English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study the Eckmann-Hilton dual of the little disks algebra structure oniterated loop spaces: With the right definitions, every $n$-fold suspension isa coalgebra over the little $n$-disks operad. This structure inducesnon-trivial cooperations on the rational homotopy groups of an $n$-foldsuspension. We describe the Eckmann-Hilton dual of the Browder bracket, whichis a cooperation that forms an obstruction for an $n$-fold suspension to be an$(n+1)$-fold suspension, i.e. if this cooperation is non-zero then the space isnot an $(n+1)$-fold suspension. We prove several results in equivariantrational homotopy theory that play an essential role in our results. Namely, weprove a version of the Sullivan conjecture for the Maurer-Cartan simplicial setof certain $L_\infty$-algebras equipped with a finite group action, and weprovide rational models for fixed and homotopy fixed points of (mapping) spacesunder some connectivity assumptions in the context of finite groups. We furthershow that by using the Eckmann-Hilton dual of the Browder operation we can usethe rational homotopy groups to detect the difference between certain spacesthat are rationally homotopy equivalent, but not homotopy equivalent.

Iterated suspensions are coalgebras over the little disks operad