# Homology calculation using Morse theory

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