Suppose that $M$ is an oriented manifold with boundary. Consider an embedding $i: M \rightarrow M$. What is the possible topology of $M \smallsetminus i(M)$? More precisely, are there examples of $M$ and a collection of $i_k: M \rightarrow M$ such that $M \smallsetminus i_k(M)$ have different homology/Betti numbers?
For example, if $M^n = B^n \cup H^3$, where $H^3$ is a 3-handle, then we can write $M = ((B^n \cup H^3) \cup H^4) \cup H^3$, where $H^4$ cancels $H^3$ so that $(B^n \cup H^3) \cup H^4= B^n$. Then $H_k(M, i(M))$ is non-zero for $k = 3, 4$.
Note that if $M$ is closed and $i$ is an embedding, then $i$ must be a diffeomorphism so only the case when $M$ has boundary is interesting.
After thinking about this problem a bit more, I think $\dim H(M \setminus i(M)) = \dim H(\partial M)$. Indeed if $M\backslash i(M)$ is a trivial cobordism, then this is true. Maybe even $H(M \setminus i(M)) \cong H(\partial M)$ is true. However, $H(M, i(M)) \ne 0$ in general.