= Integral homology of the loop space of a sphere
{title2=$H_j(\Omega S^r;\mathbb Z)=\mathbb Z\text{ if }(r-1)\mid j,\text{ and }0\text{ otherwise}$}
For $r\geq3$, 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 $2r-3$.
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