A universal branched 3-manifold W characterizes Sol 3-manifolds.
problem Characterizing Sol 3-manifolds
method Using a universal branched 3-manifold W and a regular language result A closed 3-manifold M immerses into W if and only if M admits a Sol structure. We prove a finiteness result for the ∂-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and ∂-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a…
Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.
problem Finding the minimum number of tetrahedra in triangulations of 3-manifolds.
method Computed the triangulation complexity of all elliptic and sol 3-manifolds, within a bounded error.
result Computed the triangulation complexity of all elliptic and sol 3-manifolds.
Study of η-Ricci solitons on Kenmotsu 3-manifolds.
problem Exploring η-Ricci solitons on Kenmotsu 3-manifolds. method Examined various types of η-Ricci solitons on Kenmotsu 3-manifolds, including those with specific curvature conditions. result Existence of proper η-Ricci solitons on Kenmotsu 3-manifolds. 3-manifolds are CR uniformized on spheres, proving a conjecture.
problem Uniformizing 3-manifolds with cusps using CR methods.
method Spherical CR uniformization of complex hyperbolic triangle groups.
result Magic 3-manifolds and other cusped 3-manifolds are CR uniformizable.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
In this paper, we proved that any closed orientable 3-manifold 1-dominates at most finitely many geometric 3-manifolds.
Classifies 3-manifolds with uniformly positive scalar curvature.
problem Classifying 3-manifolds with uniformly positive scalar curvature.
method Analyzes properties of 3-manifolds with mean convex boundaries and uniformly positive scalar curvature.
result 3-manifolds with uniformly positive scalar curvature are homeomorphic to sums of spherical 3-manifolds and S1imesS2. Survey article on multibranched surfaces in 3-manifolds.
problem Embedding multibranched surfaces into 3-manifolds.
method Survey and review of existing methods.
result Overview of current research and techniques.
Study describes prosoluble subgroups in 3-manifold groups.
problem Characterizing prosoluble subgroups in 3-manifold groups.
method Analyzes finitely generated prosoluble subgroups of profinite completions.
result Provides a description of these subgroups.
Survey on L^2-invariants for 3-manifolds.
problem Understanding L^2-invariants for 3-manifolds.
method Survey and focus on L^2-invariants for 3-manifolds.
result Survey results on L^2-invariants for 3-manifolds.
Matveev introduced Borromean surgery on 3-manifolds, and proved that the equivalence relation on closed, oriented 3-manifolds generated by Borromean surgeries is characterized by the first homology group and the torsion linking pairing. Massuyeau generalized this result to closed, spin 3-manifolds, and the second autho…
The paper studies entropy in branched covers of 3-manifolds.
problem Entropy of pseudo-Anosov monodromies in fibered 3-manifolds.
method Analyzes the minimal entropy of genus g surface bundles over the circle as 2-fold branched covers of 3-manifolds.
result There exist infinitely many 3-manifolds with comparable minimal entropy to 1/g.
Virtual 3-manifolds were introduced by S.V. Matveev in 2009 as natural generalizations of the classical 3-manifolds. In this paper, we introduce a notion of complexity of a virtual 3-manifold. We investigate the values of the complexity for virtual 3-manifolds presented by special polyhedra with one or two 2-co…
Classifies low-volume hyperbolic 3-manifolds with a maximal cusp.
problem Identifying hyperbolic 3-manifolds with minimal volume and maximal cusps.
method Maximal cusp volume classification and analysis of low-volume manifolds.
result Figure-8 knot complement is unique in certain volume and filling categories.
Study series invariants for plumbed 3-manifolds and their properties.
problem Understanding series invariants for plumbed 3-manifolds and their applications.
method Twisted root lattice, gluing and splitting properties, explicit description of lens spaces and Brieskorn spheres.
result Series verify gluing and splitting properties of 3-manifolds.
Paper shows unique decomposition of 3-manifolds and multiplicative property of Reidemeister torsion.
problem Unique decomposition of compact 3-manifolds.
method Study of adjoint Reidemeister torsion on disk sum decompositions.
result Adjoint Reidemeister torsion has a multiplicative property on unique decompositions.
Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.
problem Volume conjecture for Seifert fibered and graph 3-manifolds.
method Large r asymptotic behavior of Turaev-Viro invariants under gluing operation. result Volume conjecture proven for Seifert fibered and graph 3-manifolds.
Classifies involutions on spherical 3-manifolds.
problem Classifying involutions on spherical 3-manifolds.
method Geometric approach to conjugacy classification.
result Insights into topological properties of involutions.
Infinite family of hyperbolic 3-manifolds with large volumes.
problem Finding large volume 3-manifolds.
method 2-fold branched covers of 3-manifolds M.
result Existence of hyperbolic 3-manifolds with arbitrarily large volume.
The study of universal links in 3-manifolds and their properties.
problem Existence and characterization of universal links in 3-manifolds.
method Analyzing branched coverings and distinguishing between universal and complement universal links.
result Closed spherical 3-manifolds are the only ones admitting universal links.
Computes Vafa-Witten invariants of 3-manifolds.
problem Computing invariants of 3-manifolds.
method Explicit computations using Vafa-Witten theory.
result Explicit invariants computed for specific 3-manifolds.
