Search

In the middle of the last century, after hearing a talk of Mostow on one of his rigidity theorems, Borel conjectured in a letter to Serre a purely topological version of rigidity for aspherical manifolds (i.e. manifolds with contractible universal covers). The Borel conjecture is now one of the central problems of topology with many implications for manifolds that need not be aspherical. Since then, the theory of rigidity has vastly expanded in both precision and scope. This book rethinks the implications of accepting his heuristic as a source of ideas. Doing so leads to many variants of the original conjecture - some true, some false, and some that remain conjectural. The author explores this collection of ideas, following them where they lead whether into rigidity theory in its differential geometric and representation theoretic forms, or geometric group theory, metric geometry, global analysis, algebraic geometry, K-theory, or controlled topology.

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., To appear in SIAM Journal of Applied Algebra and Geometry

Parametrized motion planning algorithms \cite{CFW} have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.

We consider seriously the analogy between interpolation of nonlinear functions and manifold learning from samples, and examine the results of transferring ideas from each of these domains to the other. Illustrative examples are given in approximation theory, variational calculus (closed geodesics), and quantitative cobordism theory.

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., Comment: revision includes an appendix on fibrations of certain mapping spaces

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., 38 pages. Removed motivational material from the previous version

We compute the homology of random Cech 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 Erdi¾?s -Renyi phase transition, where the Cech 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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14-51, 2017

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., Comment: 19 pages, 2 figures; version accepted for publication in Geometric and Functional Analysis (GAFA)

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 $L$., Comment: 18 pages, 1 figure; new version after several rounds of referee reports

In this paper we address the existence of smooth manifolds proper homotopy equivalent to nonuniform arithmetic manifolds M = Γ\G/K that are not homeomorphic to it. While the manifolds M are properly rigid if rankℚ(Γ) ≤ 2, we show that the so-called virtual structure group has infinite rank as a ℚ-vector space if rankℚ(Γ) ≥ 4.© 2015 Wiley Periodicals, Inc.

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 $$n$$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 $${\mathbb {R}}^d$$Rd. 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.

We give some lower bounds on the description, sample, and computational complexities of the problems of computing dimension, homology, and topological type of a manifold, and detecting singularities for a polyhedron.

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 $Y$., 28 pages, 5 figures. Comments welcome!

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 $\pi_1 X$. 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., Comment: Minor changes. To appear in Communications on Pure and Applied Mathematics

Let M be a locally symmetric irreducible closed manifold of dimension ≥3. A result of Borel [6] combined with Mostow rigidity imply that there exists a finite group G = G ( M ) such that any finite subgroup of Homeo + ( M ) is isomorphic to a subgroup of G. Borel [6] asked if there exist M's with G ( M ) trivial and if the number of conjugacy classes of finite subgroups of 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 Homeo + ( M ) can be either one, countably many or continuum and we determine (at least for dim ⁡ M ≠ 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 ≠ 4 ) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.

We study the problem in transformation groups of realizing a manifold simple homotopy equivalent to a component of the fixed point set of a G-manifold as (a component of) the fixed set of another G-manifold equivariantly simple homotopy equivalent to the original one. We show that such replacability of (a component of) the fixed points is very often possible if the normal representation of the fixed point component is twice a complex representation (and a monodromy vanishes). In addition, we discuss for various compact groups some examples displaying topological phenomenona ranging from replacablility to rigidity.

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)., 20 pages, to appear in Algebraic and Geometric Topology

In this paper, we take a topological view of unsupervised learning. From this point of view, clustering may be interpreted as trying to find the number of connected components of any underlying geometrically structured probability distribution in a certain sense that we will make precise. We construct a geometrically structured probability distribution that seems appropriate for modeling data in very high dimensions. A special case of our construction is the mixture of Gaussians where there is Gaussian noise concentrated around a finite set of points (the means). More generally we consider Gaussian noise concentrated around a low dimensional manifold and discuss how to recover the homology of this underlying geometric core from data that do not lie on it. We show that if the variance of the Gaussian noise is small in a certain sense, then the homology can be learned with high confidence by an algorithm that has a weak (linear) dependence on the ambient dimension. Our algorithm has a natural interpretation as a spectral learning algorithm using a combinatorial Laplacian of a suitable data-derived simplicial complex.

For manifolds, topological properties such as Poincaré duality and invariants such as the signature and characteristic classes, results and techniques from complex algebraic geometry such as the Hirzebruch-Riemann-Roch theorem, and results from global analysis such as the Atiyah-Singer index theorem, worked hand in hand in the past to weave a tight web of knowledge. Individually, many of the above results are in the meantime available for singular stratified spaces as well. The 2011 Oberwolfach workshop “Stratified Spaces: Joining Analysis, Topology and Geometry” discussed these with the specific aim of cross-fertilization in the three contributing fields.

