Saturated and linear isometric transfer systems for cyclic groups of order $p^m{q}^n$

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 $C_{p^m{q}^n}$, the cyclic group of order $p^m{q}^n$ for $p,q$ distinct primes and $m,n\ge 0$. We give a complete enumeration of saturated transfer systems for $C_{p^m{q}^n}$. We also prove J. Rubin's saturation conjecture for $C_{p{q}^n}$; 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).

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