Mathematics - Algebraic Topology Publications (50)

Search

Mathematics - Algebraic Topology Publications

We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $\mathbb{S} \to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $\pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra. Read More


Pursuing the notion of ambidexterity developed by Hopkins and Lurie, we prove that the span $\infty$-category of finite n-truncated spaces is the free n-semiadditive $\infty$-category generated by a single object. Passing to presentable $\infty$-categories one obtains a description of the free presentable n-semiadditive $\infty$-category in terms of a new notion of n-commutative monoids, which can be described as spaces in which families of points parameterized by finite $n$-truncated spaces can be coherently summed. Such an abstract summation procedure can be used to give a formal definition of the finite path integrals described by Freed, Hopkins, Lurie and Teleman in the context of 1-dimensional topological field theories. Read More


We prove a Blakers-Massey theorem for the Goodwillie tower of a homotopy functor and then reprove some delooping results. The theorem is derived from a generalized Blakers-Massey theorem in our companion paper. Our main tool is fiberwise orthogonality. Read More


From a genuine $\mathbb{Z}/2$-equivariant spectrum $A$ equipped with a compatible multiplicative structure we produce a genuine $\mathbb{Z}/2$-equivariant spectrum $KR(A)$. This construction extends the real $K$-theory framework of Hesselholt-Madsen for discrete rings and the Hermitian $K$-theory framework of Burghelea-Fiedorowicz for simplicial rings. We construct a natural trace map of $\mathbb{Z}/2$-spectra $tr\colon KR(A)\to THR(A)$ to the real topological Hochschild homology spectrum, which extends the $K$-theoretic trace of B\"{o}kstedt-Hsiang-Madsen. Read More


The Dold$-$Thom theorem states that for a sufficiently nice topological space, M, there is an isomorphism between the homotopy groups of the infinite symmetric product of M and the homology groups of M itself. The crux of most known proofs of this is to check that a certain map is a quasi-fibration. It is our goal to present a more direct proof of the Dold$-$Thom theorem which does appeal to any such fact. Read More


We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions. Read More


We prove a generalization of the classical connectivity theorem of Blakers-Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system (L,R) in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization by Chach\'olski-Scherer-Werndli. Our proof is inspired by the one given in Homotopy Type Theory. Read More


Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmanian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. Read More


This paper presents a description of the fourth dimension quotient, using the theory of limits of functors from the category of free presentations of a given group to the category of abelian groups. A functorial description of a quotient of the third Fox subgroup is given and, as a consequence, an identification (not involving an isolator) of the third Fox subgroup is obtained. It is shown that the limit over the category of free representations of the third Fox quotient represents the composite of two derived quadratic functors. Read More


We prove existence results for small presentations of model categories generalizing a theorem of D. Dugger from combinatorial model categories to more general model categories. Some of these results are shown under the assumption of Vop\v{e}nka's principle. Read More


In the paper the \v{C}ech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of \v{C}ech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-\v{C}ech compactifications of metrizable spaces. Read More


We define labeling homomorphisms on the cubical homology of higher-dimensional automata and show that they are natural with respect to cubical dimaps and compatible with tensor products. We also indicate two possible applications of labeled homology in concurrency theory. Read More


We provide a nilpotency criterion for fusion systems in terms of the vanishing of its cohomology with twisted coefficients. Read More


We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic K-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy groups of pseudoisotopy spaces from the involution on rational $S^{1}$--homology group of the free loop space of a simply-connected manifold. As an application, we give explicit dimensions of the open manifolds $V$ that appear in Belegradek-Farrell-Kapovitch's work for which the spaces of complete nonnegatively curved metrics on $V$ have nontrivial rational homotopy groups. Read More


Primary cohomology operations, i.e., elements of the Steenrod algebra, are given by homotopy classes of maps between Eilenberg-MacLane spectra. Read More


Data analysis often concerns not only the space where data come from, but also various types of maps attached to data. In recent years, several related structures have been used to study maps on data, including Reeb spaces, mappers and multiscale mappers. The construction of these structures also relies on the so-called \emph{nerve} of a cover of the domain. Read More


This is a sequel to the paper [Cas]. Here, we extend the methods of Farb-Wolfson using the theory of FI_G-modules to obtain stability of equivariant Galois representations of the etale cohomology of orbit configuration spaces. We establish subexponential bounds on the growth of unstable cohomology, and then use the Grothendieck-Lefschetz trace formula to obtain results on arithmetic statistics for orbit configuration spaces over finite fields. Read More


We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of some hypersurfaces in $\mathbb{P}^n$. In the case of hyperplane arrangements and free, locally quasi-homogeneous hypersurfaces, and assuming a key conjecture, this algorithm is much faster, due to known results on the roots of their Bernstein-Sato polynomials. Our conjecture is supported by all the examples computed so far, and by some partial results. Read More


We present upper and lower bounds for symmetrized topological complexity $TC^\Sigma(X)$ in the sense of Basabe-Gonz\'alez-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of the symmetric square $SP^2(X)$. We also show that symmetrized topological complexity coincides with its monoidal version, where the path from a point to itself is required to be constant. Read More


