Let $M$ be a smooth manifold of dimension $N$, $f: M \to \mathbb R$ smooth and let $p$ be a nondegenerate critical point, $f(p) = c$. Suppose that it is the only nondegenerate critical point in $f^{-1}([c - \varepsilon, c + \varepsilon])$ for some $\varepsilon > 0$. If the Morse index of $p$ is $k$, the Handle Body Decomposition tells us that $$ M^{c + \varepsilon} \text{ has the homotopy type of } M^{c - \varepsilon} \text{ with a } k \text{- cell attached} $$
Now, Corollary 12.12 in Nonlinear Analysis and Semilinear Elliptic Problems, from Ambrosetti and Malchiodi, states that $$ H_q(M^{c + \varepsilon}, M^{c - \varepsilon}) \simeq \begin{cases} \mathbb Z \quad \text{ if } q = k \\ 0 \quad \text{ if } q \neq k \end{cases} \qquad(*) $$
How to prove this?
It is a bit intuitive. Knowing that the homology of a set relative to a deformation retract is $0$, I have tried decomposing the pair of sets $(M^{c + \varepsilon}, M^{c - \varepsilon})$ into disjoint unions involving the $k$-cell, in order to be able to apply that the homology of the union be the sum of the homologies and that $(*)$ is precisely $H_q(\overline{B_k}, S^{k - 1})$, but to no end.
Thanks in advance.
Knowing that the pair $(M^{c+\epsilon}, M^{c-\epsilon})$ is equivalent to $(M^{c-\epsilon} \cup_f D^k,M^{c-\epsilon})$, the rest is almost completely formal. Namely, $\textit{if}$ the latter is a good pair, then its relative homology is the same as the reduced homology of the quotient $M^{c-\epsilon} \cup_f D^k /M^{c-\epsilon}$ which is homeomorphic to $S^k$ from which the result follows.
So the only question is if this pair is a good pair. In fact, what we have here is called a relative CW complex (the result of attaching cells to some space), and it is a known homotopical result that these are good pairs.