Simplified proof of Cerf's theorem on 3-sphere diffeomorphisms.
problem Proving the connectedness of direct diffeomorphisms of the 3-sphere.
method Rigidity property of foliations defined by non-vanishing closed one-forms.
result Connected group of direct diffeomorphisms of the 3-sphere.
Following a line of reasoning suggested by Eliashberg, we prove Cerf's theorem that any diffeomorphism of the 3-sphere extends over the 4-ball. To this end we develop a moduli-theoretic version of Eliashberg's filling-with-holomorphic-discs method.
Study connects Morse theory with cluster variables for wall-crossing in Cerf diagrams.
problem Understanding wall-crossing in Cerf theory.
method Relates Bruhat numbers in real Morse theory to cluster variables in braid varieties.
result Provides wall-crossing coordinates in Cerf diagrams.
Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to canc…
New proof of Laudenbach and Poénaru's theorem on 4D 1-handlebodies.
problem Proving diffeomorphisms extend to 4D 1-handlebodies and compression bodies.
method Classification of Heegaard splittings and Cerf's theorem on 3-ball diffeomorphisms.
result Every diffeomorphism of the positive boundary extends to the whole compression body.
New invariants for 3-manifolds derived from equivariant Cerf theory.
problem Existence of perturbative SU(n) Casson invariants on integer homology spheres. method Equivariant Cerf theory for Morse functions, adapted to infinite-dimensional setting.
result Existence and explicit formula for SU(4) Casson invariants. In [Topology 35 (1996) 1005--1023] J H Rubinstein and M Scharlemann, using Cerf Theory, developed tools for comparing Heegaard splittings of irreducible, non-Haken manifolds. As a corollary of their work they obtained a new proof of Waldhausen's uniqueness of Heegaard splittings of S^3. In this note we use Cerf Theory …
This paper extends results of Hatcher and Vogtmann's work "Cerf Theory for Graphs" to ribbon graphs. Given an orientable, punctured and basepointed surface Sigma, we prove that the space of ribbon graphs that can be drawn in Sigma is filtered by simplicial complexes. The k-th simplicial complex is (k-1)-dimensional, (k…
Constructs pseudo-isotopies with Cerf diagrams and Hatcher-Wagoner invariants for barbell maps.
problem Understanding and constructing pseudo-isotopies for barbell maps.
method Constructs specific pseudo-isotopies with Cerf diagrams and Hatcher-Wagoner invariants.
result Every pseudo-isotopy with vanishing first Hatcher-Wagoner invariant can be isotoped to a composition of standard barbell pseudo-isotopies.
The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.
problem Understanding the homotopy types of diffeomorphism groups and sphere embeddings.
method Cerf's upgraded proof, scanning maps, canceling handles, Embedding Calculus.
result The monoid of Schoenflies spheres forms a group under connect-sum.
Extends Palais' result on diffeomorphisms to homeomorphisms and bi-Lipschitz mappings.
problem Extending diffeomorphisms to global mappings of manifolds.
method Elementary argument for diffeomorphisms, deep results for homeomorphisms and bi-Lipschitz mappings.
result Extension of Palais' result to homeomorphisms and bi-Lipschitz mappings.
New method for analyzing multiparameter persistence modules from smooth functions.
problem Analyzing multiparameter persistence modules from smooth functions.
method Generalized Morse theory applied to cobordism and Cerf theory.
result Complete description of persistence modules as direct sums of indecomposables.
New findings about twists in 4-sphere diffeomorphisms.
problem Understanding isotopy classes of diffeomorphisms in 4-sphere.
method Using Cerf theory and twists along Montesinos twins.
result The subgroup of twists along Montesinos twins is trivial or cyclic of order two.
We give an entirely geometric proof, without recourse to cellular homology, of the fact that ∂2=0 in the chain complex defined by a handle decomposition of a given manifold. Topological invariance of the resulting `handle homology' is a consequence of Cerf theory.
The paper explores how topological methods can reveal insights into electric charge distributions on knots.
problem Understanding the qualitative behavior of electric potentials on knots.
method Geometric topology techniques applied to electrostatics.
result Proved a lower bound on the size of the critical set based on knot projections.
The kernel embedding algorithm is an important component for adapting kernel methods to large datasets. Since the algorithm consumes a major computation cost in the testing phase, we propose a novel teacher-learner framework of learning computation-efficient kernel embeddings from specific data. In the framework, the h…
Link concordance equals homotopy for high-dimensional spheres.
problem Understanding when immersions of high-dimensional spheres are homotopically trivial.
method Developed stratified Morse theory for generic immersions, using gradient-like vector fields and Cerf theory.
result Every link of high-dimensional spheres is homotopically trivial, resolving a long-standing conjecture.
We consider an oriented surface S and a cellular complex X of curves on S, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of S, following the …
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…
Let X be a closed 3-manifold, MR>0 the space of metrics on X with positive scalar curvature, and Diff(X) the group of diffeomorphisms of X. Marques proves the fundamental result that MR>0/Diff(X) is path connected. Using this and the theorem of Cerf in differential…
The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.
problem Understanding knot types in bounded ropelength sublevel spaces.
method Study thick representatives in bounded ropelength sublevel spaces through lifted Reidemeister graphs.
result Define characteristic Reidemeister patterns and finite recognition length.
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
For a riemannian foliation F on a closed manifold M, it is known that F is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form κμ (relatively to a suitable riemannian metric μ) is zero. In the transversally orientable case…
Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.
problem Understanding the smooth mapping class group of the 4-sphere.
method Homomorphism from loops of 2-spheres to smooth mapping class group, generators as twists along Montesinos twins.
result Generators of the image of the homomorphism as diffeomorphisms of Montesinos twins.
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.
We develop a technique for gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4-manifolds. As an application, we describe a trisection of any closed 4-manifold which admits a Lefschetz fibration over S2 equipped with a section of square …
Around 1960, R. Palais and J. Cerf proved a fundamental result relating spaces of diffeomorphisms and imbeddings of manifolds: If V is a submanifold of M, then the map from Diff(M) to Imb(V,M) that takes f to its restriction to V is locally trivial. We extend this and related results into the context of fibered manifol…
We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is val…
We present an elementary derivation of the "intrinsic" symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group $\MCG(S^3,L)$ or $\Sym(L)$ of the pair (S3,L). Elements in this symmetry group can (and often do) fix the link and act nontrivially onl…
The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M…
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
Analyzes Saito vanishing theorem using L2 methods.
problem Proving the Saito vanishing theorem.
method Uses L2-methods to prove the theorem. result Analytic proof of the Saito vanishing theorem.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop gr…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
Proves two theorems on odd-dimensional manifolds with boundary.
problem Proving theorems on manifolds with boundaries.
method Proof of theorems using mathematical techniques.
result Proved the general Kastler-Kalau-Walze and Dabrowski-Sitarz-Zalecki type theorems.
Sharp convergence theorem for sphere submanifolds proved.
problem Sphere submanifolds in spheres.
method Proved a sharp convergence theorem.
result New differentiable sphere theorem for submanifolds in spheres.