3-Manifolds

When it's finished, this book will be a modern introduction to 3-manifolds. This is a very big subject, and the book wants to be as short as possible, so there is no hope of being comprehensive (comprehensible is another matter). Nevertheless, I've tried to be complete and rigorous (if brief) whenever mathematical reality allows.

The origin of the book is the notes I wrote for a series of graduate classes I taught at the University of Chicago over several years, some of these years being in the future (as of 1/1/2020). I always intended for these notes to become the chapters of a single book. At the moment several of these chapters are in a very rudimentary state and I haven't yet begun the serious work of integrating them. Caveat lector.

The final version will be published by Cambridge University Press, but some version of the book will remain freely downloadable after publication.

TABLE OF CONTENTS (* means not finished; ** means not started)

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• Preface
• Chapter 0: Examples
  • Building 3-manifolds **
  • Lens spaces **
  • Seifert fibered spaces **
  • Surface bundles **
• Chapter 1: Combinatorial foundations (version 12/7/2021) 56 pages
  • Dehn's Lemma and the Loop Theorem
  • Stallings' theorem on ends, and the Sphere Theorem
  • Prime and free decompositions, and the Scott Core Theorem
  • Haken manifolds
  • The Torus Theorem and the JSJ decomposition *
  • Combinatorial Algorithms *
• Chapter 2: Hyperbolic geometry (version 2/15/2019) 52 pages
  • Models of hyperbolic space
  • Building hyperbolic manifolds
  • Rigidity and the thick-thin decomposition
  • Quasiconformal deformations and Teichmuller theory *
  • Hyperbolization for Haken manifolds *
  • Tameness **
  • Ending laminations **
• Chapter 3: Minimal surfaces (version 5/15/2017) 38 pages
  • Minimal surfaces in Euclidean space
  • First variation formula
  • Second variation formula
  • Existence of minimal surfaces
  • Embedded minimal surfaces in 3-manifolds *
• Chapter 4: Taut Foliations (version 1/20/2024) 46 pages
  • Foliations *
  • Reeb components and Novikov's Theorem
  • Taut foliations *
  • Finite depth foliations and the Thurston norm *
  • Holomorphic geometry *
  • Universal circles *
  • Essential laminations *
  • RFRS and the Virtual Fibration Conjecture **
• Chapter 5: Symplectic and Contact geometry (version 11/28/2023) 31 pages
  • Symplectic geometry *
  • Contact structures *
  • J-holomorphic curves *
  • Floer homology *
• Chapter 6: Floer theories (version 2/25/2020) 41 pages
  • Classical Invariants *
  • The Casson Invariant *
  • Instanton homology *
  • Heegaard Floer Homology *
  • Proofs *
  • Computation and Examples **
• Chapter 7: Ricci Flow (version 12/10/2019) 62 pages
  • The Hamilton-Perelman program
  • Mean curvature flow: a comparison
  • Curvature evolution and pinching
  • Singularities and Limits
  • Perelman's monotone functionals
  • The Geometrization Conjecture *
• Chapter 8: Cube complexes (version 6/14/2013) 44 pages
  • The Virtual Haken Conjecture
  • Special Cube Complexes
  • Codimension 1 subgroups
  • Almost geodesic surfaces
  • Relative hyperbolicity
  • MVH and QVH
  • Proof of the VHC
• Appendix: Moise's Theorem
• References
• Index