Homotopy equivalence of Grassmannians and MacPhersonians in rank 3

TL;DR

The paper resolves the rank-3 MacPhersonian–Grassmannian conjecture by constructing a topological bridge between MacPhersonians and Grassmannians via spaces of weighted pseudocircle arrangements PsV_{3,n}. By building an O_3-equivariant open cover (hoods) around realizations of oriented matroids and applying the nerve theorem, the author shows the geometric realizations of the MacPhersonian as a nerve of contractible intersections, hence homotopy equivalent to the Grassmannian. A key innovation is the zone-based deformation scheme (zone ebb), which uses canonical parametrizations (Mox) and conformal interpolation (Douady–Earle/Radó) to deform neighborhood intersections to a point while preserving structural data. This yields not only the rank-3 MacPhersonian is homotopy equivalent to G_{3,n} but also the oriented version to the oriented Grassmannian, reinforcing the classification perspective for vector bundles via matroid bundles. The results have potential implications for combinatorial models of topological bundles and algorithmic structures for handling vector-bundle data in a purely combinatorial setting.

Abstract

We confirm a long standing conjecture in the case of rank 3 that MacPhersonians are homotopy equivalent to Grassmannians.

Figures (5)

• Figure 1: Construction of $\widetilde{C}_k$ in the proof of Lemma \ref{['lemmaSquishConformalPath']}
• Figure 2: Part of an arrangement $A \in \mathop{\mathrm{hood}}\nolimits(\{\mathcal{M}_1,\mathcal{M}_2,\mathcal{M}_3\})$ at different magnifications. Only $\mathcal{M}_1$ has a vertex covector $\sigma$ with $\{2,\dots,6\} \subseteq \sigma^0$. In both $\mathcal{M}_1$ and $\mathcal{M}_2$, the elements 3,4,5 are parallel and element 7 is a loop, but not in $\mathcal{M}_3$. Pseudocircles 2 through 6 appear to meet at a common point at low magnification, but not at medium or high. Pseudocircles 3, 4, and 5 coincide and pseudocircle 7 vanishes at low and medium magnification, but not at high.
• Figure 3: Left: Oriented matroid $\mathcal{M}$ represented by an oriented line arrangement. Right: Part of a pseudocircle arrangement, and $\mathop{\mathrm{maxcov}}\nolimits^{-1}(\sigma)$ for $\sigma = (+,0,0,0,0)$ in teal and for $\sigma = (+,+,+,0,0)$ in purple. Arrows indicate the positive side of each pseudocircle.
• Figure 4: A portion of an $\mathcal{M}$-zoning in black with an accomodated arrangement in light blue. Zones $Z(\Sigma)$ for $\Sigma \subset \{\upsilon,\sigma,\tau\}$ are labeled.
• Figure 5: $\zeta$ in the definition of $\psi_\upsilon$ where the blue pseudocircle is the greedy choice and the purple pseudocircle is not in $N_1$.

Theorems & Definitions (191)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3: dobbins2021grassmannians
• Theorem 1.4: dobbins2021grassmannians
• Remark 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3