Orlik-Solomon algebras and Tutte polynomials

TL;DR

This paper explores the relationship between Orlik-Solomon algebras and Tutte polynomials, constructing examples where different matroids share the same OS algebra but differ in Tutte polynomial, revealing nuanced topological properties.

Contribution

It introduces a method to construct infinite families of matroids with isomorphic OS algebras but distinct Tutte polynomials, and analyzes their topological implications for hyperplane arrangement complements.

Findings

Constructed pairs of matroids with isomorphic OS algebras but different Tutte polynomials.

Showed that arrangements with the same OS algebra can have non-homeomorphic complements.

Demonstrated that certain arrangements have diffeomorphic complements despite matroid differences.

Abstract

The $OS$ algebra $A$ of a matroid $M$ is a graded algebra related to the Whitney homology of the lattice of flats of $M$. In case $M$ is the underlying matroid of a hyperplane arrangement \A in $\C^r$, $A$ is isomorphic to the cohomology algebra of the complement $\C^r\setminus \bigcup \A.$ Few examples are known of pairs of arrangements with non-isomorphic matroids but isomorphic $OS$ algebras. In all known examples, the Tutte polynomials are identical, and the complements are homotopy equivalent but not homeomorphic. We construct, for any given simple matroid $M_0$, a pair of infinite families of matroids $M_n$ and $M'_n$, $n\geq 1$, each containing $M_0$ as a submatroid, in which corresponding pairs have isomorphic $OS$ algebras. If the seed matroid $ M_0$ is connected, then $M_n$ and $M'_n$ have different Tutte polynomials. As a consequence of the construction, we obtain, for any $m$, $m$ different matroids with isomorphic $OS$ algebras. Suppose one is given a pair of central complex hyperplane arrangements $\A_0$ and $\A_1$. Let $\S$ denote the arrangement consisting of the hyperplane $\{0\}$ in $\C^1$. We define the parallel connection $P(\A_0,\A_1)$, an arrangement realizing the parallel connection of the underlying matroids, and show that the direct sums $\A_0 \oplus \A_1$ and $\S\oplus P(\A_0,\A_1)$ have diffeomorphic complements.