#### 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 ${N}_{\infty }$-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
##### 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