Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
New Upsilon invariants rule out stable equivalence of knot complexes.
problem Stable equivalence of knot complexes and its invariants.
method Secondary Upsilon invariants defined by Kim and Livingston.
result Relations between Upsilon invariants do not extend to stable equivalence.
We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and l…
Bridge positions of handlebody-knots are equivalent when stable.
problem Equivalence of bridge positions in handlebody-knots.
method Demonstrated stability of bridge positions.
result Bridge positions of handlebody-knots are stably equivalent.
New homotopy theory reveals the structure of stable curves.
problem Understanding the structure of the moduli stack of stable curves.
method Using stratified homotopy theory, the category of stable curves captures the stratified homotopy type of the moduli stack.
result The category of stable curves classifies constructible sheaves via an exodromy equivalence.
Knot concordance linked to involutive knot Floer homology equivalence.
problem Understanding knot concordance through algebraic topology.
method Involutive knot Floer homology and stable equivalence.
result Concordant knots have equivalent involutive knot Floer complexes.
New infinite family of 4-manifolds with same stable properties but not homotopy equivalent.
problem Finding infinite homotopy stable classes of 4-manifolds with boundary.
method Construction of an infinite family of topological 4-manifolds with specific properties.
result Infinite family of 4-manifolds that are stably homeomorphic but not homotopy equivalent.
Let K be a knot embedded in a Heegaard surface S for a closed orientable 3-manifold M. We define K-stable equivalence between pairs (S, K) and (S', K) in M, and we prove that any two pairs are K-stably equivalent in M if they have the same surface slope.
If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth embeddings are connected by such a deformation, they will be called cusp equivalent. W…
We investigate link homology theories for stable equivalence classes of link diagrams on orientable surfaces. We apply (1+1)-dimensional unoriented topological quantum field theories to Bar-Natan's geometric formalism to define new theories for stable equivalence classes.
Paper discusses when virtual links are equivalent as twisted links.
problem Determining equivalence of virtual links as twisted links.
method Using stable equivalence classes of links in oriented thickenings of surfaces.
result Necessary and sufficient condition for virtual links to be equivalent as twisted links.
New quasimorphisms show stable commutator lengths are not equivalent.
problem Equivalence of stable commutator lengths in groups.
method Invariant quasimorphisms for groups acting on the circle.
result Stable commutator lengths are not bi-Lipschitzly equivalent.
The Poincare-Hopf theorem tells us that given a smooth, structurally stable vector field on a surface of genus g, the number of saddles is 2-2g less than the number of sinks and sources. We generalize this result by introducing a more complex combinatorial invariant. Using this tool, we demonstrate that many such struc…
The paper verifies stable handleslide triviality of some R-links and shows many are stably equivalent.
problem Stable handleslide triviality of R-links as potential counterexamples to the generalized property R conjecture.
method Implemented an algorithm to construct all R-links explicitly and verified their stable handleslide triviality.
result Many R-links are stably handleslide equivalent.
Paper proves every stable 4-sphere has a unique diffeomorphism class.
problem Identifying stable 4-spheres and their diffeomorphisms.
method Using Wall's result and properties of surface-knot spaces.
result Every stable 4-sphere has a unique orientation-preserving diffeomorphism class.
The paper defines and proves stabilization for 3-manifold decompositions with multibranched surface intersections.
problem Decomposing 3-manifolds with more than 3 handlebodies and multibranched surface intersections.
method Definition and proof of stabilization operations for these decompositions.
result Stable equivalence of handlebody decompositions with multibranched surface intersections.
New results affirmatively answer the stable converse soul question for many curved spaces.
problem Determining if vector bundles over curved spaces admit metrics with non-negative curvature.
method Topological K-theory and homotopy equivalence.
result The stable converse soul question has an affirmative answer for many curved spaces, except possibly for one specific space.
Proves stable version of Cannon Conjecture for hyperbolic groups.
problem Stability of Cannon Conjecture for hyperbolic groups and manifolds.
method Simple homotopy equivalence and unique up to homeomorphism.
result Existence of a closed manifold M from cartesian product of N and BG.
Minimal surfaces' area bounds proven equivalent, extending known results.
problem Equivalence of area bounds for minimal surfaces.
method Combining recent breakthroughs, extending known results.
result Equivalence of intrinsic and extrinsic area density bounds for minimal immersions.
Undecidability proved for DG algebras problems.
problem Stable isomorphism, quasi-isomorphism, and Morita equivalence problems for semifree DG algebras.
method Essentially autonomous solutions by Gemini Deep Think and Aletheia.
result Proved undecidability of problems for semifree DG algebras.
In view of the Segal construction each category with a coherent operation gives rise to a cohomology theory. Similarly each open stable differential relation R imposed on smooth maps of manifolds determines cohomology theories k∗ and h∗; the cohomology theory k∗ describes invariants of solutions of R, whil…
This paper characterizes stable polynomial mappings in a specific set.
problem Characterizing stable polynomial mappings in a given set.
method Analyzing polynomial mappings with specific degrees and determining topological equivalence.
result Effective determination of mappings with generic topology.
