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We give a geometric method for determining the cohomology groups of a polyhedral product under suitable freeness conditions or with coefficients taken in a field. This is done by considering first the special case for which the pairs of spaces are wedge decomposable. We derive a decomposition for these polyhedral products which resembles a Cartan formula. The theory of symmetric products is used then to generalize the result to polyhedral products involving arbitrary pairs. This leads to a direct computation of the Hilbert-Poincar\'e series and to other applications., Comment: arXiv admin note: substantial text overlap with arXiv:2009.06818

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-Poincar\'e series and to other applications. The product structure on the cohomology of the polyhedral product is computed in terms of the additive generators, labelled via the Cartan decomposition. The description given suffices to enable explicit calculations., Comment: Several important typos are corrected in this version

A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of toric topology. This field, which began as a topological approach to toric geometry and aspects of symplectic geometry, has expanded rapidly in recent years. The investigation of polyhedral products and their homotopy theoretic properties has developed to the point where they are studied in various fields of mathematics far removed from their origin. In this survey, we provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications., Comment: A serious typo in Theorem 9.3 is corrected. This article has appeared in the Handbook of Homotopy Theory

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) satisfying suitable freeness conditions.

Koschorke introduced a map from the space of closed $n$-component links to the ordered configuration space of $n$-tuples of points in $\mathbb{R}^3$, and conjectured that this map separates homotopy links. The purpose of this paper is to construct an analogous map for string links, and to prove (1) this map in fact separates homotopy string links, and (2) Koschorke's original map factors through the map constructed here together with an analogue of Markov's closure map defined on the level of certain function spaces., Comment: Added a sentence to correct the definition of $\overline{f}$ in Section 4.3, page 10. Otherwise, the content of this version agrees with the published version

Growth of the Hilbert-Poincar\"e series for the rational homology of the free loop space of a toric space is addressed. In case the toric space is a manifold, the structure of the fan dictates whether the Hilbert-Poincar\"e series has exponential growth. Applications are made to the existence of infinitely many geometrically distinct periodic geodesics.

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.

A direct proof is presented of a form of Alvise Trevisan's result, that every monomial ideal ring is represented by the cohomology of topological space. Certain of these rings are shown to be realized by polyhedral products indexed by simplicial complexes.

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., Comment: This version includes improvements and corrections as suggested by the referee. The proof of Theorem 10.5 has been shortened considerably and a new reference has been added

This paper contains an explicit computation of the KO*-ring structure of an m-fold product of CP^{\infty}, the Davis-Januszkiewicz spaces and toric manifolds which have trivial Sq^2-homology., Comment: 34 pages

The Johnson filtration of the automorphism group of a free group is composed of those automorphisms which act trivially on nilpotent quotients of the free group. We compute cohomology classes as follows: (i) we analyze analogous classes for a subgroup of the pure symmetric automorphism group of a free group, and (ii) we analyze features of these classes which are preserved by the Johnson homomorphism. One consequence is that the ranks of the cohomology groups in any fixed dimension between 1 and n-1 increase without bound for terms deep in the Johnson filtraton., Comment: Corrections; revisions to proof of main theorem

Given a family of based CW-pairs $(\underline{X},\underline{A})=\{(X;A)\}^m_{i=1}$ together with an abstract simplicial complex $K$ with $m$ vertices, there is an associated based CW-complex $Z(K;(\underline{X},\underline{A}))$ known as a generalized moment-angle complex. The decomposition theorem of \cite{bbcg}, \cite{bbcg2} splits the suspension of $Z(K; (\underline{X}, \underline{A}))$ into a bouquet of spaces determined by the full sub-complexes of $K$. Thatdecomposition theorem is used here to describe the ring structure for the cohomology of Z(K; (\underline{X}, \underline{A})). Explicit computations are made for families of suspension pairs and for the cases where $X_i$ is the cone on $A_i$. These results complement and generalize those of Davis-Januszkiewicz, Franz, Hochster as well as Panov, and Baskakov-Buchstaber-Panov. Under conditions stated below, these theorems also apply for generalized cohomology theories., Comment: An example is corrected

Let F_n denote the free group generated by n letters. The purpose of this article is to show that Hol(F_2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension of F_2 by a linear group H is linear. This result gives a large linear subgroup of Aut(F_3). A second application is that the mapping class group for genus one surfaces with two punctures is linear.

