Fundamental class
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− | For all $n$-dimensional connected compact manifolds - even if they are not orientable - one has $$ H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2, $$ and one calls the non-trivial element the $\mathbb Z/2$-fundamental class {{cite| | + | For all $n$-dimensional connected compact manifolds - even if they are not orientable - one has $$ H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2, $$ and one calls the non-trivial element the $\mathbb Z/2$-fundamental class {{cite|Dold1995|VIII Definition 4.1}}. As for the integral fundamental class (if $M$ is oriented) one gets from these classes the '''$\mathbb Z/2$-fundamental class''' of a non-connected compact manifold. Also one has a generalization of Theorem \ref{thm:K_fundamental_class} to non-compact connected manifolds, i.e. for each compact subset $K$ one has $H_n(M,M-K;\mathbb Z/2) \cong \mathbb Z/2)$ and for $K \subset K'$ the map induced by the inclusion is an ismorphism $ H_n(M,M-K';\mathbb Z/2) \to H_n(N,M-K;\mathbb Z/2)$. |
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Revision as of 18:56, 16 December 2012
An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 15:25, 6 March 2014 and the class&diff=cur&oldid=11505 changes since publication. |
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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:
Proof. If is closed then this part of [Dold95, 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 [Dold95, 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
- [Dold95] Template:Dold95
- [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 [1].
- The Wikipedia page on the fundamental class.