Pontryagin ring ($(H_{\ast }(\Omega X;R),\ast )$)

The homology ring formed from loop concatenation and the homology cross product. Homotopy associativity is enough for associativity in homology, and the constant loop is the unit. More generally the same construction applies to a homotopy-associative H-space.

For simply connected $Y$, rational Hurewicz homomorphism identifies ${\pi }_{\ast }(\Omega Y)\otimes \mathbb{Q}$ with the primitive elements of a Hopf algebra of $H_{\ast }(\Omega Y;\mathbb{Q})$ and induces an isomorphism $U({\pi }_{\ast }(\Omega Y)\otimes \mathbb{Q})\cong H_{\ast }(\Omega Y;\mathbb{Q})$ of graded Hopf algebras. The bracket on homotopy is the Samelson product. The underlying algebra theorem says that a connected graded cocommutative Hopf algebra over characteristic zero is the universal enveloping algebra of its primitive Lie algebra. These statements let a free tensor algebra of loop-space homology detect a free graded Lie algebra of homotopy.

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.

Apply the homology cross product and then the map induced by loop concatenation. This degree-additive bilinear product supplies the Pontryagin ring. The Bott–Samelson theorem computes it for looped suspensions with free homology.