I know that if two spaces $X,Y$ have the same fundamental group : $\pi_1(X,a) \cong \pi_1(Y,b) $
does not imply that they are homotopy equivalent.
But I can't find an example.
I was thinking about the Moebius strip and the circle, as both of them have the fundamental group $\mathbb Z$. I can't show that they are not homotopy equivalent, the circle is the Moebius strip boundary so it is not a deformation retract of it, but it also does not necessary means that the are not homotopy equivalent.
Can some one give me an example.
Is there a way to proof that two spaces are not homotopy equivalent or not deformation retract?
Best Answer
The $2$-sphere and the point have the same fundamental group but are not homotopy equivalent. However, to show that the $2$-sphere is not contractible you need some tools.