Let $X$ and $Y$ be two n-manifolds all of whose higher homotopy groups are trivial, and the first homotopy groups are isomorphic (but the existence of a mapping inducing an isomorphism is not assumed). Is it true that $X$ and $Y$ are homotopy equivalent?
It seems to me I read somewhere about "the triviality of higher homotopy groups makes the fundamental group a complete invariant", but I can't remember where.
Best Answer
Sorry, this was a completely classic fact (see for example http://www.map.mpim-bonn.mpg.de/Aspherical_manifolds). Should I accept this answer or delete the question?