Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization

@article{Kuperberg2019AlgorithmicHO,
  title={Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization},
  author={Greg Kuperberg},
  journal={Pacific Journal of Mathematics},
  year={2019}
}
  • G. Kuperberg
  • Published 27 August 2015
  • Mathematics
  • Pacific Journal of Mathematics
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism problem for closed, oriented, triangulated 3-manifolds. We give a self-contained proof, with several variations at each stage, that uses only the statement of the geometrization theorem, basic hyperbolic geometry, and old results from combinatorial topology and… 

Figures from this paper

On the treewidth of triangulated 3-manifolds
TLDR
It is shown that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth, then the Heegaard genus of M is at most 18(k + 1) (resp. pathwidth) k, and it is not shown that there exists an infinite family of closed 3- manifolds not admitting triangulations of bounded pathwidth or trewidth.
On the treewidth of triangulated 3-manifolds
TLDR
It is shown that if a closed, orientable, irreducible, non-Haken 3-manifold M has a triangulation of treewidth, then the Heegaard genus of M is at most 48(k+1) (resp. pathwidth) k, and it is not shown that there exists an infinite family of closed 3- manifolds not admitting triangulations of bounded pathwidth or trewidth.
Finding Non-Orientable Surfaces in 3-Manifolds
TLDR
It is proved that the complexity of finding an embedded non-orientable surface of Euler genus g in a triangulated 3-manifold is NP-hard, thus adding to the relatively few hardness results that are currently known in 3- manifold topology.
Embeddability in R3 is NP-hard
TLDR
This reduction encodes a satisfiability instance into the embeddability problem of a 3-manifold with boundary tori, and relies extensively on techniques from low-dimensional topology, most importantly Dehn fillings of manifolds with Boundary tori.
An Upper Bound on Pachner Moves Relating Geometric Triangulations
We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This
A Polynomial-Time Algorithm to Compute Turaev-Viro Invariants $\mathrm {TV}_{4, q}$ of 3-Manifolds with Bounded First Betti Number
TLDR
A fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter, which is, to the authors' knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter.
Computational Complexity of Enumerative 3-Manifold Invariants
TLDR
The results are analogs of the famous results of Freedman, Larsen and Wang establishing the quantum universality of topological quantum computing with the Jones polynomial at a root of unity.
Some Conditionally Hard Problems on Links and 3-Manifolds
  • M. Lackenby
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 2017
We show that three natural decision problems about links and 3-manifolds are computationally hard, assuming some conjectures in complexity theory. The first problem is determining whether a link in
Systole Length in Hyperbolic $n$-Manifolds
We show that the length R of a systole of a closed hyperbolic nmanifold (n ≥ 3) admitting a triangulation by t n-simplices can be bounded below by a function of n and t, namely R ≥ 1 2(nt) O(n4t) .
Combinatorial width parameters for 3-dimensional manifolds
TLDR
This thesis establishes quantitative relations between the treewidth and classical topological invariants of a 3-manifold, and shows that thetreewidth of a closed, orientable, irreducible, non-Haken 3- manifold is always within a constant factor of its Heegaard genus.
...
1
2
3
...

References

SHOWING 1-10 OF 36 REFERENCES
Volumes of hyperbolic three-manifolds
BY“hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold M of finite volume. By Mostow rigidity the volume of M is a topological invariant, indeed a homotopy invariant, of
On the triangulation of manifolds and the Hauptvermutung
1. The first author's solution of the stable homeomorphism conjecture [5] leads naturally to a new method for deciding whether or not every topological manifold of high dimension supports a piecewise
Seifert fibered spaces in 3-manifolds
Publisher Summary This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy
Algorithms for essential surfaces in 3-manifolds
In this paper we outline several algorithms to find essential surfaces in 3dimensional manifolds. In particular, the classical decomposition theorems of 3-manifolds ( Kneser-Milnor connected sum
Ricci Flow and Geometrization of 3-Manifolds
This book is based on lectures given at Stanford University in 2009. The purpose of the lectures and of the book is to give an introductory overview of how to use Ricci flow and Ricci flow with
P.L. Homeomorphic Manifolds are Equivalent by Elementary 5hellingst
  • U. Pachner
  • Computer Science, Mathematics
    Eur. J. Comb.
  • 1991
TLDR
This paper studies transformations of simplicial p.l. manifolds by elementary boundary operations (shellings and inverse shellings) and shows that a simplical p.
Algorithmic aspects of homeomorphism problems
We will describe some results regarding the algorithmic nature of homeomorphism problems for manifolds; in particular, the following theorem. Theorem 1: Every PL or smooth simply connected manifold
MOM TECHNOLOGY AND VOLUMES OF HYPERBOLIC 3-MANIFOLDS
This paper is the first in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Here we introduce Mom technology and enumerate the hyperbolic
Suspensions of homology spheres
This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. It presents the initial solutions of the fabled
Infinitesimal Rigidity of Convex Polyhedra through the Second Derivative of the Hilbert–Einstein Functional
Abstract The paper presents a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert–Einstein functional on the space of
...
1
2
3
4
...