The paper sets lower bounds for a Kirby-Thompson invariant of 4-manifolds.
problem Determining the Kirby-Thompson invariant of specific 4-manifolds.
method Using trisections, the paper establishes lower bounds and calculates the invariant for specific examples.
result The paper calculates the Kirby-Thompson invariant of the spin of L(2,1) and shows the existence of 4-manifolds with arbitrarily large invariants. This paper develops 4-manifold invariants using Hopf algebras.
problem Creating 4-manifold invariants from Hopf algebras.
method Using Hopf triplets and trisection diagrams, the authors construct 4-manifold invariants.
result Every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds.
New 4-manifold invariant defined from trisection diagrams.
problem Defining a new 4-manifold invariant from trisection diagrams.
method Algebraic data from bimodule categories and spherical fusion categories, described diagrammatically.
result Includes Hopf algebraic invariants and modular fusion category invariants.
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.
We construct an invariant of closed spinc 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a 4-manifold. We also give examples of 4-manifolds which admit positive scalar curvatu…
Detects exotic embeddings in 4-manifolds with boundary.
problem Detecting exotic diffeomorphisms in 4-manifolds with boundary.
method Defined family versions of Kronheimer-Mrowka invariants and used them to find exotic embeddings.
result First example of exotic 3-spheres in a smooth closed 4-manifold with diffeomorphic complements.
This paper constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
problem Detecting exotic diffeomorphisms on reducible 4-manifolds with odd b+.
method Using a families Bauer-Furuta invariant and pin-cobordism.
result Constructs exotic diffeomorphisms on reducible 4-manifolds with odd b+.
The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures…
The study sets constraints on 4-manifold forms linked to specific invariants.
problem Understanding the intersection forms of spin 4-manifolds bounded by Seifert rational homology 3-spheres.
method Analyzes constraints using the μ-bar and κ invariants.
result The difference between κ and -μ-bar for a Seifert rational homology 3-sphere is at most 2, and under certain conditions, it is 0.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. We give formulae for the Ozsvath-Szabo invariants of 4-manifolds X obtained by fiber sum of two manifolds M_1, M_2 along surfaces S_1, S_2 having trivial normal bundle and genus g>0. The formulae follow from a general theorem on the Ozsvath-Szabo invariants of the result of gluing two 4-manifolds along a common boundar…
New criterion for almost-complex 4-manifolds using polyhedral decompositions.
problem Bounding the genus of surfaces in almost-complex 4-manifolds.
method Polyhedral decompositions and adjunction criterion.
result Established adjunction inequality for almost-complex 4-manifolds.
We define a diffeomorphism invariant of smooth 4-manifolds which we can estimate for many smoothings of R^4 and other smooth 4-manifolds. Using this invariant we can show that uncountably many smoothings of R^4 support no Stein structure. (Gompf has constructed uncountably many smoothings of R^4 which do support Stein …
New invariants derived from link homology for 4-manifolds.
problem Creating invariants for surfaces in smooth 4-manifolds.
method Skein lasagna modules based on equivariant and deformed glN link homology. result Non-vanishing and decomposition results for deformed glN skein lasagna modules. New method classifies spin 4-manifolds using Kervaire-Milnor invariant.
problem Classifying spin 4-manifolds up to stabilisation.
method Using Kervaire-Milnor invariant to compute Arf invariants and classify.
result New stable classification of spin 4-manifolds with 2-dimensional fundamental groups.
The study examines the realizability of a 4-manifold invariant for homeomorphisms.
problem Realizing the Casson-Sullivan invariant for homeomorphisms of 4-manifolds.
method Investigation of the invariant's realizability and application to surface isotopy.
result The invariant can be realized fully after stabilizing with a single S2imesS2 for all orientable pairs of homeomorphic 4-manifolds. The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Tw…
Formula calculates Seiberg-Witten invariants after surgery.
problem Calculating Seiberg-Witten invariants for surgeries on 4-manifolds.
method Surgery formula for Seiberg-Witten invariants of 4-manifolds.
result Formula expresses invariants of the manifold after surgery.
Involutory Hopf group-coalgebras provide new invariants for 4-manifold bundles.
problem Developing invariants for flat bundles over 4-manifolds.
method Utilizing Hopf G-triplets and colored trisection diagrams. result Involutory Hopf G-triplets yield well-defined invariants of G-colored trisection diagrams. Sum formula for relative Seiberg-Witten invariants in 4-manifolds.
problem Defining and studying relative Seiberg-Witten invariants for 4-manifolds with submanifolds.
method Defined relative Seiberg-Witten invariants and proved a sum formula.
result A sum formula relating relative SW invariants of 4-manifolds and their submanifolds.
New invariant calculates 4-manifolds using trisection diagrams and combings.
problem Calculating non-semisimple 4-manifold invariants.
method Using trisection diagrams and combings of the trisection surface.
result Invariant calculated for Stein nuclei, generalizing earlier semisimple version.
New constraints on embedded spheres and projective planes in 4-manifolds from Seiberg-Witten theory.
problem Constraints on configurations of embedded spheres and real projective planes in 4-manifolds.
method Equivariant Seiberg-Witten invariants and gluing formula for relative Seiberg-Witten invariants.
result Existence of certain configurations of surfaces leads to 4-manifolds of non-simple type.
Constructs invariants from Lagrangian tori in symplectic 4-manifolds.
problem Tackles invariants of Lagrangian tori in symplectic 4-manifolds.
method Uses ECH and monopole Floer homology to construct distinguished elements.
result Repackages Gromov and Seiberg-Witten invariants of torus surgeries.
Proves simple type conjecture for mod 2 Seiberg-Witten invariants.
problem Proving simple type conjecture for mod 2 Seiberg-Witten invariants.
method Analyzes cohomology ring conditions to prove simple type for all smooth structures.
result Shows existence of topological 4-manifolds with mod 2 simple type.
