# Kunneth formula for semidirect product

I wonder if the following Kunneth formula for semidirect product is valid $$H^n(N\rtimes_\phi G;\mathbb{Z}) = \sum_{i+j=n} H^i(G; H^j(N;\mathbb{Z})),$$ where $H^*$ is the group cohomology and $G$ has a proper action on $H^j(N;\mathbb{Z})$ as induced by $\phi$. (For direct product, $G$ has no action on $H^j(N;\mathbb{Z})$ and the above reduces to the standard Kunneth formula.)

https://arxiv.org/abs/math/0406130 only showed above when $N$ has a form $\mathbb{Z}^k$.

• There's a spectral sequence with $E_2$-page the right-hand side of your formula, converging to the left-hand side. Typically it does not degenerate. See: en.wikipedia.org/wiki/… Jul 27 '18 at 1:03
• And here are some examples where it does not collapse: Totaro's paper "Cohomology of Semidirect Product Groups", and a disproof of a conjecture of Adem (arxiv.org/pdf/1105.4772.pdf) which I think is related to the arXiv paper you linked. Jul 27 '18 at 1:25
• Thanks for the refs and the counter example. In the counter example, the group $N=\mathbb{Z}^k$ is not compact. I wonder if the conjecture can be true if both $N$ and $G$ are finite or compact (like $SU(n)$). Jul 27 '18 at 1:53