This paper gives three noncompact variants of Lefschetz–Nielsen fixed-point theory parallel to developments that have occurred in surgery theory. Thus, we study when a proper map can be properly homotoped, or boundedly homotoped, or even homotoped in a C r bounded fashion to a fixed-point free map. As an example, the universal cover of a compact aspherical manifold always has a fixed-point free self-diffeomorphism C r close to the identity for all r (although this is not the case, in general, for arbitrary infinite covers of such manifolds, or general universal covers).

In this paper we address the issue of uniformly positive scalar curvature on noncompact 3-manifolds. In particular we show that the Whitehead manifold lacks such a metric, and in fact that $${\mathbb{R}^3}$$ is the only contractible noncompact 3-manifold with a metric of uniformly positive scalar curvature. We also describe contractible noncompact manifolds of higher dimension exhibiting this curvature phenomenon. Lastly we characterize all connected oriented 3-manifolds with finitely generated fundamental group allowing such a metric.

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question.

The main goal of this paper is to give the first examples of equivariant aspherical Poincare complexes, that are not realized by group actions on closed aspherical manifolds $M$. These will also provide new counterexamples to the Nielsen realization problem about lifting homotopy actions of finite groups to honest group actions. Our examples show that one cannot guarantee that a given action of a finitely generated group $\pi$ on Euclidean space extends to an action of $\Pi$, a group containing $\pi$ as a subgroup of finite index, even when all the torsion of $\Pi$ lives in $\pi$.

We prove that if [math] , [math] , is a compact ANR homology [math] –manifold, we can blow up the singularities of [math] to obtain an ANR homology [math] –manifold with the disjoint disks property. More precisely, we show that there is an ANR homology [math] –manifold [math] with the disjoint disks property and a cell-like map [math] .

Define the length of a finite presentation of a group $G$ as the sum of lengths of all relators plus the number of generators. How large can be the $k$th Betti number $b_k(G)=$ rank $H_k(G)$ providing that $G$ has length $\leq N$ and $b_k(G)$ is finite? We prove that for every $k\geq 3$ the maximum $b_k(N)$ of $k$th Betti numbers of all such groups is an extremely rapidly growing function of $N$. It grows faster that all functions previously encountered in Mathematics (outside of Logic) including non-computable functions (at least those that are known to us). More formally, $b_k$ grows as the third busy beaver function that measures the maximal productivity of Turing machines with $\leq N$ states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines. We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function. Also, we outline a construction of a finitely presented group all of whose homology groups are either ${\bf Z}$ or trivial such that its Betti numbers form a random binary sequence., To appear in Topology

We show that the rational Novikov conjecture for a group $\Gamma$ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an E$\Gamma$. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups., Comment: Preprint of Max-Planck Institut fur Mathematik

In this article, we develop a new index theory for noncompact manifolds endowed with an admissible exhaustion by compact sets. This index theory allows us to provide examples of noncompact manifolds with exotic positive scalar curvature phenomena.

We say that an arbitrary manifold (M, ∂M) is topologically rigid relative to its ends if it satisfies the following condition: If (N , ∂N ) is any other manifold with a compact subset C ⊂ N for which a proper homotopy equivalence h : (N , ∂N ) → (M, ∂M) is a homeomorphism on ∂N∪(N\C), then there is a compact subset K ⊂ N and a proper homotopy ht : (N , ∂N ) → (M, ∂M) from h to a homeomorphism such that ht and h agree on ∂N∪(N\K ) for all t ∈ [0, 1]. We say that a manifold M without boundary is properly rigid or absolutely topologically rigid if we eliminate the requirements that h be a homeomorphism on ∂N ∪ (N\C) and agree with ht on ∂N ∪ (N\K ) for all t ∈ [0, 1]. Along the lines of the classical Borel conjecture that all closed aspherical manifolds are topologically rigid, Farrell and Jones [8] provide the following important theorem

It is well known that the signature operator on a manifold defines a K-homology class which is an orientation after inverting 2. Here we address the following puzzle: what is this class localized at 2, and what special properties does it have? Our answers include the following: • the K-homology class M of the signature operator is a bordism invariant; • the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant; • the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant.

For a manifold M, the structure set S (M, rel ∂ ) is the collection of manifolds homotopy equivalent to M relative to the boundary. Siebenmann [R. C. Kirby, L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations. Princeton Univ. Press 1977] showed that in the topological category, the structure set is 4-periodic: S (M, rel ∂ ) ≅ S (MｘD 4 , rel ∂ ) up to a copy of ℤ. The periodicity has been extended to topological manifolds with homotopically stratified group actions for various representations in place of D 4, including twice any complex representation of a compact abelian group. In this paper, we extend the result to twice any complex representation of a compact Lie group. We also prove the bundle version of the periodicity.

Let K be a field. We show that every countable subgroup of GL(n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL(2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL(n,K) is exact, in the sense of C*-algebra theory.

The main goal of this paper is to show that if a finite connected CW complex admits a continuous, symmetric, and unanimous choice function for some number n>1 of agents, then the choice space is contractible. On the other hand, if one removes the finiteness, we give a complete characterization of the possible spaces; in particular, noncontractible spaces are indeed possible. These results extend earlier well-known results of Chichilnisky and Heal.