arXiv subjects

Shmuel Weinberger

Publications and source records attributed to Shmuel Weinberger.

At least 37 records · Page 2Linked to original sources

arXiv2021

Parametrized topological complexity of collision-free motion planning in the plane

Parametrized motion planning algorithms have high degrees of universality and flexibility, as they are designed to work under a variety of external conditions, which are viewed as parameters and form part of the input of the underlying motion planning problem. In this paper, we analyze the parameterized motion planning problem for the motion of many distinct points in the plane, moving without collision and avoiding multiple distinct obstacles with a priori unknown positions. This complements our prior work [arXiv:2009.06023], where parameterized motion planning algorithms were introduced, and the obstacle-avoiding collision-free motion planning problem in three-dimensional space was fully investigated. The planar case requires different algebraic and topological tools than its spatial analog.

math.AT↗

arXiv2021

Topology of parametrised motion planning algorithms

In this paper we introduce and study a new concept of parametrised topological complexity, a topological invariant motivated by the motion planning problem of robotics. In the parametrised setting, a motion planning algorithm has high degree of universality and flexibility, it can function under a variety of external conditions (such as positions of the obstacles etc). We explicitly compute the parameterised topological complexity of obstacle-avoiding collision-free motion of many particles (robots) in 3-dimensional space. Our results show that the parameterised topological complexity can be significantly higher than the standard (nonparametrised) invariant.

math.AT↗

arXiv2020

Additivity of higher rho invariants and nonrigidity of topological manifolds

Let $X$ be a closed oriented connected topological manifold of dimension $n\geq 5$. The structure group of $X$ is the abelian group of equivalence classes of all pairs $(f, M)$ such that $M$ is a closed oriented manifold and $f\colon M \to X$ is an orientation-preserving homotopy equivalence. The main purpose of this article is to prove that a higher rho invariant defines a group homomorphism from the topological structure group of $X$ to the $C^*$-algebraic structure group of $X$. In fact, we introduce a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{topological}$ manifold, and prove its additivity. This higher rho invariant map restricts to the higher rho invariant map on the topological structure group. More generally, the same techniques developed in this paper can be applied to define a higher rho invariant map on the homology manifold structure group of a closed oriented connected $\textit{homology}$ manifold. As an application, we use the additivity of the higher rho invariant map to study non-rigidity of topological manifolds. More precisely, we give a lower bound for the free rank of the $\textit{algebraically reduced}$ structure group of $X$ by the number of torsion elements in $π_1 X$. Here the algebraic reduced structure group of $X$ is the quotient of the topological structure group of $X$ modulo a certain action of self-homotopy equivalences of $X$. We also introduce a notion of homological higher rho invariant, which can be used to detect many elements in the structure group of a closed oriented topological manifold, even when the fundamental group of the manifold is torsion free. In particular, we apply this homological higher rho invariant to show that the structure group is not finitely generated for a class of manifolds.

math.KT↗

arXiv2020

Integral and rational mapping classes

Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y \to Y_{(0)}$ which is well-understood. Early on it was found that the induced map $[X,Y] \to [X,Y_{(0)}]$ on sets of mapping classes is finite-to-one. The sizes of the preimages need not be bounded; we show, however, that as the complexity (in a suitable sense) of a rational mapping class increases, these sizes are at most polynomial. This ``torsion'' information about $[X,Y]$ is in some sense orthogonal to rational homotopy theory but is nevertheless an invariant of the rational homotopy type of $Y$ in at least some cases. The notion of complexity is geometric and we also prove a conjecture of Gromov \cite{GrMS} regarding the number of mapping classes that have Lipschitz constant at most $L$.

math.AT↗

arXiv2019

On the stable Cannon Conjecture

The Cannon Conjecture for a torsionfree hyperbolic group G with boundary homeomorphic to S^2 says that G is the fundamental group of an aspherical closed 3-manifold M. It is known that then M is a hyperbolic 3-manifold. We prove the stable version that for any closed manifold N of dimension greater or equal to 2 there exists a closed manifold M together with a simple homotopy equivalence from M to the cartesian product of N and BG. If N is aspherical and pi_1(N) satisfies the Farrell-Jones Conjecture, then M is unique up to homeomorphism.

math.GT↗

arXiv2018

The Gromov--Guth--Whitney embedding theorem

This is an appendix to arXiv:1610.04888, "Quantitative null-cobordism", which improves one of the main results of that paper to a near-sharp one. It is not a self-contained paper.

math.GT↗

arXiv2018

Quantitative nullhomotopy and rational homotopy type

In \cite{GrOrang}, Gromov asks the following question: given a nullhomotopic map $f:S^m \to S^n$ of Lipschitz constant $L$, how does the Lipschitz constant of an optimal nullhomotopy of $f$ depend on $L$, $m$, and $n$? We establish that for fixed $m$ and $n$, the answer is at worst quadratic in $L$. More precisely, we construct a nullhomotopy whose \emph{thickness} (Lipschitz constant in the space variable) is $C(m,n)(L+1)$ and whose \emph{width} (Lipschitz constant in the time variable) is $C(m,n)(L+1)^2$. More generally, we prove a similar result for maps $f:X \to Y$ for any compact Riemannian manifold $X$ and $Y$ a compact simply connected Riemannian manifold in a class which includes complex projective spaces, Grassmannians, and all other simply connected homogeneous spaces. Moreover, for all simply connected $Y$, asymptotic restrictions on the size of nullhomotopies are determined by rational homotopy type.

