||
</wikitex> | </wikitex> | ||
Revision as of 20:12, 30 September 2009
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 08:32, 18 July 2013 and the changes since publication. |
Contents |
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 or half-line .
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 . To be on the safe side, we use an unambiguous term manifold of dimension 1 or 1-manifold.
2 Construction and examples
Real line , half-line , circle , closed interval .
3 Invariants
Let be a 1-manifold and let be the triple consisting of the Euler characteristic of , the number of connected components of and number of connected components of the boundary of .
- ,
- ,
- ,
- .
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 , half-line , circle , closed interval .
Any connected closed 1-manifold is homeomorphic to the circle .
Any connected compact 1-manifold with non-empty boundary is homeomorphic to the interval .
Any connected non-compact 1-manifold without boundary is homeomorphic to the line .
Any connected non-compact 1-manifold with non-empty boundary is homeomorphic to the half-line .
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.
Working inductively, we see that 1-manifolds with a finite number of components are classified by the triple .
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 is homeomorphic either to or .
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. |