Moment-angle Complexes Research Articles

Homotopy decomposition of a suspended real toric space

We give $p$-local homotopy decompositions of the suspensions of real toric spaces for odd primes $p$. Our decomposition is compatible with the one given by Bahri, Bendersky, Cohen, and Gitler for the suspension of the corresponding real moment-angle complex, or more generally, the polyhedral product. As an application, we obtain a stable rigidity property for real toric spaces.

$$\frac{n}{3}$$ n 3 -neighbourly moment-angle complexes and their unstable splittings

Given an \(\frac{n}{3}\)-neighbourly simplicial complex K on vertex set [n], we show that the moment-angle complex \(\mathcal Z_K\) is a co-H-space if and only if K satisfies a homotopy analogue of the Golod property. This gives a sufficient condition for the integral formality of \(\mathcal Z_K\).

The homotopy types of moment-angle complexes for flag complexes

We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Joellenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes Z_K which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex K the loop spaces of Z_K and DJ(K) are homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any odd prime.

Criteria for asphericity of polyhedral products: corrigenda to “right-angularity, flag complexes, asphericity”

Given a simplicial complex \(L\) with vertex set \(I\) and a family \({\mathbf A}=\{(A(i),B(i))\}_{i\in I}\) of pairs of spaces with base points \(*_i\in B(i)\), there is a definition of the “polyhedral product” \({{\mathbf A}^L}\) of \({\mathbf A}\) with respect to \(L\). Sometimes this is called a “generalized moment angle complex”. This note concerns two refinements to earlier work of the first author. First, when \(L\) is infinite, the definition of polyhedral product needs clarification. Second, the earlier paper omitted some subtle parts of the necessary and sufficient conditions for polyhedral products to be aspherical. Correct versions of these necessary and sufficient conditions are given in the present paper.

Norm minima in certain Siegel leaves

In this paper we shall illustrate that each polytopal moment-angle complex can be understood as the intersection of the minima of corresponding Siegel leaves and the unit sphere, with respect to the maximum norm. Consequently, an alternative proof of a rigidity theorem of Bosio and Meersseman is obtained; as piecewise linear manifolds, polytopal real moment-angle complexes can be smoothed in a natural way.

Moment-Angle Manifolds and Panov's Problem

We answer a problem posed by Panov, which is to describe the relationship between the wedge summands in a homotopy decomposition of the moment-angle complex corresponding to a disjoint union of |$\ell$| points and the connected sum factors in a diffeomorphism decomposition of the moment-angle manifold corresponding to the simple polytope obtained by making |$\ell$| vertex cuts on a standard |$d$|-simplex. This establishes a bridge between two very different approaches to moment-angle manifolds.

Moment-angle complexes of pairs $$(D^n,S^{n-1})$$ ( D n , S n - 1 ) and Simplicial complexes with vertex-decomposable duals

We study for a finite simplicial complex \(K\) and a CW-pair \((X,A)\) the associated CW-complex \({\mathcal Z}_K(X,A)\), known as the polyhedral product or generalized moment angle-complex. We apply discrete Morse theory to a particular CW-structure on the \(n\)-sphere moment-angle complexes \({\mathcal Z}_K(D^{n}, S^{n-1})\). For the class of simplicial complexes with vertex-decomposable duals, we show that the associated \(n\)-sphere moment-angle complexes have the homotopy type of wedges of spheres. As a corollary we show that a sufficiently high suspension of any restriction of a simplicial complex with the vertex-decomposable dual is homotopy equivalent to a wedge of spheres.

On the rational type of moment-angle complexes

In this note it is shown that the moment angle complexes Z(K;(D^2,,S^1)) which are rationally elliptic are a product of odd spheres and a disk

On the fundamental group of certain polyhedral products

Let K be a finite simplicial complex, and (X,A) be a pair of spaces. The purpose of this article is to study the fundamental group of the polyhedral product denoted ZK(X,A), which denotes the moment-angle complex of Buchstaber and Panov in the case (X,A)=(D2,S1), with extension to arbitrary pairs in [2] as given in Definition 2.2 here.For the case of a discrete group G, we give necessary and sufficient conditions on the abstract simplicial complex K such that the polyhedral product denoted by ZK(BG̲) is an Eilenberg–Mac Lane space. The fundamental group of ZK(BG̲) is shown to depend only on the 1-skeleton of K. Further special examples of polyhedral products are also investigated.Finally, we use polyhedral products to study an extension problem related to transitively commutative groups, which are given in Definition 5.2.

