Integral cohomology of an odd-sphere loop space
= Integral cohomology of an odd-sphere loop space
{title2=$H^*(\Omega S^{2n+1};\mathbb Z)=\Gamma_{\mathbb Z}(a_{2n})$}
For $n\geq1$, there is one integral cohomology generator $a_k$ in every degree $2nk$, and its products are $a_i a_j=\binom{i+j}{i}a_{i+j}$. The <Bott–Samelson theorem> supplies free polynomial <homology>; its primitive generator gives the binomial diagonal dual to this <divided power algebra>.