Showing papers in "Journal of Topology and Analysis in 2014"
TL;DR: In this paper, the authors construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the fundamental group of a graph of groups with amenable edge groups.
Abstract: This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups, we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X, Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X, Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov's Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along π1-injective boundary components with amenable fundamental group.
50 citations
TL;DR: In this paper, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1.
Abstract: We extend Vincent Lafforgue's results to Sp4. As applications, the family of expanders constructed by finite quotients of a lattice in such a group does not admit a uniform embedding in any Banach space of type > 1, and any affine isometric action of such a group, or of any cocompact lattice in it, in a Banach space of type > 1 has a fixed point.
40 citations
TL;DR: In this paper, the authors give a cohomological formula for the index of a pseudodifferential operator on a manifold with boundary, based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary.
Abstract: We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach.
20 citations
TL;DR: In this article, a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular isometric representations on reflexive Banach Spaces, was initiated.
Abstract: We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll–Nardzewski fixed point theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu–Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.
17 citations
TL;DR: In this article, a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property was identified.
Abstract: We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon–Nikodým property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the Cheeger's (1999) metric differentiation theory. The class includes the infinite diamond and both Laakso (2000) spaces. We also show that for each of these structures there is a non-RNP Banach space which does not admit its bilipschitz embedding. We prove that a dual Banach space does not have the RNP if and only if it admits a bilipschitz embedding of the infinite diamond. The paper also contains related characterizations of reflexivity and the infinite tree property.
17 citations
TL;DR: In this article, it was shown that there is no universal inequality bounding 1-width of a 2-dimensional surface in terms of its diameter, and that every 1-cycle dividing M into two regions of equal area has length > C.
Abstract: Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M, g) = 1 and every 1-cycle dividing M into two regions of equal area has length > C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Stephane Sabourau.
16 citations
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TL;DR: In this article, it was shown that for every complete Riemannian surface M diffeomorphic to a sphere with k ≥ 0 holes, there exists a Morse function f : M → ℝ, which is constant on each connected component of the boundary of M and has fibers of length no more than.
Abstract: We show that for every complete Riemannian surface M diffeomorphic to a sphere with k ≥ 0 holes, there exists a Morse function f : M → ℝ, which is constant on each connected component of the boundary of M and has fibers of length no more than . We also show that on every 2-sphere there exists a simple closed curve of length subdividing the sphere into two discs of area .
13 citations
TL;DR: In this article, the authors define Morse-Conley-Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows.
Abstract: The gradient flow of a Morse function on a smooth closed manifold generates, under suitable transversality assumptions, the Morse–Smale–Witten complex. The associated Morse homology is an invariant for the manifold, and equals the singular homology, which yields the classical Morse relations. In this paper we define Morse–Conley–Floer homology, which is an analogous homology theory for isolated invariant sets of smooth, not necessarily gradient-like, flows. We prove invariance properties of the Morse–Conley–Floer homology, and show how it gives rise to the Morse–Conley relations.
9 citations
TL;DR: In this article, a new coarse analog of paracompactness modeled on the defining characteristics of expanders is defined, which gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite nontrivial group.
Abstract: Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modeled on the defining characteristics of expanders. That analog gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite nontrivial group.
9 citations
TL;DR: The inequality for the macroscopic dimension of universal covers of almost spin n-manifolds M with positive scalar curvature was shown in this paper, where the fundamental group π 1(M) is a virtual duality group that satisfies the coarse Baum-Connes conjecture.
Abstract: We prove the inequality for the macroscopic dimension of the universal covers of almost spin n-manifolds M with positive scalar curvature whose fundamental group π1(M) is a virtual duality group that satisfies the coarse Baum–Connes conjecture.
8 citations
TL;DR: A parametrized version of Volovikov's powerful Borsuk-Ulam-Bourgin-Yang type theorem, based on a new Fadell-Husseini type ideal-valued index of G-bundles, is presented in this article.
Abstract: We present a parametrized version of Volovikov's powerful Borsuk–Ulam–Bourgin–Yang type theorem, based on a new Fadell–Husseini type ideal-valued index of G-bundles which makes computations easy. As an application we provide a parametrized version of the following waist of the sphere theorem due to Gromov, Memarian, and Karasev–Volovikov: Any map f from an n-sphere to a k-manifold (n ≥ k) has a preimage f-1(z) whose epsilon-neighborhoods are at least as large as the epsilon-neighborhoods of the equator Sn-k (if n = k we further need that f has even degree).
TL;DR: In this paper, a classification of simply-connected Riemannian manifolds with non-compact connected components is given, and it is shown that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space.
Abstract: We give a classification of many closed Riemannian manifolds M whose universal cover $\widetilde{M}$ possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that ${\rm Isom}(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.
TL;DR: In this paper, it was shown that the Equation Problem in central extensions of hyperbolic groups is solvable in polynomial time, and that the problem can be solved in a finite amount of time.
Abstract: The Equation problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions of hyperbolic groups is solvable.
TL;DR: In this paper, it was shown that simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algeses after tensoring with the universal UHF algebra.
Abstract: In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0, 1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternative way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.
TL;DR: For spin manifolds with boundary, the main result of as discussed by the authors is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary.
