New topological proof for a theorem about orbits in a specific group.
problem Characterizing orbits of dense subgroups in a specific group.
method Topological approach, using dynamics of unipotent flows.
result Orbits are either finite or dense, proving a theorem.
Two short proofs show intersection homology's stability.
problem Topological invariance of intersection homology.
method Two proofs: one short requiring support and cosupport axioms, the other longer adaptable to more perversities.
result Intersection homology's stability proven without axioms.
Simplified proof of a famous geometry result for students.
problem Mostow Rigidity in geometry and topology
method Self-contained, accessible proof for grad students
result Clean and analytically light proof accessible to students
Geometric proof shows topological invariance of handle homology.
problem Topological invariance of handle homology in manifolds.
method Entirely geometric proof using Cerf theory.
result Proof of ∂2=0 in chain complex defined by handle decomposition. Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
Simplified proof of Honda-Huang's contact convexity result.
problem Contact convexity in high dimensions
method Simplified proof of Honda-Huang's main result
result Main result from Honda-Huang's paper simplified and presented
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.
Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.
problem Proving Wolpert's formula for the Weil-Petersson symplectic form.
method Introducing a cell decomposition and groupoid cocycle on a surface to represent points in Teichmüller space.
result Topological proof of Wolpert's formula for the Weil-Petersson symplectic form.
New proof of index theorem for topological manifold bundles.
problem Index theorem for fiber bundles of compact topological manifolds.
method Use of a convenient framework for bivariant theories and recent results on the homotopy type of the topological cobordism category.
result Refinement of the assembly map for an extended A-theory characteristic.
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 …
Proof of genus formula for 3-manifolds using arithmetic topology.
problem Proving a genus formula for finite abelian branched covers over integral homology 3-spheres.
method Utilizing analogies of arithmetic topology and Hasse norm principle.
result Presentation of an Iyanaga--Tamagawa type genus formula.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
Hanner's theorem is a classical theorem in the theory of retracts and extensors in topological spaces, which states that a local ANE is an ANE. While Hanner's original proof of the theorem is quite simple for separable spaces, it is rather involved for the general case. We provide a proof which is not only short, but a…
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…
In this paper we shall give an analytic proof of the fact that the Liouville energy on a topological two sphere is bounded from below. Our proof does not rely on the uniformization theorem and the Onofri inequality, thus it is essentially needed in the alternative proof of the uniformization theorem via the Calabi flow…
The abstract discusses topological censorship with five hypotheses.
problem Proving topological censorship in spacetimes.
method Presented five hypotheses to prove a theorem.
result Proof of a topological censorship theorem.
New proof of Milnor-Wood inequality for circle bundles.
problem Proving the Milnor-Wood inequality for circle bundles.
method Using a local formula to compute the Euler class from the singularities of a quasisection, and sketching two other proofs.
result A new proof of the Milnor-Wood inequality.
Proves a special knot type is slice.
problem Characterizing slice knots.
method Proof by contradiction and algebraic topology.
result 0-shake slice knots are indeed slice.
Proofs a theorem using basic geometric tools.
problem C2-rectifiability problem
method Elementary geometric measure theory and topology
result Gives a proof of Alberti's Luzin-type theorem
Study surfaces with specific topologies, focusing on extremal properties.
problem Characterize surfaces with given topological types under extremal conditions.
method Analyzes surfaces using topological and geometric methods, focusing on extremal properties.
result Developed new proofs for extremal problems, emphasizing intuitive understanding.
Study distance functions on manifolds linking geometry to topology.
problem Understanding the relationship between curvature and topology on manifolds.
method Analyzing distance functions and their connection to manifold geometry.
result Alternative proofs of theorems linking curvature and topology.
We describe an error in the proof of a key proposition, which was necessary for the proof of the main result. Alternate proofs of the main result are given by Ozsvath-Stipsicz-Szabo and Dai-Hom-Stoffregen-Truong.
Modified proof constructs dual spheres for 4-manifolds.
problem Prove dual spheres for 4-manifolds.
method Modified proof of disc embedding theorem, geometric construction.
result Constructs geometrically dual spheres.
