$\mathcal{E}_n$–Hopf invariants

Abstract

The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1}$ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for ${\mathcal{ℰ}}_n$–operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. We will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the ${\mathcal{ℰ}}_n$–bar construction on the cochains of $X$ and the homotopy groups of $X$. This pairing gives us a set of invariants of homotopy classes of maps from $S^m$ to a simplicial set $X$; this pairing can detect more homotopy classes of maps than the classical Hopf invariant.

The second part of the paper is devoted to combining the ${\mathcal{ℰ}}_n$–Hopf invariants with the Koszul duality theory for ${\mathcal{ℰ}}_n$–operads 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 $X$. This is done by studying the suspension morphism for the ${\mathcal{ℰ}}_{\infty }$–operad, which is a morphism from the ${\mathcal{ℰ}}_{\infty }$–operad to the desuspension of the ${\mathcal{ℰ}}_{\infty }$–operad. We show that it induces a functor from ${\mathcal{ℰ}}_{\infty }$–algebras to ${\mathcal{ℰ}}_{\infty }$–algebras, which has the property that it sends an ${\mathcal{ℰ}}_{\infty }$–model for a simplicial set $X$ to an ${\mathcal{ℰ}}_{\infty }$–model for the suspension of $X$.

We use this result to give a relation between the ${\mathcal{ℰ}}_n$–Hopf invariants of maps from $S^m$ into $X$ and the ${\mathcal{ℰ}}_{n+1}$–Hopf invariants of maps from $S^{m+1}$ into the suspension of $X$. One of the main results we show here is that this relation can be used to define invariants of stable homotopy classes of maps.