I am currently reading Morse Theory from the book written by Audin and Damian. And faced the Problem.

Let $V$ be a compact connected manifold of dimension $n$ without boundary and let $D$ be a disk of dimension $n$ embedded in $V$. Show that, $$HM_n(\overline{V-D},\Bbb Z/2)=0.$$ Here $HM_n$ denotes the Morse Homology.

I want to use long exact sequence $$....\to HM_k(\partial M)\to HM_k(M)\to HM_k(M,\partial M)\to HM_{k-1}(\partial M)\to ....$$ with $M=\overline{V-D}$. Then, $\partial( \overline{V-D})=\Bbb S^{n-1}$. And so $HM_k(\overline{V-D})\simeq HM_k(\overline{V-D},\Bbb S^{n-1})$. I don't know what to do after that.


Here's more of a suggestion than a complete answer: I would try to argue as to why there exists a Morse function $f \colon V \to \mathbb{R}$ with only one (local) maximum so that the only critical point of $f$ in $D$ is precisely the said maximum.

The restriction $g \colon \overline{V \setminus D} \to \mathbb{R}, \, g(x):=f(x)$ is still a Morse function, but $$0 \to \mathrm{CM}_n(g)=0 \to \mathrm{CM}_{n-1}(g) \to \dots .$$

In particular: $$\mathrm{HM}_n(g)=0.$$

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