# Pontryagin product on the homology of cyclic groups

Consider the cyclic group $$C_{p^N}$$ of order $$p^N$$, and let $$k$$ be a field of characteristic $$p$$. I would like to know what the algebra structure on the homology $$H_*(C_{p^N};k)$$ induced by the Pontryagin product is. The homology $$H_*(C_{p^N};k)$$ is a Hopf algebra that is dual to the cohomology Hopf algebra $$H^*(C_{p^N};k)$$, so this could be determined from the coalgebra structure on group cohomology.

In the case $$p^N=2$$, we have $$BC_2\simeq \mathbb{RP}^\infty$$, and it is well-known the Pontryagin product makes $$H_*(C_2;k)$$ a divided power algebra. This is because it is dual to the cohomology which is a polynomial ring. This is documented, for instance, in Hatcher in Example 3C.11 and the paragraph thereafter.

If $$p^N>2$$, then the cohomology has a more complicated algebra structure. It is well-known that $$H^*(C_{p^N};k) \cong \Lambda[e] \otimes k[x],$$ where $$\Lambda[e]$$ denotes an exterior algebra, and the degrees are given by $$|e|=1, |x|=2$$. I have not been able to find any description of the coalgebra structure, however. Is there a reference which treats the coalgebra structure on cohomology or the Pontryagin product on homology in this case?

• The coalgebra structure is determined by the coproduct on $e$ and $x$. $e$ is clearly primitive by degree reasons, and, when $p$ is odd, so is $x$: its coproduct, viewed as an element of $H^2(C \times C)$ must be invariant under swapping the two copies of $C$, and this eliminates the possibility of $e \otimes e$ being part of the formula. (When $p=2$ more care is needed.) Nov 27, 2023 at 22:05

I recommend always looking at the canonical reference: Ken Brown's "Cohomology of Groups". Here Chapter V.5 is literally titled "The Pontraygin product" and then the very next Chapter V.6 is the application to abelian groups: Theorem V.6.6 $$H_* \cong \Lambda_*\otimes\Gamma_*$$ (exterior-algebra tensor divided-powers-algebra) holds even beyond cyclic.