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 ${}_2{H}^n(G,A,\kappa ;B)$ which we call the secondary cohomology. We give a construction that associates to a pointed topological space $(X,x_0)$ an invariant ${}_2{\kappa }^4{\in }_2{H}^4({\pi }_1(X),{\pi }_2(X),{\kappa }^3;{\pi }_3(X))$. This construction can be seen a "3-type" generalization of the classical $k$-invariant.

8 pages

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