On problems concerning moment-angle complexes and polyhedral products

On problems concerning moment-angle complexes and polyhedral products

CONNECTED SUMS OF SIMPLICIAL COMPLEXES AND EQUIVARIANT COHOMOLOGY

In this paper, we introduce the notion of a connected sum $K_{1} \#^{Z} K_{2}$ of simplicial complexes $K_{1}$ and $K_{2}$, as well as define a strong connected sum. Geometrically, the connected sum is motivated by Lerman's symplectic cut applied to a toric orbifold, and algebraically, it is motivated by the connected sum of rings introduced by Ananthnarayan--Avramov--Moore [1]. We show that the Stanley--Reisner ring of a connected sum $K_{1} \#^{Z} K_{2}$ is the connected sum of the Stanley--Reisner rings of $K_{1}$ and $K_{2}$ along the Stanley--Reisner ring of $K_{1} \cap K_{2}$. The strong connected sum $K_{1} \#^{Z} K_{2}$ is defined in such a way that when $K_{1}$, $K_{2}$ are Gorenstein, and $Z$ is a suitable subset of $K_{1} \cap K_{2}$, then the Stanley--Reisner ring of $K_{1} \#^{Z} K_{2}$ is Gorenstein, by work appearing in [1]. We also show that cutting a simple polytope by a generic hyperplane produces strong connected sums. These algebraic computations can be interpreted in terms of the equivariant cohomology of moment angle complexes and toric orbifolds.

Moment angle complexes and big Cohen–Macaulayness

Let Z_K be the moment angle complex associated to a simplicial complex K, with the canonical torus T-action. In this paper, we prove that, for any possibly disconnected subgroup G of T, G-equivariant cohomology of Z_K over the integer Z is isomophic to the Tor-module Tor_{H(BR;Z)}(Z[K],Z) as graded modules, where Z_[K] is the Stanley-Reisner ring of K. Based on this, we prove that the surjectivity of the natural map H_T(Z_K;Z) to H_G(Z_K;Z) is equivalent to the vanishing of Tor^{H(BR;Z)}_1(Z[K],Z). Since the integral cohomology of various toric orbifolds can be identified with H_G(Z_K;Z), we studied the conditions for the cohomology of a toric orbifold to be a quotient of its equivariant cohomology by linear terms.

On the Equivalence of B-Rigidity and C-Rigidity for Quasitoric Manifolds

For quasitoric manifolds and moment-angle complexes which are central objects recently much studied in toric topology, there are several important notions of rigidity formulated in terms of cohomology rings. The aim of this paper is to show that, among other things, Buchstaber-rigidity (or B-rigidity) is equivalent to cohomological-rigidity (or C-rigidity) for simple convex polytopes supporting quasitoric manifolds.

Теория инварианта Бухштабера симплициальных комплексов и выпуклых многогранников

In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The problem stated by Victor M. Buchstaber is to find a simple combinatorial description of an s-number. We describe the main properties of s(P) and study the properties of simple n-polytopes with n+3 facets. In particular, we find the value of an s-number for such polytopes, a simple formula for their h-polynomials and the bigraded cohomology rings of the corresponding moment-angle complexes

On the Rational Type of Moment–Angle Complexes

On the Rational Type of Moment–Angle Complexes

The Golod property for products and high symbolic powers of monomial ideals

We show that for any two proper monomial ideals I and J in the polynomial ring S=K[x1,…,xn] the ring S/IJ is Golod. We also show that if I is squarefree then for large enough k the quotient S/I(k) of S by the kth symbolic power of I is Golod. As an application we prove that the multiplication on the cohomology algebra of some classes of moment-angle complexes is trivial.

Cup-products for the polyhedral product functor

AbstractDavis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.

Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes

For any (n - 1)-dimensional simplicial complex, we construct a particular n-dimensional complex vector bundle over the associated Davis- Januszkiewicz space whose Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley Reisner algebra. We show that the isomorphism type of this complex vector bundle as well as of its realification are completely determined by its characteristic classes. This allows us to show that coloring properties of the simplicial complex are reflected by splitting properties of this bundle and vice versa. Similar questions are also discussed for 2n-dimensional real vector bundles with particular prescribed characteristic Pontrjagin and Euler classes. We also analyze which of these bundles admit a complex structure. It turns out that all these bundles are closely related to the tangent bundles of quasi-toric manifolds and moment angle complexes. © 2012 American Mathematical Society.

LVMB manifolds and simplicial spheres

LVM and LVMB manifolds are a large family of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a natural action of a real torus and the quotient of this action is a polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied by Buchstaber and Panov. Our aim is to generalize the polytope associated to a LVM manifold to the LVMB case and study the properties of this generalization. In particular, we show that the object we obtain belongs to a very large class of simplicial spheres. Moreover, we show that for every sphere belonging to this class, we can construct a LVMB manifold whose associated sphere is the given sphere. We use this latter result to show that many moment-angle complexes can be endowed with a complex structure (up to product with circles).

Toral rank conjecture for moment-angle complexes

We consider an operation K ↦ L(K) on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as K-powers). The main property of the doubling operation is that the moment-angle complex Open image in new window can be identified with the real moment-angle complex Open image in new window for the double L(K). By way of application, we prove the toral rank conjecture for the spaces Open image in new window by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes Open image in new window. This paper can be viewed as a continuation of the author’s previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.