Toric Topology

TL;DR

Toric topology is an interdisciplinary field exploring manifolds with torus actions, linking combinatorics, geometry, and algebra, with applications in symplectic and homotopy theories, and is rapidly evolving with many open problems.

Contribution

This work provides a comprehensive overview of toric topology, highlighting new geometric structures, categorical frameworks, and the evolving study of polyhedral products and their homotopical applications.

Findings

Introduction of moment-angle manifolds and complexes

Connections between toric topology and symplectic geometry

Development of polyhedral products in homotopy theory

Abstract

Toric topology emerged in the end of the 1990s on the borders of equivariant topology, algebraic and symplectic geometry, combinatorics and commutative algebra. It has quickly grown up into a very active area with many interdisciplinary links and applications, and continues to attract experts from different fields. The key players in toric topology are moment-angle manifolds, a family of manifolds with torus actions defined in combinatorial terms. Their construction links to combinatorial geometry and algebraic geometry of toric varieties via the related notion of a quasitoric manifold. Discovery of remarkable geometric structures on moment-angle manifolds led to seminal connections with the classical and modern areas of symplectic, Lagrangian and non-Kaehler complex geometry. A related categorical construction of moment-angle complexes and their generalisations, polyhedral products, provides a universal framework for many fundamental constructions of homotopical topology. The study of polyhedral products is now evolving into a separate area of homotopy theory, with strong links to other areas of toric topology. A new perspective on torus action has also contributed to the development of classical areas of algebraic topology, such as complex cobordism. The book contains lots of open problems and is addressed to experts interested in new ideas linking all the subjects involved, as well as to graduate students and young researchers ready to enter into a beautiful new area.