Pontryagin ring of an odd-sphere loop space ($H_{\ast }(\Omega S^{2n+1};\mathbb{Z})=\mathbb{Z}[x_{2n}]$)

For $n\ge 1$, the Bott–Samelson theorem gives a single homology generator of degree $2n$ with no word relations. Its Pontryagin product satisfies $x^i\ast x^j=x^{i+j}$. Unlike this polynomial homology product, the cohomology cup product has divided-power coefficients.