Every real 3-manifold can be turned into a real contact structure.
problem Understanding and constructing real contact structures on real 3-manifolds.
method Surgery along invariant knots and contact operations starting from the standard real S3. result Every real 3-manifold admits a real contact structure.
An (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R3 with transition maps in the affine transformation group Aff(R3). Equivalently an affine 3-manifold is a 3-manifold with a flat torsion-free affine connection. We show that a closed affine 3-mani…
New real invariants for 3-manifolds and links.
problem Defining real structures in Floer homology.
method Introducing real spin-c structures and applying monopole Floer homology.
result Invariants for links via double branched covers.
We show that the connected sum of two copies of real projective 3-space does not admit a real projective structure. This is the first known example of a connected 3-manifold without a real projective structure.
A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. In this article we study open book decompositions on smooth real 3-manifolds that are compatible with the real structure. We call them real open book decompositions. We show that each real open b…
Real bordered Floer homology computes 3-manifolds with involution.
problem Computing real Heegaard Floer homology for 3-manifolds with involution.
method Using bordered Heegaard Floer algebra modules, a practical algorithm is given.
result Computes real Heegaard Floer homology for real 3-manifolds with connected fixed set.
Study curvature loci of 3-manifolds in R^6 and R^5.
problem Characterize curvature loci of 3-manifolds in different dimensions.
method Refine affine classification of real nets of quadrics, study singularities, and analyze systems of ternary cubics.
result Obtain generic curvature loci and singularities of 3-manifolds.
Real Seiberg-Witten and monopole Floer homologies are equivalent for certain 3-manifolds.
problem Equivalence of two Floer homologies for real structures.
method Direct isomorphism proof between real Seiberg-Witten and monopole Floer homologies.
result Real Seiberg-Witten and monopole Floer homologies are isomorphic.
Complex duality for real submanifolds in complex 3-manifolds.
problem Understanding complex duality in real submanifolds of complex manifolds.
method Introducing semi-legendrian submanifolds and proving unique lifting to a 3-dimensional complex space.
result Deduction of complex duality between real submanifolds of P2(C). The paper defines equations for constructing projective structures on 3-manifolds.
problem Constructing projective structures on 3-manifolds.
method Defining a system of real valued equations and inequalities.
result These equations can be used to detect properly convex structures on 3-manifolds.
Defines flag structures on real 3-manifolds and proves null curvature models.
problem Characterizing and classifying real 3-manifolds with specific geometric structures.
method Introduces flag structures, constructs adapted connections, and defines invariants.
result Null curvature models are given by totally real submanifolds in flag space.
We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H^2xR or S^2xR is uniquely determined pointwise by its metric and its principal curvatures if and only if it is not a minimal or a prop…
Defines real link Floer homology for specific types of links.
problem Developing a new homology theory for certain types of links.
method Combining real Heegaard Floer homology and real sutured Heegaard Floer homology, using real grid diagrams in S3. result Observes structural and property properties of strongly invertible knots.
Characterizes hyperbolic links with stable maps to the plane.
problem Understanding hyperbolic links through stable maps.
method Characterization of hyperbolic links via stable maps to the plane.
result Complete characterization of hyperbolic links with specific stable maps.
Quantum dilogarithms help define invariants of 3-manifolds.
problem Defining invariants of 3-manifolds using quantum dilogarithms.
method Associate quantum dilogarithms to local fields, construct TQFTs, and use partition functions.
result Quantum dilogarithms yield invariants related to A-polynomial curves. The paper defines and analyzes a volume invariant for 3-manifolds.
problem Defining and analyzing a topological invariant for 3-manifolds.
method Definition and analysis of topological volume, refinements, bounds determination, classification of manifolds.
result Asymptotically tight upper and lower bounds for topological volume, classification of non-hyperbolic 3-manifolds.
We develop a method for preserving pseudoholomorphic curves in contact 3-manifolds under surgery along transverse links. This makes use of a geometrically natural boundary value problem for holomorphic curves in a 3-manifold with stable Hamiltonian structure, where the boundary conditions are defined by 1-parameter fam…
The paper develops a new Floer theory for 3-manifolds with involutions.
problem Constructing a Floer theory for 3-manifolds with symmetries.
method Develops a framed real monopole Floer homology for 3-manifolds with involutions and provides definitions for invariants.
result Defines Z-valued framed real Seiberg--Witten invariants for 4-manifolds with involutions. New invariants derived from a modular category for links and 3-manifolds.
problem Developing new invariants for links and 3-manifolds.
method Extracted invariants from a modular category with two simple objects.
result Invariants for 3-manifolds are related by a specific equation.
Given a real analytic function f from R4 to R2 with isolated critical point at the origin, the link Lf of the singularity is a real fibred knot in S3. From this singularities, we construct a family of real isolated suspension singularities from R6 to R2…
Study of twisted L2-torsion on 3-manifold character varieties.
problem Analyzing L2-torsion on character varieties of 3-manifolds. method Definition and study of twisted L2-torsion on character varieties, proving analyticity in hyperbolic cases. result Twisted L2-torsion is a real analytic function in a neighborhood of holonomy representations of hyperbolic 3-manifolds. We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian 3-manifold. We prove a rank rigidity theorem for complete 3-manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study 3-man…
Constructs stable maps from 3-manifolds to surfaces without cusps.
problem Creating stable maps from 3-manifolds to surfaces without problematic points.
method Visual construction of stable maps with specific properties.
result Obtains stable maps with no cusps and specific fiber structures.
