Fundamental class
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. |
The user responsible for this page is Matthias Kreck. No other user may edit this page at present. |
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.





![I=[0,1]](/images/math/1/c/a/1ca01533a825141b051c7492a6fa6427.png)
Tex syntax error, so that


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



![[M,\partial M] \in H_n(M,\partial M;\mathbb Z)](/images/math/9/c/6/9c65137bf5f40d4546f27964b7a7e049.png)


![[M,\partial M]](/images/math/f/d/e/fdeedcf7ff8b488cd51bffc2e0ad8969.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.





2 The Z/2-fundamental class









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.