Proof of general Poincaré Conjecture $\dim > 5$

Question

Given $M$ is simply connected, $\dim(M) > 5$ and homotopy equivalent to a sphere. Let $W := M - D_1 \cup D_2$ for two smoothly embedded disjoint disks.

How does one see that $W$ is simply connected and a h-cobordism?

Answer 1

$W$ is simply connected by van Kampen's theorem, applied twice: call $W' = M - \text{int}(D_1)$; then $$\pi_1(M) = \pi_1(D_1) *_{\pi_1(S^{n-1})} \pi_1(W')$$ and, because both $\pi_1(D_1) = \pi_1(S^{n-1}) = 0$, this formula shows $\pi_1(W') = \pi_1(M)$. Similarly for deleting the other disc. (You could also note that $W$ is homotopy equivalent to $M$ with two points deleted and invoke transversality to note that deleting a subset of codimension at least three doesn't change $\pi_1$.)

That it's an $h$-cobordism is similar. I'll show that the inclusion of one of the boundary spheres is an isomorphism on homology. By the Hurewicz theorem and the Whitehead theorem, because everything involved is simply connected, it's a homotopy equivalence.

First, prove with Mayer-Vietoris in a way similar to the above that $W'$ has all homology groups zero. Now let's use Mayer-Vietoris again, this time on $W \cup D^n = W'$. The long exact sequence we get is $$H_{j+1}(W') \to H_j(S^{n-1}) \to H_j(W) \to H_j(W'),$$ where the middle map is induced by inclusion; the outer two terms are always zero so the middle map is always an isomorphism, as desired.