We prove that the category of trees $\Omega$ is a test category in the sense of Grothendieck. This implies that the category of dendroidal sets is endowed with the structure of a model category Quillen-equivalent to spaces. We show that this model category structure, up to a change of cofibrations, can be obtained as an explicit left Bousfield localisation of the operadic model category structure. Read More


An embedded graph is called $z$-knotted if it contains the unique zigzag (up to reversing). We consider $z$-knotted triangulations, i.e. Read More


We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex schemes whose weight filtration in cohomology satisfies a certain purity property. This has direct applications to the formality of operads or, more generally, of algebraic structures encoded by a colored operad. We also prove a dual statement, with applications to formality in the context of rational homotopy theory. Read More


Let $\mathcal{P}$ be the class of combinatorial 3-dimensional simple polytopes $P$, different from a tetrahedron, without 3- and 4-belts of facets. By the results of Pogorelov and Andreev, a polytope $P$ admits a realisation in Lobachevsky space $\mathbb{L}^3$ with right dihedral angles if and only if $P \in \mathcal{P}$. We consider two families of smooth manifolds defined by regular 4-colourings of Pogorelov polytopes P: six-dimensional quasitoric manifolds over $P$ and three-dimensional small covers of $P$; the latter are also known as three-dimensional hyperbolic manifolds of Loebell type. Read More


There is a product decomposition of a compact connected Lie group $G$ at the prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in homology. Then in studying the multiplicative structure of the $p$-localization of $G$, the Samelson products of the factor space inclusions of the mod $p$ decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the $p$-localized exceptional Lie groups when the factor spaces are of rank $\le 2$, that is, $G$ is quasi-$p$-regular. Read More


Goerss, Hopkins and Miller have proved that the moduli stack of elliptic curves can be covered by $E_{\infty}$ elliptic spectra. It is not known whether this result can be extended to global elliptic cohomology theories and global ring spectra. Generally, it's difficult to construct the representing spectra of an elliptic cohomology. Read More


In this paper we explore a relationship between the topology of the complex hyperplane complements $\mathcal{M}_{BC_n} (\mathbb{C})$ in type B/C and the combinatorics of certain spaces of degree-$n$ polynomials over a finite field $\mathbb{F}_q$. This relationship is a consequence of the Grothendieck trace formula and work of Lehrer and Kim. We use it to prove a correspondence between a representation-theoretic convergence result on the cohomology algebras $H^*(\mathcal{M}_{BC_n} (\mathbb{C});\mathbb{C})$, and an asymptotic stability result for certain polynomial statistics on monic squarefree polynomials over $\mathbb{F}_q$ with nonzero constant term. Read More


We express the rational homotopy type of the mapping spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ of the little discs operads in terms of graph complexes. Using known facts about the graph homology this allows us to compute the rational homotopy groups in low degrees, and construct infinite series of non-trivial homotopy classes in higher degrees. Furthermore we show that for $n-m>2$, the spaces $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n^{\mathbb Q})$ and $\mathrm{Map}^h(\mathsf D_m,\mathsf D_n)$ are simply connected and rationally equivalent. Read More


We prove that the homotopy category of topological operads $P$ satisfying $P(0) = *$ forms a full subcategory of the homotopy category of all topological operads. We more precisely establish that we have a weak-equivalence of simplicial sets at the mapping space level which gives this embedding of homotopy categories when we pass to connected components. We also prove that an analogous result holds for the categories of $k$-truncated operads, which are operads defined up to arity $k$. Read More


In contrast to the case for completions at primes in the general case spaces cannot be good and bad for arbitrary solid rings in the sense of Bousfield and Kan in arbitrary combination. Read More


If $G$ is a graph with vertex set $V$, let Conf$_n^{\text{sink}}(G,V)$ be the space of $n$-tuples of points on $G$, which are only allowed to overlap on elements of $V$. We think of Conf$_n^{\text{sink}}(G,V)$ as a configuration space of points on $G$, where points are allowed to collide on vertices. In this paper, we attempt to understand these spaces from two separate, but closely related, perspectives. Read More


We determine the symmetrized topological complexity of the circle, using primarily just general topology. Read More


Building upon Hovey's work on Smith ideals for monoids, we develop a homotopy theory of Smith ideals for general operads in a symmetric monoidal category. For a sufficiently nice stable monoidal model category and an operad satisfying a cofibrancy condition, we show that there is a Quillen equivalence between a model structure on Smith ideals and a model structure on algebra maps induced by the cokernel and the kernel. For symmetric spectra this applies to the commutative operad and all Sigma-cofibrant operads. Read More


We compute the Chow-Witt rings of the classifying spaces for the symplectic and special linear groups. In the structural description we give, contributions from real and complex realization are clearly visible. In particular, the computation of cohomology with $\mathbf{I}^j$-coefficients is done closely along the lines of Brown's computation of integral cohomology for special orthogonal groups. Read More


