| Autor: |
Wierstra, Felix |
| Jazyk: |
angličtina |
| Rok vydání: |
2020 |
| Předmět: |
|
| Zdroj: |
Algebr. Geom. Topol. 20, no. 6 (2020), 2905-2956 |
| Popis: |
The classical Hopf invariant is an invariant of homotopy classes of maps from [math] to [math] , and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for [math] –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 [math] and the homotopy groups of [math] . We will give a generalization of the classical Hopf invariant by defining a pairing between the cohomology of the [math] –bar construction on the cochains of [math] and the homotopy groups of [math] . This pairing gives us a set of invariants of homotopy classes of maps from [math] to a simplicial set [math] ; 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 [math] –Hopf invariants with the Koszul duality theory for [math] –operads to get a relation between the [math] –Hopf invariants of a space [math] and the [math] –Hopf invariants of the suspension of [math] . This is done by studying the suspension morphism for the [math] –operad, which is a morphism from the [math] –operad to the desuspension of the [math] –operad. We show that it induces a functor from [math] –algebras to [math] –algebras, which has the property that it sends an [math] –model for a simplicial set [math] to an [math] –model for the suspension of [math] . ¶ We use this result to give a relation between the [math] –Hopf invariants of maps from [math] into [math] and the [math] –Hopf invariants of maps from [math] into the suspension of [math] . One of the main results we show here is that this relation can be used to define invariants of stable homotopy classes of maps. |
| Databáze: |
OpenAIRE |
| Externí odkaz: |
|
