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Book 10A

Three-Manifolds and Geometrization

What the answer should look like

A zoo of 3-manifolds, Heegaard splittings and surgery, the prime and JSJ decompositions, Thurston’s eight geometries, hyperbolic manifolds, and the min–max ideas behind finite extinction.

  • Main companionNotes on Basic 3-Manifold Topology · Allen Hatcher
  • Main companionThe Geometries of 3-Manifolds · Peter Scott
  • Main companionThree-Dimensional Geometry and Topology · William Thurston
In preparation

Contents

  1. 1
    A Zoo of Three-Manifolds

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

    In preparation

  2. 2
    Building Three-Manifolds

    Heegaard splittings and Dehn surgery.

    In preparation

  3. 3
    The Prime Decomposition

    Cutting along spheres, and why surgery is a prime decomposition done by geometry.

    In preparation

  4. 4
    Seifert Spaces and the JSJ Decomposition

    Circle fibrations, incompressible tori and graph manifolds.

    In preparation

  5. 5
    Thurston’s Eight Geometries

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

    In preparation

  6. 6
    Hyperbolic Three-Manifolds

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

    In preparation

  7. 7
    Geometrization and Ricci Flow

    The fate of each piece under the flow.

    In preparation

  8. 8
    Min–Max and Width

    Sweepouts, Birkhoff’s argument, and the width that forces extinction.

    In preparation

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