Classification of compact connected manifolds by fundamental group

Question

Every compact connected 2-manifold (I define this as a surface) is homeomorphic to a 2-sphere, a connected sum of tori or a connected sum of projective planes. Since the fundamental groups of the surfaces in this list are not isomorphic then one can say that the fundamental group determines the topological type of every surface. Is this true for manifolds of dimension greater than 2? If not can you provide a counterexample as simple as possible?

Answer 1

There are non-homeomorphic lens spaces with same fundamental group: $\pi_1(L(p,q)) = \mathbb{Z}_p$ (here), but $L(p,q_2) \cong L(p_2,q_2) \iff p = p_2$ and $(\pm q_1 q_2 \equiv_p 1$ or $\pm q_1 \equiv_p q_2 )$. I really don't remember where I studied the classification theorem, but here you can find some references to several proofs.