ℰn–Hopf invariants

Wierstra, Felix

doi:10.2140/agt.2020.20.2905

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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 $S^{4 n - 1}$ to $S^{2 n}$ , and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $ℰ_{n}$ –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 $ℰ_{n}$ –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 $ℰ_{n}$ –Hopf invariants with the Koszul duality theory for $ℰ_{n}$ –operads to get a relation between the $ℰ_{n}$ –Hopf invariants of a space $X$ and the $ℰ_{n + 1}$ –Hopf invariants of the suspension of $X$ . This is done by studying the suspension morphism for the $ℰ_{\infty}$ –operad, which is a morphism from the $ℰ_{\infty}$ –operad to the desuspension of the $ℰ_{\infty}$ –operad. We show that it induces a functor from $ℰ_{\infty}$ –algebras to $ℰ_{\infty}$ –algebras, which has the property that it sends an $ℰ_{\infty}$ –model for a simplicial set $X$ to an $ℰ_{\infty}$ –model for the suspension of $X$ .

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

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Keywords

Hopf invariant, $E_n$–algebras, suspensions

Mathematical Subject Classification 2010

Primary: 18D50, 55P40, 55P48, 55Q25

References

Publication

Received: 4 December 2018

Revised: 16 October 2019

Accepted: 17 February 2020

Published: 8 December 2020

Authors

Felix Wierstra
Department of Mathematics Stockholm University Stockholm Sweden	LAGA CNRS UMR 7539 Université Sorbonne Paris Nord Université Paris 8 Villetaneuse France

Volume 20, issue 6 (2020)