A simplicial complex L on n vertices determines a subcomplex T_L of the n-torus, with fundamental group the right-angled Artin group G_L. Given an epimorphism χ\colon G_L\to \Z, let T_L^χbe the corresponding cover, with fundamental group the Artin kernel N_χ. We compute the cohomology jumping loci of the toric complex T_L, as well as the homology groups of T_L^χwith coefficients in a field \k, viewed as modules over the group algebra \k\Z. We give combinatorial conditions for H_{\le r}(T_L^χ;\k) to have trivial \Z-action, allowing us to compute the truncated cohomology ring, H^{\le r}(T_L^χ;\k). We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy \Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.
In this paper we expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine cas…
We compute: * the cohomology with group ring coefficients of Artin groups (or actually, of their associated Salvetti complexes), Bestvina-Brady groups, and graph products of groups, * the L^2-Betti numbers of Bestvina-Brady groups and of graph products of groups, * the weighted L^2-Betti numbers of graph products of Coxeter groups. In the case of arbitrary graph products there is an additional proviso: either all factors are infinite or all are finite.(However, for graph products of Coxeter groups this proviso is unnecessary.)
We characterise integral Poincaré duality moment-angle complexes $\mathcal{Z}_{\mathcal{K}}$ in combinatorial terms of the Fan-Wang duality of the simplicial complex $\mathcal{K}$, and consequently in algebraic terms of the Gorenstein duality of the Stanley-Reisner ring $\mathbb{Z}[\mathcal{K}]$. We extend Poincaré duality results to certain polyhedral products using polyhedral join products of simplicial complexes.
Let $\mathbf{k}$ be the field $\mathbb{F}_p$ or the ring $\mathbb{Z}$. We study combinatorial and topological properties of the universal simplicial complexes $X(\mathbf{k}^n)$ and $K(\mathbf{k}^n)$ whose simplices are certain unimodular subsets of $\mathbf{k}^n$. As a main result we show that $X(\mathbf{k}^n)$, $K(\mathbf{k}^n)$ and the links of their simplicies are homotopy equivalent to a wedge of spheres specifying the exact number of spheres in the corresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz's result that $K(\mathbb{Z}^n)$ and $K(\mathbb{F}_2^n)…
In this paper we study the relationship between the moment-angle complex Z_k and the Davis-Januskiewicz space DJ(K) for a class of complexes K named missing-face complexes. If K has n vertices we consider the homotopy fibration sequence Z_k --> DJ(K) --> M where M is a product of n copies of infinite complex projective space. We observe that for such K, Z_k is homotopy equivalent to a wedge of spheres, and then show that under this equivalence the map Z_k --> DJ(K) is homotopic to a wedge sum of higher Whitehead products and iterated Whitehead products.
We compute the cohomology rings of smooth real toric varieties and of real toric spaces, which are quotients of real moment-angle complexes by freely acting subgroups of the ambient 2-torus. The differential graded algebra we present is in fact an equivariant dga model, valid for arbitrary coefficients. We deduce from our description that smooth toric varieties are M-varieties.
The fundamental Davis-Januszkiewicz construction of toric manifolds is reinterpreted in order to allow for generalization. Applications involve the simplicial wedge $J$-construction and Ayzenberg's recent identities arising from composed simplicial complexes.
We define the Coxeter cochain complex of a Coxeter group (G,S) with coefficients in a Z[G]-module A. This is closely related to the complex of simplicial cochains on the abstract simplicial complex I(S) of the commuting subsets of S. We give some representative computations of Coxeter cohomology and explain the connection between the Coxeter cohomology for groups of type A, the (singular) homology of certain configuration spaces, and the (Tor) homology of certain local Artin rings.
We extend a systolic inequality of Guth for Riemannian manifolds of maximal $\mathbb{Z}_2$ cup-length to piecewise Riemannian complexes of dimension 2. As a consequence we improve the previous best universal lower bound for the systolic area of groups for a large class of groups, including free abelian and surface groups, most of irreducible 3-manifold groups, non-free Artin groups and Coxeter groups (or more generally), groups containing an element of order 2.
