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For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E)\end{equation}For theories $h$ (such as ordinary homology or oriented bordism), are there results about when the sequence stabilizes (such as upper bounds on $r(p,q)$ such that $E^r_{p,q}=E^\infty_{p,q}$)?

What if we restrict to a class of nice spaces (such as classifying spaces $BG$ of groups)? For example, are there stability bounds for the Lyndon-Hochschild-Serre spectral sequence for group cohomology? In my limited experience, this sequence is often stable on the second page for finite abelian groups.

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2 Answers 2

up vote 7 down vote accepted

The path-loop fibration for the sphere $S^n$ is:

$$\Omega S^n \to PS^n \to S^n$$

and $PS^n$ is contractible. In singular homology, the Atiyah-Hirzebruch-Serre spectral sequence (in this case, usually just known as the Serre spectral sequence) collapses at $E^n$, and not before. That is: there is a nontrivial differential $d_n$.

Specifically: $H_* \Omega S^n$ is $\mathbb{Z}$ in dimensions a multiple of $n-1$, so the SSS is concentrated in $E^2_{jn, k(n-1)}$, where $j = 0, 1$, and $k \in 0, 1, 2, \dots$. In order for the SS to collapse to the homology of a point, it must be the case that

$$d_n: E^2_{n, (k-1)(n-1)} \to E^2_{0, k(n-1)}$$

is an isomorphism. So the SSS degenerates in arbitrarily high degrees.

Playing the same game for this fibration with any other homology theory $h$ will yield equally bad results.

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Regarding the question where the base is $BG$: if $G$ has homological dimension $n$, then it is certainly the case that the SS will degenerate by $E^n$, since there are no possible differentials of length greater than $n$, regardless of what $h_*(F)$ is. Of course there are groups of arbitrarily high homological dimension, though... –  Craig Westerland Jun 26 at 2:16

In general, you can't hope for a nice collapsing result. For instance, the AHSS $$ H^*(K(\mathbb Z/2,2);\; K(1)^*) \Longrightarrow K(1)^*(K(\mathbb Z/2,2)) = 0 $$ does not collapse at any finite stage (this is a fun exercise). Here $K(1)$ denotes mod-$2$ periodic complex $K$-theory.

In "The Order of the Differentials in the Atiyah-Hirzebruch Spectral Sequence" (K-Theory 6, 1992), Arlettaz proves some bounds on the additive order of differentials in the combined Atiyah-Hirzebruch-Serre spectral sequence you're interested in. Under very special circumstances, this can imply that the SS collapses, you may want to check out if this applies to the cases you're interested in.

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