$N_\infty$-operads and associahedra

Scott Balchin, David Barnes and Constanze Roitzheim

Vol. 315 (2021), No. 2, 285–304

DOI: 10.2140/pjm.2021.315.285

Abstract

We provide a combinatorial approach to studying the collection of $N_{\infty }$-operads in $G$-equivariant homotopy theory for $G$ a finite cyclic group of prime power order. In particular, we show that for $G=C_{p^n}$ the natural order on the collection of $N_{\infty }$-operads is in bijection with the poset structure of the $\left(n+1\right)$-associahedron. We further provide a lower bound for the number of possible $N_{\infty }$-operads for any finite cyclic group $G$. As such, we have reduced an intricate problem in equivariant homotopy theory to a manageable combinatorial problem.

Keywords

operads, equivariant spectra, ring spectra, associahedra, Catalan numbers

Mathematical Subject Classification 2010

Primary: 18D50, 55P91

Secondary: 06A07, 52B20, 55N91

Milestones

Received: 9 March 2020

Revised: 10 January 2021

Accepted: 8 July 2021

Published: 19 January 2022