Secondary Cohomology and k-invariants

Abstract

For a triple $(G,A,\kappa)$ (where $G$ is a group, $A$ is a $G$-module and $\kappa:G^3\to A$ is a 3-cocycle) and a $G$-module $B$ we introduce a new cohomology theory $_2H^n(G,A,\kappa;B)$ which we call the secondary cohomology. We give a construction that associates to a pointed topological space $(X,x_0)$ an invariant $_2\kappa^4\in_2H^4(\pi_1(X),\pi_2(X),\kappa^3;\pi_3(X))$. This construction can be seen a "3-type" generalization of the classical $k$-invariant.