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.



$RO(Q)$-graded homotopy fixed point spectral sequence
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 $[A\_k/A\_{k-1},X]^G\_\*=\pi\_{k+\*}(X)$.


