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I have the following problem. Let $Q$ denote $\mathbb{Z}/2$ as an abelian group and let $X$ be a $Q$-spectrum. If I want to compute the homotopy groups of $X^{hQ}$, homotopy fixed points spectrum, I can use the homotopy fixed points spectral sequence (HFPSS). It's construction is briefly described in Neil's answer to this question: $RO(Q)$-graded homotopy fixed point spectral sequence.

In particular, it's $E_1$ page is given by $E_1^{p,q}=[Q_+\wedge\mathbb{S}^p, X]^{p+q}_Q$. By work of Greenlees and May this is isomorphic to $Hom_{Q}(H_p(Q_+\wedge \mathbb{S}^p), \pi_{-q}(i^*X))$ (I hope signs are correct here). The $d_1$ differential is then induced by the "geometric boundary", i.e. a composition $Q_+\wedge\mathbb{S}^p\to\Sigma\mathbb{S}((p-1)\sigma)_+\to Q_+\wedge\mathbb{S}^{p-1}$. Both maps here are coming from cofiber sequences of the form $\mathbb{S}((k-1)\sigma)_+\to\mathbb{S}(k\sigma)_+\to Q_+\wedge\mathbb{S}^k$. Of course, $H_p(Q_+\wedge\mathbb{S}^p)\cong\mathbb{Z}[Q]$.

Question: Why the map induced by the geometric boundary in $H_p$ is the map arising in the classical $\mathbb{Z}[Q]$-resolution of $\mathbb{Z}$, i.e. either $1+g$ or $1-g$?

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    $\begingroup$ You have constructed an explicit $C_2$-CW-structure on $\mathbb S(\infty)$, which is a model for $EC_2$ (contractible with free action). Namely, you have one cell $C_2\times D^n$ in each dimension, which are the north and south hemisphere attached to the equator. The differential is given by the degrees of the attaching maps; to get the signs, note that this CW structure comes from the realization of the nerve of the category $C_2//C_2$ (two isomorphic objects). In fact, the HFPSS does not depend on the genuine $C_2$-structure, and is the Serre SS of the fibration $EC_2\times_{C_2}X\to BC_2$. $\endgroup$
    – Bertram Arnold
    Commented Nov 17, 2020 at 21:21
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    $\begingroup$ I made a mistake in the last part of the comment: the Serre SS I described is the spectral sequence converging to homotopy groups of the homotopy orbits $X_{hG}$. For fixed points, you essentially do the same construction of filtering $EG$ with an equivariant CW structure, coming eg from the realization of the nerve of $G//G$; the $E_1$-page is the standard cobar complex computing group cohomology with values in $\pi_*(X)$. Again, this only depends on the naive equivariant structure, i.e. the functor $BG\to Sp$, and the HFPSS is the Serre-Atiyah-Hizebruch SS computing twisted $X$-cohomology. $\endgroup$
    – Bertram Arnold
    Commented Nov 17, 2020 at 22:34
  • $\begingroup$ Of course, I see now. Thanks! $\endgroup$
    – Igor Sikora
    Commented Nov 18, 2020 at 11:38

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