Fundamental class
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− | {{Authors|Matthias Kreck | + | {{Authors|Matthias Kreck}} |
== The integral fundamental class == | == The integral fundamental class == | ||
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If $M$ is connected and non-orientable then $H_n(M,\partial M;Z)$ is zero. | If $M$ is connected and non-orientable then $H_n(M,\partial M;Z)$ is zero. | ||
{{endthm}} | {{endthm}} | ||
− | {{beginproof}} If $M$ is closed then this is part of \cite[VIII Corollary 3.4]{Dold1995}; see also \cite[22.26]{Greenberg&Harper1981}. | + | {{beginproof}} If $M$ is closed then this is part of \cite[VIII Corollary 3.4]{Dold1995}; see also \cite[22.26]{Greenberg&Harper1981}. If $M$ has a boundary $\partial M$, then the inclusion $\partial M=\partial M \times \{1\} \subset M$ extends to an embedding $\partial M \times I \subset M$ of a collar, where $I=[0,1]$ \cite[Proposition 3.42]{Hatcher2002}. Let $M_0={\rm cl.}(M-\partial M \times I)$, so that $M = M_0 \cup_{\partial M} (\partial M \times I)$. By excision |
− | If $M$ has a boundary $\partial M$, then the inclusion | + | |
− | $\partial M=\partial M \times \{1\} \subset M$ extends to an embedding | + | |
− | $\partial M \times I \subset M$ of a collar, where $I=[0,1]$ \cite[Proposition 3.42]{Hatcher2002}. | + | |
− | Let $M_0={\rm cl.}(M-\partial M \times I)$, so that | + | |
− | $M = M_0 \cup_{\partial M} (\partial M \times I)$. | + | |
− | By excision | + | |
$$H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong | $$H_n(M, \partial M; \mathbb Z) \cong H_n(M, \partial M \times I; \mathbb Z) \cong | ||
H_n(M -\partial M \times \{ 1 \}, \partial M \times [0, 1); \mathbb Z).$$ | H_n(M -\partial M \times \{ 1 \}, \partial M \times [0, 1); \mathbb Z).$$ |
Latest revision as of 17:25, 6 March 2014
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Contents |
1 The integral fundamental class
For compact manifolds one can characterize the orientability by the existence of a certain homology class called the fundamental class. The background is the following theorem. Recall that if is an -dimensional topological manifold (possibly with boundary), then for each in the interior of , one has , [Greenberg&Harper1981, 22.1].
Theorem 1.1. Let be an -dimensional compact topological manifold (possibly with boundary). If is connected and orientable then for each x in the interior of , the map induced by the inclusion
is an isomorphism. In particular, . If is connected and non-orientable then is zero.
Tex syntax error, so that . By excision
Now apply [Dold1995, VIII Corollary 3.4] to the open manifold and the closed subset .
Theorem 1.2 [Greenberg&Harper1981, 22.24] . Let be a connected oriented -dimensional manifold. Then for each compact subset there is a class such that the following hold.
- If is another compact subset, then maps to under the map induced by the inclusion.
- For each the class is the local orientation of .
- The classes are uniquely characterized by these properties.
2 The Z/2-fundamental class
3 References
- [Dold1995] A. Dold, Lectures on algebraic topology, Springer-Verlag, 1995. MR1335915 (96c:55001) Zbl 0872.55001
- [Greenberg&Harper1981] M. J. Greenberg and J. R. Harper, Algebraic topology, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, 1981. MR643101 (83b:55001) Zbl 0498.55001
- [Hatcher2002] A. Hatcher, Algebraic topology, Cambridge University Press, 2002. MR1867354 (2002k:55001) Zbl 1044.55001
- [Kreck2010] M. Kreck, Differential algebraic topology, Graduate Studies in Mathematics, 110, American Mathematical Society, 2010. MR2641092 (2011i:55001) Zbl 05714474
4 External links
- The Encylopedia of Mathematics article on the fundamental class.
- The Wikipedia page on the fundamental class.