arXiv subjects
Publications and source records attributed to Frederick R. Cohen.
We give a refinement of the stable Snaith splitting of the double loop space of a Moore space and use it to construct infinite $v_1$-periodic families of elements of order $p^{r+1}$ in the homotopy groups of mod $p^r$ Moore spaces. For odd primes $p$, our splitting implies that the homotopy groups of the mod $p^{r+1}$ Moore spectrum are summands of the unstable homotopy groups of each mod $p^r$ Moore space.
We prove that the order of the canonical vector bundle over the configuration space is $2$ for a general planar graph, and is $4$ for a nonplanar graph.
The equivariant cohomology of the classical configuration space $F(\mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(\mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojević, Lück \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $\ell$-skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.
The purpose of this article is to give an exposition of topological properties of spaces of homomorphisms from certain finitely generated discrete groups to Lie groups $G$, and to describe their connections to classical representation theory, as well as other structures. Various properties are given when $G$ is replaced by a small category, or the discrete group is given by a right-angled Artin group.
The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\mathbb Z^n,G)$, with the order of the Weyl group inverted.
A continuous map C^d -> C^N is a complex k-regular embedding if any k pairwise distinct points in C^d are mapped by f into k complex linearly independent vectors in C^N. Our central result on complex k-regular embeddings extends results of Cohen & Handel (1978), Chisholm (1979) and Blagojevic, Lück & Ziegler (2013) on real k-regular embeddings: We give new lower bounds for the existence of complex k-regular embeddings. These are obtained by modifying the framework of Cohen & Handel (1978) and a study of Chern classes of complex regular representations. The main technical result, used for the study of the Chern classes, is an upper bound for the height of the cohomology of an unordered configuration space Furthermore, we give similar lower bounds for the existence of complex l-skew embeddings C^d -> C^N, for which we require that the images of the tangent spaces at any l distinct points are skew complex affine subspaces of C^N.
In this article, we give a combinatorial approach to the exponents of the Moore spaces. Our result states that the projection of the $p^{r+1}$-th power map of the loop space of the $(2n+1)$-dimensional mod $p^r$ Moore space to its atomic piece containing the bottom cell $T^{2n+1}\{p^r\}$ is null homotopic for $n>1$, $p>3$ and $r>1$. This result strengthens the classical result that $ΩT^{2n+1}\{p^r\}$ has an exponent $p^{r+1}$.
The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group $Σ_n$, the Lie module $\mathsf{Lie}(n)$ has attracted a great deal of interest in recent years. We prove here that the complexity of $\mathsf{Lie}(n)$ in characteristic $p$ is $t$ where $p^t$ is the largest power of $p$ dividing $n$, thus proving a conjecture of Erdmann, Lim and Tan. The proof uses work of Arone and Kankaanrinta which describes the homology $\operatorname{H}_\bullet(Σ_n, \mathsf{Lie}(n))$ and earlier work of Hemmer and Nakano on complexity for modules over $Σ_n$ that involves restriction to Young subgroups.
We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $μ$--invariants as homotopy periods of $\text{Conf}(n)$.
Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.
A statistical framework is introduced for a broad class of problems involving synchronization or registration of data across a sensor network in the presence of noise. This framework enables an estimation-theoretic approach to the design and characterization of synchronization algorithms. The Fisher information is expressed in terms of the distribution of the measurement noise and standard mathematical descriptors of the network's graph structure for several important cases. This leads to maximum likelihood and approximate maximum-likelihood registration algorithms and also to distributed iterative algorithms that, when they converge, attain statistically optimal solutions. The relationship between optimal estimation in this setting and Kirchhoff's laws is also elucidated.
In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\Rep(\Z^n, SU)$, $\Rep(\Z^n,U)$ and $\Rep(\Z^n, Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.
In this paper the space of commuting elements in the central product $G_{m,p}$ of $m$ copies of the special unitary group $SU(p)$ is studied, where $p$ is a prime number. In particular, a computation for the number of path connected components of these spaces is given and the geometry of the moduli space $\Rep(\mathbb Z^n, G_{m,p})$ of flat principal $G_{m,p}$--bundles over the $n$--torus is completely described for all values of $n$, $m$ and $p$.
The main result of this paper is an application of the topology of the space $Q(X)$ to obtain results for the cohomology of the symmetric group on $d$ letters, $Σ_d$, with `twisted' coefficients in various choices of Young modules and to show that these computations reduce to certain natural questions in representation theory. The authors extend classical methods for analyzing the homology of certain spaces $Q(X)$ with mod-$p$ coefficients to describe the homology $\HH_{\bullet}(Σ_d, V^{\otimes d})$ as a module for the general linear group $GL(V)$ over an algebraically closed field $k$ of characteristic $p$. As a direct application, these results provide a method of reducing the computation of $\text{Ext}^{\bullet}_{Σ_{d}}(Y^λ,Y^μ)$ (where $Y^λ$, $Y^μ$ are Young modules) to a representation theoretic problem involving the determination of tensor products and decomposition numbers. In particular, in characteristic two, for many $d$, a complete determination of $\Hs Y^λ)$ can be found. This is the first nontrivial class of symmetric group modules where a complete description of the cohomology in all degrees can be given. For arbitrary $d$ the authors determine $\HH^i(Σ_d,Y^λ)$ for $i=0,1,2$. An interesting phenomenon is uncovered--namely a stability result reminiscent of generic cohomology for algebraic groups. For each $i$ the cohomology $\HH^i(Σ_{p^ad}, Y^{p^aλ})$ stabilizes as $a$ increases. The methods in this paper are also powerful enough to determine, for any $p$ and $λ$, precisely when $\HH^{\bullet}(\sd,Y^λ)=0$. Such modules with vanishing cohomology are of great interest in representation theory because their support varieties constitute the representation theoretic nucleus.
This article records basic topological, as well as homological properties of the space of homomorphisms Hom(L,G) where L is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If L is a free abelian group of rank equal to n, then Hom(L,G) is the space of ordered n-tuples of commuting elements in G. If G=SU(2), a complete calculation of the cohomology of these spaces is given for n=2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting.
The purpose of this article is to analyze several Lie algebras associated to "orbit configuration spaces" obtained from a group G acting freely, and properly discontinuously on the upper 1/2-plane H^2. The Lie algebra obtained from the descending central series for the associated fundamental group is shown to be isomorphic, up to a regrading, to (1) the Lie algebra obtained from the higher homotopy groups of "higher dimensional arrangements" modulo torsion, as well as (2)the Lie obtained from horizontal chord diagrams for surfaces. The resulting Lie algebras are similar to those studied in [13, 14, 15, 2, 7, 8, 6]. The structure of a related graded Poisson algebra defined below and obtained from an analogue of the infinitesimal braid relations parametrized by G is also addressed.
Let G be the fundamental group of the complement of a K(G,1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group (as defined in the paper). The subgroup of elements in the complex K-theory of BG which arises from complex unitary representations of G is shown to be trivial. In the case of real K-theory, this subgroup is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. Furthermore, an orthogonal representation of G gives rise to a trivial bundle if and only if the representation factors through the spinor groups. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles.
Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive elements in the homology of the loop space for the complement of the associated subspace arrangement. Furthermore, this last Lie algebra is given by the homotopy groups modulo torsion of the loop space of the complement of the subspace arrangement. Looping further yields an associated Poisson algebra, and generalizations of the "universal infinitesimal Poisson braid relations."