Let $X_{\bullet}$ denote a simplicial space. The purpose of this note is to record a decomposition of the suspension of the individual spaces $X_n$ occurring in $X_{\bullet}$ in case the spaces $X_n$ satisfy certain mild topological hypotheses and where these decompositions are natural for morphisms of simplicial spaces. In addition, the summands of $X_n$ which occur after one suspension are stably equivalent to choices of filtration quotients of the geometric realization $|X_{\bullet}|$. The purpose of recording these decompositions is that they imply decompositions of the single suspension of certain spaces of representations as well as other varieties and are similar to decompositions of suspensions of moment-angle complexes which appear in a different context.

This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants of pure braids. Natural related questions are posed at the end of this article., Comment: This is the version published by Geometry & Topology Monographs on 22 February 2008

The purpose of this article is to 1. define M(t,k) the t-fold center of mass arrangement for k points in the plane, 2. give elementary properties of M(t,k) and 3. give consequences concerning the space M(2,k) of k distinct points in the plane, no four of which are the vertices of a parallelogram. The main result proven in this article is that the classical unordered configuration of k points in the plane is not a retract up to homotopy of the space of k unordered distinct points in the plane, no four of which are the vertices of a parallelogram. The proof below is homotopy theoretic without an explicit computation of the homology of these spaces. In addition, a second, speculative part of this article arises from the failure of these methods in the case of odd primes p. This failure gives rise to a candidate for the localization at odd primes p of the double loop space of an odd sphere obtained from the p-fold center of mass arrangement. Potential consequences are listed., Comment: This is the version published by Geometry & Topology Monographs on 14 November 2007

A topological group is constructed which is homotopy equivalent to the pointed loop space of a path-connected Riemannian manifold $M$ and which is given in terms of "composable small geodesics" on $M$. This model is analogous to J. Milnor's free group construction \cite{Milnor} which provides a model for the pointed loop space of a connected simplicial complex. Related function spaces are constructed from "composable small geodesics" which provide models for the free loop space of $M$ as well as the space of continuous maps from a surface to $M$.

This article gives a natural decomposition of the suspension of generalized moment-angle complexes or {\it partial product spaces} which arise as {\it polyhedral product functors} described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in Goresky-MacPherson \cite{goresky.macpherson}, Hochster\cite{hochster}, Baskakov \cite{baskakov}, Panov \cite{panov}, and Buchstaber-Panov \cite{buchstaber.panov}. Since the splitting is geometric, an analogous homological decomposition for a generalized moment-angle complex applies for any homology theory. This decomposition gives an additive decomposition for the Stanley-Reisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter \cite{porter} and Ganea \cite{ganea}. The spirit of the work here follows that of Denham-Suciu in \cite{denham.suciu}.

A discrete group which admits a faithful, finite dimensional, linear representation over a field $\mathbb F$ of characteristic zero is called linear. This note combines the natural structure of semi-direct products with work of A. Lubotzky on the existence of linear representations to develop a technique to give sufficient conditions to show that a semi-direct product is linear. Let $G$ denote a discrete group which is a semi-direct product given by a split extension $1 \to \pi \to G \to \Gamma \to 1$. This note defines an additional type of structure for this semi-direct product called a stable extension below. The main results are as follows: 1. If $\pi$ and $\Gamma$ are linear, and the extension is stable, then $G$ is also linear. Restrictions concerning this extension are necessary to guarantee that $G$ is linear as seen from properties of the Formanek-Procesi "poison group". 2. If the action of $\Gamma$ on $\pi$ has a "Galois-like" property that it factors through the automorphisms of certain natural "towers of groups over $\pi$" (to be defined below), then the associated extension is stable and thus $G$ is linear. 3. The condition of a stable extension also implies that $G$ admits filtration quotients which themselves give a natural structure of Lie algebra and which also imply earlier results of Kohno, and Falk-Randell on the Lie algebra attached to the descending central series associated to the fundamental groups of complex hyperplane complements. The methods here suggest that a possible technique for obtaining new linearity results may be to analyze automorphisms of towers of groups., Comment: 26 pages

