We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal{R}_K$, to be a one-relator group; and for the Pontryagin algebra $H_*(Ω\mathcal{Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterisation of these properties. For $RC_K'$, it is given by a condition on the homology group $H_2(\mathcal{R}_K)$, whereas for $H_*(Ω\mathcal{Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal{Z}_K$.
We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only if $\mathscr{K}$ is either a $p$-cycle with $p\geqslant4$ or has the form ($p$-cycle)$*Δ^q$ with $p\geqslant4$ and $g\geqslant0$, where $Δ^q$ is a $q$-simplex.
We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial $n$-sphere, we always have such an identity, and for the $(n-1)$-skeleton of a $(n+1)$-simplex, the identity is the Jacobi identity of Whitehead products ($n=1$) and Hardie's identity of higher Whitehead products ($n\ge 2$).
This article presents a machinery based on polyhedral products that produces faithful representations of graph products of finite groups and direct products of finite groups into automorphisms of free groups $\rm Aut(F_n)$ and outer automorphisms of free groups $\rm Out(F_n)$, respectively, as well as faithful representations of products of finite groups into the linear groups $\rm SL(n,\mathbb Z)$ and $\rm GL(n,\mathbb Z)$. These faithful representations are realized as monodromy representations.
We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.
We prove that certain polyhedral products, including the moment-angle complexes, for the Alexander duals of shellable and sequentially Cohen-Macaulay complexes decompose into wedges of explicitly given suspension spaces, which implies the properties of the Stanley-Reisner rings, known as Golod, of theses complexes.
The polyhedral product constructed from a collection of pairs of cones and their bases and a simplicial complex $K$ is studied by investigating its filtration called the fat wedge filtration. We give a sufficient condition for decomposing the polyhedral product in terms of the fat wedge filtration of the real moment-angle complex for $K$, which is a desuspension of the decomposition of the suspension of the polyhedral product due to Bahri, Bendersky, Cohen, and Gitler. We show that the condition also implies a strong connection with the Golodness of $K$, and is satisfied when $K$ is dual seque…
The purpose of this paper is to exhibit fine structure for polyhedral products Z(K;(X,A) and polyhedral smash products $\widehat{Z}(K;(X,A)$. (Moment-angle complexes are special cases for which (X,A) = (D^2,S^1)). There are three main parts. The first defines a natural filtration of the polyhedral product and derives properties of the resulting spectral sequence. This is followed with applications. The second part uses the first to give a homological decomposition of the polyhedral smash product. Finally there are applications to the ring structure of H*(Z(K;(X,A))) for CW-pairs (X,A) satisfyi…
In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral join complexes. Then we compute the universal algebra of polyhedral join complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.
In this paper, we define and prove basic properties of complement polyhedral product spaces, dual complexes and polyhedral product complexes. Then we compute the universal algebra of polyhedral product complexes under certain split conditions and the Alexander duality isomorphism on certain polyhedral product spaces.
The polyhedral product is a space constructed from a simplicial complex and a collection of pairs of spaces, which is connected with the Stanley Reisner ring of the simplicial complex via cohomology. Generalizing the previous work Grbic and Theriault, Grujic and Welker, and the authors, we show a decomposition of polyhedral products for a large class of simplicial complexes including the ones whose Alexander duals are shellable or sequentially Cohen-Macaulay. This implies the property, called Golod, of the corresponding Stanley-Reisner rings proved by Herzog, Reiner and Welker.
The construction of a simplicial complex given by polyhedral joins (introduced by Anton Ayzenberg), generalizes Bahri, Bendersky, Cohen and Gitler's $J$-construction and simplicial wedge construction. This article gives a cohomological decomposition of a polyhedral product over a polyhedral join for certain families of pairs of simplicial complexes. A formula for the Hilbert-Poincaré series is given, which generalizes Ayzenberg's formula for the moment-angle complex.
In this paper, we compute the cohomology ring of all homology split polyhedral product spaces and the cohomology algebra over a field of all polyhedral product spaces. As an application, we give two polyhedral product spaces such that all the cohomology homomorphisms induced by inclusion map are the same, but the cohomology rings of the two polyhedral product spaces are not isomorphic.
We prove that for a simplicial complex $K$ whose Taylor resolution for the Stanley-Reisner ring is minimal, the following four conditions are equivalent: (1) $K$ satisfies the strong gcd-condition; (2) $K$ is Golod; (3) the moment-angle complex $\mathcal{Z}_K$ is homotopy equivalent to a wedge of spheres; (4) the decomposition of the suspension of the polyhedral product $\mathcal{Z}_K(C\underline{X},\underline{X})$ due to Bahri, Bendersky, Cohen, and Gitler desuspends.
In this paper, we compute the homology group and cohomology algebra of various polyhedral product objects uniformly from the point of view of diagonal tensor product. As applications, we introduce the polyhedral product method into commutative algebra and show that the homotopy types of polyhedral product spaces depend on not only the homotopy type of each summand pair but also on the character coproduct of the pair.
