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Maximal compatibility of disklike $G$-transfer systems

David DeMark, Michael A. Hill, Yigal Kamel, Nelson Niu, Kurt Stoeckl, Danika Van Niel, Guoqi Yan

Abstract

Transfer systems are a combinatorial model for $N_{\infty}$-operads, which encode commutative structures in equivariant homotopy theory. Blumberg--Hill and Chan gave criteria for when two transfer systems are a compatible pair, meaning they encode the additive transfers and multiplicative norms of a ring-type structure. In this paper, given a transfer system encoding an additive structure, we give explicit formulae for the maximal transfer system it is compatible with. Our formulae simplify for disklike transfer systems, which typically encode additive structures. Further, we prove that (maximal) compatibility is functorial with respect to the inflation map induced by a quotient of groups, letting us compute maximal compatible transfer systems as inflations of connected transfer systems.

Maximal compatibility of disklike $G$-transfer systems

Abstract

Transfer systems are a combinatorial model for -operads, which encode commutative structures in equivariant homotopy theory. Blumberg--Hill and Chan gave criteria for when two transfer systems are a compatible pair, meaning they encode the additive transfers and multiplicative norms of a ring-type structure. In this paper, given a transfer system encoding an additive structure, we give explicit formulae for the maximal transfer system it is compatible with. Our formulae simplify for disklike transfer systems, which typically encode additive structures. Further, we prove that (maximal) compatibility is functorial with respect to the inflation map induced by a quotient of groups, letting us compute maximal compatible transfer systems as inflations of connected transfer systems.

Paper Structure

This paper contains 14 sections, 41 theorems, 49 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Transfer Systems
  4. Compatible Pairs
  5. Characterizing the Maximal Compatible Transfer System
  6. Maximal Compatible Formulae
  7. Consequences and Examples
  8. Disklike Maximal Compatible Formulae
  9. Disklike Consequences and Examples
  10. Disklike Conjectured Characterization
  11. Maximal Compatibility is Preserved by Inflation
  12. Inflation Formulae
  13. Preservation
  14. Consequence and Counterexamples

Key Result

Theorem 1.1

Let $\mathcal{O}$ be a $G$-transfer system. The maximal compatible transfer system $\mathrm{M}({\mathcal{O}})$ admits the following recursive formula where $r<e$ means, that in the poset of restrictions in $\mathcal{O}$, the transfer $e$ restricts into $r$, and $r<^S e$ means that this restriction satisfies the compatibility condition (def:compatibility successes and failures). Furthermore, if $\

Figures (1)

  • Figure 1: The Hasse diagram of the $C_{pq}$-transfer systems ordered by containment, $p$ and $q$ distinct primes. Each transfer system $\mathcal{O}$ is paired with its maximal compatible transfer system $\mathrm{M}({\mathcal{O}})$ via a blue circling.

Theorems & Definitions (97)

  • Theorem 1.1: Theorems \ref{["thm:r' can be assumed to be in M(O) 2"]} and \ref{['thm:maximal compatible disk-like via cover relations']}
  • Theorem 1.2
  • Corollary 1.3: \ref{['cor:universal-transfer']}
  • Conjecture 1.4: \ref{['conj:maximal compatible disk-like via single restriction']}
  • Definition 2.1
  • Definition 2.3
  • Proposition 2.4: RubinOperadicLifts
  • Definition 2.5
  • Definition 2.6: RubinDetectingOperads
  • Proposition 2.7: RubinDetectingOperads
  • ...and 87 more