Let A be a semi-simplicial space and $k^*$ be a generalised cohomology theory as in This paper proposition 5.1. Using the natural filtration of the realisation of $A$ and then using the staircase diagramme of long exact sequences of pairs, we get the first page and exact couple which gives the second page and so on. Now we have the following relations (!)$${E_1^p}^q=k^{p+q}(\Delta^pA,\Delta^{p-1}A)\cong k^{p+q}(\Sigma^p(A_p)_+,\Sigma^p(A^d_p)_+) \cong k^q(A_p,A^d_p)\rightarrow k^q(A_p)$$ (please see ~5 ,page 109 for notations) where the arrow indicates the **natural** map. So we can make this replacement in the staircase diagramme.

**Question:** How can you calculate $E_2^{pq}$ from $k^q(A)$ ?

Using induction on the cardinality of $S$ and using the fact that any epimorphish between finite ordinals has a section we get(right?) the following $$k^q(A(S))\cong\bigoplus \limits_Tk^q(A(T),A^d(T)) $$ where $T$ runs through the quotient (i.e through all the epimorphisms from S). Now putting $S=[P]$, we get that $k^q(A_p,A_p^d)$ is a direct summand in $k^q(A_p)$ as stated there.

**But how** does it helps us to get $E_2^{pq}$? We also see that the **natural** map commutes with differential.

**Question**:For fix $q$, are the cohomology groups of the complexes $k^q(A_p,A^d_p)$ and $k^q(A_p)$ same? **Q:** What does he want to mean by **semi-simplicial (so called simplicial) cochain complex $k^q(A)$?** Can we say something about the convergence of the stated **right-half space** spectral sequence?