This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follow…
3-manifolds with two abelian-filled spaces.
problem Finding 3-manifolds with multiple abelian-filled spaces.
method Constructing 3-manifolds with specific embedding properties.
result 3-manifolds with at least two abelian-filled spaces.
It is well-known that Heegaard genus is additive under connected sum of 3-manifolds. We show that Heegaard genus of contact 3-manifolds is not necessarily additive under contact connected sum. We also prove some basic properties of the contact genus (a.k.a. open book genus) of 3-manifolds, and compute this invariant fo…
Deciding 3-manifolds fibering over the circle is in NP
problem Deciding if a 3-manifold fibres over the circle
method Show fibredness lies in NP
result Problem is NP-complete
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…
Surveying recent progress on hyperbolic 3-manifold rigidity.
problem Profinite rigidity of hyperbolic 3-manifolds.
method Review of profinite completion and rigidity of groups, evidence from other types of 3-manifolds, and existing ideas.
result Positive evidence for profinite rigidity in hyperbolic 3-manifolds.
Classifies 3-manifolds from cube identifications.
problem Classifying non-orientable 3-manifolds.
method Identifying cube faces to form manifolds, classifying based on surface-complexity.
result Identifies four flat non-orientable 3-manifolds with surface-complexity one.
3-manifolds study Hasse norm principle, akin to number fields.
problem Topological analog of Hasse norm principle for 3-manifolds.
method Analogy with number fields and finite cyclic coverings.
result Showed topological Hasse norm principle for 3-manifolds.
Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
Surfaces in 3-manifolds concentrate at curvature critical points.
problem Understanding concentration of surfaces in 3-manifolds.
method Proving surfaces concentrate at critical points of scalar curvature.
result Simply connected H-surfaces concentrate at curvature critical points.
Equivalent categories of groups and risandles for 3-manifolds.
problem Constructing invariants of 3-manifolds.
method Introducing risandles and showing equivalence with groups, categorifying coloring invariants.
result Equivalence of fundamental groups and risandles for infinitely many 3-manifolds.
3-manifolds can be virtually dominated by maps of degree 8.
problem Understanding virtual domination of 3-manifolds.
method Proving existence of finite covers with specific map properties.
result Virtual 8-dominance of 3-manifolds.
First the title could be also understood as ``3-manifolds related by non-zero degree maps" or "Degrees of maps between 3-manifolds" for some aspects in this survey talk. The topology of surfaces was completely understood at the end of 19th century, but maps between surfaces kept to be an active topic in the 20th centur…
Study tight contact structures on specific 3-manifolds.
problem Counting and constructing tight contact structures on plumbed 3-manifolds.
method Algorithm to construct stein diagrams for tight structures without Giroux torsion.
result Explicit algorithm to construct tight contact structures.
Proves existence of strongly overtwisted contact structures on 3-manifolds.
problem Existence of overtwisted contact structures on 3-manifolds.
method Complete h-principle for overtwisted disks and Legendrians.
result Homotopy type of overtwisted disk embeddings is determined.
Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
problem Identifying Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.
method Comprehensive classification of commensurability classes of cusped, arithmetic, and non-arithmetic complex hyperbolic 2-manifolds.
result Some Nil 3-manifolds are cross-sections in every commensurability class, while others are cross-sections in only one.
We describe theoretical backgrounds for a computer program that recognizes all closed orientable 3-manifolds up to complexity 8. The program can treat also not necessarily closed 3-manifolds of bigger complexities, but here some unrecognizable (by the program) 3-manifolds may occur.
Study of transformations in 3-manifolds with boundary and their equivalence classes.
problem Understanding transformations in 3-manifolds with boundary and their equivalence classes.
method Investigation of extended Andrews-Curtis transformations and equivalence classes of simple balanced 3-manifolds.
result Every balanced 3-manifold in the trivial equivalence class admits a simplifier to a trivial balanced 3-manifold.
New method to calculate 3-manifold invariants via skew-racks.
problem Calculating invariants of 3-manifolds.
method Introducing skew-racks with good involution and Property FR, defining cocycle invariants.
result Established new approach to obtain 3-manifold invariants via Dehn surgery.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
Proves graph 3-manifold groups have two specific properties.
problem Understanding fundamental groups of graph 3-manifolds.
method Constructing sequences of covers to prove properties.
result Graph 3-manifold groups are virtually poly-free and in Lex family.
The simplicial volume of non-R^3 contractible 3-manifolds is infinite.
problem Characterizing contractible 3-manifolds based on their simplicial volume.
method Analyzing the simplicial volume of contractible 3-manifolds and open 3-manifolds.
result The Euclidean space is the unique contractible 3-manifold with vanishing minimal volume.
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.
New examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.
Proof of Tholozan's volume formula for anti-de-Sitter 3-manifolds.
problem Volume calculation for closed anti-de-Sitter 3-manifolds.
method Chern-Simons theory approach.
result Proof of Tholozan's volume formula.
Random hyperbolic 3-manifolds can be obtained via Dehn surgery.
problem Understanding the generic properties of hyperbolic 3-manifolds.
method Counting model on links and Dehn surgeries.
result Random hyperbolic 3-manifolds can be obtained via Dehn surgery.