Fundamental class
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== The integral fundamental class == | == The integral fundamental class == | ||
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− | 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 | + | 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 $M$ is an $n$-dimensional topological manifold (possibly with boundary), then for each $x$ in the interior of $M$, one has $H_n(M,M-x;\Zz)\cong \Zz$, \cite{Greenberg&Harper1981|22.1}. | ||
− | {{beginthm|Theorem}}\label{thm:fundamental_class}Let $M$ be | + | {{beginthm|Theorem}}\label{thm:fundamental_class} |
+ | Let $M$ be an $n$-dimensional topological manifold (possibly with boundary). | ||
+ | If $M$ is connected and orientable then for each x in the interior of $M$, the map induced by the inclusion | ||
+ | $$ | ||
+ | H_n(M,\partial M;\Zz) \to H_n((M,M-x;\Zz), | ||
+ | $$ | ||
+ | is an isomorphism. In particular, $H_n(M,\partial M;\Zz)\cong \Zz$. | ||
+ | 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 part of \cite{Dold1995|VIII Corollary 3.4}; see also \cite{Greenberg&Harper1981|22.26}. If $M$ has a boundary $\partial M$, then write | {{beginproof}} If $M$ is closed then this part of \cite{Dold1995|VIII Corollary 3.4}; see also \cite{Greenberg&Harper1981|22.26}. If $M$ has a boundary $\partial M$, then write |
Revision as of 12:26, 27 May 2013
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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 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.
Proof. If is closed then this part of [Dold1995, VIII Corollary 3.4]; see also [Greenberg&Harper1981, 22.26]. If has a boundary , then write where and is a collar of the boundary of . By excision
Now apply [Dold1995, VIII Corollary 3.4] to the 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
- [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.