Why is a simply connected homology sphere a topological sphere?

Question

I post this for a friend who currently doesn’t have access to this site.

It is about an implication in the last paragraph of the following paper:

• KATSUHIRO SHIOHAMA and HONGWEI XU, The topological sphere theorem for complete submanifolds, Compositio Mathematica 107 (1997) 221–232, https://doi.org/10.1023/A:1000189116072

The question is about the last sentence:

Since $\tilde{M}$ is simply connected, it is also a topological sphere.

Why is this true? Is it easy to work out a proof of this by hand ? If not may I ask a reference from which the question simply follows?

Sorry if it is a well-known fact in algebraic topology (if so may you point out a reference?)

Thanks a lot !

Answer 1

There are few steps.

The first one. A simply connected homology sphere is a homotopy sphere actually. It follows from the combination of the Whitehead and Hurewicz theorems. By the Hurewicz theorem, $\pi_n(X) \cong H_n(X) \cong \mathbb Z$. Therefore, there is a map inducing homology isomorphism. And by the Whitehead theorem it is a homotopy equivalence.

The second one. Any homotopy sphere is actually topological sphere. It follows from classification theorem in dimension 2. It follows from Poincare conjecture in dimension 3. And finally, in dimension n $\geqslant 4$ it follows from the generalized Poincaré conjecture, proved by Smale 1960-61 for n ≥ 5 and ca. 1982 by Michael Freedman for n = 4.