In this set-up, let M be a compact n-dimensional manifold with boundary $\partial M \neq \emptyset$.
Assume that M is orientable, and that $[M] \in H_n(M,\partial M;R)$ is the fundamental class of M. If you haven't seen orientability / fundamental classes for manifolds with boundary before, the definition is the same, you just consider it for $M – \partial M$ instead. Now,one can consider the image of $[M]$ under the boundary map $\partial: H_n(M, \partial M;R) \rightarrow H_{n-1}(\partial M ;R)$ . I want to show that $\partial [M]$ is also a fundamental class of $\partial M$, and preferably in a manner that generalies to other homology theories, so a proof just using the Eilenberg-Steenrod axioms together with maybe something else would be nice. I can't seem to find one, so any help would be welcome!
Best Answer
By the definition, an orientation on a manifold $M$ is a class $[M]\in h(M,\partial M)$ such that for any (interior) point its image under $h(M,M\setminus U_\epsilon(x))$ is a generator of $h(M,M\setminus U)\cong \tilde h(S^n)$ (considered as an $h(pt)$-module). Let's write, say, $[M]\bigr|_U$ for this element.
OK, let's move to the proof. Consider a small neighborhood $U$ of a point in $\partial M$. We need to show that $\partial$ maps $[M]\bigr|_U$ to $[\partial M]\bigr|_{U\cap\partial M}$. So everything reduces to the statement for $M=D^n$, $\partial M=S^{n-1}$ — that is, to well-known suspension isomorphism.
Maybe all this is clearer on a commutative diagram ($V:=U\cap\partial M$): $$\require{AMScd}\begin{CD} [M]\in h(M,\partial M) @>{\partial}>> h(\partial M);\\ @VVV @VVV \\ [M]\bigr|_{U\setminus V}\in h(M,M\setminus (U\setminus V)) @. h(\partial M,\partial M\setminus V);\\ @| @| \\ [D^n]\in h(D^n,S^{n-1})=\tilde h(S^n) @>{\partial}>>\tilde h(S^{n-1})\ni [S^{n-1}]. \end{CD}$$
(Summary: being an orientation is a local condition, so it can be changed locally — i.e. essentially just for $M=D^n$.)
(As Tedar points out) all this is explained in May's Concise Course in Algebraic Topology (see the proof of «Proposition. An R-orientation of M determines an R-orientation of ∂M» on p. 169 + the proof of the next one) — the proof is essentially the same, but of course May's writing is better.