Torus actions, combinatorial topology and homological algebra

TL;DR

This paper explores the connections between combinatorial topology, homological algebra, and torus actions, introducing a new approach using moment-angle complexes to study polytopes, complexes, and manifolds.

Contribution

It develops a novel algebraic framework based on moment-angle complexes to analyze combinatorial and topological properties of polytopes and manifolds.

Findings

Expressed invariants of complexes via cohomology rings

Linked toric varieties to moment-angle complexes

Provided solutions to classical topological problems

Abstract

The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular decomposition. In the framework of this theory, the well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. We express different invariants of simplicial complexes and related combinatorial-geometrical objects in terms of the bigraded cohomology rings of the corresponding moment-angle complexes. Finally, we show that the new relationships between combinatorics, geometry and topology result in solutions to some well-known topological problems.