Golodness of 2-dimensional simplicial complexes is studied through polyhedral products, and combinatorial and topoogical characterization of Golodness of surface triangulations is given. An answer to the question of Berglund is also given so that there is a 2-dimensional simplicial complex which is rationally Golod but is not Golod over $\mathbb{Z}/p$.
To a Coxeter group $W$ one associates a quandle $X_W$ from which one constructs a group $\mathrm{Ad}(X_W)$. This group turns out to be an intermediate object between $W$ and the associated Artin group. By using a result of Akita, we prove that $\mathrm{Ad}(X_W)$ is given by a pullback involving $W$, and by using this pullback, we show that the classifying space of $\mathrm{Ad}(X_W)$ is given by a space called a polyhedral product whenever $W$ is right-angled. Two applications of this description of the classifying space are given.
We give a geometric method for determining the cohomology groups and the product structure of a polyhedral product, under suitable freeness conditions or with coefficients taken in a field. This is done by considering first a special class of CW pairs for which we derive a decomposition of the polyhedral product resembling a Cartan formula. The result is then generalized to arbitrary CW pairs of finite type. This leads to a direct computation of the Hilbert-Poincaré series and to other applications. The product structure on the cohomology of the polyhedral product is computed in terms of the a…
In this paper we present unpublished work by David Stone on polyhedral smash products. He proved that the polyhedral smash product of the CW-pair $(D^2, S^1)$ over a simplicial complex $K$ is homeomorphic to an iterated suspension of the geometric realization of $K$. Here we generalize his technique to the CW-pair $(D^{k+1}, S^{k})$, for an arbitrary $k$. We generalize the result further to a set of disks and spheres of different dimensions.
A polyhedral product is a natural subspace of a Cartesian product, which is specified by a simplicial complex K. The automorphism group Aut(K) of K induces a group action on the polyhedral product. In this paper we study this group action and give a formula for the fixed point set of the polyhedral product for any subgroup H of Aut(K). We use the fixed point data to compute examples of Bredon cohomolohgy for small non-Abelian groups such as $D_8$ and $Σ_4$.
In this note we prove that the integer cohomology ring of polyhedral products with space pairs $(D^1, S^0)$ can also be described explicitly by a multiplicative Hochster's formula.
If $K$ is a simplicial complex on $m$ vertices the flagification of $K$ is the minimal flag complex $K^f$ on the same vertex set that contains $K$. Letting $L$ be the set of vertices, there is a sequence of simplicial inclusions $L\to K\to K^f$. This induces a sequence of maps of polyhedral products $(\underline X,\underline A)^L\stackrel g\longrightarrow(\underline X,\underline A)^K\stackrel f\longrightarrow (\underline X,\underline A)^{K^f}$. We show that $Ωf$ and $Ωf\circΩg$ have right homotopy inverses and draw consequences. For a flag complex $K$ the polyhedral product of the form $(\unde…
Moore's Conjecture is shown to hold for generalized moment-angle complexes and a criterion is proved that determines when a polyhedral product is elliptic or hyperbolic.
Representation stability in the sense of Church-Farb is concerned with stable properties of representations of sequences of algebraic structures, in particular of groups. We study this notion on objects arising in toric topology. With a simplicial $G$-complex $K$ and a topological pair $(X, A)$, a $G$-polyhedral product $(X, A)^K$ is associated. We show that the homotopy decomposition [2] of $Σ(X, A)^K$ is then $G$-equivariant after suspension. In the case of $Σ_m$-polyhedral products, we give criteria on simplicial $Σ_m$-complexes which imply representation stability of $Σ_m$-representations …
Two homotopy decompositions of supensions of spaces involving polyhedral products are given. The first decomposition is motivated by the decomposition of suspensions of polyhedral products by Bahri, Bendersky, Cohen, and Gitler, and is a generalization of the retractile argument of James. The second decomposition is on the union of an arrangement of subspaces called diagonal subspaces, and generalizes the result of Labbasi.
We construct "higher" motion planners for automated systems whose space of states are homotopy equivalent to a polyhedral product space $Z(K,\{(S^{k_i},\star)\})$, e.g. robot arms with restrictions on the possible combinations of simultaneously moving nodes. Our construction is shown to be optimal by explicit cohomology calculations. The higher topological complexity of other families of polyhedral product spaces is also determined.
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.
The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik-Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if X is a simply-connected space then the weak cocategory of X equals the homotopy nilpotency class of the based loops on X. This answers a fifty year old question posed by Ganea. The methods are applied to determine the homotopy nilpotency class of quasi-p-regular exceptional Lie groups and sporadic p-compa…
This paper aims to find the most general combinatorial conditions under which a moment-angle complex $(D^2,S^1)^K$ is a co-$H$-space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment-angle complex over a Golod simplicial complex is a co-$H$-space. Our main tool is a certain generalisation of the theory of labelled configuration spaces.
