The paper introduces new 4D twists and exotic families from a plug, with applications to knot surgeries.
problem Exploring new 4-dimensional twists and exotic families from a plug.
method Defined an infinite order plug (P,φ) and used it to create families Yn, Zn of exotic enlargements. result Introduced a new method to produce 2-bridge knot or link surgeries.
By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spinc manifolds. Especially, we get twisted Rokhlin congruences for 8k+4 dimensional spi…
Algorithm constructs Kirby diagrams for 4D open books.
problem Constructing Kirby diagrams for 4D open books.
method Algorithm using Heegaard diagrams of pages.
result Diffeomorphic open books constructed with different pages and monodromies.
In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boun…
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the J-twist of the Dirac operator on spin manifolds.
method Lichnerowicz type formula and Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator on 3D and 4D almost product Riemannian spin manifolds with boundary.
The paper proves a theorem for a twisted Dirac operator on specific manifolds.
problem Analyzing the Dirac operator with torsion on spin manifolds.
method Develops a Lichnerowicz type formula and proves a Kastler-Kalau-Walze type theorem.
result Proves a Kastler-Kalau-Walze type theorem for the J-twist of the Dirac operator with torsion on 4D and 6D almost product Riemannian spin manifolds. New relation found in 4D symplectic mapping class group.
problem Relation between Dehn twists in symplectic 4-manifolds.
method Holomorphic curve techniques, symplectic isotopy problem solution.
result Relation between two products of Dehn twists.
Study Lefschetz fibrations with 4D fibers using Seiberg-Witten theory.
problem Constraints on the topology of Lefschetz fibrations with 4D fibers.
method Using the family Bauer-Furuta invariant and framed bordism class of 1D Seiberg-Witten moduli spaces.
result New obstructions to the smooth isotopy of compositions of Dehn twists.
Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.
problem Linking fibered classes in 3-manifolds via Alexander polynomials.
method Applies twisted Alexander polynomials to fibered classes in ribbon homology cobordisms.
result Fibered classes of Y+ map to those of Y−. By 2-twist-spinning the knotted graph that represents the knotted handlebody 52, we obtain a knotted foam in 4-dimensional space with a non-trivial quandle cocycle invariant.
Develops a theorem for a 6D manifold with boundary.
problem No specific problem stated; focuses on extending a theorem.
method Extends a Kastler-Kalau-Walze type theorem to 6D almost product Riemannian spin manifolds with boundary.
result Proves a Kastler-Kalau-Walze type theorem for 6D manifolds.
Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.
problem Proving a theorem for a specific type of Dirac operator on various manifolds.
method Extending previous results to even-dimensional almost product Riemannian spin manifolds.
result Established the general Kastler-Kalau-Walze type theorem for even-dimensional manifolds.
Develops obstruction theory for a specific 4-manifold index.
problem Computing the Z2-index of 4-manifolds with free involution. method Uses spectral sequences and cohomology with twisted coefficients.
result Computes the Z2-index for various examples. Study of torus surgeries on knot traces, finding exotic surfaces and traces.
problem Understanding exotic surfaces and traces through torus surgeries.
method Realizing annulus twisting as torus surgery, using key technical insight.
result Exotic elliptic surfaces and traces discovered, improving known geography.
We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized Kähler structures. By considering the commutator Q of the two associated almost complex structures J±, we prove that if either the manifold is 4-dimensional or the distribution ${Im} …
Two special moves relate any two open book decompositions of a 3-manifold.
problem Relating any two open book decompositions of a 3-manifold.
method Using 4-dimensional Lefschetz fibrations and 3-dimensional contact topology, including Harer's twisting and Giroux-Goodman stable equivalence theorem.
result A complete set of two moves suffices to relate any two open book decompositions of a given 3-manifold.
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
problem Detecting exotic smooth structures in 4-manifolds.
method Proving field theories lead to stable invariants, distinguishing only homeomorphic and homotopy equivalent manifolds.
result Semisimple 4D field theories can't distinguish homotopy equivalent 4-manifolds.
New infinite order corks found that cannot twist exotic 4-manifolds.
problem Finding new infinite order corks to twist exotic 4-manifolds.
method Proved necessary conditions for generating 4-manifolds by infinite order corks and found non-contractible examples.
result First example of infinite order corks not generating exotic 4-manifolds.
New method computes Yamabe invariants for specific 4D manifolds.
problem Computing Yamabe invariants for new 4D manifolds.
method Using twisted Seiberg-Witten equations, Pin(2)-monopole equations.
result Computed Yamabe invariants for a new class of 4D manifolds.
The paper studies invariant generalized complex structures on flag manifolds.
problem Classifying invariant generalized complex structures on flag manifolds.
method Analyzing invariant 4-dimensional generalized almost complex structures restricted to each root space, and studying the Nijenhuis operator for a triple of roots. result Classification of integrable and Ω-integrable generalized complex structures. In 1987 S Cappell and J Shaneson constructed an s-cobordism H from the quaternionic 3-manifold Q to itself, and asked whether H or any of its covers are trivial product cobordism? In this paper we study H, and in particular show that its 8-fold cover is the product cobordism from S^3 to itself. We reduce the triviality…
The paper quantizes hybrid topological-holomorphic field theories on RmimesCn.
problem Quantizing hybrid topological-holomorphic field theories rigorously.
method Constructing perturbative, one-loop quantizations on RmimesCn. result The one-loop obstruction to quantization vanishes when m≥1. From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and …
We prove that for any integer n there exist infinitely many different knots in S3 such that n-surgery on those knots yields the same 3-manifold. In particular, when ∣n∣=1 homology spheres arise from these surgeries. This answers Problem 3.6(D) on the Kirby problem list. We construct two families of examples, t…
The paper defines and calculates stabilization distances between surfaces in 4-manifolds.
problem Calculating the minimal number of 1-handle stabilizations for surfaces to become isotopic.
method Using homology of cyclic covers and metabelian twisted homology.
result For every nonnegative integer m, there exist pairs of surfaces with a specific stabilization distance.