We prove that the arrow category of a monoidal model category, equipped with the pushout product monoidal structure and the projective model structure, is a monoidal model category. This result confirms a belief expressed by Hovey in his work on Smith ideals. As illustrations we include examples of non-cofibrantly generated monoidal model categories, including chain complexes, small categories, and topological spaces. Read More


We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichm\"{u}ller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichm\"{u}ller group acts nontrivially on $\overline{\mathcal{M}}_{0,\bullet+1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little 2-disks operad is formal. Read More


On the category of compact metric spaces an exact homology theory was defined and its relation to the Vietoris homology theory was studied by N. Steenrod [S]. In particular, the homomorphism from the Steenrod homology groups to the Vietoris homology groups was defined and it was shown that the kernel of the given homomorphism are homological groups, which was called weak homology groups [S], [E]. Read More


A small cover is a closed smooth manifold of dimension $n$ having a locally standard $\mathbb{Z}_2^n$-action whose orbit space is isomorphic to a simple polytope. A typical example of small covers is a real projective toric manifold (or, simply, a real toric manifold), that is, a real locus of projective toric manifold. In the paper, we classify small covers and real toric manifolds whose orbit space is isomorphic to the dual of the simplicial complex obtainable by a sequence of wedgings from a polygon, using a systematic combinatorial method finding toric spaces called puzzles. Read More


The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. Read More


Let $D(M,N)$ be the set of integers that can be realized as the degree of a map between two closed connected orientable manifolds $M$ and $N$ of the same dimension. In this paper, we determine the set $D(M,N)$ where $M$ and $N$ are closed $3$-manifolds with $S^3$-geometry. Read More


In extension theory, in particular in dimension theory, it is frequently useful to represent a given compact metrizable space X as the limit of an inverse sequence of compact polyhedra. We are going to show that, for the purposes of extension theory, it is possible to replace such an X by a better metrizable compactum Z. This Z will come as the limit of an inverse sequence of triangulated polyhedra with simplicial bonding maps that factor in a certain way. Read More


The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$, $[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in this paper, the spectral and strong homology groups of continuous maps were defined and studied $[B_6]$, $[Be]$, $[Be-Tu]$. Read More


We introduce a new `geometric realization' of an (abstract) simplicial complex, inspired by probability theory. This space (and its completion) is a metric space, which has the right (weak) homotopy type, and which can be compared with the usual geometric realization through a natural map, which has probabilistic meaning : it associates to a random variable its probability mass function. This `probability map' function is proved to be a (Serre) fibration and a (weak) homotopy equivalence. Read More


We use combinatorial group theory methods to extend the definition of a classical James-Hopf invariant to a simplicial group setting. This allow us to realize certain coalgebra idempotents at sSet -level and obtain a functorial decomposition of the spectral sequence, associated with the lower p-central series filtration on the free simplicial group. Read More


Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gerstenhaber bracket, on $\mathcal{D}^*(A,A)$. Read More


Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(\pi_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(\pi)$ denotes the cohomological dimension of a group $\pi$ and ${\rm hd}(X)$ denotes the homotopy dimension of $X$. Furthermore, there is a well-known inequality of Grossman,~\cite{G}: \[ {\rm cat} X\leq \Bigl\lceil\frac{{\rm hd} (X)}{k+1}\Bigr\rceil \text{ if } \pi_i(X)=0 \text{ for } i\leq k. Read More


Given a fiber bundle, we construct a differential graded Lie algebra model for the classifying space of the monoid of homotopy equivalences of the base covered by a fiberwise isomorphism of the total space. Read More


Special subgroups of singular cochains and chains for a topological space called alernative cochains and chains are defined. A modified cup product for alternative cochains is introduced having the same algebraic properties of the wedge product in differential forms. It is shown that alternative chains and cochains induce the corresponding homology and cohomology groups which are isomorphic to ordinary singular homology and cohomology respectively. Read More


In this paper we explain how to convert discrete invariants into stable ones via what we call hierarchical stabilization. We illustrate this process by constructing stable invariants for multi parameter persistence modules with respect to so called simple noise systems. For one parameter we recover the standard bar code information. Read More


In this paper, we study representation homology of topological spaces, that is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology in terms of classical (abelian) homological algebra. Our construction is parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by showing that the representation homology of the (reduced) suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. Read More


We construct a chain map from the normalized bar resolution to the tensor resolution for a given finite abelian group, then we provide a unified formulae for the normalized $4$-cocycles and a formulae for a part of the normalized $n$-cocycles on arbitrary finite abelian groups. As an application of these formulae, we give a formula for the Dijkgraaf-Witten invariant of the $n$-torus for all $n$ in the case that the finite group $G$ in the definition of that invariant is abelian and compute its exact number in the special case that $G$ is the product of at most $n$ cyclic groups. Finally we obtain the formula for the dimension of the irreducible projective representations of an abelian group $G$ as a by-product. Read More