$RO(Q)$-graded homotopy fixed point spectral sequence

Question

I am trying to understand some part of J. Greenlees's "Four approaches to cohomology theories with reality": https://arxiv.org/abs/1705.09365

I have a problem with understanding $RO(Q)$-graded homotopy fixed point spectral sequence. Namely: 1. In section 2.C, proof of Lemma 2.1 - how filtration on $EQ_+$ helps us in describing $E_1$-page of this spectral sequence? And why it looks as described?

Which leads to seceond question:

2. How this spectral sequence arises?

Any help or reference would be strongly appreciated.

Answer 1

For a based $G$-space or $G$-spectrum $X$, the homotopy fixed point object $X^{hG}$ is by definition $F_G(EG_+,X)$. Suppose we write $EG$ as the colimit of a sequence of $G$-subspaces $A_k$. This then gives a tower of objects $F_G((A_k)_+,X)$ whose inverse limit is $X^{hG}$, and the fibres of the maps in the tower are $F_G(A_k/A_{k-1},X)$. There is a standard spectral sequence for the homotopy groups of the inverse limit of any tower, with the $E^1$ term being given by the homotopy groups of the fibres. In particular, we have a spectral sequence for $\pi_*(X^{hG})$ with $E^1$ term given by $[A_k/A_{k-1},X]^G_*$. In the example in question, $G$ has order $2$, so we can take $EG$ to be $S(\mathbb{R}^\infty)$ with the antipodal action, or $S(\infty\sigma)$ in the notation of Greenlees. We can then take $A_k=S((k+1)\sigma)$ and check that $A_k/A_{k-1}\simeq G_+\wedge S^k$ (with trivial action on $S^k$). This gives $[A_k/A_{k-1},X]^G_*=\pi_{k+*}(X)$.