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?


Segal is assuming the reader has knowledge of a basic fact about chain complexes formed from simplicial (or cosimplicial) abelian groups: the sub (or quotient) complex of `degenerate' chains is acyclic.

He points out that the $E_1$ term of the spectral sequence he is describing is naturally mapping to a complement of the degenerate part of the chain complex for the cosimplicial abelian group $k^q(A)$. His big diagram on page 110 of the paper you linked to is checking that the differential is right: note that the composite of the bottom line is the usual alternating sum of face maps, as he points out.

It may help to note that in the special case when $A$ is a simplicial set, rathen than a general simplicial space, the spectral sequence will be the Atiyah--Hirzebruch spectral sequence.

  • $\begingroup$ @ Nicholas Kuhn ,now the questions about cohomology groups are clear to me from the first paragraph of your post and the stack project .So, we get the terms of the second page as desired. Can we say something more about on the differential on the second page ? Note that the stated natural map is not a map between spectral sequences etc .(right ?) $\endgroup$ – Aloke Mar 17 at 11:20
  • $\begingroup$ @Aloke One can't really say much that is useful about the 2nd differential, except that it is natural in the data. At the end of his paper, Segal is pretty much pointing out that, if the cohomology theory has a cup product then one will be getting a spectral sequence of differential graded algebras. $\endgroup$ – Nicholas Kuhn Mar 17 at 15:36

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