Mathematics - Algebraic Topology Publications (50)


Mathematics - Algebraic Topology Publications

Almost-flat manifolds were defined by Gromov as a natural generalisation of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilising this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Read More

We explicitely construct an SO(2)-action on a skeletal version of the 2-dimensional framed bordism bicategory. By the 2-dimensional Cobordism Hypothesis for framed manifolds, we obtain an SO(2)-action on the core of fully-dualizable objects of the target bicategory. This action is shown to coincide with the one given by the Serre automorphism. Read More

In this paper examples are constructed to illustrate that homotopically rigid spaces are not as rare as they initially were thought to be. These examples are then used as building blocks to forge highly connected rational spaces with prescribed finite group of self-homotopy equivalences. Finally, they are exploited to: (i) provide inflexible and strongly chiral manifolds; and (ii) compare Lusternik-Schnirelmann category to another numerical invariant. Read More

Let $M$ be a compact orientable surface equipped with a volume form $\omega$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $\omega$. Let also $\mathcal{Z}_{\omega}(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,\omega)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $\omega$ and $f$. We construct a canonical map $\varphi: \mathcal{Z}_{\omega}(f) \to \mathcal{S}_{\mathrm{id}}(f,\omega)$ being a homeomorphism whenever $f$ has at least one saddle point, and an infinite cyclic covering otherwise. Read More

We consider for two based graphs $G$ and $H$ the sequence of graphs $G_k$ given by the wedge sum of $G$ and $k$ copies of $H$. These graphs have an action of the symmetric group $\Sigma_k$ by permuting the $H$-summands. We show that the sequence of representations of the symmetric group $H_q(\mathrm{Conf}_n(G_\bullet); \mathbf{Q})$, the homology of the ordered configuration space of these spaces, is representation stable in the sense of Church and Farb. Read More

The idea is to demonstrate the beauty and power of Alexandrov geometry by reaching interesting applications with a minimum of preparation. The applications include 1. Estimates on the number of collisions in billiards. Read More

We use a coarse version of the fundamental group first introduced by Barcelo, Capraro and White to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations $(N_i)$ and $(M_i)$ of a free group $F$ such that $M_i>N_i$ for all $i$ with $[M_i:N_i]$ uniformly bounded, but with $\Box_{(N_i)}F$ not coarsely equivalent to $\Box_{(M_i)}F$. Read More

Given an infinity category C, one can construct an infinity category of families of objects in C indexed by infinity groupoids. In this note, we define and study the fundamental infinity groupoids of such parametrized families of objects. Read More

A tower is a sequence of simplicial complexes connected by simplicial maps. We show how to compute a filtration, a sequence of nested simplicial complexes, with the same persistent barcode as the tower. Our approach is based on the coning strategy by Dey et al. Read More

The paper surveys topological problems relevant to the motion planning problem of robotics and includes some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible for discontinuities in algorithms for robot navigation. Then we present explicit motion planning algorithms for coordinated collision free control of many particles moving in Euclidean spaces or on graphs. Read More

This note addresses the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the Matrix Structural Theorem. Any of the various algorithms for computing persistent homology constitutes a constructive proof of the Matrix Structural Theorem. Read More

We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in $R^3$ using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate students. It may be viewed as an illustration and exercise in multivariate calculus and a motivation to go deeper into the fields of geometry and topology. Read More

We give a survey of the ideas of descent and nilpotence. We focus on examples arising from chromatic homotopy theory and from group actions, as well as a few examples in algebra. Read More

This paper shows that generalizations of operads equipped with their respective bar/cobar dualities are related by a six operations formalism analogous to that of classical contexts in algebraic geometry. As a consequence of our constructions, we prove intertwining theorems which govern derived Koszul duality of push-forwards and pull-backs. Read More

We introduce cannibalistic classes for string bundles with values in $TMF$ with level structures. This allows us to compute the Morava $E$-homology of all maps from the bordism spectrum $MString$ to $TMF$ with level structures. Read More

Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(\Lambda)$, with coefficients in chains of the based loop space of $\Lambda$ and study its relation to the Floer cohomology $CF^{\ast}(L)$ of $L$. Using the augmentation induced by $L$, $CE^{\ast}(\Lambda)$ can be expressed as the Adams cobar construction $\Omega$ applied to a Legendrian coalgebra, $LC_{\ast}(\Lambda)$. Read More

