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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 $\mathbb{Z}^k$.

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    $\begingroup$ 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/… $\endgroup$ – Daniel Litt Jul 27 '18 at 1:03
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    $\begingroup$ 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. $\endgroup$ – Chris Gerig Jul 27 '18 at 1:25
  • $\begingroup$ 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 $SU(n)$). $\endgroup$ – Xiao-Gang Wen Jul 27 '18 at 1:53
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Just to collect the references I wrote in comments, and more. They each give (in)finite (non)abelian counterexamples, as well as general explanations for failure of collapse of the LHS spectral sequence with semi-direct products:

  1. Charlap, L. S.; Vasquez, A. T., The cohomology of group extensions, Bull. Am. Math. Soc. 69, 815-817 (1963). ZBL0122.02803.

  2. Benson, D. J.; Feshbach, M., On the cohomology of split extensions, Proc. Am. Math. Soc. 121, No. 3, 687-690 (1994). ZBL0819.20058.

  3. Totaro, Burt, Cohomology of semidirect product groups, J. Algebra 182, No. 2, 469-475 (1996). ZBL0862.20038.

  4. Siegel, Stephen F., On the cohomology of split extensions of finite groups, Trans. Am. Math. Soc. 349, No. 4, 1587-1609 (1997). ZBL0951.20039.

See also references within.

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