Mazur, Kapranov, Reznikov, and others developed ``Arithmetic Topology,'' a theory describing some surprising analogies between 3-dimensional topology and number theory, which can be summarized by saying that knots are like prime numbers. We extend their work by proving several formulas concerning branched coverings of …
We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Sch{ö}nflies theorem. As an application, we show that one can read off oriented matroids from arrangement…
The study of tiling homology on flat surfaces, proving impossibility of certain tilings.
problem Proving the non-existence of polyomino tilings on specific square-tiled surfaces.
method Study of homology groups for topological tilings, using coloring proofs.
result Several results about the non-existence of polyomino tilings on certain square-tiled surfaces.
New proof shows special surfaces have finite type.
problem Characterizing surfaces with finite topological type.
method Shorter proof of Huber's theorem.
result Special surfaces have finite topological type.
Simple condition proves when 2-bridge knots are quasipositive.
problem Characterizing quasipositive two-bridge knots.
method Continued fraction expansion of p/q to prove a necessary and sufficient condition.
result Simple condition proves when 2-bridge knots are quasipositive.
A key result in four dimensional black hole physics, since the early 1970s, is Hawking's topology theorem asserting that the cross-sections of an "apparent horizon", separating the black hole region from the rest of the spacetime, are topologically two-spheres. Later, during the 1990s, by applying a variant of Hawking'…
Two proofs show that removing a loop from a plane circuit splits the plane.
problem Proving the Weak Jordan Theorem about plane circuits.
method Detailed presentation of Thomassen's and Filippov's proofs.
result The complement of any loop in a plane circuit is disconnected.
The controlled end and h-cobrodism theorems (Ends of maps I, 1979) are used to give quick proofs of the Top/PL and PL/DIFF product structure theorems.
New proofs of h-principles in contact 3-manifolds.
problem Characterizing contractible bypasses in contact 3-manifolds.
method Topological characterization and disjoint bypass construction.
result New proofs of h-principles in overtwisted contact 3-manifolds.
AI models help solve a symplectic topology problem.
problem Lagrangian smoothability question from Abouzaid et al.
method Used large language models (LLM) to approach the problem.
result Suggested new constructions in polyhedral symplectic topology.
Proof of K(π,1) conjecture for spherical Artin groups.
problem Proving the K(π,1) conjecture for Artin groups of spherical type. method Combining combinatorial topology with methods from the original proof of the spherical case.
result Proof of the K(π,1) conjecture for Artin groups of spherical type. The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part w…
We provide an alternative, simpler proof of the existence of thick triangulations for noncompact C1 manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also…
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the Melvin-Morton-Rozansky conjecture, and a connection between entropy of braids and quantum repr…
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.
Simplified proof classifies surfaces using normal curves.
problem Classifying surfaces using algebraic topology invariants.
method Using normal curves and analogy to 3-manifolds classification.
result Simple proof without using algebraic topology invariants.
Proof that SU(2) character variety of genus 2 surface is CP3.
problem Character variety structure of genus 2 surface.
method Differential topology, algebraic topology, SU(2) representations. result Character variety is homeomorphic to CP3. The paper corrects and improves previous assertions in digital topology.
problem Incorrect or poorly proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
Any 2-dim Riemannian manifold with spherical topology can be embedded isometrically into a lightcone of the Minkowski spacetime. We apply this fact to give a proof of the Kazdan-Warner identity.
Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.
problem Normal forms for Lagrangian submanifolds in multisymplectic geometry.
method Detailed, self-contained proof of normal form theorem, including necessary results in foliated differential topology.
result Geoffrey Martin's theorem provides a normal form for Lagrangian submanifolds in multisymplectic geometry.
Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.
problem The Goeritz group of a genus g Heegaard splitting of the 3-sphere.
method Proof relies on the topological minimality of Heegaard surfaces and their disk complexes.
result The Goeritz group is generated by four specific elements for g ≥ 3.
Stable actions of hyperbolic groups on their boundaries.
problem Stability of group actions on boundaries.
method Dynamical coding and semi-conjugacy analysis.
result Topological stability of actions on hyperbolic group boundaries.
By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The present paper is devoted to a new approach to the results of W. Vautaw about Dehn multi-twists in Torelli groups and abelian subgroups of Torelli groups. The new proofs are more tran…
The paper defines a stratification for Lie groupoids in a tame topology context.
problem Presenting a tame topology counterpart to canonical stratification of Lie groupoids.
method Using Shiota's isotopy lemma and approximation theorem, the paper defines a canonical Whitney stratification of definable Lie groupoids into invariant strata.
result A canonical Whitney stratification of the Lie groupoid into definable strata invariant under the groupoid action.