Fundamental class

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An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print.

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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 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, [Greenberg&Harper1981, 22.1].

Theorem 1.1. Let M be an n-dimensional compact 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

\displaystyle  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.

Proof. If M is closed then this is part of [Dold1995, VIII Corollary 3.4]; see also [Greenberg&Harper1981, 22.26]. 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] [Hatcher2002, Proposition 3.42]. Let
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, so that M = M_0 \cup_{\partial M} (\partial M \times I). By excision
\displaystyle 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).

Now apply [Dold1995, VIII Corollary 3.4] to the open manifold X : = M - \partial M \times \{ 1 \} and the closed subset M_0 \subset X.

Theorem 1.1 implies that a connected compact manifold M is orientable if and only if
\displaystyle H_n(M,\partial M;\mathbb Z) \cong \mathbb Z.
A choice of a generator is then called a fundamental class [M,\partial M] \in H_n(M,\partial M;\mathbb Z) for M. 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 M. In other words a connected compact manifold together with the choice of a fundamental class [M,\partial M] is the same as an oriented manifold. If M is not connected, then M 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 M 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 M be a connected oriented n-dimensional manifold. Then for each compact subset K \subset M there is a class [M]_K \in H_n(M, M-K) such that the following hold.

  1. If K \subset K' is another compact subset, then [M]_K' maps to M_K under the map induced by the inclusion.
  2. For each x \in M the class [M]_x is the local orientation of M.
  3. The classes M_K are uniquely characterized by these properties.
Using this one can use the Mayer-Vietoris sequence to "glue" together the local orientations inductively over a finite oriented atlas together to construct M_K. The inductive construction is rather indirect. If one defines the homology of a space X as the bordism classes of certain stratified spaces \mathcal S together with a continuous map to X, e.g. stratifolds, then the the fundamental class is easy to obtain, it is a tautology. Then the fundamental class of a closed manifold is the bordism class represented by the identity map
\displaystyle id:M \to M.
For this see [Kreck2010, Chapter 7, Section 1].

2 The Z/2-fundamental class

For all n-dimensional connected compact manifolds - even if they are not orientable - one has
\displaystyle  H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2,
and one calls the non-trivial element the \mathbb Z/2-fundamental class [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 1.2 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).

3 References

4 External links

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