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
![\displaystyle H_n(M,\partial M;\Zz) \to H_n(M,M-x;\Zz),](/images/math/b/4/1/b41c32672ede9ec8a5b0a29a736d6d02.png)
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].
Tex syntax error, so that
.
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).](/images/math/2/3/3/233c449a623e5155b074640bb35ee8ad.png)
Now apply [Dold1995, VIII Corollary 3.4] to the open manifold and the closed subset
.
![\square](/images/math/7/b/3/7b39786e26d837954085516609e12817.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle H_n(M,\partial M;\mathbb Z) \cong \mathbb Z.](/images/math/5/d/0/5d019344d9e66596841be7a472a81ebb.png)
![[M,\partial M] \in H_n(M,\partial M;\mathbb Z)](/images/math/9/c/6/9c65137bf5f40d4546f27964b7a7e049.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![[M,\partial M]](/images/math/f/d/e/fdeedcf7ff8b488cd51bffc2e0ad8969.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
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.
![M_K](/images/math/8/2/4/82445669aaddbb28e620b10ae7d3e1f3.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![\mathcal S](/images/math/e/f/e/efe3e683a5a73971b13b0390854d2b76.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![\displaystyle id:M \to M.](/images/math/4/7/9/4793cda41dd554c973ad9c7e1f34505f.png)
2 The Z/2-fundamental class
![n](/images/math/e/4/a/e4a3f5f7a18b1ed0ee22a93864ad15d8.png)
![\displaystyle H_n(M,\partial M;\mathbb Z/2) = \mathbb Z/2,](/images/math/e/c/2/ec2329b2f67cbeab4e9f76de9e46e371.png)
![\mathbb Z/2](/images/math/e/2/6/e26d67ac2c46a7d6bf885294d11449a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\mathbb Z/2](/images/math/e/2/6/e26d67ac2c46a7d6bf885294d11449a6.png)
![K](/images/math/8/6/7/8673dabfddfa6347e822f593dfa1d916.png)
![H_n(M,M-K;\mathbb Z/2) \cong \mathbb Z/2)](/images/math/1/f/8/1f863e0e84d5d0fbd6b12c947082c196.png)
![K \subset K'](/images/math/4/8/a/48ae00396e3a80a7b7b145f62498f844.png)
![H_n(M,M-K';\mathbb Z/2) \to H_n(N,M-K;\mathbb Z/2)](/images/math/c/1/e/c1e6f6ce811405e419f90a33675c86d6.png)
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.