Study connects geodesic sums to area constant.
problem Understanding geodesic sums in Teichmüller space.
method Relates trimmed sums of twists to area Siegel-Veech constant.
result Established connection between geodesic sums and area constant.
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48 results for “twisted connected sums”
New G_2-manifolds created from quotients of twisted connected sums.
problem Creating new G_2-manifolds with specific topological properties.
method Generalizing Kovalev's twisted connected sums by taking quotients of pieces before gluing.
result Realization of a wider range of topological types of G_2-manifolds.
The paper studies relationships between twisted torsion and connected sums of 3-manifolds.
problem Understanding the topology and geometry of 3-manifolds through twisted torsion.
method Investigates the multiplicative property of twisted Reidemeister torsion and its connection to the connected sum operation.
result Establishes a relationship between the multiplicative property of twisted torsion and connected sums of 3-manifolds.
Method constructs G2-instantons over twisted connected sums.
problem Constructing G2-instantons over complex manifolds.
method Gluing G2-instantons from holomorphic bundles over building blocks.
result Explicit examples from semi-Fano 3-folds.
We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of t…
Using earlier work of Sá Earp and the author [SEW13] we construct an irreducible unobstructed -instanton on an -bundle over a twisted connected sum recently discovered by Crowley-Nordström [CN14].
We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.
New invariant shows Dehn twist on connected sum of homology tori is not isotopic to identity.
problem Determining when Dehn twists on connected sums of homology tori are isotopic to identity.
method Generalized Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds and constructed a refinement.
result Dehn twist on is not isotopic to identity if determinants are odd.
The study explores dimensions for connected sums of almost complex manifolds and extends results to rational homology spheres.
problem Understanding dimensions for connected sums of almost complex manifolds and extending results to rational homology spheres.
method Obstruction theory and Yang's results on almost complex structures were used to answer questions about dimensions. The index of the twisted spin^c Dirac operator was applied to extend results to rational homology spheres.
result The study partially extends Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres.
Distinguishing G_2-manifoldsmath.DG
In this survey, we describe invariants that can be used to distinguish connected components of the moduli space of holonomy G_2 metrics on a closed 7-manifold, or to distinguish G_2-manifolds that are homeomorphic but not diffeomorphic. We also describe the twisted connected sum and extra-twisted connected sum construc…
Study on realizing subgroup twists in 3-manifolds.
problem Realizing subgroups of twist groups in 3-manifolds.
method Analyzing Nielsen realization problem for Twist(M) subgroups, applying to Burnside problem.
result Nontrivial subgroups of Twist(M) are realized by diffeomorphisms if and only if they are cyclic and M is a connected sum of lens spaces.
Study G_2-manifolds from glued circles and Calabi-Yau manifolds.
problem Analyse G_2-manifolds from glued circles and Calabi-Yau manifolds.
method Express nu-invariant in terms of fixpoints and gluing contributions, involving Dedekind sums and eta-function.
result Existence of compact G_2-manifolds that are not G_2-nullbordant.
When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo and the twi…
Every shadow can be a knot or a connected sum of trefoils.
problem Understanding the structure of shadows and their relation to knots.
method Analyzing reduced shadows and their relationship to specific knots.
result Every shadow corresponds to a torus knot, twist knot, or connected sum of trefoils.
The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
problem Preserving the diffeomorphism type of satellite 2-knots under the Gluck twist.
method Using new descriptions of satellite 2-knots, the paper shows that the Gluck twist does not change the diffeomorphism type of certain satellite 2-knots in three ways.
result The Gluck twist preserves the diffeomorphism type of certain satellite 2-knots.
Example of non-smooth isotopy using K3 surfaces.
problem Non-smooth isotopy examples in 4-manifolds.
method Dehn twist along a 3-sphere in K3 surfaces' connected sum.
result First example of self-diffeomorphisms isotopic to identity but not smoothly.
The goal of this paper is the construction of a compact manifold with G holonomy and nodal singularities along circles using twisted connected sum method. This paper finds matching building blocks by solving the Calabi conjecture on certain asymptotically cylindrical manifolds with nodal singularities. However, by …
We introduce a method to construct -instantons over compact -manifolds arising as the twisted connected sum of a matching pair of building blocks [Kov03,KL11,CHNP12]. Our construction is based on gluing -instantons obtained from holomorphic bundles over the building blocks via the first named author's wo…
We propose a method to construct G_2-instantons over a compact twisted connected sum G_2-manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end (see arXiv:1310.7933). In our example, the moduli spaces of the ingredient instantons are non-trivial, and their…
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 .
result No non-cyclic subgroup of twist subgroup can be realized by diffeomorphisms.
Construct M-Theory lifts of type IIA orientifolds.
problem Lift type IIA orientifolds to M-Theory.
method Construct M-Theory on twisted connected sum G2 manifolds.
result Two building blocks correspond to open and closed string sectors.
New contactomorphisms found via Dehn twists on 3-manifold sums.
problem Detecting exotic contactomorphisms in 3-manifold sums.
method Combining Kronheimer-Mrowka invariant and h-principle for convex spheres.
result First examples of infinite-order exotic contactomorphisms.
Paper generalizes Satoh's result on knots and projective planes.
problem Understanding connected sums of knots and projective planes.
method Generalizes Satoh's result to odd natural twist spun spheres and projective planes.
result Connected sum of -twist spun sphere of a knot and an unknotted projective plane is equivalent to the projective plane.
