Vol. 315, No. 2, 2021

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$N_\infty$-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-operads in G-equivariant homotopy theory for G a finite cyclic group of prime power order. In particular, we show that for G = Cpn the natural order on the collection of N-operads is in bijection with the poset structure of the (n + 1)-associahedron. We further provide a lower bound for the number of possible N-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
Authors
Scott Balchin
Max Planck Institute for Mathematics
Bonn
Germany
David Barnes
Mathematical Sciences Research Centre
Queen’s University Belfast
United Kingdom
Constanze Roitzheim
School of Mathematics, Statistics and Actuarial Science
University of Kent
Canterbury
United Kingdom