New smoothing techniques for topological surfaces in 4-manifolds.
problem Topological isotopy of surfaces in smooth 4-manifolds.
method Combining Quinn's smoothing theory with Gabai's light bulb theorem and other developments.
result Proves topological = smooth results for certain disks and spheres.
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
Kontsevich's classes distinguish smooth structures on fiber bundles.
problem Distinguishing smooth structures on fiber bundles.
method Using Kontsevich's characteristic classes and real blow-up construction.
result Kontsevich's classes are determined by the topology of the 2-point configuration space bundle.
Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.
problem Understanding smooth fibrations of compact Spin(7)-manifolds by Cayley submanifolds.
method Geometric and topological constraints from Spin(7)-structure, spinnability criterion, gauge-theoretic input.
result Rules out smooth Cayley fibrations on all known compact torsion-free Spin(7)-manifolds.
We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…
We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot 12a255. These also provide the first known examples of alternating knots for which the smooth and topological genera differ.
Computes mapping class groups of 4-manifolds with boundary.
problem Computing mapping class groups for 4-manifolds with boundary.
method Topological and smooth methods applied to compact, simply connected 4-manifolds.
result Description of topological and stable smooth mapping class groups.
The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
problem The minimality of compact-open topology on diffeomorphism and homeomorphism groups.
method Analyzing the compact-open topology on diffeomorphism and homeomorphism groups of smooth manifolds.
result The compact-open topology is minimal on diffeomorphism and homeomorphism groups of most smooth manifolds.
Proves every equivariant vector bundle over toric manifolds is a Klyachko bundle.
problem Characterizing equivariant vector bundles over toric manifolds.
method Analyzes topological and smooth equivariant vector bundles over toric manifolds.
result Every equivariant vector bundle is a Klyachko bundle.
Entropy data replaces classical charts for smooth manifolds.
problem Establishing smooth structures on topological manifolds.
method Using entropy data to define admissible coordinate functions and reconstruct smooth atlases.
result Entropy-smooth structures are equivalent to classical smooth structures and stable under perturbations.
Algebras of smooth functions help reconstruct bulk topological types.
problem Reconstructing the smooth topological type of a compact manifold from its boundary.
method Introducing subalgebras of boundary functions and proving their tensor product reconstruction of the bulk algebra.
result The topological algebras A(v) and B(f) allow for the recovery of the smooth topological type of the bulk X. Construct divide knots with specific genus properties.
problem Understanding the difference between smooth and topological four-genus for knots.
method Construct divide knots with controlled smooth and topological four-genus ratios.
result For strongly quasipositive fibred knots, the ratio between smooth and topological four-genus can be made arbitrarily close to zero.
The paper extends properties of smooth functions to closed sets and maps.
problem Properties of smooth functions on closed sets and maps.
method Extending properties of smooth functions to closed sets and maps, proving isomorphisms with natural topologies.
result Bornological isomorphisms of function spaces are established.
Improved proofs for topological and smooth pseudo-isotopies of simply connected 4-manifolds.
problem Proving topological and smooth pseudo-isotopies of simply connected 4-manifolds.
method Provided different arguments that bypass the replacement criterion, thus completing Quinn's proofs.
result Corrected and completed Quinn's proofs of both topological and stable smooth pseudo-isotopy theorems.
This paper constructs real algebraic maps that are topologically special generic maps.
problem Constructing smooth maps in differential topology and real algebraic geometry.
method Constructs real algebraic maps that are topologically special generic maps.
result Real algebraic maps are topologically special generic maps.
Study on constraints for topological and smooth realizations of line arrangements and configurations.
problem Investigating constraints on topological and smooth realizations of combinatorial line arrangements and (nk)-configurations. method Exploring constraints via locally-flatly or smoothly embedded 2-spheres, using Furuta's 10/8-Theorem, and G-signature theorem.
result Established a new lower bound for (nk)-configurations, showing n≥k2−5 for topological realizations. This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology …
We establish a number of results about smooth and topological concordance of knots in S1×S2. The winding number of a knot in S1×S2 is defined to be its class in H1(S1×S2;Z)≅Z. We show that there is a unique smooth concordance class of knots with winding number one. …
Topologies on algebraic and equational theories are used to define germ determined, near-point determined, and point determined rings of smooth functions, without requiring them to be finitely generated. It is proved, that any commutative algebra morphism (without requiring continuity) between near-point determined rin…
Generic smooth plane-to-plane map germs are topologically equivalent to cones of mappings of the circle. We carry out a complete topological classification of smooth stable mappings of the circle and show how this classification leads, via the result mentioned above, to a topological classification of finitely determin…
Paper proves Whitney stratified spaces can be given a conically smooth structure.
problem Proving Whitney stratified spaces can be given a conically smooth structure.
method Introduced conically smooth structure by Ayala, Francis, and Tanaka. Proved conjecture that any Whitney stratified space admits a canonical conically smooth structure.
result Established a connection between Whitney stratified spaces and conically smooth spaces.
