Let $M$ be a compact connected (smooth) $n$ dimensional manifold, and let $D$ be an $n$-dimensional disk embedded in $M$. Let $W:=\overline{M\setminus D}$, then $H_{n}(W;Z/2)=0$.
I see the following proof - triangulate $M$ so that $\partial D$ is a union of open simplices of dimension $<n$. Then it is quite easy to see that the fundamental class $[W]$ is the class of the sum of two chains $a+b$, contained in $W,D$ respectively, the sum of whose boundaries is $0$. In other words, the connecting map (last) in the following Mayer vietoris sequence is an isomorphism: $0\to H_n(W;Z/2)\bigoplus H_n(D;Z/2)\to H_n(M;Z/2)\to H_{n-1}(S^{n-1};Z/2)$, which is sufficient. Here are my problems:
- I triangulated stuff to see that the above map is an isomorphism. How can one see that it is an isomorphism without triangulation? That is, how would I generalize my proof for non-smooth manifolds, where you can't always triangulate?
- Let us check what the non-orientable Poincare duality theorem will give us. As far as I understand, it should state that $H_n(W;Z/2)\simeq H_{0}(W,\partial D;Z/2)$, but wouldn't the latter vector space be $1$-dimensional? It is just spanned by the class of a point, I don't see how the relative homology affects this...
Any corrections or suggestions will be very much appreciated!