Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
Study shows simplicial volumes add for certain manifold gluings.
problem Understanding simplicial volumes in manifold gluings.
method Proves additivity of locally finite simplicial volume and Lipschitz simplicial volume for connected sums and specific boundary components.
result Additivity of simplicial volumes in dimension 3 and above for certain gluings.
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
We provide an explicit section for a mapping class group sequence.
problem Mapping class group of connect sums of S2imesS1. method Provided an explicit section for the split exact sequence.
result Explicit section s for the exact sequence. Spatial graphs are decomposed into planar forests and braids.
problem Understanding the structure of spatial graphs in 3-space.
method Decomposition of spatial graphs into planar forests and braids.
result Every finite spatial graph is a connected sum of a planar graph and a braid.
Proves a formula for a special invariant of 4-manifolds.
problem Calculating the Bauer-Furuta invariant for connected sums of 4-manifolds.
method Uses a finite dimensional approximation of the Seiberg-Witten monopole map to derive a formula for the families Bauer-Furuta invariant of a fibrewise connected sum.
result Derives a general connected sum formula for the families Bauer-Furuta invariant.
Study open 3-manifolds as sums of closed ones, finding a classification.
problem Classifying open 3-manifolds that are sums of closed 3-manifolds.
method Introduced topological invariants, classified when finitely many summands up to diffeomorphism.
result Unified classification of open 3-manifolds and closed 3-manifolds.
We analyze the obstruction to metrics of positive scalar curvature within a given bounded distortion class of metrics. This obstruction lives in a non-Hausdorff cohomology group Poincare dual to the uniformly finite homology studied by Block and Weinberger. One of the applications is a converse to their theorem on infi…
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.
For most positive integer pairs (a,b), the topological space $#a{\mathbb C \mathbb P}^2#b{\bar{\mathbb C \mathbb P^2}}$ is shown to admit infinitely many inequivalent smooth structures which dissolve upon performing a single connected sum with S2×S2. This is then used to construct infinitely many non-equiva…
Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but that…
Study shows non-cyclic groups of diffeomorphisms can't act on certain 3-manifolds.
problem Realization of finite groups as diffeomorphisms on specific 3-manifolds.
method Analysis of group actions on connected sums of S2imesS1. result No non-cyclic subgroup of twist subgroup can be realized by diffeomorphisms.
Gluing instantons on 4-manifolds with group action.
problem Gluing invariant ASD connections on 4-manifolds with group action.
method Equivariant gluing theory for instanton moduli spaces.
result Construction of Γ-invariant ASD connections on connected sums. Novel analysis of EFP for finite-sum problems in neural networks.
problem Optimization of two-layer neural networks in the mean-field regime.
method Primal-dual analysis of entropic fictitious play (EFP) for finite-sum problems.
result Established global convergence guarantees for EFP dynamics.
Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.
problem Restricting Morse indices of saddles in gradient-like flows.
method Analyzing invariant manifolds and their intersections for gradient-like flows.
result Morse indices of saddles are either 1 or n-1, no other indices possible.
We give a short proof that if a non-trivial band sum of two knots results in a tight fibered knot, then the band sum is a connected sum. In particular, this means that any prime knot obtained by a non-trivial band sum is not tight fibered. Since a positive L-space knot is tight fibered, a non-trivial band sum never yie…
Proves properties of instanton knot Floer homology and connected sum formula.
problem Properties of instanton knot Floer homology.
method General properties of sutured Floer theories, Heegaard Floer and monopole Floer settings.
result Connected sum formula for instanton knot Floer homology.
Classifies 3D spaces with circle actions, showing they are sums of manifolds and projective planes.
problem Classifying 3D Alexandrov spaces with circle actions.
method Topological and equivariant classification.
result Spaces are homeomorphic to sums of manifolds and finitely many suspensions of the real projective plane.
The paper extends a connected-sum inequality to calculate λ-Yamabe invariants of certain manifolds.
problem Calculating λ-Yamabe invariants for specific compact manifolds with boundary.
method Generalizing Kobayashi's connected-sum inequality and applying it to specific manifolds.
result The paper proves that certain manifolds have the same λ-Yamabe invariants as the hemi-sphere.
Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the A^-class to obstruct such metrics. In this note…
Examples of smooth maps with finitely many critical points in dimensions (4,3), (8,5) and (16,9)math.GT We consider manifolds M2n which admit smooth maps into a connected sum of S1×Sn with only finitely many critical points, for n∈{2,4,8}, and compute the minimal number of critical points.
Knots connected via a trivial band sum to connected sum.
problem Conditions for band-connected sum to equal connected sum.
method Analyzing knots and bands to determine conditions for equality.
result A band is trivial if and only if a band-connected sum equals a connected sum.
Classifies T2-bundles over closed surfaces up to isomorphisms.
problem Classifying T2-bundles over closed orientable surfaces. method Using group homomorphisms and mapping class groups.
result Any orientable T2-bundle over Σg with g≥1 is isomorphic to the fiber connected sum of g pieces of T2-bundles over T2. On a smooth closed oriented 4-manifold M with a smooth action of a finite group G on a Spinc structure, G-monopole invariant is defined by "counting" G-invariant solutions of Seiberg-Witten equations for any G-invariant Riemannian metric on M. We compute G-monopole invariants on some G-manifolds. F…
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf G-algebras and coloring Kirby diagrams. result Invariants defined for 4-manifolds and connections.
Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S) provided the K-rank of G is 1. This shows that the general upper bound for the fini…
Study cobordism distances between 3-braid links and trefoil knots.
problem Understanding the geometric relationship between 3-braid links and trefoil knots.
method Determined cobordism distances between 3-braid links and trefoil knots, and explored limits of Coxeter's braid group result.
result Found cobordism distances between 3-braid links and trefoil knots, up to a constant error.
The homology of 4-manifold moduli spaces can be infinitely generated.
problem The homology of moduli spaces of 4-manifolds is not finitely generated.
method The proof uses characteristic classes from Seiberg-Witten theory.
result There exist simply-connected closed smooth 4-manifolds whose homology groups are not finitely generated.
We prove that many simply connected symplectic four-manifolds dissolve after connected sum with only one copy of S2×S2. For any finite group G that acts freely on the three-sphere we construct closed smooth four-manifolds with fundamental group G which do not admit metrics of positive scalar curvature, bu…
Connected sum affects crossing numbers of flat virtual knots.
problem Understanding how connected sum impacts the crossing numbers of flat virtual knots.
method Analyzing minimal crossing diagrams and using super-additivity properties.
result Crossing number of flat virtual knots is super-additive under connected sum.
On a smooth closed oriented 4-manifold M with a smooth action by a compact Lie group G, we define a G-monopole class as an element of H2(M;Z) which is the first Chern class of a G-equivariant Spinc structure which has a solution of the Seiberg-Witten equations for any G-invariant Riemannian metri…
The paper solves the Nielsen realization problem for high degree del Pezzo surfaces.
problem Which finite subgroups of the mapping class group of a del Pezzo surface lift to the diffeomorphism group?
method Classification and partial answers for d≥7, equivariant connected sum for d=6. result Complete classification for d≥7, partial answer for d=6. Knots can only be fibered if each component is fibered and their genus is at least the connected sum.
problem Understanding the fibered nature of band-connected sums of knots.
method Analyzing the genus of fibered knots and their connected sums.
result The genus of a fibered knot is greater than or equal to the genus of its connected sum components.
Classifies exceptional Legendrian realizations of Hopf link connected sums.
problem Classifying exceptional Legendrian realizations of Hopf link connected sums.
method Complete coarse classification using Legendrian knot theory.
result First classification result about exceptional Legendrian representatives for Hopf link connected sums.
Connected sums of quasipositive links are quasipositive.
problem Understanding when connected sums of links are quasipositive.
method Filling disk technique
result Connected sums of quasipositive links are quasipositive.
Study knot Floer homology for connected sums, proving a formula and constructing a homomorphism.
problem Understanding knot Floer homology for connected sums of knots.
method Proved a formula for the conjugation action on knot Floer complexes of connected sums, constructed a homomorphism from smooth concordance group.
result Computed involutive invariants on large surgeries of connected sums of knots using the connected sum formula.
Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.
problem Behavior of Morse-Novikov number under knot operations.
method Additivity under connected sum and invariance under cabling.
result Morse-Novikov number is additive under connected sum and unchanged by cabling.
Quantifies the crossing number of knots based on genus and braid index.
problem Estimating the crossing number of knots given their genus and braid index.
method Quantitative Birman-Menasco finiteness theorem applied to crossing numbers.
result Estimates the crossing number of knots in terms of genus and braid index.
Defines a new method for combining curves and describes how certain properties behave.
problem Combining curves and understanding their properties.
method Generalized connected sum for plane curves.
result Describes behavior of Arnold invariants under the generalized connected sum.
New corks found with Stein structures and hyperbolic boundaries.
problem Existence of boundary-sum irreducible finite order corks with Stein structures.
method Proved for any positive integer n, constructed corks with Stein structures and verified hyperbolic boundaries.
result Found boundary-sum irreducible Zn-corks with Stein structures and verified hyperbolic boundaries. Proofs knot homology connected sums using grid complexes.
problem Proving Künneth formula for knot Floer homology of connected sums.
method Constructs a quasi-isomorphism of grid chain complexes.
result Functorial behavior of Legendrian and transverse invariants under connected sum.
New method proves Jones Polynomial's connect sum property.
problem Jones Polynomial's behavior under connect sums.
method Trip matrix method for calculating Jones Polynomial.
result Jones Polynomial is multiplicative under connect sums.
Contact connected sums do not increase support genus.
problem Understanding how support genus changes under contact connected sums.
method Analyzing the support genus of contact connected sums of 3-manifolds.
result The support genus of contact connected sums is at most the maximum of the summands' support genera.
In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot concordance group.
Odd m-fold connected sums of complex projective spaces admit almost complex structures.
problem Existence of almost complex structures on connected sums of complex projective spaces.
method Analyzing the m-fold connected sum m#CP2n for m odd or even. result Only odd m-fold connected sums of complex projective spaces admit almost complex structures.
Proves a general connected sum formula for families Seiberg-Witten invariants.
problem Limited connected sum formulae for families Seiberg-Witten theory.
method Develops a general connected sum formula incorporating previous results.
result Proves a new connected sum formula for Seiberg-Witten families.
In this paper we produce families of complete non compact Riemannian metrics with positive constant σk-curvature by performing the connected sum of a finite number of given n-dimensional Delaunay type solutions, provided 2≤2k<n. The problem is equivalent to solve a second order fully nonlinear elliptic eq…