The Calabi invariant and The Lyndon-Hochschild-Serre spectral sequence

TL;DR

This paper explores the relationship between the Calabi invariant, the Lyndon-Hochschild-Serre spectral sequence, and group extensions, providing explicit formulas and clarifying their interconnections.

Contribution

It introduces a formula for the extension class of a group extension using connection cochains, linking it to the LHS spectral sequence.

Findings

Derived a formula for the extension class using connection cochains

Clarified the relation among connection cochains, extension classes, and the LHS spectral sequence

Provided insights into the structure of group extensions and their invariants

Abstract

Let $G$ be a group and $N$ be a normal subgroup of $G$. There exists the group extension $G$ of $G/N$ by $N$. For a $G$-module $A$ which $N$ acts on trivially and a $G$-invariant homomorphism on $N$ to $A$, we obtain a central extension of $G/N$ by $A$. By using connection cochains, we exhibit the formula of its extension class such that clarify the relation among connection cochains, extension classes and the LHS spectral sequence.