A modern perspective on Tutte's homotopy theorem

TL;DR

This work reframes Tutte's classical homotopy theory of matroids in a modern lattice- and pasture-based language, showing how 'foundations' of matroids can be presented by generators (universal cross-ratios) and relations derived from Tutte's path and extended homotopy theorems. It provides an explicit, self-contained route from the lattice of flats to a universal algebraic object FM that encodes all rescaling classes of representations, and demonstrates how these foundations yield classical excluded-minor results and Dressian structure for matroids without large uniform minors. The extended-homotopy framework refines the classification of elementary cycles, enabling a more direct presentation for FM and facilitating practical computations via the hyperplane-incidence graph. Beyond consolidating known results, the paper sketches a rich program toward higher (2D, 3D, etc.) homotopy theorems, with preliminary constructions and computations suggesting a deep link between homotopical structure and syzygies among universal cross-ratios. Overall, the approach unifies combinatorial matroid theory with algebraic and topological methods, offering new computational tools and conceptual insights into matroid representations and their foundational relations.

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.

Figures (26)

• Figure 1: The subposet structure in \ref{["thm: Tutte's path theorem"]}
• Figure 2: Some important graphs for the current section and \ref{['subsection:extended-homotopy-theorem']}
• Figure 3: The Tutte constellations and elementary Tutte paths of the first kind
• Figure 4: Tutte constellations and elementary Tutte paths of the second kind
• Figure 5: The Tutte constellation and elementary Tutte path of the third kind

Theorems & Definitions (165)

• Definition 1
• Proposition 1
• Proposition 2
• Lemma 1
• Definition 2
• Example 1
• Definition 3
• Theorem 1.1: Tutte's path theorem Tutte58a
• proof
• Remark 1