math.GT↗

arXiv2017

Quantitative null-cobordism

For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. This construction relies on another of independent interest. Take $X$ and $Y$ to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose $Y$ is simply connected and rationally homotopy equivalent to a product of Eilenberg-MacLane spaces: for example, any simply connected Lie group. Then two homotopic L-Lipschitz maps $f, g : X \rightarrow Y$ are homotopic via a $CL$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $Y$.

math.GT↗

arXiv2016

Convergence of the Reach for a Sequence of Gaussian-Embedded Manifolds

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold $M$ into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of $M$. Roughly speaking, the reach is a measure of a manifold's departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.

math.PR↗

arXiv2016

On the Vanishing of Homology in Random Čech Complexes

We compute the homology of random Čech complexes over a homogeneous Poisson process on the d-dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős-Rényi phase transition, where the Čech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups.

math.PR↗

arXiv2016

A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity

Let $M$ be a locally symmetric irreducible closed manifold of dimension $\ge 3$. A result of Borel [Bo] combined with Mostow rigidity imply that there exists a finite group $G = G(M)$ such that any finite subgroup of $\text{Homeo}^+(M)$ is isomorphic to a subgroup of $G$. Borel [Bo] asked if there exist $M$'s with $G(M)$ trivial and if the number of conjugacy classes of finite subgroups of $\text{Homeo}^+(M)$ is finite. We answer both questions: (1) For every finite group $G$ there exist $M$'s with $G(M) = G$, and (2) the number of maximal subgroups of $\text{Homeo}^+(M)$ can be either one, countably many or continuum and we determine (at least for $\dim M \neq 4$) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dim$M\neq 4$) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.

math.GR↗

arXiv2013

Finite part of operator K-theory for groups finitely embeddable into Hilbert space and the degree of non-rigidity of manifolds

In this paper, we study lower bounds on the K-theory of the maximal $C^*$-algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator K-theory and give a lower bound that is valid for a large class of groups, called the "finitely embeddable groups". The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov's monster groups, virtually torsion free groups (e.g. $Out(F_n)$), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. It is an open question if every countable group is finitely embeddable. We apply this result to measure the degree of non-rigidity for any compact oriented manifold $M$ with dimension $4k-1$ $(k>1)$. We derive a lower bound on the rank of the structure group $S(M)$ in this case. For a compact Riemannian manifold $M$ with dimension greater than or equal to 5 and positive scalar curvature metric, there is an abelian group $P(M)$ that measures the size of the space of all positive scalar curvature metrics on $M$. We obtain a lower bound on the rank of the abelian group $P(M)$ when the compact smooth spin manifold $M$ has dimension $2k-1$ $(k>2)$ and the fundamental group of $M$ is finitely embeddable.

math.OA↗

arXiv2013

Topological Classification of Multiaxial U(n)-Actions

A U(n)-manifold is multiaxial if the isotropy groups are always conjugate to unitary subgroups. The classification and the concordance of such manifolds have been studied by Davis, Hsiang and Morgan under much more strict conditions. We show that in general, without much extra condition, the homotopy classification of multiaxial manifolds can be split into a direct sum of the classification of pairs of adjacent strata, which can be computed by the classical surgery theory. Moreover, we also compute the homotopy classification for the case of the standard representation sphere. We also present the result for the similar multiaxial Sp(n)-manifolds.

math.GT↗

arXiv2013

Closed Aspherical Manifolds with Center

We show that in all dimensions >7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.

math.GT↗

arXiv2013

Crackle: The Persistent Homology of Noise

We study the homology of simplicial complexes built via deterministic rules from a random set of vertices. In particular, we show that, depending on the randomness that generates the vertices, the homology of these complexes can either become trivial as the number $n$ of vertices grows, or can contain more and more complex structures. The different behaviours are consequences of different underlying distributions for the generation of vertices, and we consider three illustrative examples, when the vertices are sampled from Gaussian, exponential, and power-law distributions in $\R^d$. We also discuss consequences of our results for manifold learning with noisy data, describing the topological phenomena that arise in this scenario as `crackle', in analogy to audio crackle in temporal signal analysis.

math.PR↗

arXiv2012

Homological aperiodic tilings of 3-dimensional geometries

We construct the first aperiodic tiles for two amenable 3-dimensional Lie groups: Sol and the Heisenberg group. Our construction relies on the use of higher-dimensional uniformly finite homology. In particular, we settle completely the existence of aperiodic tiles for all of the non-compact geometries of 3-manifolds appearing in the geometrization conjecture.

math.GR↗

arXiv2010

Relative systoles of relative-essential 2-complexes

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8 of sys(X,phi)^2. Combining our results with a method of Larry Guth, we obtain new quantitative results for certain 3-manifolds: in particular for Sigma the Poincare homology sphere, we have sys(Sigma)^3 < 24 vol(Sigma).

math.DG↗

arXiv2010

Replacement of fixed sets for compact group actions: The 2ρtheorem

If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally linear actions on topological or PL manifolds by compact Lie groups, the replacement is always possible if the normal bundle of the fixed set is twice of a complex bundle over a 1-skeleton of the fixed set. Moreover, we also study some specific examples, where the answer to the replacement problem ranges from always possible to the rigidity.

math.GT↗