On the second cohomology group of a simplicial group

TL;DR

This paper provides an algebraic proof that the second cohomology group of a simplicial group can be computed using its homotopy groups and Postnikov invariant, utilizing crossed module extensions.

Contribution

It offers a new algebraic proof of a classical result relating second cohomology to homotopy groups and Postnikov invariants of simplicial groups.

Findings

Algebraic proof of Eilenberg-Mac Lane's result

Explicit computation method for second cohomology group

Application of crossed module extensions in cohomology

Abstract

We give an algebraic proof for the result of Eilenberg and Mac Lane that the second cohomology group of a simplicial group G can be computed as a quotient of a fibre product involving the first two homotopy groups and the first Postnikov invariant of G. Our main tool is the theory of crossed module extensions of groups.