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 !