Paper · Oct 8, 2016

An Introduction to Geometric Topology

Bruno Martelli

math.GTarXiv:1610.02592v3
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Abstract

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem.

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Submitted
Oct 8, 2016
Last revised
Apr 5, 2022 (v3)
Subjects
math.GT, math.DG

Comments: 486 pages, more than 200 colour figures. Second version of April 2022, with minor corrections

Geometrization3-manifoldsStability & uniqueness

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 Ricci flow on cone surfaces and a three-dimensional expanding soliton (Daniel Ramos, 2017)5Ramos 2017 Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow (Paul Bryan, Mohammad N. Ivaki and Julian Scheuer, 2019)2Bryan 2019 Uniform models and short curves for random 3-manifolds (Peter Feller, Alessandro Sisto and Gabriele Viaggi, 2019)6Feller 2019 How to see the eight Thurston geometries (Tiago Novello, Vinícius da Silva, Luiz Velho et al., 2020)7Novello 2020 Classical Poincaré conjecture via 4D topology (Akio Kawauchi, 2021)8Kawauchi 2021 Large-scale geometry of the Universe (Yassir Awwad and Tomislav Prokopec, 2022)4Awwad 2022 Combinatorial Ricci Flow and Thurston's Triangulation Conjecture (Feng Ke and Ge Huabin, 2025)3Ke 2025 Geometrisation of 3-manifolds (Bruno Martelli, 2026)1Martelli 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometrisation of 3-manifolds

    Bruno Martelli

    Topics Geometrization, 3-manifolds · Same author Bruno Martelli · Shared terms “geometrisation”, “Thurston”, “describe”

  2. Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow

    Paul Bryan, Mohammad N. Ivaki and Julian Scheuer

    Topics Geometrization, 3-manifolds, Stability & uniqueness · Shared terms “Thurston Geometrisation”

  3. Combinatorial Ricci Flow and Thurston's Triangulation Conjecture

    Feng Ke and Ge Huabin

    Topics Geometrization, 3-manifolds, Stability & uniqueness · Shared terms “Thurston”, “decomposition”

  4. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”, “Perelman”

  5. Ricci flow on cone surfaces and a three-dimensional expanding soliton

    Daniel Ramos

    Topics Geometrization, 3-manifolds, Stability & uniqueness · Shared terms “Perelman”

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

    Peter Feller, Alessandro Sisto and Gabriele Viaggi

    Topics Geometrization, 3-manifolds · Shared terms “topological”, “Perelman”

  7. How to see the eight Thurston geometries

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

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”

  8. Classical Poincaré conjecture via 4D topology

    Akio Kawauchi

    Topics Geometrization, 3-manifolds · Shared terms “Thurston”, “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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.