Both homotopy and homology capture information about holes in a topological space, but they're not the same topological invariant (Poincaré Homology Spheres have the same homology groups but not the same homotopy groups). There are also examples of spaces with the same homotopy groups but not the same homology. But I want to know if there's a geometric intuition for how these two invariants count holes in space. Are there any two surfaces with the same homology groups but different fundamental groups? By surface, I mean an orientable 2-manifold embedded in an euclidean space $\mathbb{R}^3$.
If these invariants are the same for surfaces, does anyone know where I can find a proof of it?
Thanks in advance!