Tutte’s Homotopy Theory in Matroid Foundations

• Tutte’s Homotopy Theory is a combinatorial framework defining strong connectivity in Tutte graphs through modular cuts in matroid lattices.
• It classifies elementary cycles into distinct types, enabling cycle deformations to trivial ones via finite insertions and deletions.
• The theory links cycle decomposition with algebraic structures like the matroid foundation and universal cross-ratios, impacting representation theory and tropical geometry.

Tutte’s homotopy theory is a collection of theorems and constructions describing strong connectivity and homotopy properties of certain graphs associated with a matroid and a modular cut in its lattice of flats. Originating in the classical study of matroids and developed further by Baker, Jin, and Lorscheid, the theory provides a combinatorial and algebraic framework for analyzing cycles in the resulting graph, decomposing them into elementary pieces, and relating these combinatorial features to the algebraic structure known as the matroid foundation. Modern treatments extend Tutte's original results, refining the classification of elementary cycles and connecting the theory deeply to matroid representation, universal cross-ratios, and higher homotopical phenomena (Baker et al., 5 Jan 2026).

1. Construction of the Tutte Graph

Given a matroid $M$ on ground set $E$, with lattice of flats $\Lambda_M$, a nonempty modular cut $\Gamma \subset \Lambda$ is fixed (equivalently, a single-element extension of $M$). The set of hyperplanes “off” $\Gamma$ is

$$\mathcal H = \{ H \in \Lambda : \mathrm{rank}(H) = \mathrm{rank}(M) - 1, H \notin \Gamma \}.$$

For a corank–2 flat $L \in \Lambda^{(2)}$, $L$ is called indecomposable if $M/L$ is connected.

The Tutte graph $G_{M, \Gamma}$ is the simple graph with vertex set $\mathcal H$. Edges correspond to unordered pairs $\{H_1, H_2\} \subset \mathcal H$ for which $L = H_1 \cap H_2$ is an indecomposable corank–2 flat. Thus, adjacency encodes combinatorially strong “intersection connectivity” in the lattice of hyperplanes, filtered by the modular cut.

2. Tutte’s Path Theorem

A central result of the theory is Tutte’s Path Theorem: If $M$ is connected, then $G_{M,\Gamma}$ is also connected. Explicitly, for any distinct $H, H' \in \mathcal H$, there exists a sequence

$H = H_0, H_1, \dots, H_k = H'$

such that each intersection $H_i \cap H_{i+1}$ is an indecomposable corank–2 flat. The proof uses induction on corank, constructing intermediate indecomposable flats

$S \supsetneq U \supsetneq T$

to find a hyperplane $X$ "transversal" to both $H$ and $H'$; the inductive hypothesis then constructs a path from $H$ to $H'$ via $X$ inside a smaller contraction.

This strong connectivity result underpins later theorems by ensuring that local combinatorial manipulations can be globalized across the Tutte graph (Baker et al., 5 Jan 2026).

3. Tutte’s Homotopy Theorem and Elementary Cycles

A closed Tutte path in $G_{M,\Gamma}$ is a cycle

$\gamma = (H_1, H_2, \dots, H_k, H_1)$

such that each consecutive pair meets in an indecomposable corank–2 flat. Tutte’s Homotopy Theorem asserts that every such cycle can be "deformed" (by adjunction and deletion of certain small cycles) to the trivial one-vertex cycle.

3.1. Elementary Cycle Types

Historically, Tutte identified four kinds of elementary cycles:

1. First kind (rank–2, $U_{2,2}$): $(H, H', H)$ with $H \neq H'$, $H \cap H'$ indecomposable.
2. Second kind (rank–3, $U_{2,3}$ or $U_{3,3}$): $(H_1,H_2,H_3,H_1)$ with $$\mathrm{rank}(H_1\cap H_2\cap H_3)=\mathrm{rank}(M)-2$$.
3. Third kind (rank–4, $U_{3,4}$): $(H_{12}, H_{13}, H_{34}, H_{24}, H_{12})$.
4. Fourth kind (rank–5, $M(K_{2,3})$): $(H_{1245}, H_{126}, H_{1346}, H_{456}, H_{1245})$.

The modern extension [Baker–Jin–Lorscheid] refines these into nine types, indexed by isomorphism types of the relevant sublattices (e.g., $U_{2,3}$, parallel-extensions, variants of $U_{3,4}$, $M(K_4)$, $M(K_{2,3})$), each inducing a cycle of length $3$, $4$, $5$, or $6$. A complete list of these nine types is constructed, and by explicit and finite computations, the theory shows that insertions/deletions of these types suffice to trivialize any cycle.

3.2. Homotopy Theorem

The modern statement:

Every closed Tutte cycle in $G_{M, \Gamma}$ is homotopic to the trivial cycle by a finite sequence of insertions/deletions of elementary cycles of the nine types above.

This establishes an explicit presentation of the fundamental group of the Tutte graph in terms of local combinatorial data (Baker et al., 5 Jan 2026).

