# Moment-angle Complexes, Monomial Ideals and Massey Products

@article{Denham2007MomentangleCM, title={Moment-angle Complexes, Monomial Ideals and Massey Products}, author={Graham C. Denham and Alexander I. Suciu}, journal={Pure and Applied Mathematics Quarterly}, year={2007}, volume={3}, pages={25-60} }

Associated to every finite simplicial complex K there is a "moment-angle" finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth, compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study the cohomology ring, the homotopy groups, and the triple Massey products of a moment-angle complex, relating these topological invariants to the algebraic combinatorics of the underlying simplicial complex. Applications to the study of non-formal manifolds and…

## 130 Citations

### Non-trivial higher Massey products in moment-angle complexes

- MathematicsAdvances in Mathematics
- 2021

### Massey products, toric topology and combinatorics of polytopes

- MathematicsIzvestiya: Mathematics
- 2019

In this paper we introduce a direct family of simple polytopes such that for any there are non-trivial strictly defined Massey products of order in the cohomology rings of their moment-angle…

### Homotopy theory in toric topology

- Mathematics
- 2016

In toric topology one associates with each simplicial complex on vertices two key spaces, the Davis–Januszkiewicz space and the moment-angle complex , which are related by a homotopy fibration . A…

### MOMENT-ANGLE COMPLEXES, GOLODNESS AND SEQUENTIALLY

- Mathematics
- 2009

Last years the construction of generalized moment-angle complexes attracted a lot of interest. The connections with many other famous constructions were found, such as subspace arrangements, toric…

### The rational homology of real toric manifolds

- Mathematics
- 2013

This is an extended abstract for a talk given at the mini-workshop "Cohomology rings and fundamental groups of hyperplane arrangements, wonderful compactifications, and real toric varieties", held in…

### Operations on polyhedral products and a new topological construction of infinite families of toric manifolds

- Mathematics
- 2010

A combinatorial construction is used to analyze the properties of polyhedral products and generalized moment-angle complexes with respect to certain operations on CW pairs including exponentiation.…

### LS-category of moment-angle manifolds, Massey products, and a generalisation of the Golod property

- Mathematics
- 2016

This paper is obtained as as synergy of homotopy theory, commutative algebra and combinatorics. We give various bounds for the Lusternik-Schnirelmann category of moment-angle complexes and show how…

### On Higher Massey Products and Rational Formality for Moment—Angle Manifolds over Multiwedges

- MathematicsProceedings of the Steklov Institute of Mathematics
- 2019

We prove that certain conditions on multigraded Betti numbers of a simplicial complex K imply the existence of a higher Massey product in the cohomology of a moment-angle complex \({{\cal Z}_K}\),…

### Geometric structures on moment-angle manifolds

- Mathematics
- 2013

A moment-angle complex is a cell complex with a torus action constructed from a finite simplicial complex . When this construction is applied to a triangulated sphere or, in particular, to the…

### Massey Products in the Cohomology of the Moment-Angle Manifolds Corresponding to Pogorelov Polytopes

- MathematicsMathematical Notes
- 2019

Nontrivial Massey products in the cohomology of the moment-angle manifolds corresponding to polytopes in the Pogorelov class are constructed. This class includes the dodecahedron and all fullerenes,…

## References

SHOWING 1-10 OF 84 REFERENCES

### Algebraic invariants for right-angled Artin groups

- Mathematics
- 2005

A finite simplicial graph Γ determines a right-angled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation υw=wυ for each pair of adjacent vertices. We compute the…

### The cohomology rings of complements of subspace arrangements

- Mathematics
- 2001

Abstract. The ring structure of the integral cohomology of complements of real linear subspace arrangements is considered. While the additive structure of the cohomology is given in terms of the…

### Small rational model of subspace complement

- Mathematics
- 1998

This paper concerns the rational cohomology ring of the complement M of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for…

### On Davis-Januszkiewicz homotopy types I; Formality and rationalisation

- Mathematics
- 2005

For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose inte- gral cohomology rings are isomorphic to the Stanley-Reisner…

### Real quadrics in Cn, complex manifolds and convex polytopes

- Mathematics
- 2006

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special…

### Categorical aspects of toric topology

- Mathematics
- 2007

We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle…

### Colimits, Stanley-Reisner algebras, and loop spaces

- Mathematics
- 2002

We study diagrams associated with a finite simplicial complex Kin various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and…

### On the integral cohomology of smooth toric varieties

- Mathematics
- 2003

Abstra t. Let X be a smooth, not ne essarily ompa t tori variety. We show that a ertain omplex, de ned in terms of the fan , omputes the integral ohomology of X , in luding the module stru ture over…

### Homotopy Lie algebras, lower central series and the Koszul property

- MathematicsGeometry & Topology
- 2004

Let X and Y be finite-type CW–complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k–rescaling of the rational cohomology ring of X. Assume H ∗ (X, Q) is a…