RTG-Lecture: Handling the Poincaré conjecture

Marc Kegel

Winter term 2023/24

Lecture: The lectures will take place at the RTG days. For details on the dates and times see here. The first lecture takes place on 24. October, 10:00 - 11:30 in Kleiner Hörsaal INF 231 COS.

Content: Poincaré conjectured in 1904 that a manifold that has the same algebraic topology as a sphere is already a sphere. Since then the various versions of this conjecture have turned out to be the leading questions in geometric and differential topology. In the last century, most of the progress in geometric topology can in some way be traced back to an attempt to prove or disprove one version of the Poincaré conjecture. Up to today a total of seven fields medals were awarded (or offered) for contributions that are connected to the Poincaré conjecture:

1962 to Milnor for showing that there exist exotic 7-spheres, i.e. 7-manifolds that are homeomorphic but not diffeomorphic to a 7-sphere.
1966 to Smale for his proof of the topological Poincaré conjecture for manifolds of dimension larger than 4.
1982 to Thurston for his geometrization program of 3-manifolds.
1986 to Freedman for his proof of the topological Poincaré conjecture in dimension 4.
1986 to Donaldson for applying gauge theory to show that there exist simple exotic 4-manifolds.
1990 to Jones for the introduction of new polynomial invariants in knot theory.
2006 to Perelman for the proof of the geometrization conjecture which also implies the Poincaré conjecture in dimension 3.

While some of these results are obviously extremely difficult, some others are surprisingly simple (once you get the main idea). In this series of six lectures, I will survey about the parts of these results that I understand and also discuss some other results that were not awarded fields medals but are still very interesting.

The lectures are intended for Ph.D. stundents of the RTG 2229 (Asymptotic Invariants and Limits of Groups and Space). But I hope to make these lectures interesting for any mathematician (from bachelor students to professors) interested in learning about these landmark results. I will only assume that the audience is familiar with some of the basic notions of differential topology (smooth manifolds, homotopy equivalences, diffeomorphism) and algebraic topology (fundamental group, homology groups) that are normally covered in an introductory course on topology. The different lectures are obviously related, but I will try my best to make every lecture as self-contained as possible, so that it will be possible without much problems to miss a lecture and still understand the rest (or just come to a single lecture if one is only interested in a single topic).

Table of Contents: (tentative)

Lecture 1: Overview and dimension 2
1.1. Topological and smooth manifolds
1.2. The Poincaré conjecture
1.3. Handle decompositions
1.4. Kirby calculus of surfaces and the classification of 2-manifolds

Lecture 2: 3-manifolds and Heegaard splittings
2.1. Heegaard splittings
2.2. Heegaard diagrams
2.3. Lens spaces
2.4. How not to prove the Poincaré conjecture

Lecture 3: High dimensional manifolds
3.1. Construction of exotic spheres
3.2. The h-cobordism theorem and Smale's proof of the Poincare conjecture
3.3. The s-cobordism theorem and surgery theory

Lecture 4: Knot theory
4.1. Seifert surfaces and knot genera
4.2. Linking numbers
4.3. The Alexander polynomial
4.4. The Jones polynomial
4.5. Categorification and Khovanov homology
4.6. Slice knots and the s- and tau-invariants

Lecture 5: The wild world of 4-manifolds
5.1. The intersection form of 4-manifolds and Freedman's theorem
5.2. Gauge theory and Donaldson's theorem
5.3. Knot traces
5.4. Constructions of potential exotic 4-spheres
5.5. Construction of an exotic R⁴

Lecture 6: The geometrization theorem
6.1. Geometrization of 2-manifolds
6.2. Geometrization of 3-manifolds and the 3-dimensional Poincaré conjecture
6.3. Hyperbolic geometry of 3-manifolds
6.4. Algorithms in 3-manifold topology

The lectures will not follow a single source and it will not be necessary to read additional material. People that are interested in learning the details, are adviced to contact me for further reading.