4. Foundations of Matroids and Universal Cross-Ratios

Baker and Lorscheid associate to each matroid $M$ a universal object $F_M$, called the foundation, in the category of pastures, with the characterization: $\mathrm{Hom}(F_M, P) \cong \{ \text{rescaling-classes of %%%%47%%%%–representations of %%%%48%%%%} \}$ for every pasture $P$.

For each Tutte cycle $\gamma = (H_1,H_2,H_3,H_4)$, a canonical element

$r_{H_1 H_2 H_3 H_4} \in F_M^\times$

is defined (the universal cross-ratio). For any $P$-representation $\varphi$ and corresponding map $\psi: F_M \to P$,

$$r_{H_1H_2H_3H_4}(\varphi) = \frac{\varphi_{H_1}(a)\,\varphi_{H_2}(b)}{\varphi_{H_1}(b)\,\varphi_{H_2}(a)} = \psi\left(r_{H_1H_2H_3H_4}\right)$$

for $a\in H_3 - H_1\cap H_2$ and $b\in H_4 - H_1\cap H_2$.

A key theorem is that $F_M^\times$ is generated by the finite set of all universal cross-ratios $r_{H_1H_2H_3H_4}$. The proof is inductive, employing the Path Theorem to reduce to cases with fewer elements, ultimately expressing all new "matrix entries" in terms of existing ones and a single new cross-ratio (Baker et al., 5 Jan 2026).

5. Algebraic Relations and the Fundamental Presentation

By the Homotopy Theorem, each closed Tutte cycle yields an algebraic relation among universal cross-ratios. The entire set of such relations, as induced by cycles of types 1–9, suffices to present the foundation $F_M$. Let $\{ r_\gamma \mid \gamma \in \Theta_M \}$ be the set of all symbols for universal cross-ratios; the fundamental presentation imposes relations in the free pasture generated by them:

Relation	Description
$(R-)$	$-1 + 1 = 0$ if $F_7$ or $F_7^*$ is a minor
$(R0)$	$r_\gamma = 1$ for degenerate $\gamma$
$(R\sigma)$	Symmetry among $r_{H_1H_2H_3H_4}$
$(R1)$	$r_{H_1H_2H_4H_3}=r_{H_1H_2H_3H_4}$
$(R2)$	$r_{1234}\,r_{1342}\,r_{1423}=-1$ (from $U_{2,4}$)
$(R3)$	$r_{12,34}\,r_{12,45}\,r_{12,53}=1$ (from $U_{2,5}$)
$(R4)$	$r_{13,46}\,r_{12,45}\,r_{23,56}=1$ (from $M(K_{2,3})$)
$(R+)$	$r_{H_1H_2H_3H_4} + r_{H_1H_3H_2H_4} = 1$

The resulting pasture is isomorphic to $F_M$. Each generating cycle imposes one of these relations, providing a rigorous foundation for the algebraic structure of matroid representations (Baker et al., 5 Jan 2026).

6. Applications in Matroid Representation Theory

The fundamental presentation yields transparent proofs and recoveries of classical results:

• Excluded-minor characterizations:
  • Regular matroid iff no $U_{2,4}, F_7, F_7^*$ minor
  • Binary iff no $U_{2,4}$ minor
  • Ternary iff no $U_{2,5}, U_{3,5}, F_7, F_7^*$ minor
• Lift-theorems: If $M$ is orientable (resp., positively orientable), then $M$ is representable over the hyperfield $\mathbb S$ (resp., the sign-hyperfield $\mathbb S_+$), recovering and extending results of Lee–Scobee (1999).
• Dressians and realization spaces: For matroids with no $U_{2,5}$ or $U_{3,5}$ minor, the Dressian (tropical realization space) decomposes as a product of $[0,\infty)$, $\R$, and tropical lines (Baker et al., 5 Jan 2026).

The presentation provides a systematic combinatorial and algebraic toolkit for analyzing matroid representability and related questions in tropical and algebraic geometry.

7. Higher Tutte Homotopy Theorems and Future Directions

Tutte’s Path and Homotopy Theorems verify that the $1$- and $2$-skeleta of a simplicial complex of flats, associated to a matroid, are $0$- and $1$-connected, respectively. Extensions of the theory define a $3$-skeleton $\Sigma_3$, obtained by adjoining $3$-simplices for each class of "minimal" $4$-tuple configurations that obstruct the vanishing of $\pi_2$, and so forth for higher skeleta.

Preliminary computer-assisted search in the poset of small subconstellations aims to identify the finite list of $3$-simplices ("Type 3a–3d", etc.) necessary to kill $H_2$. The guiding conjecture is that there exists a finite universal list $\Sigma_k$ whose $H_{k-1}$ always vanishes, giving rise to a tower of higher Tutte homotopy theorems. Notably, the first novel cases beyond the classical four arise for $4$ atoms, producing $8$- or $12$-cycles whose second homology is nonzero unless one attaches an appropriate $3$-cell.

Determining the full list and interpreting these higher cycles as "syzygies" among universal cross-ratios is an open and promising field for further exploration (Baker et al., 5 Jan 2026).