Moment-angle Complexes Research Articles

Möbius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the Möbius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes \(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}\), including the moment-angle complex \(\mathcal{Z}_{K}\) and the real moment-angle complex \(\mathbb{R}\mathcal {Z}_{K}\) as special examples. We show that \(H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k})\) has the same graded k-module structure as Tor k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for \(\mathcal{Z}_{K}\) (resp. \(\mathbb{ R}\mathcal{Z}_{K}\)) under the restriction of the natural T m-action on \(\mathcal{Z}_{K}\) (resp. (ℤ2)m-action on \(\mathbb{ R}\mathcal{Z}_{K}\)).

Buchstaber invariants of skeleta of a simplex

A moment-angle complex $\mathcal{Z}_{K}$ is a compact topological space associated with a finite simplicial complex $K$. It is realized as a subspace of a polydisk $(D^{2})^{m}$, where $m$ is the number of vertices in $K$ and $D^{2}$ is the unit disk of the complex numbers $\mathbb{C}$, and the natural action of a torus $(S^{1})^{m}$ on $(D^{2})^{m}$ leaves $\mathcal{Z}_{K}$ invariant. The Buchstaber invariant $s(K)$ of $K$ is the largest integer for which there is a subtorus of rank $s(K)$ acting on $\mathcal{Z}_{K}$ freely. The story above goes over the real numbers $\mathbb{R}$ in place of $\mathbb{C}$ and a real analogue of the Buchstaber invariant, denoted $s_{\mathbb{R}}(K)$, can be defined for $K$ and $s(K)\leqq s_{\mathbb{R}}(K)$. In this paper we will make some computations of $s_{\mathbb{R}}(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find $s_{\mathbb{R}}(K)$ and the latter one turns out to be a problem of integer linear programming and of independent interest.

Moment-angle complexes from simplicial posets

Abstract We extend the construction of moment-angle complexes to simplicial posets by associating a certain T m-space Z S to an arbitrary simplicial poset S on m vertices. Face rings ℤ[S] of simplicial posets generalise those of simplicial complexes, and give rise to new classes of Gorenstein and Cohen-Macaulay rings. Our primary motivation is to study the face rings ℤ[S] by topological methods. The space Z S has many important topological properties of the original moment-angle complex Z K associated to a simplicial complex K. In particular, we prove that the integral cohomology algebra of Z S is isomorphic to the Tor-algebra of the face ring ℤ[S]. This leads directly to a generalisation of Hochster’s theorem, expressing the algebraic Betti numbers of the ring ℤ[S] in terms of the homology of full subposets in S. Finally, we estimate the total amount of homology of Z S from below by proving the toral rank conjecture for the moment-angle complexes Z S.

Гипотеза о торическом ранге для момент-угол комплексов

We consider an operation K \to L(K) on the set of simplicial complexes, which we call the "doubling operation". This combinatorial operation has been recently brought into toric topology by the work of Bahri, Bendersky, Cohen and Gitler on generalised moment-angle complexes (also known as K-powers). The crucial property of the doubling operation is that the moment-angle complex Z_K can be identified with the real moment-angle complex RZ_L(K) for the double L(K). As an application we prove the toral rank conjecture for Z_K by estimating the lower bound of the cohomology rank (with rational coefficients) of real moment-angle complexes RZ_K$. This paper extends the results of our previous work, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.

The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or partial product spaces which arise as polyhedral product functors described below. The geometrical decomposition presented here provides structure for the stable homotopy type of these spaces including spaces appearing in work of Goresky–MacPherson concerning complements of certain subspace arrangements, as well as Davis–Januszkiewicz and Buchstaber–Panov concerning moment-angle complexes. Since the stable decompositions here are geometric, they provide corresponding homological decompositions for generalized moment-angle complexes for any homology theory.

Graph products of spheres, associative graded algebras and Hilbert series

Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show that the Hilbert series of this algebra is the inverse of the clique polynomial of the graph. Using this result it easy to recognize if the ideal is inert, from which strong results on the algebra follow. Non-commutative Grobner bases play an important role in our proof. There is an interesting application to toric topology. This algebra arises naturally from a partial product of spheres, which is a special case of a generalized moment-angle complex. We apply our result to the loop-space homology of this space.

Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces.

This article gives a natural decomposition of the suspension of a generalized moment-angle complex or partial product space which arises as the polyhedral product functor described below. The introduction and application of the smash product moment-angle complex provides a precise identification of the stable homotopy type of the values of the polyhedral product functor. One direct consequence is an analysis of the associated cohomology. For the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in earlier work of others as described below. Because the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. Implied, therefore, is a decomposition for the Stanley-Reisner ring of a finite simplicial complex, and natural generalizations.

Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the Stanley―Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the Eilenberg―Moore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.

Moment-angle Complexes, Monomial Ideals and Massey Products

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 subspace arrangements are given.

On the homotopy groups of toric spaces

Given a certain class of simple polyhedral complexes P and the associated Borel space BTP we compute the E2-term of the Unstable Adams Novikov Spectral Sequence for BTP through a range. As a result, through a range, the higher homotopy groups of BTP are isomorphic to the homotopy groups of a wedge of spheres whose dimensions depend on the combinatorics of P. This paper provides a unified approach to attacking the problem of computing the higher homotopy groups of complements of arbitrary complex coordinate subspace arrangements. We extend all higher homotopy group computations in the cases where the homotopy type of a complement of a complex coordinate subspace arrangement is unknown. If K is a simplicial complex that defines a triangulation of a sphere that is dual to a simple convex polytope P, then, in many cases, the homotopy groups of the quasi-toric manifold M 2n (‚) can be computed through a range that was previously unknown. As an application, the homotopy type of a family of moment angle complexes ZK will be determined.

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

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. This allows for the construction of infinite families of toric manifolds, associated to a given one, in a way which simplifies the combinatorial input and consequently, the presentation of the cohomology rings. The new input is the interaction of a purely combinatorial construction with natural associated geometric constructions related to polyhedral products and toric manifolds. Applications of the methods and results developed here have appeared in literature.