Study of 3-handle attachments in 4D manifolds using Kirby calculus.
problem Inclusion of 3-handles in Kirby diagrams for 4D manifolds.
method Developed moves to extend Kirby calculus, established homological criterion.
result Identified geometric basis of 3-handle attachments, proved uniqueness theorem.
New 4-manifolds without 1- and 3-handles are created from knots.
problem Creating 4-manifolds without specific handles. method Knot surgery with specific knots.
result Knot surgery 4-manifolds E(n)K have a handle decomposition without 1- and 3-handles. New 4-manifolds found without 1- and 3-handles.
problem Finding 4-manifolds without specific handles.
method Perfect Morse functions and handle decompositions.
result Infinitely many 4-manifolds homeomorphic but not diffeomorphic to standard manifolds.
Starting with the Dolgachev surface E(1)_{2,3} we construct an infinite family of distinct exotic copies of the rational surface E(1), each of which admits a handlebody decomposition without 1- and 3- handles, and we draw these handlebodies.
We give new rational blowdown constructions of exotic CP^2#n(-CP^2) (5\leq n\leq 9) without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for 7\leq n\leq 9. A strategy for rational blowdown constructions of exotic CP^2#n(-CP^2) (1\leq n\leq 4) is also…
Study shows knots surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.
problem Understanding handle decompositions of knots surgered elliptic surfaces.
method Using Kirby diagrams derived from Lefschetz fibrations.
result Knot surgered elliptic surfaces admit handle decompositions without 1- and 3-handles.
Harer, Kas and Kirby have conjectured that every handle decomposition of the elliptic surface E(1)2,3 requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as E(1)2,3 and admits neither 1- nor 3-handles, by using rational blow-downs and K…
Formulae for 1-3 handle attachments in 4-manifolds.
problem Understanding handle attachments in 4-dimensional manifolds.
method Developed general formulae for 1-3 handle attachments using skein modules.
result Derived complete description of gluing homomorphism on skein modules.
We prove that a positive definite smooth four-manifold with b2+≥2 and having either no 1-handles or no 3-handles cannot admit a symplectic structure.
A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle deco…
New method constructs Seifert solids from bridge trisections.
problem Constructing Seifert solids from bridge trisections.
method Adapting Seifert's algorithm to tri-plane diagrams.
result Classification results on surface decomposability and unknottedness.
We prove that the Dolgachev surface E(1)_{2,3} admits a handlebody decomposition without 1- and 3- handles, and we draw the explicit picture of this handlebody. We also locate a "cork" inside of E(1)_{2,3}, so that E(1)_{2,3} is obtained from E(1) by twisting along this cork.
Harer-Kas-Kirby conjectured that every handle decomposition of the elliptic surface E(1)_{2,3} requires both 1- and 3-handles. We prove that the elliptic surface E(n)_{p,q} has a handle decomposition without 1-handles for n≥1 and (p,q)=(2,3),(2,5),(3,4),(4,5).
D.Nash defined a family of homotopy 4-spheres in [11]. Proving that his manifolds Sm,n,m′,n′ are all real S4, we find that they have handle decomposition with no 1-handles, two 2-handles and two 3-handles. The handle structures give new potential counterexamples of Property 2R conjecture.
Study of contractible manifolds and their twists to determine if they are S4.
problem Determine if a specific manifold is S4 using twists and handlebody descriptions. method Analyze two contractible manifolds, one Stein and one non-Stein, with non-trivial boundary mapping classes. Use a specific homotopy sphere to test if it is S4. result Show that a specific manifold is indeed S4. We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on …
Study handles in 4-manifolds with cyclic fundamental group.
problem Understanding handle decompositions for specific 4-manifolds.
method Constructed handle decompositions based on fundamental group and Betti numbers.
result Handle decomposition formula for specific 4-manifolds.
