New approach connects pseudoisotopy theory to algebraic K-theory.
problem Understanding pseudoisotopy theory using algebraic K-theory.
method Constructing a zig-zag from pseudoisotopy space to algebraic K-theory space, using homological computations.
result Maps are p-locally (2n-4)-connected for large primes p.
Study pseudoisotopies in 4-manifolds, finding specific elements of obstruction.
problem Understanding pseudoisotopies in 4-manifolds.
method Explicit construction of pseudoisotopies for certain elements of the second obstruction.
result Explicit construction of pseudoisotopies for specific elements of the second obstruction.
Study involutions on spaces to compute nonnegatively curved metrics dimensions.
problem Computing dimensions of spaces of nonnegatively curved metrics.
method Involution on pseudoisotopy spaces and free loop spaces.
result Explicit dimensions of manifolds with nontrivial rational homotopy groups.
Let V be an open manifold with complete nonnegatively curved metric such that the normal sphere bundle to a soul has no section. We prove that the souls of nearby nonnegatively curved metrics on V are smoothly close. Combining this result with some topological properties of pseudoisotopies we show that for many V the s…
The paper establishes a new pseudoisotopy result for embedding spaces, leading to computations of homotopy groups of long knots.
problem Computing homotopy groups of spaces of long knots in high codimension.
method Using pseudoisotopy results and algebraic K-theory, the paper describes the difference in homotopy types of block and ordinary embeddings of a codimension at least three embedding.
result The homotopy type of spaces of long knots of codimension at least 3 is determined explicitly, including torsion information.
Computes homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
problem Determining homotopy groups of diffeomorphism spaces for high-dimensional manifolds.
method Computes rational homotopy groups of classifying spaces of diffeomorphisms.
result Determines rational pseudoisotopy stable range for compact spin manifolds.
The paper constructs a nontrivial diffeomorphism in 4-manifold topology.
problem Constructing a nontrivial diffeomorphism in 4-manifold topology.
method Using lens-shaped models and the second obstruction to pseudoisotopy.
result The constructed diffeomorphism is not isotopic to the identity.
The study finds non-pseudoisotopic diffeomorphisms in certain 4-manifolds.
problem Identifying diffeomorphisms that are homotopic but not pseudoisotopic.
method Examples of diffeomorphisms in specific 4-manifolds.
result No non-pseudoisotopic diffeomorphisms in orientable 4-manifolds with free fundamental groups.
The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A rather striking amount of work on pseudoisotopy theory [Hatcher, …
We show that the group ${\Cal D}(M)$ of pseudoisotopy classes of diffeomorphisms of a manifold of dimension ≥5 and of finite fundamental group is commensurable to an arithmetic group. As a result π0(DiffM) is a group of finite type.
We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reform…
We study the Fibered Isomorphism Conjecture of Farrell and Jones in L-theory for groups acting on trees. In several cases we prove the conjecture. This includes wreath products of abelian groups and free metabelian groups. We also deduce the conjecture in pseudoisotopy theory for these groups. Finally in B of Theorem 1…
In this paper we generalize the notion of strongly poly-free group to a larger class of groups, we call them strongly poly-surface groups and prove that the Fibered Isomorphism Conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for any virtually strongly poly-surface g…
In this paper we show that the fibered isomorphism conjecture of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of complex manifolds. A consequence of this result is that the Whitehead group, reduced projective class groups and the neg…
The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example for discrete subgroups of Lie groups, virtually poly-infinite cyclic groups, Artin braid groups, a class of virtually poly-surface groups and virtually solvable lin…
We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions the conjecture is true for groups acting on trees when the stabilizers satisfy the conjecture. These conditions are satisfied in several cases of the conjecture. We prove some general resul…
Theta graph diffeomorphism shows nontrivial mapping class of 4-sphere.
problem Identifying nontrivial elements in the smooth mapping class group of 4-sphere.
method Diagrammatic calculus for smooth mapping class group of 4-sphere, Watanabe's clasper surgery construction.
result Theta graph diffeomorphism is isotopic to a nontrivial element of (1,2)-subgroup.
New results show metrics with positive scalar curvature can cancel on certain 4-manifolds.
problem Understanding cancellation of metrics with positive scalar curvature.
method Cobordism theory and rigidity properties of diffeomorphism groups.
result Metrics from Ruberman's family map to the same path component on product manifolds.
We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. …
This article has two purposes. In \cite{R3} (math.KT/0405211) we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by \cal C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental …
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
(1,1) non-L-space knots are foliar in 3D space.
problem Proving (1,1) non-L-space knots are foliar.
method Analyzing (1,1) non-L-space knots in S3 and lens spaces. result (1,1) non-L-space knots are persistently foliar.
Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
Defines strongly Gauduchon spaces and their properties.
problem Generalization of strongly Gauduchon manifolds in complex spaces.
method Definition and properties of strongly Gauduchon spaces and class SG.
result Similarities and properties of strongly Gauduchon spaces to Kahlerian spaces.
