# Buchstaber Invariant of Simple Polytopes

## TL;DR

This paper introduces a new combinatorial invariant for simple polytopes derived from toric topology, analyzes its properties, and computes it for polytopes with n+3 facets, including related algebraic invariants.

## Contribution

It defines the s-invariant for simple polytopes, explores its properties, and provides explicit calculations for polytopes with n+3 facets, linking combinatorics and topology.

## Key findings

- s(P) is characterized for polytopes with n+3 facets
- Derived a formula for h-polynomials of these polytopes
- Computed the bigraded cohomology rings of associated moment-angle complexes

## Abstract

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The problem stated by Victor M. Buchstaber is to find a simple combinatorial description of an s-number. We describe the main properties of s(P) and study the properties of simple n-polytopes with n+3 facets. In particular, we find the value of an s-number for such polytopes, a simple formula for their h-polynomials and the bigraded cohomology rings of the corresponding moment-angle complexes