Paper · Mar 30, 2021

Classical Poincaré conjecture via 4D topology

Akio Kawauchi

math.GTarXiv:2103.16001v6
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Abstract

The classical Poincaré conjecture that every homotopy 3-sphere is diffeomorphic to the 3-sphere is confirmed by Perelman in arXiv papers solving Thurston's program on geometrizations of 3-manifolds. A new confirmation of this conjecture is given by a method of 4D topology. For this proof, the spun torus-knot of every knot in every homotopy 3-sphere is observed to be a ribbon torus-knot in the 4-sphere, where Smooth 4D Poincaré Conjecture and Ribbonness of a sphere-link with (not necessarily meridian-based) free fundamental group are used. By examining a disk-chord system of a ribbon solid torus bounded by the spun torus-knot,it is proved that the knot belongs to a 3-ball in the homotopy 3-sphere. Then by Bing's result, it is confirmed that the homotopy 3-sphere is diffeomorphic to the 3-sphere.

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Submitted
Mar 30, 2021
Last revised
Mar 31, 2024 (v6)
Subjects
math.GT

Comments: Many explanations and notations are revised

Geometrization3-manifolds4-manifolds

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 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)2Cao 2006 An Introduction to Geometric Topology (Bruno Martelli, 2016)4Martelli 2016 Uniform models and short curves for random 3-manifolds (Peter Feller, Alessandro Sisto and Gabriele Viaggi, 2019)5Feller 2019 Canonical geometrization of orientable 3-manifolds defined by vector-colourings of 3-polytopes (Nikolai Erokhovets, 2020)8Erokhovets 2020 On 4-dimensional smooth Poincare conjecture (Selman Akbulut, 2022)3Akbulut 2022 Geometrization in Geometry (Izabella Muraro de Freitas and Álvaro Krüger Ramos, 2022)1de Freitas 2022 Geometrisation of 3-manifolds (Bruno Martelli, 2026)7Martelli 2026 A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture (Yuan Liao, Ziyang Qin, Jui-Hui Chung et al., 2026)6Liao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

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

  2. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture

    Huai-Dong Cao and Xi-Ping ZhuThe canon

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

  3. On 4-dimensional smooth Poincare conjecture

    Selman Akbulut

    Topics 4-manifolds · Shared terms “Poincaré conjecture”

  4. An Introduction to Geometric Topology

    Bruno Martelli

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

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

    Peter Feller, Alessandro Sisto and Gabriele Viaggi

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

  6. 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 “Thurston”, “Perelman”

  7. Geometrisation of 3-manifolds

    Bruno Martelli

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

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

    Nikolai Erokhovets

    Topics Geometrization · 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) 4-manifolds Four is the first dimension where the singularities of Ricci flow are not classified. It is also the dimension of the still-open smooth Poincaré… Start with: The topology of four-dimensional manifolds (Freedman, 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.