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Topics in 3-manifold topology

V5D2 - Selected Topics in Topology - Topics in 3-manifold topology
Winter 2024/25, University of Bonn
Study program: Master of Science in Mathematics
Link to BASIS
Time and place: Wednesdays, 14:15 - 16:00 in Übungssaal II, Nussallee 17 (second floor)
First lecture: October 9th, 2024
↪ Announcements
🕮 Lecture notes and summaries
🖉 Exercise sheets
↪ Literature
↪ Course description
↪ Audience and prerequisites

Announcements ⤴
  • Preliminary exam information: Oral exams will be held on February 10-14, 2025 and March 17-19, 2025 (see also all lecture course exam dates). The exact dates and times will be determined at the end of the semester. Please let me know in advance (by the end of January at the latest) if you have special needs for the exam or if there are any time conflicts with other exams you're taking. The exams will cover both the lectures and the exercises.
  • Thank you for your feedback in the course evaluation!
  • The list of references (literature) has been updated regarding lens spaces. I also added a book by Farb and Margalit on mapping class groups (see "other literature").
  • 🖉 Exercise sheet 4 has been posted here. I also added some links regarding bonus exercise 6 (Seifert surfaces) to the references.
  • 🕮 My handwritten lecture notes for Lecture 8 and a corrected version of the lecture notes for Lecture 7 are now online.
🕮 Lecture notes and summaries ⤴
  • Lecture 1, Oct 9, 2024 - Introduction: examples and overview
  • Lecture 2, Oct 16, 2024 - Smooth vs. PL/triangulated vs. topological manifolds (also for \(n\neq 3\)); definition and examples of n-dimensional k-handles
  • Lecture 3, Oct 23, 2024 - Isotopic attaching maps give rise to diffeomorphic manifolds (proof via Isotopy Extension Thm and Collar Nbhd Thm), existence of handle decompositions, reordering of handles, handle cancellation, dual handle decomposition, unique handles of dimension 0 and n
  • Lecture 4, Oct 30, 2024 - Orientation, orientation-preserving smooth map, handlebodies, boundary connected sum \(\natural\), handlebodies are diffeomorphic to \(\natural_g (S^1 \times D^2)\), existence of Heegaard splittings via handle decompositions, (planar) Heegaard diagrams, examples
  • Lecture 5, Nov 6, 2024 - From Heegaard diagrams to Heegaard splittings to 3-manifolds, Heegaard moves: isotopies, handle slides, (de-)stabilization, Reidemeister-Singer theorem
  • Lecture on Nov 13 cancelled
  • Lecture 6, Nov 20, 2024 - Definition of Heegaard genus \(g(M)\), \(g(M)=0 \Leftrightarrow M \cong S^3\), \(g(S^1 \times S^2)=1\), \(g(T^3)=3\); Theorem: \(g(M)=1 \Leftrightarrow M \cong L_{p,q} \not\cong S^3\) or \(M \cong S^1\times S^2\); classification of lens spaces (up to homeomorphism and homotopy equivalence); proof of "\(\Leftarrow\)": \(g(L_{p,q})=1\); simple closed curves on the torus \(T^2\): meridian, longitude, characterization lemma
  • Lecture 7, Nov 27, 2024 (corrected version) (given by Arunima Ray) - Mapping class group of \(T^2\) (second part of characterization lemma), Heegaard diagrams of lens spaces, proof of "\(\Rightarrow\)" of above Theorem; knots and links, isotopy of links, linking number
  • No lecture on Dec 4 (Dies academicus)
  • Lecture 8, Dec 11, 2024 - tubular neighborhood, meridian and longitude of a knot; Dehn surgery along a knot, examples; Theorem 1: every closed, orientable, connected 3-manifold arises as Dehn surgery along a link; Kirby moves, Rolfsen twist; Theorem 2 by Kirby, Fenn-Rourke, Rolfsen
  • Lecture 9, Dec 18, 2024 - Notes to be posted - examples, in particular \(L_ {p,q}\) again (corrected version); Dehn twists and example, Theorem 3: mapping class group of surfaces generated by Dehn twists; proof of Theorem 1 using Theorem 3
🖉 Exercise sheets ⤴
Literature ⤴

The course will mainly follow the following introductory books on 3-manifolds.
🔎 This symbol indicates that the book is available at the Fachbibliothek Mathematik, Endenicher Allee 60.

  • J. Schultens, Introduction to 3-Manifolds, AMS, 2014. 🔎
  • V. Prasolov and A. Sossinsky, Knots, Links, Braids and 3-Manifolds: An Introduction to the New Invariants in Low-Dimensional Topology, AMS, 1997. 🔎

For the first part of the course on handle decompositions and Heegaard splittings, the books by Matsumoto, Rolfsen, and Gompf-Stipsicz mentioned below also contain a lot of helpful material.

