The typical algebraic shifting of a surface

TL;DR

This work investigates how Kalai's exterior algebraic shifting behaves under random triangulations of a fixed space. It introduces a universality principle for edge contractions of Delaunay triangulations and proves concentration results: in dimension 1, the shifted complex concentrates to a homology lex-segment $\\Delta(X,n)$ (or $\\Delta(|G|,n)$); in dimension 2, random Delaunay triangulations concentrate to $\\Delta(S,n)$ for orientable surfaces and certain field characteristics, while uniform disc triangulations do not concentrate. The shifted complexes are shown to be homology lex-segment complexes with explicit forms depending only on $n$ and Betti numbers or genus, providing a probabilistic, structural understanding of algebraic shifting in triangulations. Overall, the paper connects combinatorial shifting, geometric triangulations, and topological invariants to characterize typical behavior in random settings.

Abstract

We initiate a statistical study of Kalai's exterior algebraic shifting, focusing on concentration phenomena for random triangulations of a fixed space. First, for a uniform $n$-vertex refinement of any given graph $G$, we show that asymptotically almost-surely (a.a.s.) its exterior algebraic shifting is an explicit shifted graph depending only on $n$ and the Betti numbers of $G$. Next, for any given compact connected Riemannian surface $S$, sample $n$ points independently at random according to the volume measure, and consider the resulted a.a.s. unique Delaunay triangulation. We prove that a.a.s. its exterior algebraic shifting is an explicit shifted complex depending only on $n$ and the genus of $S$. In both results the expected shifted complex is a homology lex-segment complex, a notion we define combinatorially and characterize numerically a lá Björner-Kalai. As a tool to prove the result on surfaces, we prove a universality result on edge contractions: for every fixed surface triangulation $K$, every dense enough point set in the surface yields a Delaunay triangulation that edge contracts to $K$.

Figures (4)

• Figure 1: (1) An irreducible triangulation of the torus on $7$ vertices. (2) Triangulation of the torus on $10$ vertices whose exterior algebraic shifting is a homology lex-segment. (3) Triangulation of the projective plane on $7$ whose exterior algebraic shifting is a homology lex-segment over fields with characteristic $0$ and $2$.
• Figure 2: (1) Embedding of the flat torus with intersection in a neighborhood of a vertex. (2) Embedding in the flat torus with straightening of edges adjacent to vertex $8$. (3) A transversal embedding in the flat torus with respect to a layered vertex cover.
• Figure 3: (1) The Delaunay triangulation in a neighborhood of the vertex $v$, where layered vertex cover of $v$ is depicted. The vertices of $\partial \mathop{\mathrm{Del}}\nolimits_{B"_v,B_v}$ (marked in red) and its edges (marked in blue) are disjoint from $B_v$. The embedding $\gamma_{u,v}$ is shown in light blue. (2) The triangulation obtained after contracting edges inside $D_v$. The resulting simplicial complex is a cone. Note that the vertices on the boundary of $D_v$ form a strict subset of those on the boundary of $\mathop{\mathrm{Del}}\nolimits_{B"_v,B_v}$. (3) The embedding of the edge $\gamma_{u,v}$ is modified: after intersecting $\partial D_v$, it continues along the shortest path to the vertex $v$.
• Figure 4: Edge approximation in a Delaunay triangulation inside the edge neighborhood $\mathop{\mathrm{Del}}\nolimits_e$.

Theorems & Definitions (62)

• Theorem 1.1: Universality for edge contractions
• Theorem 1.2: Concentration of exterior algebraic shifting for Random Delaunay on surfaces
• Theorem 1.3: Concentration of exterior algebraic shifting for uniform triangulations in dimension 1
• Example 1.4
• Conjecture 1.5
• Theorem 2.1
• Corollary 2.2
• proof
• Corollary 2.3
• proof