Integral homology of the loop space of a sphere ($H_j(\Omega S^r;\mathbb{Z})=\mathbb{Z}\text{ if }(r-1)\mid j,\text{ and }0\text{ otherwise}$)

For $r\ge 3$, the path-loop Serre fibration has contractible total space. The relative homology of a fibration over a sphere gives $H_j(\Omega S^r)\cong H_{j-r+1}(\Omega S^r)$ for $j>0$, with negative homology zero and $H_0=\mathbb{Z}$. This proves the displayed groups. The unit map $S^{r-1}\to \Omega S^r$, adjoint to the identity of $S^r$, represents the generator of the first nonzero homotopy group. Naturality of the Hurewicz theorem shows that it induces an integral homology isomorphism in all degrees through $2r-3$.