I am interested in understanding the connection between homology and homotopy. In particular, I know that Hurewicz theorem states that for path connected topological spaces we can always find a homomorphism between $H_n(G)$ and $\pi_n(G)$, with $G$ a path connected topological space. If we restrict to $n = 1$, I also know that $H_1(G)\cong \pi^{\rm ab}_1(G)$, i.e. the first homology group is isomorphic to the abelianization of the first homotopy group. My questions are
- Given Hurewicz theorem, are all homotopy invariants captured by homology? (if G is path connected? What if it isn't?)
- Given $n = 1$, if $\pi_1(G)$ is either trivial or abelian, does it mean that the induced map of the first homology is exactly associated with the elements of that group? (Sorry for the improper language)
- Is the winding number of the determinant map of $G$, the induced map of the first homology?
To make it easier, we can consider a map $f:S^1\to U(N)$. I know that $\pi_1(U(N)) \cong \mathbb{Z}$. I also know that $\pi^{\rm ab}_1(U(N)) = \pi_1(U(N))\cong \mathbb{Z}$, since $\mathbb{Z}$ is abelian. This also mean that $H_1(U(N))\cong \mathbb{Z}$. Therefore, is $f_*:H_1(S^1)\to H_1(U(N))$ the winding number of $det\circ f$?
I apologize if I am not using the correct terminology and/or if I am being not so rigirous. I am sort of new to this topic and I would like a deeper understanding but I can not find direct help in book about topology.