The paper extends Stone duality to topological convexity spaces.
problem Understanding the relationship between topological convexity spaces and sup-lattices.
method Extending Stone duality to topological convexity spaces using preconvexity spaces.
result An adjunction between topological convexity spaces and sup-lattices.
New findings on GRW space-times with constant scalar curvature.
problem Understanding GRW space-times in different subspaces.
method Analyzing orthogonal subspaces of Gray's decomposition.
result Generalized quasi-Einstein GRW space-times reduce to known types of space-times.
No Einstein hypersurfaces found in Damek-Ricci spaces.
problem Existence of Einstein hypersurfaces in symmetric spaces.
method Analyzing properties of Damek-Ricci spaces and proving no Einstein hypersurface exists.
result No Einstein hypersurfaces in Damek-Ricci spaces.
Generalizes balanced manifolds to complex spaces, studying their properties and relations.
problem Generalizing balanced manifolds to complex spaces.
method Defined class B and balanced spaces, compared with Kahlerian spaces, and studied properties and relations.
result Properties and relations between balanced spaces and class B are obtained.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
New kernels defined for various spaces, including measures.
problem Defining kernels on non-standard spaces like measures.
method Integrally strictly positive definite and characteristic kernels on Hilbert, Banach, and metric spaces.
result Explicit classes of kernels on Lp spaces and sets of measures. In a paper (math.DG/0403528) we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, …
Metric spaces uniquely split into Hilbert and non-line-split parts.
problem Understanding the structure of metric spaces.
method Proved unique decomposition into Hilbert and non-line-split parts.
result Metric spaces have a unique decomposition into a Hilbert space and a non-line-split part.
A metric defined on body moduli space.
problem Defining a metric on the moduli space of bodies.
method Quotient space by integral affine transformations.
result Identifies moduli space of Delzant polytopes with symplectic toric manifolds.
Study geometry of tetrahedra in complex hyperbolic space and Hilbert spaces.
problem Understanding geometric relationships between complex hyperbolic spaces and Hilbert spaces.
method Use a complex analog of the cosine of a vertex angle as a novel technical tool.
result Describe possible triangular faces of tetrahedra in hyperbolic space and three-dimensional subspaces in Hilbert spaces with Pick kernels.
New curvature positivity helps classify spherical spaces and complex projective spaces.
problem Classifying spherical space forms and complex projective spaces.
method Introducing a new positivity notion for curvature.
result Characterizations for spherical space forms and complex projective spaces.
Study of complete space-like self-expanders in Minkovski space.
problem Characterize complete space-like self-expanders in Minkovski space.
method Use of maximum principle of Omori-Yau type to prove rigidity theorems.
result Classification of 2-dimensional complete space-like self-expanders with constant squared norm of the second fundamental form.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
Study on convergence of transformed metric spaces as dimensions grow.
problem Conditions for convergence of transformed metric spaces.
method Clarifying conditions for convergence of transformed spaces from original sequence and vice versa.
result Spheres and projective spaces converge to Gaussian space and its quotient as dimensions increase.
Berwald spaces with non-zero flag curvature are Riemannian.
problem Understanding the conditions under which Berwald spaces become Riemannian.
method Analyzing the flag curvature of Berwald spaces and proving their rigidity under certain conditions.
result Berwald spaces with non-zero flag curvature are Riemannian.
Characterizes when almost smooth spaces become RCD spaces.
problem Understanding conditions for almost smooth spaces to be RCD spaces.
method Characterizations via local volume doubling and Poincaré inequality.
result Characterizes Einstein 4-orbifolds.
Stability of Wasserstein spaces under various convergence types.
problem Stability and finiteness of Wasserstein spaces over singular and non-singular spaces.
method Gromov--Hausdorff convergence and equivariant Gromov--Hausdorff convergence.
result Analogue of Perelman's stability theorem on Wasserstein spaces.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
New quasi space forms solve Thurston's geometrical space form problem.
problem Solving Thurston's geometrical space form problem.
method Introducing quasi space forms as non-real space forms with specific geometric properties.
result Quasi space forms offer a metrical, local geometrical solution to Thurston's problem.
Sphere theorems extended to RCD spaces and improved for Einstein stratified spaces.
problem Generalizing sphere theorems to new types of spaces.
method Proved sphere theorems for RCD(n-1, n) spaces and Einstein stratified spaces.
result Extended sphere theorems to RCD spaces and improved results for Einstein stratified spaces.
This work describes compactifications of metric spaces and vector spaces using asymmetric norms.
problem Compactifying metric spaces and vector spaces using asymmetric norms.
method Nonstandard methods, ultrapowers of the spaces at hand.
result Polyhedral compactifications of vector spaces with stratified structure.
Introduces new types of homogeneous spaces and their properties.
problem Defining and understanding new types of homogeneous spaces.
method Introducing and analyzing (strongly) (Θ-)discrete homogeneous spaces. result Discovers relationships between new and existing homogeneous space types.
Classifies compact spaces by shape, finite spaces by weak homotopy.
problem Classifying compact Hausdorff spaces and finite topological spaces.
method Constructs a category that classifies spaces by shape and weak homotopy.
result Classifies compact spaces by shape, finite spaces by weak homotopy.
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.