The third cohomology group classifies crossed module extensions

2009/11/15 by Thomas, Sebastian
#18D35 #20J06 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.0911.2861

Abstract

We give an elementary proof of the well-known fact that the third cohomology group H³(G, M) of a group G with coefficients in an abelian G-module M is in bijection to the set Ext²(G, M) of equivalence classes of crossed module extensions of G with M.

Related

• Crossed modules and symmetric cohomology of groups
• Group cohomology with coefficients in a crossed-module
• On the nonabelian cohomology with coefficients in a crossed module
• Abelian extensions and crossed modules of Hom-Lie algebras
• Inverse semigroup cohomology and crossed module extensions of semilattices of groups by inverse semigroups