Study rules out exotic S4 and #nCP2 construction using zero surgery homeomorphisms.
problem Tackles the possibility of constructing exotic S4 or #nCP2 using zero surgery homeomorphisms. method Uses zero surgery homeomorphisms to relate slice properties of knots stably after a connected sum with a 4-manifold.
result Rules out the possibility of constructing exotic S4 or #nCP2 using zero surgery homeomorphisms. Proof outlined for 4D smooth Poincaré conjecture.
problem 4-dimensional smooth Poincaré conjecture.
method Outline of proof.
result Proof of 4D smooth Poincaré conjecture.
We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitel…
The paper calculates transformation operators and proves a theorem on 4D manifolds.
problem Computing transformation operators and proving theorems on 4D manifolds.
method Computed lower-dimensional volumes and applied Kastler-Kalau-Walze type theorems.
result Proved a Kastler-Kalau-Walze type theorem on 4D compact manifolds with boundary.
The study classifies 331 specific 4D polytopes with 7 facets.
problem Classifying finite-volume hyperbolic Coxeter 4D polytopes.
method Complete classification through exhaustive search.
result 331 unique polytopes with 7 facets identified.
New modules derived from Khovanov homology for links.
problem Constructing new link homology modules from Khovanov homology.
method Functoriality proof for cobordisms in 4D relative 1-handlebody complements.
result Functoriality of Rozansky-Willis's homology for cobordisms.
Study non-orientable surfaces in gauge theories, matching fluxes with homotopy classes.
problem Understanding non-orientable surfaces in gauge theory and their electric-magnetic duality.
method Reduction of twisted N=4 gauge theory on non-orientable manifolds, matching fluxes with homotopy classes.
result Verification of mirror symmetry of branes as predicted by S-duality.
We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.
We give an elementary proof of the fact that any 4-dimensional para-Hermitian manifold admits a unique para-Kaehler--Weyl structure. We then use analytic continuation to pass from the para-complex to the complex setting and thereby show any 4-dimensional pseudo-Hermitian manifold also admits a unique Kaehler--Weyl stru…
A weakly Einstein manifold is a generalization of a 4-dimensional Einstein manifold, which is defined as an application of a curvature identity derived from the generalized Gauss-Bonnet formula for a 4-dimensional compact oriented Riemannian manifold. In this paper, we shall give a characterization of a weakly Einstein…
Classifies metrics on specific Lie groups.
problem Classifying Riemannian metrics on nonunimodular Lie groups.
method Automorphism classification of inner products on Lie algebras.
result Classification of metrics on 4D nonunimodular Lie groups.
The goal of this article is to study the pinching problem proposed by S.-T. Yau in 1990 replacing sectional curvature by one weaker condition on biorthogonal curvature. Moreover, we classify 4-dimensional compact oriented Riemannian manifolds with nonnegative biorthogonal curvature. In particular, we obtain a partial a…
Maps from 4D handlebodies to their boundaries have sections.
problem Understanding mappings between 4D handlebodies and their boundaries.
method Induced map analysis and section construction.
result Existence of sections for maps from BHomeo(Hg) to BHomeo(Ug). The paper proves UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
problem Proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories.
method Rigorously proving UV finiteness and vanishing anomalies for hybrid topological-holomorphic field theories on Rd′imesCd. result Proves vanishing anomalies for hybrid topological-holomorphic field theories, allowing for the definition of a factorization algebra structure for quantum observables.
4D theorem for disks, generalizing previous work.
problem Properly embedded disks in 4-manifolds.
method Generalization of Gabai and Kosanovic-Teichner's work, new geometric interpretation of Dax invariant.
result Homotopic disks are smoothly isotopic under certain conditions.
The paper characterizes and contrasts knots with high 4D clasp numbers.
problem Characterizing knots with high 4-dimensional clasp numbers.
method Topological category analysis and construction of counterexamples.
result Characterization and contrast of knots with high 4D clasp numbers.
Global regularity proved for 4D Ricci flow with scalar curvature integral bound.
problem Global regularity of 4D Ricci flow with integral scalar curvature bound.
method Extended Ge-Jiang's result to include integral bound on scalar curvature.
result Global ε-regularity for 4D Ricci flow with integral scalar curvature bound. New procedures connect braid charts, triplane diagrams, and braid movies for knotted surfaces.
problem Understanding the braid index and bridge index of knotted surfaces in 4D.
method Introducing rainbow diagrams and new procedures for passing among triplane diagrams, braid movies, and braid charts.
result Inequalities relating braid index and bridge index of 2-knots are obtained.
The paper studies the Pontrjagin dual of 4D Spin bordism.
problem Understanding the Pontrjagin dual of 4-dimensional Spin bordism.
method Using cochains on X to describe the dual group and its pairing with Spin manifolds.
result Described the dual group G(X) and its pairing with Spin manifolds.
We investigate proper biharmonic hypersurfaces with at most three distinct principal curvatures in space forms. We obtain the full classification of proper biharmonic hypersurfaces in 4-dimensional space forms.
In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the first time in 4-dimensional Euclidean space. Then we obtain the conditio…
In this paper we will construct a Weierstrass type representation for minimal surfaces in 4-dimensional Lorentzian Damek-Ricci spaces and we give some examples of such surfaces.
Withdrawn May 2005. There is an error in the even-dimensional case of the proof in the April 2005 version. The hoped-for 4-dimensional applications are unlikely to survive the repairs.