Saturated and linear isometric transfer systems for cyclic groups of order pmqn

Abstract

Transfer systems are combinatorial objects which classify N∞ operads up to homotopy. By results of A. Blumberg and M. Hill [2], 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 Cpmqn, the cyclic group of order pmqn for p,q distinct primes and m,n≥0. We give a complete enumeration of saturated transfer systems for Cpmqn. We also prove J. Rubin's saturation conjecture for Cpqn; 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).