Abstract: For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the metric is extended over a handle of codimension at least two attached at the boundary. Applications of this result include the construction of non-isotopic metrics with invertible Dirac operator, and a concordance existence and classification theorem.
TL;DR: In this article, it was shown that for all n = 4k - 2, k ≥ 2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that is nontrivial.
Abstract: In this paper we prove that for all n = 4k - 2, k ≥ 2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that is nontrivial. denotes the Teichmuller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity. Gromov–Thurston branched cover manifolds provide examples of negatively curved manifolds that do not have the homotopy type of a locally symmetric space. These manifolds will be used in this paper to prove the above stated result.
TL;DR: In contrast, discrete subgroups of the group Diff+([0, 1]) are much less studied as mentioned in this paper, and very little is known in this area especially in comparison with the very rich theory of discrete sub groups of Lie groups which has started in the works of F. Klein and H. Poincare in the 19th century, and has experienced enormous growth in the work of A. Selberg, A. Borel, G. Margulis and many others.
Abstract: In recent decades, many remarkable papers have appeared which are devoted to the study of finitely generated subgroups of Diff+([0, 1]) (see [8, 15, 16, 19–23, 29, 30, 39, 40] only for some of the most recent developments). In contrast, discrete subgroups of the group Diff+([0, 1]) are much less studied. Very little is known in this area especially in comparison with the very rich theory of discrete subgroups of Lie groups which has started in the works of F. Klein and H. Poincare in the 19th century, and has experienced enormous growth in the works of A. Selberg, A. Borel, G. Mostow, G. Margulis and many others in the 20th century. Many questions which are either very easy or have been studied a long time ago for (discrete) subgroups of Lie groups remain open in the context of the infinite-dimensional group Diff+([0, 1]) and its relatives.
TL;DR: In this paper, Toda, Selick and Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration.
Abstract: To help study the double suspension when localised at a prime p, Selick filtered Ω2S2n+1 by H-spaces which geometrically realise a natural Hopf algebra filtration of H*(Ω2S2n+1;ℤ/p). Later, Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration . In this paper we correspondingly filter BWn in a manner compatible with Selick's filtration and the homotopy fibration , study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S2n+1. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BWn.
TL;DR: In this paper, a connection between Morse theory and Gromov-Witten/Floer theory for monotone symplectic manifolds (M, ω) is made.
Abstract: Following [16], we develop here a connection between Morse theory for the (positive) Hofer length functional L : ΩHam(M, ω) → ℝ, with Gromov–Witten/Floer theory, for monotone symplectic manifolds (M, ω). This gives some immediate restrictions on the topology of the group of Hamiltonian symplectomorphisms (possibly relative to the Hofer length functional), and a criterion for non-existence of certain higher index geodesics for the Hofer length functional. The argument is based on a certain automatic transversality phenomenon which uses Hofer geometry to conclude transversality and may be useful in other contexts. Strangely the monotone assumption seems essential for this argument, as abstract perturbations necessary for the virtual moduli cycle, decouple us from underlying Hofer geometry, causing automatic transversality to break.
TL;DR: In this paper, Atiyah, Harada, Landweber and Sjamaar showed that a T-equivariant KK-group is G-equivariant if and only if it is annihilated by an ideal of divided difference operators.
Abstract: Let G be a compact connected Lie group with a maximal torus T. Let A, B be G-C*-algebras. We define certain divided difference operators on Kasparov's T-equivariant KK-group KKT(A, B) and show that KKG(A, B) is a direct summand of KKT(A, B). More precisely, a T-equivariant KK-class is G-equivariant if and only if it is annihilated by an ideal of divided difference operators. This result is a generalization of work done by Atiyah, Harada, Landweber and Sjamaar.
TL;DR: In this article, the existence of a n + 1 periodic billiard trajectory inside an n-dimensional regular simplex in the hyperbolic space is established, which hits the interior of every facet exactly once.
Abstract: In this note we establish the existence of a n + 1 periodic billiard trajectory inside an n-dimensional regular simplex in the hyperbolic space, which hits the interior of every facet exactly once.
TL;DR: In this article, it was shown that if B is a C*-algebra then the differential is given explicitly in terms of an enhanced Samelson product with the clutching map of the principal bundle.
Abstract: Assume that given a principal G bundle ζ : P → Sk (with k ≥ 2) and a Banach algebra B upon which G acts continuously. Let denote the associated bundle and let denote the associated Banach algebra of sections. Then π*GLAζ⊗B is determined by a mostly degenerate spectral sequence and by a Wang differential We show that if B is a C*-algebra then the differential is given explicitly in terms of an enhanced Samelson product with the clutching map of the principal bundle. Analogous results hold after localization and in the setting of topological K-theory. We illustrate our technique with a close analysis of the invariants associated to the C*-algebra of sections of the bundle constructed from the Hopf bundle ζ : S7 → S4 and by the conjugation action of S3 on M2 = M2(ℂ). We compare and contrast the information obtained from the homotopy groups π*(U◦Aζ⊗M2), the rational homotopy groups π*(U◦Aζ⊗M2) ⊗ ℚ and the topological K-theory groups K*(Aζ⊗M2), where U◦B is the connected component of the unitary group of the C*-algebra B.