Describes the space of spherical triangles on a smooth 3-manifold.
problem Understanding the geometric structure of spherical triangles.
method Analyzes the homotopy and analytic properties of the space.
result The space is a smooth 3-manifold embedded in R^6.
It is known by A. Loi and R. Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L' to L such that the union of the links …
We determine the contributions of isolated singularities of spin V 4-manifolds to the index of the Dirac operator over them. From these data we derive certain constraints on the intersection forms of spin 4-manifolds bounded by spherical 3-manifolds, and also on the embeddings of the real projective planes into 4-manif…
The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.
problem Understanding which 3-manifold groups can have convex co-compact representations.
method Analyzing representations of 3-manifold groups into projective general linear group, focusing on convex co-compactness.
result Fundamental groups of closed irreducible orientable 3-manifolds can only admit convex co-compact representations if they are geometric or hyperbolic.
Machine learning identifies 3-manifold triangulations using isomorphism signatures.
problem Differentiating and classifying 3-manifolds and their Dehn surgeries.
method Training machine learning models on isomorphism signatures derived from 3-manifold triangulations and Pachner graphs.
result Gradient saliency analysis reveals key parts of the language-like encoding scheme.
New lattices in higher rank contain a fixed 3-manifold group with increasing systole.
problem Finding lattices with a fixed 3-manifold group and large systole.
method Constructing arithmetic lattices in SL(8,R) with specific properties. result Existence of lattices with large systole containing a fixed 3-manifold group.
Study zippers in hyperbolic 3-manifolds, proving fixed point dichotomy.
problem Existence and behavior of elements in hyperbolic 3-manifold groups.
method Action analysis on minimal zippers, proving fixed point dichotomy.
result Every nontrivial element either fixes a unique point in each tree or acts freely on both.
Guts determine the leading coefficients of L2-Alexander torsions for 3-manifolds.
problem Determining the leading coefficient of L2-Alexander torsions for 3-manifolds. method Using a new criterion for the convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on guts of 3-manifolds.
result The leading coefficient equals the relative L2-torsion of the guts associated to the cohomology class. Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary. Consider any element φin the first cohomology of M with integral coefficients. Then one can define the φ-twisted L^2-torsion function of the universal covering which is a function from the set of positive real nu…
Reconstructing 3D manifolds from boundary electromagnetic data.
problem Reconstructing a compact, connected, real-analytic Riemannian 3-manifold from tangential electric and magnetic fields on its boundary.
method Factorizing Maxwell's equations and using an isometric transform to reconstruct the metric.
result The electromagnetic Dirichlet-to-Neumann map uniquely determines all derivatives of electromagnetic parameters on the boundary.
A flow defined by a nonsingular smooth vector field X on a closed manifold M is said to be parameter rigid if given any real valued smooth function f on M, there are a smooth funcion g and a constant c such that f=X(g)+c holds. We show that the parameter rigid flows on closed orientable 3-manifolds are sm…
The paper disproves a conjecture about 3D manifolds using even lattice points.
problem Thurston's Euler class one conjecture for fillable contact structures.
method Analyzing finite covers of hyperbolic 3-manifolds and properties of their dual Thurston norm unit balls.
result Found counter-examples to the conjecture using even lattice points on boundary.
The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.
problem Determining the virtual Euler class one in hyperbolic 3-manifolds.
method Analyzing Alexander polynomials and constructing taut foliations.
result Constructing examples of hyperbolic 3-manifolds with virtual Euler class one.
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
We propose an approach to find constant curvature metrics on triangulated closed 3-manifolds using a finite dimensional variational method whose energy function is the volume. The concept of an angle structure on a tetrahedron and on a triangulated closed 3-manifold is introduced following the work of Casson, Murakami …
New proof eliminates definite fold in higher dimensions.
problem Eliminating definite fold in higher dimensions.
method Homotopy and algorithmic procedures for constructing examples.
result Non-existence of singular Legendre fibrations on 3-manifolds.
We prove that for every compact, connected, differentiable 3--manifold M there is a compact complex manifold X which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that M is diffeomorphic to the set of real points of X. By earlier results, such an X can almost n…
The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic 3-manifold M, the inclusion from the isometry group of M to the diffeomorphism group of M…
Invariant of 3-manifolds using Hopf algebra elements.
problem Constructing an invariant for closed 3-manifolds.
method Using involutory Hopf algebra elements and normal o-graphs.
result Invariant values in cyclic quotient of Heisenberg double.
Cohomology fractals are visual representations of cohomology classes on hyperbolic 3-manifolds.
problem Visualizing cohomology classes on hyperbolic 3-manifolds.
method Cohomology fractals are images associated to cohomology classes. They are related to limit sets of Kleinian groups but differ in key aspects. An implementation using ideal triangulations and ray-casting is presented.
result Cohomology fractals allow for real-time zooming in any direction at arbitrary depth.
We prove that any real analytic strictly pseudoconvex CR 3-manifold is the boundary (at infinity) of a unique selfdual Einstein metric defined in a neighborhood. The proof uses a new construction of twistor space based on singular rational curves.
An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformati…
We give a simple and uniform construction of essentially all known deformation classes of gravitational instantons with ALF, ALG or ALH asymptotics and nonzero injectivity radius. We also construct new ALH Ricci flat metrics asymptotic to the product of a real line with a flat 3-manifold.
In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric s…