Extends calculus to topological manifolds using generalized functions.
problem Proving the existence of non-singular generalized tangent vector fields on spheres.
method Develops a theory of generalized functions and applies it to continuous maps between topological spaces.
result Shows coherence between non-existence of smooth vector fields on spheres and existence of generalized ones.
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
Study bounds topological entropy of toroidal attractors.
problem Bounding entropy of toroidal attractors.
method Analyzing topological properties of toroidal sets to bound entropy.
result Entropy of toroidal attractors is bounded from below.
Graph Neural Networks (GNNs) have achieved promising performance on a wide range of graph-based tasks. Despite their success, one severe limitation of GNNs is the over-smoothing issue (indistinguishable representations of nodes in different classes). In this work, we present a systematic and quantitative study on the o…
Two types of 4-manifolds disagree on minimum Euler characteristic.
problem Disagreement between topological and smooth 4-manifolds.
method Comparison of topological and smooth 4-manifolds with a fixed fundamental group.
result Minimum Euler characteristic differs between topological and smooth 4-manifolds.
Example shows smooth vs topological isotopy in a 4-manifold.
problem Smooth vs topological isotopy in a specific 4-manifold.
method Constructing specific embeddings and isotopies.
result Found a smooth embedding not isotopic to a topological one.
We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists …
Study shows infinite kernels in topological monodromy for curve families.
problem Understanding kernels of topological monodromy representations.
method Extending Kuno's arguments and using Carlson-Toledo techniques.
result Kernels are infinite for certain linear systems on surfaces.
New knots found with tough, unsliceable discs.
problem Finding tough knots that can't be sliced smoothly.
method Constructed infinitely many knots with non-approximable slice discs.
result Smoothly sliceable knots have non-approximable slice discs.
Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the D-topology. However, the D-topology has not yet been studied seriously in the existing literature. In this paper, we…
Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
Extends Gelfand duality to various geometric and analytical categories.
problem Generalizing Gelfand duality to different types of manifolds and bundles.
method Unified cohomological argument for manifolds and suitable classes of functions for bundles.
result Gelfand duality extended to real analytic and Stein manifolds, and to vector, affine, and jet bundles.
New examples show not all homology fiber bundles are topological.
problem Disprove conjecture about homology fiber bundles.
method Construct flat, projective morphisms that are Z-homology fiber bundles. result Disprove conjecture about homology fiber bundles.
Study on topological rigidity of ALE vector bundles with specific conditions.
problem Classifying ALE vector bundles with asymptotically conical total spaces.
method Topological classification and geometric analysis of ALE vector bundles.
result Only 2-sphere, projective plane, and open contractible manifolds admit ALE tangent bundles.
Global and local blowups of manifolds are proven equivalent.
problem Equivalence of global and local blowups in differential topology.
method Proof of equivalence between global and local constructions of blowups.
result Global and local constructions of blowups are shown to be equivalent.
The Mazur pattern acts by the identity up to topological concordance.
problem Understanding the action of the Mazur pattern up to topological concordance.
method Comparing satellite operators and using topological concordance properties.
result Evidence that the Mazur pattern acts by the identity up to topological concordance.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in S3. result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
We prove that smooth 1-dimensional topological field theories over a manifold are equivalent to vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism category, which encodes cutting laws rather than gluing laws. We make this idea precise through a smooth version of Rezk'…
We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite simple subgroups of diffeomorphism groups of compact smooth topological manifolds.
The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite sub…
In this article we prove that any unitary, axiomatic topological quantum field theory in four-dimensions can not detect changes in the smooth structure of M, a simply connected, closed (compact without boundary), oriented smooth manifold. However, as Donaldson-Witten theory (a topological quantum field theory but not a…
A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of S2xS2x[0,1]'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-di…
New topology defined from spacetime paths, reconstructing spacetime structure.
problem Reconstructing spacetime structure from path homotopy classes.
method Defining a topology on spacetime based on timelike and causal homotopy classes.
result The topology on spacetime is reconstructed from the space of homotopy classes.
New Einstein metrics found on manifolds with opposite curvature signs.
problem Finding Einstein metrics with opposite curvature signs on manifolds.
method Reviewing and extending previous work on high-dimensional smooth closed manifolds.
result Proved various related results, including new Einstein metrics.
This paper contains the results of efforts to determine values of the smooth and the topological slice genus of 11- and 12-crossing knots. Upper bounds for these genera were produced by using a computer to search for genus one concordances between knots. For the topological slice genus further upper bounds were produce…
We review the basic definition of a stack and apply it to the topological and smooth settings. We then address two subtleties of the theory: the correct definition of a ``stack over a stack'' and the distinction between small stacks (which are algebraic objects) and large stacks (which are generalized spaces).