Paper · Nov 23, 2020

Canonical geometrization of orientable -manifolds defined by vector-colourings of -polytopes

Nikolai Erokhovets

math.GTarXiv:2011.11628v2
View PDF View HTML arXiv page

As on arXiv

Abstract

In short geometrization conjecture of W.\,Thurston (finally proved by G. Perelman) says that any oriented -manifold can be canonically partitioned into pieces, which have a geometric structure of one of the eight types. In the seminal paper (1991) M.\,W.\,Davis and T.\,Januszkiewicz introduced a wide class of -dimensional manifolds – small covers over simple -polytopes. We give a complete answer to the following problem: to build an explicit canonical decomposition for any orientable -manifold defined by a vector-colouring of a simple -polytope, in particular for a small cover. The proof is based on analysis of results in this direction obtained before by different authors.

View PDF View HTML arXiv page
Submitted
Nov 23, 2020
Last revised
Aug 4, 2021 (v2)
Subjects
math.GT, math.AT

Comments: 40 pages, 1 figure. In the new version main results are generalized to 3-manifolds defined by vector-colourings, and the exposition is clarified

Geometrization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Geometrization of the local Langlands correspondence: an overview (Laurent Fargues, 2016)4Fargues 2016 Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local (Laurent Fargues, 2017)5Fargues 2017 Etale cohomology of diamonds (Peter Scholze, 2017)7Scholze 2017 Higgs bundles and geometric structures on manifolds (Daniele Alessandrini, 2018)3Alessandrini 2018 Uniform models and short curves for random 3-manifolds (Peter Feller, Alessandro Sisto and Gabriele Viaggi, 2019)8Feller 2019 How to see the eight Thurston geometries (Tiago Novello, Vinícius da Silva, Luiz Velho et al., 2020)6Novello 2020 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)2de Freitas 2022 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)1Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “geometrization conjecture”, “Thurston”, “Perelman”

  2. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization · Shared terms “Thurston”, “Perelman”

  3. Higgs bundles and geometric structures on manifolds

    Daniele Alessandrini

    Topics Geometrization · Shared terms “geometric structure”, “geometrization conjecture”, “Thurston”

  4. Geometrization of the local Langlands correspondence: an overview

    Laurent Fargues

    Topics Geometrization · Shared terms “geometrization conjecture”

  5. Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

    Laurent Fargues

    Topics Geometrization · Shared terms “geometrization conjecture”

  6. How to see the eight Thurston geometries

    Tiago Novello, Vinícius da Silva, Luiz Velho et al.

    Topics Geometrization · Shared terms “geometrization conjecture”, “Thurston”, “dimensional manifolds”

  7. Etale cohomology of diamonds

    Peter Scholze

    Topics Geometrization · Shared terms “geometrization conjecture”

  8. Uniform models and short curves for random 3-manifolds

    Peter Feller, Alessandro Sisto and Gabriele Viaggi

    Topics Geometrization · Shared terms “geometrization conjecture”, “Perelman”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.