Abstract
AbstractWe study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.
Highlights
Commutative ring spectra are of fundamental importance in stable homotopy theory
Every genuine commutative ring G-spectrum R is equipped with a family of twisted products
We begin our incremental analysis of linear isometries operads by specializing G to finite cyclic groups. This reduces the problem to a pleasant puzzle in modular arithmetic (Proposition 5.15), and we identify two cases where our original hopes about linear isometries operads are met
Summary
Commutative ring spectra are of fundamental importance in stable homotopy theory. They represent cohomology theories, which are equipped with power operations akin to the usual Steenrod operations. The homotopy types of Steiner and linear isometries G-operads are determined by the representation theory of G over the reals, but the translation to the algebra of indexing systems is surprisingly bad. Given a finite group G, identify extra algebraic conditions on indexing systems that characterize the images of the Steiner operads and linear isometries operads under the map A : Ho(N∞-OpG) → Ind(G).
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