We show the homotopy spheres Σn=−W⌣fnW, formed by doubling the infinite order loose-cork (W,f) by iterates of the cork diffeomorphism f:∂W→∂W is S4. To do this we first show that Σn are obtained by Gluck twistings of S4; then from this we show how to cancel 3-han…
Elliptic surfaces without 1-handles proven for specific cases.
problem Proving elliptic surfaces have no 1-handles.
method Analyzing handle decompositions of elliptic surfaces E(n)p,q for specific n. result Proven existence of handle decompositions without 1-handles for specific elliptic surfaces.
New definition of skein lasagna module for specific 4-manifolds.
problem Defining a new homology theory for specific 4-manifolds.
method Elementary definition using diagrams in the 4-manifold.
result Explicit calculations for disk bundles over S^2.
Determines conditions for ribbon cobordisms between lens spaces.
problem Conditions for ribbon rational homology cobordisms between lens spaces.
method Analyzes ribbon cobordisms and uses properties of lens spaces and linear lattices.
result If a lens space admits a ribbon rational homology cobordism to a different lens space, it must be homeomorphic to L(n,1), up to orientation-reversal. Heegaard diagrams for 5-manifolds help in understanding their structure.
problem Understanding the structure of 5-dimensional manifolds.
method Introducing a version of Heegaard diagrams for 5-dimensional cobordisms and manifolds, showing diffeomorphisms through specific moves.
result Every smooth 5-manifold can be represented by a Heegaard diagram, and diagrams representing diffeomorphic manifolds are related by certain moves.
Review of gem theory's interactions with Kirby diagrams and trisections.
problem Representing and understanding PL 4-manifolds.
method Combining gem theory with Kirby diagrams and trisections.
result New gems representing 4-manifolds requiring 3-handles.
Simple branched coverings established between certain 4-manifolds.
problem Establishing simple branched coverings between specific 4-manifolds.
method Using handle decompositions and isometric embeddings of intersection lattices.
result Existence and construction of d-fold simple branched coverings. New hyperbolic 4-manifolds found with special functions.
problem Finding hyperbolic 4-manifolds with specific circle-valued Morse functions.
method Constructing hyperbolic 4-manifolds with only index 2 critical points.
result Existence of infinitely many hyperbolic 4-manifolds with bounded Betti numbers.
Fintushel and Stern showed that the Brieskorn sphere Σ(2,3,7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…
We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings…
Study compact PL 4-manifolds with special handle decompositions.
problem Existence of special handlebody decompositions for simply-connected closed PL 4-manifolds.
method Investigate colored triangulations inducing handle decompositions without 1-handles or 1- and 3-handles.
result Detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary that admit such decompositions.
We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology 3-sphere can be obtained from the standard Heegaard decomposition of S3 by regluing…
Proposes a new framework for learning image augmentations to improve classification performance.
problem Improving classification performance with a given class of predictors.
method Transformed Risk Minimization (TRM) framework that optimizes both predictive models and data transformations.
result Performance of TRM with SCALE algorithm compares favorably to prior methods on CIFAR10/100.
The paper connects Kirby diagrams and 5-colored graphs to represent 4-manifolds.
problem Representing compact 4-manifolds using Kirby diagrams and graphs.
method Algorithmically constructing 5-colored graphs from Kirby diagrams to represent PL 4-manifolds.
result Upper bounds for gem-complexity and regular genus derived from Kirby diagrams.
Well-quasi-orders proved on embedded planar graphs.
problem Proving well-quasi-orders on embedded planar graphs.
method Careful analysis and extensions of classical methods for embedded minor relations.
result Embedded minor relations are well-quasi-orders on various classes of embedded planar graphs.
The paper generalizes trisections to compact PL 4-manifolds with boundary and introduces gem-induced trisections.
problem Generalizing trisections to compact PL 4-manifolds with boundary and any 5-colored graph encoding simply-connected 4-manifolds.
method Introducing gem-induced trisections and analyzing their properties for compact PL 4-manifolds with and without boundary.
result Conditions for gem-induced trisections to realize the G-trisection genus and direct determination of the genus from the graph.