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

TL;DR

This paper explores the structure of transfer systems on lattices, introducing characteristic functions and matchstick games, and applies these concepts to classify transfer systems for rank two elementary Abelian groups.

Contribution

It establishes a connection between Hill's characteristic function and interior operators, develops the theory of saturated transfer systems on modular lattices, and introduces a matchstick game for classification.

Findings

Characteristic function $ ext{chi}$ surjects onto interior operators.

Unique maxima of fibers are saturated transfer systems.

Full lattice of transfer systems for rank two elementary Abelian groups determined.

Abstract

We prove that Hill's characteristic function $\chi$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $\chi$ 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.