Non-homotopy equivalent spaces with isomorphic fundamental groups 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?

The standard way to prove two spaces are not homotopy equivalent is to find some homotopy invariant that distinguishes them. Since they are going to have the same fundamental group, the obvious candidates are homology groups and higher homotopy groups (either of which will tell you a sphere is not homotopy equivalent to a point, as in the answer above).
If you don't have those tools available, you're going to have to do some hard work.
