\(\newcommand{\from}{\colon}\)\(\newcommand{\cross}{\times}\)\(\newcommand{\cover}[1]{{\widetilde{#1}}}\)Question
handed in by a student on 2017-11-13.

Lecture -

1. If \(X, Y\) are topological spaces with fundamental group the same
finitely generated \(G\).  Are \(X, Y\) homeomorphic necessarily? Or
otherwise equivalent?

Answer: They are not necessarily homeomorphic, or homotopy equivalent.
The examples to consider on one hand are the point, the line, the
plane (not homeomorphic, but all homotopy equivalent) and on the other
hand the higher dimensional spheres (for example the two-sphere).

All of these have trivial fundamental group, but no two of them are
homeomorphic.  Futhermore the two-sphere is not contractible, so it is
not homotopy equivalent to a point.  This last claim is harder to
prove than the others!