A differential in the Lyndon-Hochschild-Serre spectral sequence

Ian J. Leary
Faculty of Mathematical Studies, University of Southampton SO17 1BJ,
England.

J. Pure and Appl. Alg. 88 (the Karl Gruenberg 65th birthday issue) (1993), 
155-168. 

MR94m:20102. 

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.