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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.
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
- [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