Regarding lens spaces, besides the book by Prasolov and Sossinsky, you can find many equivalent characterizations in the book by Rolfsen (Chapter 9B). There are also handwritten notes (unpublished undergraduate thesis!) by Matthew Watkins, available on his website.

Below is a (long!) list of books/references if you would like to read more about certain topics. I don't expect you to read them! However, I will refer to them for details not covered in class.

Introductory books on 3-manifolds
  • B. Martelli, An Introduction to Geometric Topology. Available online on his website.
  • A. Hatcher, Notes on Basic 3-Manifold Topology. Available online on his website.
Handle decompositions
  • H. Geiges, How to depict 5-dimensional manifolds, Jahresbericht der DMV, 2017. Available online at the ArXiv.
  • R. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus, AMS, 1999. 🔎
Morse theory
  • Y. Matsumoto, An Introduction to Morse Theory, AMS, 2002. 🔎
  • J. Milnor, Lectures on the H-Cobordism Theorem, Princeton Univ. Press, 1965/2015. 🔎 (also online)
Differential topology
  • J. M. Lee, Introduction to Smooth Manifolds, Springer, 2013. 🔎
  • C. T. C. Wall, Differential Topology, Cambridge University Press, 2016. 🔎
  • M. W. Hirsch, Differential Topology, Springer, 2010. 🔎
  • A. Kosinski, Differential Manifolds, Chantilly: Elsevier Science, 1992. 🔎
  • J. Munkres, Elementary Differential Topology, Princeton University Press, 1963. 🔎 (online)
  • V. Guillemin and A. Pollack, Differential Topology, AMS, 2010. 🔎
Algebraic topology
  • S. Friedl, Lecture notes for Algebraic Topology I-IV. Available online here or on his website.
  • A. Hatcher, Algebraic topology, Cambridge Univ. Press, 2010. 🔎 Also available online on his website.
⚝ Knot theory
  • D. Rolfsen, Knots and Links, AMS, 2003. 🔎
  • W. B. R. Lickorish, An Introduction to Knot Theory, Springer, 1997. 🔎
⚝ Seifert surfaces

Some nice visualizations of Seifert surfaces for oriented links (oriented, connected, compact surfaces smoothly embedded in \(S^3\) with oriented boundary the link; see also Lecture 8 and Exercise Sheet 4) can be found e.g. here (I'm sure there's more such sources online):

Other literature mentioned in the lecture notes/in class
  • W. P. Thurston and S. Levy, Three-Dimensional Geometry and Topology, Volume 1, Princeton University Press, 1997. 🔎
  • S. Behrens, B. Kalmár, M. H. Kim, M. Powell, A. Ray, The Disc Embedding Theorem, Oxford University Press, 2021. 🔎
  • B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Univ. Press, 2012. 🔎 (also online)
Course description ⤴

This course is about 3-manifolds, i.e. topological spaces locally modeled on 3-dimensional Euclidean space. Our goal is to introduce several classes of examples of 3-manifolds and to study them via embedded 1- and 2-dimensional submanifolds (knots, links and surfaces). For example, we will define and study lens spaces, Seifert fibered spaces and knot exteriors. We will discuss several important structure/decomposition theorems for 3-manifolds. For example, every 3-manifold admits a Heegaard splitting along a surface and can be obtained by Dehn surgery along a link. We will see that there is a prime decomposition theorem for 3-manifolds and that irreducible 3-manifolds can be further decomposed along embedded tori (so-called JSJ decomposition), and we will talk about other fundamental results from 3-dimensional topology such as the Schönflies/Alexander theorem or Dehn's lemma with important consequences for knot theory.

Audience and prerequisites ⤴

The lecture is aimed at mathematics students (advanced bachelor, master and PhD students) with a basic knowledge and interest in topology. Non-specialist researchers are also welcome.
Please note that credit points for this lecture can only be awarded to Master of Science Mathematics students.

Prerequisites are point set topology and some algebraic topology (as covered in Introduction to Geometry and Topology/Einführung in die Geometrie und Topologie; see e.g. this summer's course website), in particular (from the module handbook) metric and topological spaces and their construction, concepts of connectedness, compactness, covering spaces and fundamental groups. Some knowledge of singular homology theory is helpful but not strictly necessary, so it should be possible to follow Topology I/Topologie I in parallel. Some experience with differential topology, and manifolds in particular, will be useful but not strictly necessary for understanding the lecture. We will summarize and recall important definitions and results.

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Last updated on December 18, 2024.