I have to prove that two different topological spaces $X,Y$ have the same homotopy type. I've been able to prove so far that $\pi_1(X)=\pi_1(Y)$ but I don't know if this is enough to say that $X$ and $Y$ are isomorphic in the category $\mathcal{HoTop}$.
I also know that two spaces with the same fundamental group do not have to be homeomorphic (i.e. an isomorphism in the category $\mathcal{Groups}$ does not imply that the objects are isomorphic in the category $\mathcal{Top}$), but I was wondering what is the situation in $\mathcal{HoTop}$. Are two spaces with the same fundamental group homotopy equivalent?
I don't know if the notation that I'm using is usual or not, so if anyone needs clarification do not hesitate of commenting.
Best Answer
No. As lulu already pointed in a comment, a point and $\mathbb{S}^2$ have the same fundamental group but are not homotopically equivalent ($\mathbb{S}^2$ is not contractile, say).
There is a result, the Whitehead Theorem, which states that if $X$ and $Y$ have CW complex structures, and $f\colon X \to Y$ is a map such that $f_\ast\colon \pi_n(X,x_0) \to \pi_n(Y,f(x_0))$ is an isomorphism for all $n \geq 1$, then $X$ and $Y$ have the same homotopy type.
Here $\pi_n(X,x_0)$ is the space of homotopy equivalence classes of hyperloops $\alpha: \mathbb{S}^n \to X$ such that $\alpha(1,0,\cdots,0) = x_0$.