A differential in the Lyndon-Hochschild-Serre spectral sequence

TL;DR

This paper analyzes differentials in the Lyndon-Hochschild-Serre spectral sequence for certain group extensions, providing new formulas for higher differentials and applications to cohomology of finite groups.

Contribution

It introduces a description for the fourth differential in the spectral sequence and relates it to Massey products, extending understanding of cohomological computations.

Findings

Derived a formula for the fourth differential in the spectral sequence.

Connected higher differentials to Massey triple products in cohomology.

Determined Poincare series for mod-3 cohomology of specific 3-groups.

Abstract

We consider the Lyndon-Hochschild-Serre spectral sequence with mod-p coefficients for a central extension with kernel cyclic of order a power of p and arbitrary discrete quotient group. For this spectral sequence the second and third differentials are known, and we give a description for the fourth differential. Using this result we deduce a similar formula for the Serre spectral sequence for a principal fibration with fibre the classifying space of a cyclic p-group. The differential from odd rows to even rows involves a Massey triple product, so we describe the calculation of such products in the cohomology of a finite abelian group. As an example we determine the Poincare series for the mod-3 cohomology of various 3-groups. Remarks. 1) My definition of the higher differentials $d_i$ for $i\geq 2$ in the spectral sequence for a double chain complex differs from the usual one by a factor of $(-1)^{i+1}$. Both conventions are consistent, but the usual definition has the advantage of agreeing with the ``obvious'' definition of the differentials in the spectral sequence for the associated filtered chain complex. All of the theorems in this paper remain true exactly as stated if the more usual definition of $d_i$ is taken. 2) Carles Broto found a small mistake in this paper: the result for fibrations with fibre the classifying space of a cyclic group is stated for arbitrary fibrations, although it is only proved for principal fibrations. Since it is apparent from the first sentence of the proof that only principal fibrations are being considered, I have not bothered to publish an erratum.