The purpose of this article is to \begin{enumerate} \item define $M(t,k)$ the $t$-fold center of mass arrangement for $k$ points in the plane, \item give elementary properties of $M(t,k)$ and \item give consequences concerning the space $M(2,k)$ of $k$ distinct points in the plane, no four of which are the vertices of a parallelogram. \end{enumerate} The main result proven in this article is that the classical unordered configuration of $k$ points in the plane is not a retract up to homotopy of the space of $k$ unordered distinct points in the plane, no four of which are the vertices of a parallelogram. The proof below is homotopy theoretic without an explicit computation of the homology of these spaces. In addition, a second, speculative part of this article arises from the failure of these methods in the case of odd primes $p$. This failure gives rise to a candidate for the localization at odd primes $p$ of the double loop space of an odd sphere obtained from the $p$-fold center of mass arrangement. Potential consequences are listed., Comment: 11 pages

Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the integers. In particular among them are the automorphisms chi_{k,i} which conjugate the generator x_k by the generator x_i leaving the x_j fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by chi_{k,i} for i

Poly-free groups are constructed as iterated semidirect products of free groups. The class of poly-free groups includes the classical pure braid groups, fundamental groups of fiber-type hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The purpose of this article is to compute centralizers of certain natural Lie subalgebras of the Lie algebra obtained from the descending central series of poly-free groups G including some of the geometrically interesting classes of groups mentioned above. The main results here extend results of Cohen and Prassidis for such groups. These results imply that a homomorphism out of G is faithful, essentially, if it is faithful when restricted to the level of Lie algebras obtained from the descending central series for the product F x Z, where F is the "top" free group in the semidirect products of free groups and Z is the center of G. The arguments use a mixture of homological, and Lie algebraic methods applied to certain choices of extensions. The limitations of these methods are illustrated using the "poison groups" of Formanek and Procesi, poly-free groups whose Lie algebras do not have certain properties considered here., Comment: minor revisions

This article gives certain fibre bundles associated to the braid groups which are obtained from a translation as well as conjugation on the complex plane. The local coefficient systems on the level of homology for these bundles are given in terms of the determinant of the Burau representation. De Concini, Procesi, and Salvetti [Topology 40 (2001) 739--751] considered the cohomology of the n-th braid group B_n with local coefficients obtained from the determinant of the Burau representation, H^*(B_n;Q[t^{+/-1}]). They show that these cohomology groups are given in terms of cyclotomic fields. This article gives the homology of the stable braid group with local coefficients obtained from the determinant of the Burau representation. The main result is an isomorphism H_*(B_infty; F[t^{+/-1}])-->H_*(Omega^2S^3<3>; F) for any field F where Omega^2S^3<3> denotes the double loop space of the 3-connected cover of the 3-sphere. The methods are to translate the structure of H_*(B_n;F[t^{+/-1}]) to one concerning the structure of the homology of certain function spaces where the answer is computed., Comment: This is the version published by Geometry & Topology Monographs on 29 January 2007

The purpose of this article is to compare the two self-maps of $\Omega^kS^{2n+1}$ given by $\Omega^k[2]$ the $k$-fold looping of a degree 2 map and $\Psi^k(2)$ the H-space squaring map. The main results give that in case $2n+1 \neq 2^j-1$, these maps are frequently not homotopic and also that their homotopy theoretic fibres are not homotopy equivalent. The methods are a computation of an unstable secondary operation constructed by Brown and Peterson in the first case and the Nishida relations in the second case. One question left unanswered here is whether the maps $\Omega^{2n+1}[2]$ and $\Psi^{2n+1}(2)$ are homotopic on the level of $\Omega^{2n+1}_0S^{2n+1}$. A natural conjecture is that these two maps are homotopic.

The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno, and provides connections between these various topics. Two consequences are as follows: 1) the homotopy groups of spheres are identified as "natural" sub-quotients of free products of pure braid groups, and 2) an axiomatization of certain simplicial groups arising from braid groups is shown to characterize the homotopy types of connected $CW$-complexes.

The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie algebraic condition which guarantees that a homomorphism out of the classical pure braid group is faithful. However, it is unclear whether the methods here can be applied to any open cases such as the Gassner representation., Comment: Change in Context