I know that if $h : (X,x_0) \longrightarrow (Y,y_0)$ is a homeomorphism then that induces an isomorphism $h_{*} : \pi_1(X,x_0) \longrightarrow \pi_1(Y,y_0)$ defined by $$h_{*} ([f]) = [h \circ f].$$ Now my question is "Does the converse of this hold?" i.e. Suppose I have an isomorphism $h : \pi_1(X,x_0) \longrightarrow \pi_1(Y,y_0)$. Does that necessarily induce a homeomorphism $h^{*} : (X,x_0) \longrightarrow (Y,y_0)$? If it is true then can we go further i.e. can we also say that the isomorphism induced by $h^{*}$ is same as that of $h$? Please help me in understanding this concept.
Thank you very much.
Best Answer
Definitely not. Among other things, this would imply that any two simply connected spaces are homeomorphic.
More generally, the fact that homeomorphisms between topological spaces induce isomorphisms of their fundamental groups shows that the fundamental group of a space is a topological invariant. If the converse were true, then we could completely characterize topological spaces by their fundamental groups.