\(\mathcal{E}_n\)-Hopf invariants (Q2663387)

The classical Hopf invariant is defined as an invariant of maps \(f : \mathbb{S}^{4n-1}\to \mathbb{S}^{4n}\) and was initially used to show that the Hopf fibration is not homotopic to the constant map, but found later many more applications. One way of defining the classical Hopf invariant is by defining a pairing \[\big<-,-\big> : H^\ast(B_{Ass}C^\ast(X,\mathbb{K}))\otimes \pi_\ast(X)\to\mathbb{K}\] between the cohomology of the associative bar construction \( B_{Ass}C^\ast(X,\mathbb{K})\) on the cochains \(C^\ast(X,\mathbb{K})\) of a space \(X\) and the homotopy groups \(\pi_\ast(X)\) of \(X\), where \(X\) is a simply connected space and \(\mathbb{K}\) is a commutative ring. The goal of this paper is to generalize this pairing by using the fact that the cochains \(C^\ast(X,\mathbb{K})\) are not just an associative algebra, but are an \(\mathcal{E}_\infty\)-algebra. The author generalizes the classical Hopf invariant and defines \(\mathcal{E}_n\)-invariants of a space \(X\), by considering a pairing between the cohomology of the \(\mathcal{E}_n\)-bar construction on \(C^\ast(X,\mathbb{K})\) and \(\pi_\ast(X)\). Then, to get a relation between the \(\mathcal{E}_n\)-Hopf invariants of a space \(X\) and the \(\mathcal{E}_{n+1}\)- Hopf invariants of the suspension of \(X\), the \(\mathcal{E}_n\)-Hopf invariants are combined with Fresse's theory of Koszul duality theory [\textit{B. Fresse}, Sel. Math., New Ser. 17, No. 2, 363--434 (2011; Zbl 1248.55003)] for \(\mathcal{E}_n\)-operads. This result is used to give a relation between the \(\mathcal{E}_n\)-Hopf invariants of maps from \(\mathbb{S}^m\) into \(X\) and the \(\mathcal{E}_{n+1}\)-Hopf invariants of maps from \(\mathbb{S}^{m+1}\) into the suspension of \(X\). It is shown that this relation can be used to define invariants of stable homotopy classes of maps.