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 $Z_K(X,A)$, which denotes the moment-angle complex of Buchstaber-Panov in the case $(X,A) = (D^2, S^1)$, 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 $Z_K(\underline{BG})$ is an Eilenberg-Mac Lane space. The fundamental group of $Z_K(\un…
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…
In this paper, we construct Adams-Hilton models for the polyhedral products of spheres $(\underline{S})^{\mathcal{K}}$ and Davis-Januszkiewicz spaces $\left(\mathbb{C} P^{\infty}\right)^{\mathcal{K}}$. We show that in these cases the Adams-Hilton model can be chosen so that it coincides with the cobar construction of the homology coalgebra. We apply the resulting models to the study of iterated higher Whitehead products in $\left(\mathbb{C} P^{\infty}\right)^{\mathcal{K}}$. Namely, we explicitly construct a chain in the cobar construction representing the homology class of the Hurewicz image o…
A generalised Postnikov tower for a space $X$ is a tower of principal fibrations with fibres generalised Eilenberg-MacLane spaces, whose inverse limit is weakly homotopy equivalent to $X$. In this paper we give a characterisation of a polyhedral product $Z_K(X,A)$ whose universal cover either admits a generalised Postnikov tower of finite length, or is a homotopy retract of a space admitting such a tower. We also include $p$-local and rational versions of the theorem. We end with a group theoretic application.
This paper studies the map between polyhedral products $\mathcal{Z}_K(C\underline{X},\underline{X})\to\mathcal{Z}_K(Σ\underline{X},*)$ induced from the pinch maps $(CX_i,X_i)\to(ΣX_i,*)$, which is the higher order Whitehead product if $K$ is the boundary of a simplex. When $K$ is a shifted complex, a wedge decomposition of $\mathcal{Z}_K(C\underline{X},\underline{X})$ is given by the authors. Based on this decomposition, when $K$ is shifted, the induced pinch map is explicitly described as a wedge of iterated Whitehead products each of which includes at most one higher product. As a corollary,…
The equivariant cohomology for actions of compact connected abelian groups and elementary abelian p-groups have been widely studied in the last decades. We study some of these results on actions of finite cyclic groups over a field of positive characteristic. In particular, we study the module structure of the equivariant cohomology over the cohomology of the classifying space and we provide a criterion for freeness over this ring and the polynomial subring of it. We apply these results to canonical actions of finite cyclic groups on polyhedral products arising from the boundary of a polygo
We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices, X_1,..., X_n are spaces and CX_i is the cone on X_i, then the polyhedral product determined by K and the pairs (CX_i,X_i) is homotopy equivalent to a wedge of suspensions of smashes of the X_i's. This generalises earlier work of the two authors in the special case where each X_i is a loop space. Connections are made to toric topology, combinatorics, and classical homotopy theory.
We compute the topological complexity of a polyhedral product $\mathcal{Z}$ defined in terms of an LS-logarithmic family of locally compact connected CW topological groups. The answer is given by a combinatorial formula that involves the LS category of the polyhedral-product factors. As a by-product, we show that the Iwase-Sakai conjecture holds true for $\mathcal{Z}$. The proof methodology uses a Fadell-Husseini viewpoint for the monoidal topological complexity (MTC) of a space, which, under mild conditions, recovers Iwase-Sakai's original definition. In the Fadell-Husseini context, the stasi…
In this paper we establish a connection between the loop space homology of the generalization of wedge defined by a simplicial complex K (so called polyhedral product) and the homology of certain diagonal arrangements associated with K. We illustrate these results by finding the presentations of those loop homology algebras for certain K generalizing results of Panov-Ray, Papadima-Suciu, Lemaire. Finally, we show that in the case when the functor is applied to suspensions, this homology splitting comes from the stable homotopy splitting of the loop spaces.
For a graph $Γ$, let $K(H_Γ,1)$ denote the Eilenberg-Mac Lane space associated to the right-angled Artin (RAA) group $H_Γ$ defined by $Γ$. We use the relationship between the combinatorics of $Γ$ and the topological complexity of $K(H_Γ,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$, we construct a graph $\mathcal{O}_n$ whose TC-generating function has polynomial numerator of degree $n$. Additionally, motivated by the fac…
A product of cochains in a polyhedral complex is constructed. The multiplication algorithm depends on the choice of a parameter. The parameter is a linear functional on the ambient space. Cocycles form a subring of the ring of cochains, cobounds form an ideal of the ring of cocycles, and the quotient ring is a ring of cohomologies. If the complex is simplicial, then the algorithm reduces to the well known algorithm of ${\rm\breve{C}}$ech. Similar algorithms are used in geometry of polyhedra for multiplying of cocycles taking values in the exterior algebra of the ambient space. Therefore, we as…
