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Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games

Linus Bao, Christy Hazel, Tia Karkos, Alice Kessler, Austin Nicolas, Kyle Ormsby, Jeremie Park, Cait Schleff, Scotty Tilton

Abstract

We prove that Hill's characteristic function $χ$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $χ$ have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a ``matchstick game'' that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups.

Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games

Abstract

We prove that Hill's characteristic function for transfer systems on a lattice surjects onto interior operators for . Moreover, the fibers of have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a ``matchstick game'' that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups.
Paper Structure (8 sections, 32 theorems, 30 equations, 4 figures)

This paper contains 8 sections, 32 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation, context, and results
  3. Transfer systems
  4. Outline of the paper
  5. Characteristic functions for transfer systems
  6. Modular lattices and matchstick games
  7. Categorical properties of transfer systems
  8. Transfer systems for rank two elementary Abelian groups

Key Result

Theorem 1

Let $p$ be a prime number. There are exactly $2^{p+2}+p+1$ transfer systems for $C_p\times C_p$, and the lattice of transfer systems has an explicit decomposition in terms of two Boolean lattices on $p+1$ elements along with $p+1$ intermediate elements (see prop:interatedfusion for lattice structure

Figures (4)

  • Figure 1: Transfer systems for the lattice $[2]$.
  • Figure 2: Saturated covers for the lattice $[2]^{*3}\cong \operatorname{Sub}(C_2\times C_2)$.
  • Figure 3: The subgroup lattice of $C_p\times C_p$
  • Figure 4: The Hasse diagram for $\mathop{\mathrm{Tr}}\nolimits([2]^{*3})\cong \mathop{\mathrm{Tr}}\nolimits(C_2\times C_2)$. Compare with the Hasse diagram for saturated covers in \ref{['fig:satcovers']}.

Theorems & Definitions (74)

  • Theorem : \ref{['thm:ranktwo']}
  • Theorem : \ref{['thm:maxsat']} and \ref{['thm:minchi']}
  • Definition 1
  • Proposition 1
  • Theorem 1: BlumbergHillRubin
  • Definition 2
  • Theorem 2: Rubin
  • Definition 3
  • Proposition 2: Rubin
  • Proposition 3
  • ...and 64 more