Saturated and linear isometric transfer systems for cyclic groups of order $p^mq^n$

TL;DR

This paper classifies saturated and linear isometric transfer systems for cyclic groups of order p^mq^n, providing a complete enumeration and confirming Rubin's conjecture for certain cases, advancing understanding of N_infinity operads.

Contribution

It offers a complete enumeration of saturated transfer systems for cyclic groups of order p^mq^n and proves Rubin's saturation conjecture for specific cases, linking transfer systems to linear isometries operads.

Findings

Complete enumeration of saturated transfer systems for C_{p^mq^n}

Proof of Rubin's saturation conjecture for C_{pq^n} with large primes

Establishment of correspondence between saturated transfer systems and linear isometries operads

Abstract

Transfer systems are combinatorial objects which classify $N_\infty$ operads up to homotopy. By results of A. Blumberg and M. Hill, every transfer system associated to a linear isometries operad is also saturated (closed under a particular two-out-of-three property). We investigate saturated and linear isometric transfer systems with equivariance group $C_{p^mq^n}$, the cyclic group of order $p^mq^n$ for $p,q$ distinct primes and $m,n\ge 0$. We give a complete enumeration of saturated transfer systems for $C_{p^mq^n}$. We also prove J. Rubin's saturation conjecture for $C_{pq^n}$; this says that every saturated transfer system is realized by a linear isometries operad for $p,q$ sufficiently large (greater than $3$ in this case).