Secondary Cohomology and k-invariants

Abstract

For a triple $(G,A,κ)$ (where $G$ is a group, $A$ is a $G$-module and $κ:G^3\to A$ is a 3-cocycle) and a $G$-module $B$ we introduce a new cohomology theory $_2H^n(G,A,κ;B)$ which we call the secondary cohomology. We give a construction that associates to a pointed topological space $(X,x_0)$ an invariant $_2κ^4\in_2H^4(π_1(X),π_2(X),κ^3;π_3(X))$. This construction can be seen a "3-type" generalization of the classical $k$-invariant.