Graded $p$-polar rings and the homology of ${\Omega }^n{\Sigma }^{n}X$

Abstract

As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the homology of the free $E_n$-algebra $H^{\ast }({\Omega }^n{\Sigma }^{n}X;{\mathbf{F}}_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^{\ast }(X;{\mathbf{F}}_p)$ in a functorial way.

25 pages. Improved exposition. Published in Homotopy, Homology and Applications 27 (2025), pp. 355--382. Partial overlap with arXiv:2106.01845