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…
We introduce an invariant of tuples of commutative diffeomorphisms on a 4-manifold using families of Seiberg-Witten equations. This is a generalization of Ruberman's invariant of diffeomorphisms defined using 1-parameter families of Seiberg-Witten equations. Our invariant yields an application to the homotopy groups of…
New inequality for links in 3D using 4D invariants.
problem Understanding links in 3D using 4D invariants.
method Using a Bauer--Furuta-type invariant for 4-manifolds with contact boundary.
result Generalized Thurston--Bennequin inequality for links in S3. We find the shape of the Donaldson invariants of a 4-manifold with b_1=0 and b^+>1, which may be not of simple type. The invariants appear as the q^0 coefficient of a expression given in terms of modular forms (as was predicted by Moore and Witten). We re-express the formula using complete elliptic integrals to prove a…
Study BF invariants using simple type concepts.
problem Understanding Bauer--Furuta invariants for 4-manifolds.
method Extend simple type concept to BF blowup and BF homogeneous types, prove gluing formulae and adjunction inequality.
result Determine Bauer--Furuta invariant of a specific 4-manifold and give constraints on gluing decompositions.
We introduce a diffeomorphism invariant of 4-manifolds, the Pin−(2)-monopole invariant, defined by using the Pin−(2)-monopole equations. We compute the invariants of several 4-manifolds, and prove gluing formulae. By using the invariants, we construct exotic smooth structures on the connecte…
Study cohomotopy classes for 4-manifolds using complex spin structures.
problem Understanding cohomotopy classes for families of 4-manifolds with complex spin structures.
method Using Bauer--Furuta invariants in parametrised stable homotopy theory.
result Definition of characteristic cohomotopy classes on Thom spectra.
The abstract introduces a method to compute surface invariants in 4-manifolds.
problem Computing invariants of surfaces embedded in 4-manifolds.
method Using a nested pair of ribbon fusion categories and banded-link presentations.
result The method provides an invariant sensitive to both genus and knotting.
Paper extends Miyazawa's construction to more 4-manifolds, finding exotic involutions and embeddings.
problem Finding exotic involutions and embeddings in 4-manifolds.
method Real Seiberg-Witten theory.
result Many infinite families of exotic involutions and embeddings.
New invariant detects non-homotopy equivalent 4-manifolds.
problem Detecting non-homotopy equivalent 4-manifolds.
method Extending Kreck and Schafer's doubling construction to 2-complexes with finite fundamental group.
result Existence of k closed smooth 4-manifolds that are stably diffeomorphic but not homotopy equivalent. New invariant for 4-manifolds with framed links, stronger than existing invariants.
problem Distinguishing 4-manifolds with framed links.
method Introducing a new invariant called KLS lasagna homotopy type.
result The new invariant is stronger than existing invariants.
We use a 1-parameter version of gauge theory to investigate the topology of the diffeomorphism group of 4-manifolds. A polynomial invariant, analogous to the Donaldson polynomial, is defined, and is used to show that the diffeomorphism group of certain simply-connected 4-manifolds has infinitely generated π_0.
The Hausmann-Weinberger invariant of a group G is the minimal Euler characteristic of a closed orientable 4-manifold M with fundamental group G. We compute this invariant for finitely generated free abelian groups and estimate the invariant for all finitely generated abelian groups.
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
A formula is given for the Seiberg-Witten invariants of a 4-manifold that is cut along certain kinds of 3-dimensional tori. The formula involves a Seiberg-Witten invariant for each of the resulting pieces.
Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.
problem Classifying and understanding Whitney towers in 4-manifolds.
method Local manipulations of surfaces, definitions of Whitney towers and trees, geometric Jacobi identities, classification of twisted Whitney towers.
result Classification of order n twisted Whitney towers in the 4-ball and related invariants.
The main results of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold which admits a nontrivial free S^1-action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over S^1. We also describe a 3-manifold which is not the orbit space…
The paper studies special crystallizations of 4-manifolds to minimize certain PL-invariants.
problem Minimizing combinatorially defined PL-invariants in crystallizations of compact 4-manifolds.
method Analysis of semi-simple and weak semi-simple crystallizations to minimize regular genus, Gurau degree, gem-complexity, and trisection genus.
result An original theorem on the minimization of PL-invariants for compact 4-manifolds with weak semi-simple crystallizations.
Proves surface embedding theorem for 4-manifolds with good fundamental group.
problem Surface embedding in 4-manifolds with good fundamental group.
method Introduces dual spheres and Kervaire-Milnor invariant for computation.
result Combinatorial formula for Kervaire-Milnor invariant.
By using the gluing formulae of the Seiberg-Witten invariant, we show the nonexistence of Einstein metric on manifolds obtained from a 4-manifold with nontrivial Seiberg-Witten invariant by performing sufficiently many connected sums or appropriate surgeries along circles or homologically trivial 2-spheres with closed …
New proof shows exotic 4-manifolds exist without complex calculations.
problem Existence of exotic 4-manifolds in topology.
method Used Beliakova and Wehrli's s-invariant for links in S3 and Stošić's induction scheme to simplify computations. result Existence of exotic compact, orientable 4-manifolds proven without skein lasagna modules.
We investigate the Pin−(2)-monopole invariants of symplectic 4-manifolds and Kähler surfaces with real structures. We prove the nonvanishing theorem for real symplectic 4-manifolds which is an analogue of Taubes' nonvanishing theorem of the Seiberg-Witten invariants for symplectic 4-manifolds. Further…
New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.
We make use of F-structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature bounded from below. À la Gompf, we show that these invariants vanish on symplecti…