Homology, Homotopy and Applications

Volume 12 (2010)

Number 2

On the second cohomology group of a simplicial group

Pages: 167 – 210

DOI: https://dx.doi.org/10.4310/HHA.2010.v12.n2.a6

Author

Sebastian Thomas (Lehrstuhl D für Mathematik, RWTH Aachen University, Aachen, Germany)

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.

Keywords

cohomology, simplicial groups, crossed modules

2010 Mathematics Subject Classification

18D35, 18G30, 20J06, 55U10

Published 1 January 2010