Paper · Sep 17, 2022

On 4-dimensional smooth Poincare conjecture

Selman Akbulut

math.GTarXiv:2209.09968v8
View PDF View HTML arXiv page

As on arXiv

Abstract

Here we outline a proof for the 4-dimensional smooth Poincare Conjecture.

View PDF View HTML arXiv page
Submitted
Sep 17, 2022
Last revised
Jul 30, 2024 (v8)
Subjects
math.GT

Comments: 8 pages, 10 figures

Poincaré conjecture4-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 2021, 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 and the Poincare Conjecture (John W. Morgan and Gang Tian, 2006)1Morgan 2006 Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture (Huai-Dong Cao and Xi-Ping Zhu, 2006)3Cao 2006 Completion of the Proof of the Geometrization Conjecture (John Morgan and Gang Tian, 2008)8Morgan 2008 Product structure and regularity theorem for totally nonnegative flag varieties (Huanchen Bao and Xuhua He, 2022)7Bao 2022 Rigor with Machine Learning from Field Theory to the Poincaré Conjecture (Sergei Gukov, James Halverson and Fabian Ruehle, 2024)5Gukov 2024 Moving Manifolds and the Poincare Conjecture (David V. Svintradze, 2025)4Svintradze 2025 Equations in Products of Free Groups and 3-Manifold Groups, I (Olga Kharlampovich and Alina Vdovina, 2026)6Kharlampovic 2026 A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture (Ziyang Qin, Yuan Liao, Ayush Khaitan et al., 2026)2Qin 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow and the Poincare Conjecture

    John W. Morgan and Gang TianThe canon

    Topics Poincaré conjecture

  2. A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture

    Ziyang Qin, Yuan Liao, Ayush Khaitan et al.

    Topics Poincaré conjecture

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

    Huai-Dong Cao and Xi-Ping ZhuThe canon

    Topics Poincaré conjecture

  4. Moving Manifolds and the Poincare Conjecture

    David V. Svintradze

    Topics Poincaré conjecture

  5. Rigor with Machine Learning from Field Theory to the Poincaré Conjecture

    Sergei Gukov, James Halverson and Fabian Ruehle

    Topics Poincaré conjecture

  6. Equations in Products of Free Groups and 3-Manifold Groups, I

    Olga Kharlampovich and Alina Vdovina

    Topics Poincaré conjecture

  7. Product structure and regularity theorem for totally nonnegative flag varieties

    Huanchen Bao and Xuhua He

    Topics Poincaré conjecture

  8. Completion of the Proof of the Geometrization Conjecture

    John Morgan and Gang TianThe canon

    Topics Poincaré conjecture

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.

Poincaré conjecture Every closed, simply connected 3-manifold is homeomorphic to the 3-sphere. Poincaré posed the question in 1904, and Perelman proved it in 2002–03… Start with: Generalized Poincare's Conjecture in Dimensions Greater Than Four (Smale, 1961) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982)

In the guidebook

  1. 7A.12
    The Poincaré Conjecture, PreciselyBook 7A · Topology and the Fundamental Group

    Every word of the statement, and why each hypothesis is needed.

  2. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.

  3. 10A.6
    Necks, Caps and Their TopologyBook 10A · Three-Manifolds and Geometrization

    What an ε-neck is, how necks overlap and chain into tubes, and which manifolds necks and caps can build.