2
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.