We consider the coefficients in the series expansion at zero of the Weierstrass sigma function \[ \sigma(z) = z \sum_{i, j \geqslant 0} {a_{i,j} \over (4 i + 6 j + 1)!} \left({g_2 z^4 \over 2}\right)^i \left(2 g_3 z^6\right)^j. \] We have $a_{i,j} \in \mathbb{Z}$. We present the divisibility Hypothesis for the integers $a_{i,j}$ \begin{align*} \nu_2(a_{i,j}) &= \nu_2((4i + 6j + 1)!) - \nu_2(i!) - \nu_2(j!) - 3 i - 4 j, & \nu_3(a_{i,j}) &= \nu_3((4i + 6j + 1)!) - \nu_3(i!) - \nu_3(j!) - i - j. Read More

Given a smooth manifold $M$ and a Lie group $G$, we consider parallel transport maps --groupoid homomorphisms from a path groupoid in $M$ to $G$-- as an alternative description of principal $G$-bundles with smooth connections on them. Using a cellular decomposition $\mathscr{C}$ of $M$, and a system of paths associated to $\mathscr{C}$, we define a homotopical equivalence relation of parallel transport maps, leading to the concept of an extended lattice gauge (ELG) field. A lattice gauge field, as used in Lattice Gauge Theory, is part of the data contained in an ELG field, but the latter contains additional topological information of local nature, sufficient to reconstruct a principal $G$-bundle up to equivalence, in the spirit of Barrett. Read More

In previous works, we have introduced the blown-up intersection cohomology and used it to extend Sullivan's minimal models theory to the framework of pseudomanifolds, and to give a positive answer to a conjecture of M. Goresky and W. Pardon on Steenrod squares in intersection homology. Read More

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. Read More

We analyse the homotopy types of gauge groups of principal U(n)-bundles associated to pseudo Real vector bundles in the sense of Atiyah. We provide satisfactory homotopy decompositions of these gauge groups into factors in which the homotopy groups are well known. Therefore, we substantially build upon the low dimensional homotopy groups as provided in a paper by I. Read More

We find boundaries of Borel-Serre compactifications of locally symmetric spaces, for which any filling is incompressible. We prove this result by showing that these boundaries have small singular models and using these models to obstruct compressions. We also show that small singular models of boundaries obstruct $S^1$-actions (and more generally homotopically trivial $\mathbb Z/p$-actions) on interiors of aspherical fillings. Read More

We prove that the Real Johnson-Wilson theories ER(n) are homotopy associative and commutative ring spectra up to phantom maps. We further show that ER(n) represents an associatively and commutatively multiplicative cohomology theory on the category of (possibly non-compact) spaces. Read More

In this paper, we prove determinant formulas for the $K$-theory classes of the structure sheaves of degeneracy loci classes associated to vexillary permutations in type $A$. As a consequence we obtain determinant formulas for Lascoux-Sch\"utzenberger's double Grothendieck polynomials associated to vexillary permutations. Furthermore, we generalize the determinant formula to algebraic cobordism. Read More

Let $\Sigma$ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds $M_+, M_-$ in $\Sigma$ are called dual to each other if the complement $\Sigma - M_+$ strongly homotopy retracts onto $M_-$ or vice-versa. In this paper we will give a complete answer of which integral triples $(n; m_+, m_-)$ can appear, where $n=dim \Sigma -1$, $m_+={codim}M_+ -1$ and $m_-={codim}M_- -1$. Read More

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of R^n and various types of automorphisms of 2-connected manifolds. Read More

This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. Read More

We study vector bundles over Lie groupoids and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and prove the Morita invariance of VB-cohomology, with implications to deformation cohomology. We also discuss applications to Poisson geometry via Marsden-Weinstein reduction and the integration of Dirac structures. Read More

We determine the second, third, and fourth cohomology groups of Alexander $f$-quandles of the form $\mathbb{F}_q[T,S]/ (T-\omega, S-\beta)$, where $\mathbb{F}_q$ denotes the finite field of order $q$, $\omega \in \mathbb{F}_q\setminus \{0,1\}$, and $\beta \in \mathbb{F}_q$ Read More

We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicial complex. To accomplish this, a new notion of critical object of a discrete Morse function is presented, which generalizes the usual concept of critical simplex (in the sense of R. Forman). Read More

