This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equivalent to the finite fundamental group case of the Geometrization Conjecture, and the other is equivalent to the union of the Geometrization Conjecture and Thurston's Virtual Bundle Conjecture.

1999-06-23math.GRmath.GT

Completion of the Proof of the Geometrization Conjecture

This article is a sequel to the book `Ricci Flow and the Poincare Conjecture' by the same authors. Using the main results of that book we establish the Geometrization Conjecture for all compact, orientable three-manifolds following the approach indicated by Perelman in his preprints on the subject. This approach is to study the collapsed part of the manifold as time goes to infinity in a Ricci flow with surgery. The main technique for this study is the theory of Alexandrov spaces. This theory gives local models for the collapsed part of the manifold. These local models can be glued together to…

2008-09-23math.DGmath.GT

Geometrization in Geometry

So far, the most magnificent breakthrough in mathematics in the 21st century is the Geometrization Theorem, a bold conjecture by William Thurston (generalizing Poincaré's Conjecture) and proved by Grigory Perelman, based on the program suggested by Richard Hamilton. In this survey article, we will explain the statement of this result, also presenting some examples of how it can be used to obtain interesting results in differential geometry.

2022-10-18math.DGmath.GT

Geometrization of 3-dimensional Coxeter orbifolds and Singer's conjecture

Associated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $Σ_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $Σ_L$ admits a cellulation under which the nerve of each vertex is $L$. It follows that if $L$ is a triangulation of $\mathbb{S}^{n-1}$, then $Σ_L$ is a contractible $n$-manifold. In this case, the orbit space, $K_L:=Σ_L/W$, is a \emph{Coxeter orbifold}. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into t…

2008-10-28math.ATmath.GRmath.GT

A counterexample to Batson's conjecture

We show that the torus knot $T_{4,9}$ bounds a smooth Möbius band in the $4$-ball, giving a counterexample to Batson's non-orientable analogue of Milnor's conjecture on the smooth slice genera of torus knots.

2019-05-31math.GT

An infinite family of counterexamples to Batson's conjecture

Batson's conjecture is a non-orientable version of Milnor's conjecture, which states that the 4-ball genus of a torus knot $T(p,q)$ is equal to $\frac{(p-1)(q-1)}{2}$. Batson's conjecture states that the nonorientable 4-ball genus is equal to the pinch number of a torus knot, i.e. the number of a specific type of (nonorientable) band surgeries needed to obtain the unknot. The conjecture was recently proved to be false by Lobb. We will show that Lobb's counterexample fits into an infinite family of counterexamples.

2020-10-30math.GT

更多相关论文收起更多论文

How to see the eight Thurston geometries

In this expository paper, we present a survey about the history of the geometrization conjecture and the background material on the classification of Thurston's eight geometries. We also discuss recent techniques for immersive visualization of relevant three-dimensional manifolds in the context of the Geometrization Conjecture.

2021-09-14cs.GRmath.GTmath.HO

环带数猜想的一个反例

A counterexample to the wrapping number conjecture

本文构造了一个环带数为四的环带纽结，其Kauffman括号环带度至多二，从而否证了环带数猜想。

2026-09-15math.GT

对Luo的三个猜想的一个反例

A Counterexample to Three Conjectures of Luo

本文针对Luo提出的三个猜想，构造了一个透镜空间L(3,1)的极小两个四面体三角剖分反例，通过四边形匹配方程等方法完成证明，否定了相关猜想。

2026-10-14math.GT

A counterexample to the Bernhard-Jablan unknotting conjecture

We show that there is a knot satisfying the property that for each minimal crossing number diagram of the knot and each single crossing of the diagram, changing the crossing results in a diagram for a knot whose unknotting number is at least that of the original knot, thus giving a counterexample to the Bernhard-Jablan Conjecture.

2017-12-31math.GT

Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to (S^1 x S^2) # (S^1 x S^2). We exhibit a family of such links that are probably counterexamples to General…

2011-03-08math.GT

Allen猜想的反例

Counterexamples to Allen's conjectures

本文构造了环面纽结$T(2,5)$和$T(2,9)$在4球中的光滑Möbius带，其双分支覆盖负定，反驳了Allen的两个猜想。

2024-07-02math.GT

A Counterexample to the Generalisation of Witten's Conjecture

This note provides a counterexample to a conjecture by Marché about the structure of the Kauffman bracket skein module for closed compact oriented 3-manifolds over the ring of Laurent polynomials.

2022-05-03math.GTmath.QA

A Counterexample to a Conjecture about Positive Scalar Curvature

Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based on the counterexample to the unstable Gromov-Lawson-Rosenberg conjecture given in Schick: "A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture".

2013-08-16math.ATmath.GTmath.KT

HK猜想的一个低维反例

A Low-Dimensional Counterexample to the HK-Conjecture

本文构造了HK猜想在平坦流形里程计中的低维反例，证明维度至少为4且最小，并讨论了其对Smale空间群胚的影响。

2025-07-07math.GTmath.KTmath.OA