The following was posted to math.stackexchange to no avail: https://math.stackexchange.com/questions/908756/an-exercise-in-homology-computation-what-is-the-geometric-fixed-points-of-an-e

The question I want to ask has a reasonably elementary formulation and I think there is a good chance it can be answered in this form (by someone more computationally skilled than me, or perhaps by someone recognising the construction). It is, however, motivated by equivariant stable homotopy theory and certainly insights can be drawn from this picture as well. I'm going to state the elementary form first, and then explain background and motivation.

For a pointed simplicial set $X_\bullet$ I shall write $\tilde{\mathbb{Z}}[X]$ for the chain complex with $\tilde{\mathbb{Z}}[X]_n = \mathbb{Z}\{X_n\}/*$ by which I mean the free abelian group on $X_n,$ except that we treat the base point as zero. Now let $G$ be a finite group. I write $X^{\wedge G}$ for the simplicial $G$-set $X \wedge X \wedge \dots \wedge X$ ($|G|$ copies of $X$) with $G$ acting by permuting factors. Note that if $X_\bullet$ is a pointed simplicial $G$-set then $\tilde{\mathbb{Z}}[X]$ has a natural $G$-action. Finally let $S^n = \Delta^n/\partial \Delta^n$ be the $n$-sphere (naturally pointed!).

**Question:** What is $H_{n,k} := H_{n+k}\left( \left[\tilde{\mathbb{Z}}[(S^n)^{\wedge G} \right]^G \right)$? Here the outer-most $[]^G$ means $G$-invariants in the chain complex, and $H_*$ is just homology groups.

One may show that there is a stabilisation map $H_{n,k} \to H_{n+1,k}$. Write $H_{\infty, k}$ for the colimit. This is the group I'm really after.

**Some background:** Recall that there is a tensor triangulated category $SH(G)$, the $G$-equivariant stable homotopy category. This has two structures I am interested in here: firstly there is a tensor triangulated functor $\Phi^G: SH(G) \to SH$, the *geometric fixed points* functor. Secondly, the category has a natural $t$-structure whose heart is equivalent to the category of mackey functors. If we write $\underline{\mathbb{Z}}$ for the constant mackey functor with value $\mathbb{Z}$ and $H\underline{\mathbb{Z}}$ for the corresponding Eilenberg-MacLane spectrum, then what I would really like to understand is $\Phi^G(H\underline{\mathbb{Z}}).$ If I interpret correctly example 2.13 and paragraph 7.3 in [1], then $H_{\infty, k} = \pi_k \Phi^G(H\underline{\mathbb{Z}})$, hence my question.

**Some results:** Let $H$ be a proper subgroup of $G.$ Then $\Phi^G (G/H_+ \wedge S_G) = 0$. The transfer $H\underline{\mathbb{Z}} \to G/H_+ \wedge H\underline{\mathbb{Z}} \to H\underline{\mathbb{Z}}$ is multiplication by $|G:H|.$ Hence multiplication by $|G:H|$ is zero on $\Phi^G(H\underline{\mathbb{Z}})$ and so also on $H_{\infty, k}.$ In particular $H_{\infty, k} = 0$ unless $G$ is a $p$-group, in which case $H_{\infty, k}$ is $p$-torsion.

One may work out $\tilde{\mathbb{Z}}[(S^n)^{\wedge G}]$ fairly explicitly, but the combinatorics of the resulting chain complex is a big mess. Using some standard results about lifting homotopies, one may prove that in the definition of $H_{n,k}$ we can replace $\left[\tilde{\mathbb{Z}}[(S^n)^{\wedge G} \right]^G$ by $(C_\bullet^{\otimes G})^G$ for any simplicial free abelian group $C_\bullet$ with $H_* C = H_* S^n,$ for example the Dold-Kan inverse to the chain complex $\mathbb{Z}[n]$, i.e. $C_k = 0$ for $k < n$, $C_n = \mathbb{Z}$ and the higher degree groups are just freely added degeneracies. Hence in particular $H_{n,k} = 0$ for $k < 0.$ But the resulting complex still does not seem particularly amenable to computation (it is related to the orbits of $G$ on $G$-sets of the form $X^{\times G}$ where $X$ has trivial $G$ action). I believe I can use this description to show that $H_{n,0} = \mathbb{Z}/p.$

Some wishful thinking led me to ask in the stackexchange post if in fact $H_{\infty, k} = 0$ for all $k > 0.$ I'm thinking now that this may be wrong. In fact using paper [2] on "derived mackey functors" I seem to have convinced myself that for $G = C_2$ (the unique group of order two), $H_{\infty,*} = \mathbb{Z}/2[t]$ where $t$ is placed in degre $2$ (i.e. the stabilised homology is two-periodic). Better take this computation with a grain of salt, though.

[1] http://www.math.uni-bonn.de/people/schwede/equivariant.pdf