We find families of graphs $G$ and subgraphs $H$ of $G$ such that the number of edge colorings of $G$ avoiding a monochromatic coloring of $H$ is determined by lattice point counts or a Hodge structure on the cohomology of a certain toric variety. In general, this gives a class of ``anti-Ramsey theory problems'' with a geometric structure. For example, we find one for Ramsey numbers of classes of such graphs. The key observation is that our previous result expressing simplicial chromatic polynomials in terms of $h$-vectors of auxiliary simplicial complexes can be reinterpreted as one on edge c…
A real toric space is a topological space which admits a well-behaved $\mathbb{Z}_2^k$-action. Real moment-angle complexes and real toric varieties are typical examples of real toric spaces. A real toric space is determined by a pair of a simplicial complex $K$ and a characteristic matrix $Λ$. In this paper, we provide an explicit $R$-cohomology ring formula of a real toric space in terms of $K$ and $Λ$, where $R$ is a commutative ring with unity in which $2$ is a unit. Interestingly, it has a natural $(\mathbb{Z} \oplus \operatorname*{row} Λ)$-grading. As corollaries, we compute the cohomolog…
Let W be a finite irreducible Coxeter group and let X_W be the classifying space for G_W, the associated Artin group. If A is a commutative unitary ring, we consider the two local systems L_q and L_q' over X_W, respectively over the modules A[q,q^{-1}] and A[[q,q^{-1}]], given by sending each standard generator of G_W into the automorphism given by the multiplication by q. We show that H^*(X_W,L_q') = H^{*+1}(X_W,L_q) and we generalize this relation to a particular class of algebraic complexes. We remark that H^*(X_W,L_q') is equal to the cohomology with trivial coefficients A of the Milnor fi…
The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and more generally, polyhedral products. In this paper we extend the analysis to include toric orbifolds. Our main results yield infinite families of toric orbifolds, derived from a given one, whose integral cohomology is free of torsion and is concentrated in even degrees, a property which might be termed \emph{integrally equivariantly formal}. In all cases, it …
We develop a framework to construct geometric representations of finite groups $G$ through the correspondence between real toric spaces $X^{\mathbb R}$ and simplicial complexes with characteristic matrices. We give a combinatorial description of the $G$-module structure of the homology of $X^{\mathbb R}$. As applications, we make explicit computations of the Weyl group representations on the homology of real toric varieties associated to the Weyl chambers of type $A$ and $B$, which show an interesting connection to the topology of posets. We also realize a certain kind of Foulkes representatio…
The result of this paper is the determination of the cohomology of Artin groups of type A_n, B_n and \tilde{A}_{n} with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B_n with coefficients over the module \Q[q^{\pm 1},t^{\pm 1}]. Here the first (n-1) standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapi…
In this paper we study the Buchstaber invariant of simplicial complexes, which comes from toric topology. With each simplicial complex $K$ on $m$ vertices we can associate a moment-angle complex $\mathcal Z_K$ with a canonical action of the compact torus $T^m$. Then $s(K)$ is the maximal dimension of a toric subgroup that acts freely on $\mathcal Z_K$. We develop the Buchstaber invariant theory from the viewpoint of the set of minimal non-simplices of $K$. It is easy to show that $s(K)=1$ if and only if any two and any three minimal non-simplices intersect. For $K=\partial P^*$, where $P$ is a…
We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to certain combinatorially defined subcomplexes of the toric Salvetti category in the complexified case, and use a technical argument in order to extend the results to full generality. As a byproduct we obtain: -a "combinatorial" version of Brieskorn's lemma in terms of Salvetti complexes of complexified arrangements, -a uniqueness result for realizations of a…
In this paper, we discuss the connected sum K_1#^Z K_2 of simplicial complexes K_1 and K_2, as well as define the notion of 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. 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 the intersection of K_1 and K_2. The strong connected sum K_1 #^Z K_2 is de…
In toric topology, to a simplicial complex $K$ with $m$ vertices, one associates two spaces, the moment-angle complex $\mathcal{Z}_K$ and the Davis-Januszkiewicz space $DJ_K$. These spaces are connected by a homotopy fibration $\mathcal{Z}_K\to DJ_K\to(\mathbb{C}P^\infty)^m$. In this paper, we show that the map $\mathcal{Z}_K\to DJ_K$ is identified with a wedge of iterated (higher) Whitehead products for a certain class of simplicial complexes $K$ including dual shellable complexes. We will prove the result in a more general setting of polyhedral products.
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 complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as …
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 s…
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 coef…