Method constructs rigid associative submanifolds in twisted G2-manifolds.
problem Constructing rigid associative submanifolds in twisted G2-manifolds.
method Introducing a gluing theorem for asymptotically cylindrical associative submanifolds in ACyl G2-manifolds.
result Yields many new topological types of rigid associative submanifolds.
We simplify and prove a splitting theorem for mapping class groups of connect sums of .
problem Understanding the structure of mapping class groups of specific 3-manifolds.
method We prove a splitting theorem for the mapping class group of connect sums of , simplifying Laudenbach's original proof.
result The mapping class group of connect sums of is the semidirect product of Out by .
The paper generalizes a result on smooth mapping class groups and proves a property of Dehn twists in 4-manifolds.
problem Properties of Dehn twists in smooth 4-manifolds.
method Generalization of a previous result and alternative proof of a consequence.
result Dehn twists along the boundary of simply-connected 4-manifolds are trivial after connected sums.
Diagrams and Reidemeister moves for links in a twisted S^1-bundle over an unorientable surface are introduced. Using these diagrams, we compute the Kauffman Bracket Skein Module (KBSM) of the connected sum of two projective spaces. In particular, we show that it has torsion. We also present a new computation of the KBS…
In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact struc…
Recently, at least 50 million of novel examples of compact holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. The purpose of this paper is to study mirror symmetry for compactifications of Type II superstrings in this context. We focus on …
We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G_2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU…
We prove that the moduli space of holonomy G_2-metrics on a closed 7-manifold is in general disconnected by presenting a number of explicit examples. We detect different connected components of the G_2-moduli space by defining an integer-valued analytic refinement of the nu-invariant, a Z/48-valued defect invariant of …
A generalization of Rasmussen's invariant, with applications to surfaces in some four-manifoldsmath.GT
Extends Rasmussen's invariant to new surfaces in four-manifolds.
problem Computing genus bounds for surfaces in specific four-manifolds.
method Extending Khovanov-Lee homology and using Hochschild homology.
result Proves inequalities relating the invariant to surface genus.
M-theory on compact eight-manifolds with -holonomy is a framework for geometric engineering of 3d gauge theories coupled to gravity. We propose a new construction of such -manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four…
We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular…
We define a Z/48-valued homotopy invariant nu of a G_2-structure on the tangent bundle of a closed 7-manifold in terms of the signature and Euler characteristic of a coboundary with a Spin(7)-structure. For manifolds of holonomy G_2 obtained by the twisted connected sum construction, the associated torsion-free G_2-str…
New example of non-smooth isotopy after stabilization in 4-manifolds.
problem Non-smooth isotopy of Dehn twist after stabilization.
method Pin(2)-equivariant Bauer-Furuta invariant.
result Dehn twist is not smoothly isotopic even after stabilization.
Nontrivial boundary Dehn twist found on K3#K3 manifold.
problem Proving nontriviality of a Dehn twist on a specific 4-manifold.
method Algebraic criterion and equivariant topological K-theory to show non-isotopy.
result Boundary Dehn twist is nontrivial in the smooth mapping class group.
The paper proves properties of branched covers of specific knots and tori.
problem Investigating the smoothness and diffeomorphism of specific 4-manifolds.
method Analyzing double branched covers of twist-roll spun knots and turned twisted tori, applying techniques to show diffeomorphism.
result Proves that certain 4-manifolds are diffeomorphic to standard manifolds.
In this article we construct a family of knot surgery -manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface using connected sums of fibered knots obtained by Stallings twist from a…
Prove integrality of genus- indices with adjoint Reidemeister torsions for twist knots and meridians.
problem Prove integrality of genus- indices with adjoint Reidemeister torsions for twist knots and meridians.
method Consider the sum of the adjoint Reidemeister torsions and prove integrality for twist knots and meridians.
result Prove integrality of genus- indices with adjoint Reidemeister torsions for twist knots and meridians.
Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.
problem Constructing non-trivial Cayley fibrations with conical singularities.
method Using gluing methods and stability results for weak and usual fibrations.
result Construction of examples of Cayley fibrations on twisted connected sum manifolds.
The paper classifies manifolds with free torus actions and positive Ricci curvature.
problem Classifying manifolds with specific geometric properties.
method Using circle bundles and free torus actions on connected sums of products of spheres.
result Infinitely-many manifolds with positive Ricci curvature and torus actions are classified.
Boundary Dehn twists become trivial after abelianization.
problem Understanding the behavior of boundary Dehn twists in smooth mapping classes.
method Constructing diffeomorphisms and showing commutators represent boundary Dehn twists.
result Boundary Dehn twists become trivial after abelianization.
The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…
A short proof for a theorem about composite knots.
problem Proving a theorem about composite knots with symmetric union presentations.
method Presenting a concise proof of Tanaka's theorem.
result Composite knots with symmetric union presentations have non-trivial connected summands.
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.
We show that for a suitable class of ``Dirac-like'' operators there holds a Gluing Theorem for connected sums. More precisely, if and are closed Riemannian manifolds of dimension together with such operators, then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that the spectrum…
Study of codimension-1 embeddings in 3-manifolds using twist maps and push maps.
problem Embedding 3-manifolds in specific topological spaces.
method Sphere twist maps and push maps to construct codimension-1 spun embeddings.
result Every closed orientable 3-manifold admits a codimension-1 spun embedding in specific spaces.