arXiv subjects
Publications and source records attributed to Shmuel Weinberger.
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices, and, in most detail, the algebraic topology of the excursion sets of random fields.
Royden proved that any isometry of Teichmuller space in the Teichmuller metric must be an element of the extended mapping class group M(S). He also proved that the Teichmuller metric is not symmetric at any point. In this paper we give extensions of Royden's theorems from the Teichmuller metric to an arbitrary complete, finite covolume, M(S)-invariant Finsler (e.g. Riemannian) metric on Teichmuller space. In particular this gives a new mechanism behind Royden's original theorem.
Let G be a torsion-free hyperbolic group and let n > 5 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n-1)-dimensional sphere.
It is a deep fact that the homotopy classification of topological manifolds is convariantly functorial. In other words, a map from a topological manifold M to another N naturally induces a map from the structure set S(M) to S(N). We extend the fact to the isovariant structure set S_G(M, rel M_s) of G-equivariant topological manifolds isovariantly homotopy equivalent to M and restricts to homormorphism on the singular part M_s, consisting of those points fixed by some non-trivial elements of G. We further explain that the structure set S_G(M, rel M_s) is the fibre of the assembly map for the generalized homology theory with the L-spectrum as the coefficient. This relates our result to the Farrell-Jones Conjecture for L-theory.
We study optimal curvature-free inequalities of the type discovered by C. Loewner and M. Gromov, using a generalisation of the Wirtinger inequality for the comass. Using a model for the classifying space BS^3 built inductively out of BS^1, we prove that the symmetric metrics of certain two-point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds. We point out the unexpected role played by the exceptional Lie algebra E_7 in systolic geometry, via the calculation of Wirtinger constants. Using a technique of pullback with controlled systolic ratio, we calculate the optimal systolic ratio of the quaternionic projective plane, modulo the existence of a Joyce manifold with Spin(7) holonomy and unit middle-dimensional Betti number.
We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of all contractible Riemannian manifolds covering both compact and (noncompact, complete) finite volume manifold. There are also applications to the Hopf Conjecture, a new proof of Kazhdan's Conjecture (Frankel's Theorem) in complex geometry, etc. Ideas in the proof come from Lie theory, the homological theory of transformation groups, harmonic maps, and large-scale geometry. An extension to the non-aspherical case is also given.
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.
This paper attempts to provide an analogue of the Novikov conjecture for algebraic (or Kähler) manifolds. Inter alia, we prove a conjecture of Rosenberg's on the birational invariance of higher Todd genera. We argue that in the algebraic geometric setting the Novikov philosophy naturally includes non-birational mappings.
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 $π$ on Euclidean space extends to an action of $Π$, a group containing $π$ as a subgroup of finite index, even when all the torsion of $Π$ lives in $π$.
A "hidden symmetry" of a Riemannian manifold M is an isometry of a d-sheeted, 1<d<\infty, Riemannian cover of M which is not the lift of any isometry. In this paper we characterize the locally symmetric metric(s) on a closed, arithmetic manifold as the unique metric with infinitely many hidden symmetries.
We prove that, if M is a compact oriented manifold of dimension 4k+3, where k>0, such that pi_1(M) is not torsion-free, then there are infinitely many manifolds that are homotopic equivalent to M but not homeomorphic to it. To show the infinite size of the structure set of M, we construct a secondary invariant tau_(2): S(M)-->R that coincides with the rho-invariant of Cheeger-Gromov. In particular, our result shows that the rho-invariant is not a homotopy invariant for the manifolds in question.
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the classification of neighborhood germs, the construction of exotic stratifications, a multiparameter isotopy extension theorem and an h-coborsism extension theorem.
This paper surveys the significant progress over the past couple of decades in the theory of stratified spaces through the application of controlled methods as well as through the application of intersection homology.
The rho-invariant is an invariant of odd-dimensional manifolds with finite fundamental group, and lies in the representations modulo the regular representations (after tensoring with Q). It is a fundamental invariant that occurs in classifying lens spaces, their homotopy analogues, and is intimately related to the eta-invariant for the signature operator. The goal of this note is to use some of the technology developed in studying the Novikov higher signature conjecture to define an analogous invariant for certain situations with infinite fundamental group.
In this paper we introduce a new approach to variational problems on the space Riem(M^n) of Riemannian structures (i.e. isometry classes of Riemannan metrics) on any fixed compact manifold M^n of dimension n >= 5. This approach often enables one to replace the considered variational problem on Riem(M^n) (or on some subset of Riem(M^n)) by the same problem but on spaces Riem(N^n) for every manifold N^n from a class of compact manifolds of the same dimension and with the same homology as M^n but with the following two useful properties: (1) If νis any Riemannian structure on any manifold N^n from this class such that Ric_(N^n,ν) >= -(n-1), then the volume of (N^n,ν) is greater than one; and (2) Manifolds from this class do not admit Riemannian metrics of non-negative scalar curvature. As a first application we prove a theorem which can be informally explained as follows: Let M be any compact connected smooth manifold of dimension greater than four, M et(M) be the space of isometry classes of compact metric spaces homeomorphic to M endowed with the Gromov-Hausdorff topology, Riem_1(M) in M et(M ) be the space of Riemannian structures on M such that the absolute values of sectional curvature do not exceed one, and R_1(M) denote the closure of Riem_1(M) in M et(M ). Then diameter regarded as a functional on R_1(M) has infinitely many "very deep" local minima.
We will describe some results regarding the algorithmic nature of homeomorphism problems for manifolds; in particular, the following theorem. Theorem 1: Every PL or smooth simply connected manifold M^n of dimension n at least 5 can be recognized among simply connected manifolds. That is, there is an algorithm to decide whether or not another simply connected manifold is Top, PL or Diff isomorphic to M. Moreover, an analogous statement is true for embeddings in codimension at least three: one can algorithmically recognize any given embedding of one simply connected manifold in another up to isomorphism of pairs, or up to isotopy, if the codimension of the embedding is not two.
We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized manifolds and present a program for understanding their topology.