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If $G$ is a finite group, what do we know of the natural «restriction» map $$H^\bullet(G,\mathbb Z)\to\left(\bigoplus_{g\in G}H^\bullet(Z(g),\mathbb Z)\right)^G,$$ with $Z(g) $ the centralizer of $g $. In particular, can we describe the kernel and cokernel, or fit it into an exact sequence?

Mariano Suárez-Álvarez
  • 46,795
  • (the kernel and cokernel can be described using Ext-groups from the aumentation ideal with its adjoint action, but that is rather opaque) – Mariano Suárez-Álvarez Mar 04 '14 at 22:09
  • Are you allowing $g=1$ on the right hand side? If so, then your map is injective, and the cokernel is isomorphic to the sum of the terms with $g\neq 1$.

    It is also worth noting that the right hand side is the cohomology of the free loop space of the classifying space $BG$.

     – Neil Strickland Mar 04 '14 at 22:52
  • I meant to sum over conjugacy closed subset, really. I'll fix this when I get hold of a real keyboard. – Mariano Suárez-Álvarez Mar 04 '14 at 22:56

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