[Paper Review] Toric Topology

Taras Panov, Victor Matveevich Buchstaber|arXiv (Cornell University)|Jul 15, 2015

Advanced Combinatorial MathematicsMathematics232 references152 citations

TL;DR

This paper introduces toric topology as an interdisciplinary field linking equivariant topology, algebraic and symplectic geometry, combinatorics, and commutative algebra through the study of moment-angle manifolds and their generalizations, polyhedral products. It establishes foundational connections to symplectic, Lagrangian, and non-Kaehler complex geometry, while positioning polyhedral products as a universal framework in homotopical topology and advancing classical areas like complex cobordism.

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.

Motivation & Objective

• To establish toric topology as a unifying framework connecting equivariant topology, algebraic and symplectic geometry, and combinatorics.
• To investigate the geometric and topological properties of moment-angle manifolds defined through combinatorial data.
• To develop polyhedral products as a universal construction in homotopical topology with broad applications.
• To reveal new connections between torus actions and classical algebraic topology invariants such as complex cobordism.
• To provide a comprehensive resource with open problems for experts and early-career researchers entering the field.

Proposed method

• Utilizing combinatorial constructions to define moment-angle manifolds with torus actions.
• Applying techniques from algebraic and symplectic geometry to study geometric structures on moment-angle manifolds.
• Employing categorical constructions to formalize polyhedral products as a universal model in homotopy theory.
• Leveraging the interplay between quasitoric manifolds and toric varieties to explore topological invariants.
• Introducing a new perspective on torus actions to reframe classical problems in complex cobordism.
• Using polyhedral products to generalize fundamental constructions in homotopical topology.

Experimental results

Research questions

• RQ1 How do moment-angle manifolds with torus actions unify concepts from symplectic and complex geometry?
• RQ2 What is the role of polyhedral products in providing a universal framework for constructions in homotopical topology?
• RQ3 How do geometric structures on moment-angle manifolds connect to non-Kaehler complex geometry?
• RQ4 In what ways do torus actions on moment-angle manifolds advance classical invariants like complex cobordism?
• RQ5 What are the key open problems linking toric topology to algebraic topology and combinatorics?

Key findings

• Moment-angle manifolds provide a combinatorially defined class of manifolds with torus actions that link to toric varieties and quasitoric manifolds.
• Remarkable geometric structures on moment-angle manifolds reveal deep connections to symplectic, Lagrangian, and non-Kaehler complex geometry.
• Polyhedral products emerge as a universal construction in homotopical topology, generalizing fundamental topological invariants.
• The study of polyhedral products has evolved into a distinct area of homotopy theory with strong interdisciplinary relevance.
• A new perspective on torus actions contributes to the advancement of classical algebraic topology, particularly in complex cobordism.
• The paper identifies numerous open problems, positioning toric topology as a vibrant and evolving field for future research.