What is the image of the Homology group of the boundary of a space under the map induced by inclusion?

Question

Suppose $M$ is a compact manifold of dimension $n+1$ and $\delta M$ its boundary. What can we say about the image of $H_n(\delta M)$ under the inclusion $H_n(\delta M)\to H_n(M)$?
I was originally asked this question in terms of Morse Homology. In Morse Homology, this statement translates to showing that every critical point of index $n$ in $\delta M$ that connects to each critical point of index $n-1$ in $\delta M$ with an even number of trajectories, must be connected to some critical point of index $n+1$ in $M$ via an odd number of trajectories.

Answer 1

I'm going to assume that the manifold $M$ is orientable, so that the methods of Poincare duality can be applied.

Let's also assume that $M$ is not closed, and so $H_{n+1}(M) \approx 0$ (this is a general theorem about $n$-manifolds, proved as part of the suite of Poincare duality theorems). There is a relative version of Poincare duality which says that $H_{n+1}(M,\partial M) \approx \mathbb Z$. Let's list the components of the boundary as $\partial M = N_1 \cup \cdots \cup N_k$, and so $H_n(N_k) \approx \mathbb Z$ (again by Poincare duality). Let's write out a few terms of the long exact sequence of pairs: $$\underbrace{H_{n+1}(M)}_{0} \mapsto \underbrace{H_{n+1}(M,\partial M)}_{\mathbb Z} \mapsto H_n(\partial M) = \underbrace{H_n(N_1)}_{\mathbb Z} \oplus\cdots\oplus \underbrace{H_n(N_k)}_{\mathbb Z} \mapsto H_{n}(M) $$ There is an additional conclusion of the whole Poincare duality suite of theorems in this situation: the fundamental class $[M] \in H_{n+1}(M,\partial M)$ maps to the sum of fundamental classes $[N_1] + ... +[N_k] \in H_n(N_1) \oplus \cdots \oplus H_n(N_k) \approx H_n(\partial M)$.

In follows that the kernel of the inclusion induced map $H_n(\partial M) \to H_n(M)$ is generated by $[N_1] + ... +[N_k]$. The image is therefore isomorphic to $\mathbb Z^{k-1}$. In the special case that $\partial N=N_1$ is connected, the inclusion induced map $H_n(\partial M) \to H_n(M)$ is therefore trivial.

This conclusion, as discussed in the comments, is transparent in the case of simplicial homology. For the case of singular homology, I would look up one of the standard proofs of Poincare duality. I have always liked the proof that Milnor wrote out in the appendix to his book Characteristic Classes, although nowadays one can also find pretty much the same proof in Hatcher's book Algebraic Topology. Frankly I'm unsure whether the exact statement that I gave above, regarding the homomorphism $H_{n+1}(M,\partial M) \to H_n(\partial M)$, can be found in Hatcher's book, but if it's not then I'll just say that it should or ought to be part of the suite of Poincare duality theorems, and anyway it is a good exercise once one understands the proof of Poincare duality.