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:

The topological sphere theorem for complete submanifolds

by
KATSUHIRO SHIOHAMA and HONGWEI XU  Compositio Mathematica 107: 221–232

https://link.springer.com/article/10.1023/A:1000189116072

The question is about the last sentence: 

Since ~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 !