Minimal crossing virtual links have minimal supporting genus.
problem Understanding the relationship between the number of crossings and the genus of virtual links.
method Developed a new parity theory for virtual links to prove minimal crossing implies minimal genus.
result Minimal crossing virtual links have minimal supporting genus.
Real vector bundles are determined by their Dirac indices on specific spin manifolds.
problem Determining real vector bundles using Dirac indices.
method Mapping spin or spinh manifolds into a compact smooth manifold and using Dirac indices. result Real vector bundles are uniquely determined by their Dirac indices on prescribed manifolds.
Proves equivalence of two types of representations of free groups in hyperbolic spaces.
problem Equivalence of two types of representations of free groups in hyperbolic spaces.
method Independent proof of equivalence for free groups of rank two in Gromov-hyperbolic spaces.
result Set of Bowditch representations equals set of primitive-stable representations.
Generalizes ribbonness result for surface-links.
problem Characterizing ribbon surface-links.
method Analyzes handle-irreducible summands and uses equivalences.
result Every stable-ribbon surface-link is a ribbon surface-link.
This paper shows that certain knots are equivalent under a specific relation.
problem Understanding the equivalence classes of genus one knots.
method Using knot Floer complexes and Heegaard Floer theory invariants.
result Any genus one knot is ν+-equivalent to a trefoil, its mirror, or the unknot. Stable solutions to a specific equation are one-dimensional.
problem Stability and dimensionality of solutions to the Allen-Cahn equation.
method Analysis of stable solutions with bounded energy density.
result Stable solutions to the Allen-Cahn equation are one-dimensional.
In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.
Study on stable vector bundles over Gauduchon manifolds.
problem Existence and stability of vector bundles over Gauduchon manifolds.
method Uhlenbeck--Yau's continuity method for approximate Hermitian--Einstein structures.
result Equivalence of semi-stability and existence of Hermitian--Einstein structures.
For a germ of a smooth map f and a subgroup G_V of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form V in the source or in the target we study the G_V-moduli space of f that parameterizes the G_V-orbits inside the G-orbit of f. We find, for example, that this modu…
We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tange…
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
Paper proves contractible fake surfaces up to complexity 6 are deformable.
problem Stable Andrews-Curtis conjecture and contractible fake surfaces.
method Induction scheme proving contractibility up to complexity 6.
result Contractible fake surfaces up to complexity 6 are 3-deformable.
Investigates stock models using tempered stable processes for option pricing.
problem Analyzing option pricing in stock models driven by tempered stable processes.
method Investigates exponential stock models driven by tempered stable processes, providing existence of equivalent martingale measures and pricing formulae.
result Existence of equivalent martingale measures and pricing formulae for European call options.
The note proves a metric equivalence for stable bundles on surfaces.
problem Understanding stability conditions and metrics on complex projective surfaces.
method Analyzing stability in the large scaling limit and proving equivalence with deformed Hermitian-Yang-Mills metrics.
result Equivalence of stability and deformed Hermitian-Yang-Mills metrics for smooth projective surfaces.
We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other well-known low-dimensional singularities. We also give an application to stable p…
Tillmann introduced two infinite loop space structures on the plus construction of the classifying space of the stable mapping class group, each with different computational advantages. The first one uses disjoint union on a suitable cobordism category, whereas the second uses an operad which extends the pair of pants …
Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…
A calculus modifies flow categories without changing their homotopy type.
problem Modifying flow categories without altering their homotopy type.
method A calculus of moves to modify framed flow categories.
result Two flow categories with stable homotopy type give move equivalent categories.
We compute the Picard group of a stable b-symplectic manifold M by introducing a collection of discrete invariants Gr which classify M up to Morita equivalence.
Ribbon knots with isomorphic quandles are stably equivalent.
problem Determining when ribbon knots with isomorphic quandles are equivalent.
method Using quandle isomorphism to show stable equivalence after finite connected sums.
result Ribbon knottings with isomorphic quandles are stably equivalent.
Paper proves non-equivalence of RKHS stability and kernel absolute summability.
problem Equivalence of RKHS stability and kernel absolute summability.
method Analyzes Reproducing Kernel Hilbert spaces and positive semidefinite kernels.
result Stable RKHSs can be induced by non-absolutely summable kernels.
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…
Study stabilizes components of Galois cover moduli spaces.
problem Decide equivalence and stable equivalence of monodromy maps.
method Develop algebraic framework to study equivalence classes of monodromy maps.
result Recover a homological invariant that distinguishes equivalence classes.
Stable and Morse subgroups coincide in mapping class groups.
problem Understanding subgroup properties in mapping class groups.
method Analyzing stability and Morse properties in mapping class groups.
result Stability and Morse properties coincide for subgroups of infinite index in mapping class groups.
Decomposes J-energy into simpler intersection numbers for stability analysis.
problem Analyzing J-stability in algebraic geometry.
method Proves a decomposition formula for J-energy and shows equivalence of stability conditions.
result Equivalence of J-stability and K-stability for surfaces under pseudoeffective conditions.
This paper introduces new Lagrangian branes in stable generalized complex manifolds.
problem Understanding stable generalized complex manifolds and their properties.
method Using log symplectic geometry and Floer theory techniques.
result Lagrangian branes with boundary are introduced and their properties are studied.