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Coinvariant stresses, Lefschetz properties and random complexes

Thiago Holleben

TL;DR

The paper develops a cohesive framework linking inverse systems, coinvariant stresses, and Lefschetz properties to study the $h$- and $g$-vectors of simplicial spheres. It provides an explicit top coinvariant stress for orientable homology spheres via a Macaulay dual generator built from Vandermonde-type pieces, and uses this to explain WLP failures for monomial almost complete intersections. It further derives $f$-vector inequalities for squarefree monomial ideals, connecting combinatorial topology with algebraic Lefschetz properties through basic double linkage. Finally, it investigates probabilistic behavior in Linial–Meshulam random complexes, showing that Lefschetz failures occur with high probability in certain regimes, thus linking randomness in topology to algebraic properties of monomial reductions.

Abstract

Lefschetz properties and inverse systems have played key roles in understanding the $h$-vector of simplicial spheres. In 1996, Lee established connections between these two algebraic tools and rigidity theory, an area often used in the study of motions of geometric complexes. One of the key ideas, is to translate geometric information about a complex, coming from vertex coordinates, to the algebraic notion of a linear system of parameters. In this paper, we explore similar connections in the nonlinear case, by using recent results of Herzog and Moradi (2021) where they prove that a subset of the elementary symmetric polynomials is always a system of parameters for the Stanley-Reisner ideal of a complex. We investigate connections to the study of Lefschetz properties of monomial ideals. Using this perspective, we recover and extend the well known result of Migliore, Miró-Roig and Nagel on the failure of the WLP of monomial almost complete intersections, by showing that, with one simple exception, every homology sphere has a monomial artinian reduction failing the weak Lefschetz property. Finally, we state probabilistic consequences of our results under a model introduced by Linial and Meshulam. We prove that there exists an open interval for the probability parameter where failure of Lefschetz properties of monomial ideals should be expected.

Coinvariant stresses, Lefschetz properties and random complexes

TL;DR

The paper develops a cohesive framework linking inverse systems, coinvariant stresses, and Lefschetz properties to study the - and -vectors of simplicial spheres. It provides an explicit top coinvariant stress for orientable homology spheres via a Macaulay dual generator built from Vandermonde-type pieces, and uses this to explain WLP failures for monomial almost complete intersections. It further derives -vector inequalities for squarefree monomial ideals, connecting combinatorial topology with algebraic Lefschetz properties through basic double linkage. Finally, it investigates probabilistic behavior in Linial–Meshulam random complexes, showing that Lefschetz failures occur with high probability in certain regimes, thus linking randomness in topology to algebraic properties of monomial reductions.

Abstract

Lefschetz properties and inverse systems have played key roles in understanding the -vector of simplicial spheres. In 1996, Lee established connections between these two algebraic tools and rigidity theory, an area often used in the study of motions of geometric complexes. One of the key ideas, is to translate geometric information about a complex, coming from vertex coordinates, to the algebraic notion of a linear system of parameters. In this paper, we explore similar connections in the nonlinear case, by using recent results of Herzog and Moradi (2021) where they prove that a subset of the elementary symmetric polynomials is always a system of parameters for the Stanley-Reisner ideal of a complex. We investigate connections to the study of Lefschetz properties of monomial ideals. Using this perspective, we recover and extend the well known result of Migliore, Miró-Roig and Nagel on the failure of the WLP of monomial almost complete intersections, by showing that, with one simple exception, every homology sphere has a monomial artinian reduction failing the weak Lefschetz property. Finally, we state probabilistic consequences of our results under a model introduced by Linial and Meshulam. We prove that there exists an open interval for the probability parameter where failure of Lefschetz properties of monomial ideals should be expected.
Paper Structure (8 sections, 30 theorems, 107 equations)

This paper contains 8 sections, 30 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inverse systems and (coinvariant) stresses
  4. The (coinvariant) Macaulay dual generator of homology spheres
  5. Lefschetz properties in the monomial setting and inverse systems
  6. $f$-vector inequalities and the WLP of squarefree monomial ideals
  7. Sometimes failure should be expected
  8. Concluding remarks and future work

Key Result

Theorem 1.1

Let $\Delta$ be a $d$-dimensional homology sphere with facets $F_1, \dots, F_s$ and orientation $\varepsilon$. Set $x_{F_i} = \prod_{j \in F_i} x_j$. Then the unique top coinvariant stress of $\Delta$ is (up to multiplication by scalars): where $V(F_i) = \prod_{j < k, \{j,k\} \subset F_i} (x_j - x_k)$ is the Vandermonde determinant on variables $\{x_j \colon j \in F_i\}$.

Theorems & Definitions (68)

  • Theorem 1.1: The unique top coinvariant stress of a homology sphere
  • Theorem 1.2: Nonvanishing homology and the failure of the WLP
  • Corollary 1.3: The Gorenstein property in the monomial setting
  • Theorem 1.4: Sometimes failure should be expected
  • Definition 2.1
  • Example 2.2: The projective plane: a non-orientable pseudomanifold
  • Theorem 2.3: S1996B
  • Example 2.4: The pinched torus: A non-CM orientable pseudomanifold
  • Definition 3.1: Inverse systems
  • Theorem 3.2: Macaulay inverse system duality, macaulaybookinversesystems
  • ...and 58 more