McGibbon and Saumell studied the higher homotopy commutativity of $p$-localized Lie groups in the sense of Williams. We study the higher homotopy commutativity of $p$-localized Lie groups in the sense of Sugawara, which has a deep connection with higher homotopy associativity and L-S category. We prove that the $p$-localized Lie group $G$ decomposes into the product of spheres as an $A_n$-space for sufficiently large $p$. Read More

We prove the Burghelea Conjecture for all groups with finite virtual cohomological dimension. In particular, we proved the Burghelea Conjecture in its original formulation Read More

We use equivariant surgery to classify all involutions on closed surfaces, up to isomorphism. Work on this problem is classical, dating back to the nineteenth century, but some questions seem to have been left unanswered. We give a modern treatment that leads to a complete classification. Read More

We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional space. In this context we show that the involutions satisfy a remarkable duality, and we investigate several numerical invariants. Read More

This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set $X\subset \mathbb{R}^d$, a random variable $\omega_Q\in[0,1]$ is assigned for each elementary cube $Q$ in $X$, and a random cubical set $X(t)$ is defined by the sublevel set of $X$ consisting of elementary cubes with $\omega_Q\leq t$ for each $t\in[0,1]$. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. Read More

The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching $\mu$ on a finite regular CW complex $X$, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of $X$. Read More

The persistent homology of a stationary point process on ${\bf R}^N$ is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. Read More

We show that the homology of configuration spaces of finite graphs is torsion free. Furthermore, we give a concrete generating set for the homology of configuration spaces of finite trees. Read More

We prove that shifted cotangent stacks carry a canonical shifted symplectic structure. We also prove that shifted conormal stacks carry a canonical Lagrangian structure. These results were believed to be true but no written proof was available in the Artin case. Read More

We consider the category whose objects are filtered, or complete, $L_\infty$-algebras and whose morphisms are $\infty$-morphisms which respect the filtrations. We then discuss the homotopical properties of the Getzler-Hinich simplicial Maurer-Cartan functor which associates to each filtered $L_\infty$-algebra a Kan simplicial set, or $\infty$-groupoid. In previous work with V. Read More

Motivated by the operad built from moduli spaces of Riemann surfaces, we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions. We prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces. The recent, strong homological stability results of Galatius and Randal-Williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability. Read More

In this paper we will define an invariant $mc_{\infty}(f)$ of maps $f:X \rightarrow Y_{\mathbb{Q}}$ between a finite CW-complex and a rational space $Y_{\mathbb{Q}}$. We prove that this invariant is complete, i.e. Read More

The main theorem of Church-Ellenberg [arXiv:1506.01022] is a sharp bound on the homology of FI-modules, showing that the Castelnuovo-Mumford regularity of FI-modules over Z can be bounded in terms of generators and relations. We give a new proof of this theorem, inspired by earlier proofs by Li-Yu and Li. Read More

As an application of Behrens and Rezk's spectral algebra model for unstable v_n-periodic homotopy theory, we give explicit presentations for the completed E-homology of the Bousfield-Kuhn functor on odd-dimensional spheres at chromatic level 2, and compare them to the level 1 case. The latter reflects earlier work in the literature on K-theory localizations. Read More

The classical Tucker lemma is a combinatorial version of the Borsuk - Ulam theorem. In this paper we consider several generalizations of this theorem for G-spaces that yield Tucker's type lemmas for G-simplicial complexes and manifolds. Read More

We define a Cayley transform on Stiefel manifolds. Applications to the Lusternik-Schnirelmann category and optimisation problems are presented. Read More

The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the rational level and by considering Lie groups as a starting point for defining each of the higher structures, making close connection to $p_i$-structures. We indicatively call these (rational) Spin-Fivebrane and Spin-Ninebrane structures. Read More

We show that the theory of quasicategories embeds in that of prederivators, in that there exists a simplicial functor from quasicategories to prederivators and strict morphisms which is an equivalence onto its image. Thus no information need be lost in the passage from quasicategories to prederivators, in contrast to the apparent lesson of previous work on the subject, and a certain class of prederivators can serve as an axiomatization of homotopy theories or $(\infty,1)$-categories. Read More

Allegedly, Brouwer discovered his famous fixed point theorem while stirring a cup of coffee and noticing that there is always at least one point in the liquid that does not move. In this paper, based on a talk in honour of Brouwer at the University of Amsterdam, we will explore how Brouwer's ideas about this phenomenon spilt over in a lot of different areas of mathematics and how this eventually led to an intriguing geometrical theory we now know as mirror symmetry. Read More