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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 S4n1 to S2n, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for 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 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 Sm 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 n–Hopf invariants with the Koszul duality theory for n–operads to get a relation between the n–Hopf invariants of a space X and the n+1–Hopf invariants of the suspension of X. This is done by studying the suspension morphism for the –operad, which is a morphism from the –operad to the desuspension of the –operad. We show that it induces a functor from –algebras to –algebras, which has the property that it sends an –model for a simplicial set X to an –model for the suspension of X.

We use this result to give a relation between the n–Hopf invariants of maps from Sm into X and the n+1–Hopf invariants of maps from Sm+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
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