本书聚焦于环拓扑这一全新数学领域，它作为等变拓扑、代数几何与辛几何、组合学和交换代数的边缘交叉学科于20世纪90年代末兴起，随后迅速发展成为一个非常活跃的领域，与其他数学领域有着许多密切联系，并持续吸引着来自不同领域的专家。
环拓扑中的关键角色是矩-角（moment-angle）流形，它是一类以组合术语定义、具有环面作用的流形。矩-角流形的构造通过准环面（quasitoric）流形的概念与环簇的组合几何和代数几何相关联。人们在矩-角流形上发现了显著的几何结构，这使得辛几何、Lagrange几何和非Kahler复几何的古典与现代领域产生重要关联。矩-角复形和多面体乘积的相关分类构造为同伦拓扑的许多基本构造提供了通用框架。多面体乘积的研究已经发展成为同伦理论的一个独立主题。而对环面作用的新视角也促进了复配边等代数拓扑经典领域的发展。
本书包含许多未解决的问题，适合对将所有相关学科联系起来的新思想感兴趣的专家，以及准备进入这一优美的全新领域的研究生和年轻研究人员研读和学习。

Introduction
Chapter guide
Acknowledgements
Chapter 1.Geometry and Combinatorics of Polytopes
1.1.Convex polytopes
1.2.Gale duality and Gale diagrams
1.3.Face vectors and Dehn-Sommerville relations
1.4.Characterising the face vectors of polytopes
Polytopes: Additional Topics
1.5.Nestohedra and graph-associahedra
1.6.Flagtopes and truncated cubes
1.7.Differential algebra of combinatorial polytopes
1.8.Families of polytopes and differential equations
Chapter 2.Combinatorial Structures
2.1.Polyhedral fans
2.2.Simplicial complexes
2.3.Barycentric subdivision and flag complexes
2.4.Alexander duality
2.5.Classes of triangulated spheres
2.6.Triangulated manifolds
2.7.Stellar subdivisions and bistellar moves
2.8.Simplicial posets and simplicial cell complexes
2.9.Cubical complexes
Chapter 3.Combinatorial Algebra of Face Rings
3.1.Face rings of simplicial complexes
3.2.Tor-algebras and Betti numbers
3.3.Cohen–Macaulay complexes
3.4.Gorenstein complexes and Dehn-Sommerville relations
Face rings of simplicial posets
3.5.Face Rings: Additional Topics
3.6.Cohen-Macaulay simplicial posets
3.7.Gorenstein simplicial posets
3.8.Generalised Dehn_Sommerville relations
Chapter 4.Moment-Angle Complexes
4.1.Basic definitions
4.2.Polyhedral products
4.3.Homotopical properties
4.4.Cell decomposition
4.5.Cohomology ring
4.6.Bigraded Betti numbers
4.7.Coordinate subspace arrangements
Moment-Angle Complexes: Additional Topics
4.8.Free and almost free torus actions on moment-angle complexes
4.9.Massey products in the cohomology of moment-angle complexes
4.10.Moment-angle complexes from simplicial posets
Chapter 5.Toric Varieties and Manifolds
5.1.Classical construction from rational fans
5.2.Projective toric varieties and polytopes
5.3.Cohomology of toric manifolds
5.4.Algebraic quotient construction
5.5.Hamiltonian actions and symplectic reduction
Chapter 6.Geometric Structures on Moment-Angle Manifolds
6.1.Intersections of quadrics
6.2.Moment-angle manifolds from polytopes
6.3.Symplectic reduction and moment maps revisited
6.4.Complex structures on intersections of quadrics
6.5.Moment-angle manifolds from simplicial fans
6.6.Complex structures on moment-angle manifolds
6.7.Holomorphic principal bundles and Dolbeault cohomology
6.8.Hamiltonian-minimal Lagrangian submanifolds
Chapter 7.Half-Dimensional Torus Actions
7.1.Locally standard actions and manifolds with corners
7.2.Toric manifolds and their quotients
7.3.Quasitoric manifolds
7.4.Locally standard T-manifolds and torus manifolds
7.5.Topological toric manifolds
7.6.Relationship between different classes of T-manifolds
7.7.Bounded flag manifolds
7.8.Bott towers
7.9.Weight graphs
Chapter 8.Homotopy Theory of Polyhedral Products
8.1.Rational homotopy theory of polyhedral products
8.2.Wedges of spheres and connected sums of sphere products
8.3.Stable decompositions of polyhedral products
8.4.Loop spaces, Whitehead and Samelson products
8.5.The case of flag complexes
Chapter 9.Torus Actions and Complex Cobordism
9.1.Toric and quasitoric representatives in complex bordism classes
9.2.The universal toric genus
9.3.Equivariant genera, rigidity and fibre multiplicativity
9.4.Isolated fixed points: localisation formulae
9.5.Quasitoric manifolds and genera
9.6.Genera for homogeneous spaces of compact Lie groups
9.7.Rigid genera and functional equations
Appendix A.Commutative and Homological Algebra
A.1.Algebras and modules
A.2.Homological theory of graded rings and modules
A.3.Regular sequences and Cohen-Macaulay algebras
A.4.Formality and Massey products
Appendix B.Algebraic Topology
B.1.Homotopy and homology
B.2.Elements of rational homotopy theory
B.3.Eilenberg-Moore spectral sequences
B.4.Group actions and equivariant topology
B.5.Stably complex struc