# Fundamental class

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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.

**Proof.**If is closed then this is part of [Dold1995, VIII Corollary 3.4]; see also [Greenberg&Harper1981, 22.26]. If has a boundary , then the inclusion extends to an embedding of a collar, where [Hatcher2002, Proposition 3.42]. Let

Tex syntax error, so that . By excision

Now apply [Dold1995, VIII Corollary 3.4] to the open manifold and the closed subset .

**fundamental class**for . The fundamental class determines by the isomorphism above a continuous choice of local orientations and in turn the fundamental class is determined by a homological orientation of . In other words a connected compact manifold together with the choice of a fundamental class is the same as an oriented manifold. If is not connected, then is orientable if and only all components are orientable. If the components are oriented the fundamental classes of the components give the

**fundamental class of**under the isomorphism which decomposes the homology groups into the homology groups of the components. Thus for oriented manifolds again one has a fundamental class which corresponds to a orientation as in the connected case. The construction of the fundamental class of an oriented closed manifold is done inductively over an atlas (similarly for manifolds with boundary). Namely one has the following generalization of Theorem 1:

**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

**-fundamental class**of a non-connected compact manifold. Also one has a generalization of Theorem 1.2 to non-compact connected manifolds, i.e. for each compact subset one has and for the map induced by the inclusion is an ismorphism .

## 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.