#### Volume 20, issue 6 (2020)

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$\mathcal{E}_n$–Hopf invariants

### Felix Wierstra

Algebraic & Geometric Topology 20 (2020) 2905–2956
DOI: 10.2140/agt.2020.20.2905
##### 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 ${\mathsc{ℰ}}_{n}$–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 ${\mathsc{ℰ}}_{n}$–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 ${\mathsc{ℰ}}_{n}$–Hopf invariants with the Koszul duality theory for ${\mathsc{ℰ}}_{n}$–operads to get a relation between the ${\mathsc{ℰ}}_{n}$–Hopf invariants of a space $X$ and the ${\mathsc{ℰ}}_{n+1}$–Hopf invariants of the suspension of $X$. This is done by studying the suspension morphism for the ${\mathsc{ℰ}}_{\infty }$–operad, which is a morphism from the ${\mathsc{ℰ}}_{\infty }$–operad to the desuspension of the ${\mathsc{ℰ}}_{\infty }$–operad. We show that it induces a functor from ${\mathsc{ℰ}}_{\infty }$–algebras to ${\mathsc{ℰ}}_{\infty }$–algebras, which has the property that it sends an ${\mathsc{ℰ}}_{\infty }$–model for a simplicial set $X$ to an ${\mathsc{ℰ}}_{\infty }$–model for the suspension of $X$.

We use this result to give a relation between the ${\mathsc{ℰ}}_{n}$–Hopf invariants of maps from ${S}^{m}$ into $X$ and the ${\mathsc{ℰ}}_{n+1}$–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.

##### Keywords
Hopf invariant, $E_n$–algebras, suspensions
##### Mathematical Subject Classification 2010
Primary: 18D50, 55P40, 55P48, 55Q25