A modern perspective on Tutte's homotopy theorem

TL;DR

This paper revisits Tutte's homotopy theorem for graphs associated with matroids, extends it with a refined classification of elementary cycles, and explores implications for matroid foundations and potential higher-dimensional generalizations.

Contribution

It provides a more detailed classification of elementary cycles in Tutte's homotopy theorem and offers a self-contained proof of the fundamental presentation of matroid foundations.

Findings

Extended classification of elementary cycles

Foundation of a matroid generated by universal cross-ratios

Preliminary ideas on higher Tutte homotopy theorem

Abstract

We begin with a review of Tutte's homotopy theory, which concerns the structure of certain graph associated to a matroid (together with some extra data). Concretely, Tutte's path theorem asserts that this graph is connected, and his homotopy theorem asserts that every cycle in the graph is a composition of ''elementary cycles'', which come in four different flavors. We present an extended version of the homotopy theorem, in which we give a more refined classification of the different types of elementary cycles. We explain in detail how the path theorem allows one to prove that the foundation of a matroid (in the sense of Baker--Lorscheid) is generated by universal cross-ratios, and how the extended homotopy theorem allows one to classify all algebraic relations between universal cross-ratios. The resulting ''fundamental presentation'' of the foundation was previously established in [Baker--Lorscheid], but the argument here is more self-contained. We then recall a few applications of the fundamental presentation to the representation theory of matroids. Finally, in the most novel but also the most speculative part of the paper, we discuss what a ''higher Tutte homotopy theorem'' might look like, and we present some preliminary computations along these lines.