Secondary cohomology and \(k\)-invariants
Let \((X,x_0)\) be a pointed topological space. The realization problem asks when a family of isomorphisms \(\phi_i : \pi_i(X) \to \pi_i(Y)\) is induced by a morphism \(f: X\to Y\). \textit{S. Eilenberg} and \textit{S. MacLane} [Proc. Natl. Acad. Sci. USA, 32, 277--280 (1946; Zbl 0061.40707)] gave a solution to this problem in terms of the \(k\)-invariant in the case \(\pi_i(X)=0\) for \(i>2\). Inspired by their work, the author constructs an invariant called the secondary \(k\)-invariant. This is an element of the secondary cohomology group. The body of the paper is devoted to showing that the secondary \(k\)-invariant is indeed an invariant, first in the case when \(X\) is simply connected and then in the general case. After describing the third cohomology group of \(X\) in terms that may be amenable to defining higher order \(k\)-invariants, the author concludes with a discussion of potential generalizations as well as how the invariant may be used to study other problems of interest.
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