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 $\mathbb{Z}^k$.

  • 9
    $\begingroup$ 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
  • 4
    $\begingroup$ 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

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.