Universität Wien
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250150 VO Low dimensional topology (2025S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Tuesday 04.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 05.03. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 11.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 18.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.03. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 25.03. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.03. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 01.04. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 02.04. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 08.04. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 09.04. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 29.04. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 30.04. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 06.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 07.05. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 14.05. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 20.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 21.05. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 27.05. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 28.05. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 03.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 04.06. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 10.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 11.06. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 17.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 18.06. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock
  • Tuesday 24.06. 11:30 - 13:00 Seminarraum 7 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 25.06. 09:45 - 11:15 Seminarraum 9 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

Low dimensional topology studies (smooth) 3- and 4-dimensional manifolds (=spaces that can be modelled on the Euclidian 3- or 4-space). A classical way of studying 3-manifolds is by understanding embedded submanifolds: knots and surfaces.
Thus as a warm-up, we will spend some lectures on knots in the 3-space.
We will then briefly introduce notions and constructions from Differential Topology concentrating on Morse theory and its consequences. And use surfaces (2-dimensional manifolds) as basic examples.
Then we move on to the study of 3-manifolds by first giving some constructions and then understanding the specifics of the Algebraic Topological invariants of 3-manifolds. Next, we will decompose 3-manifolds into simple pieces first along spheres and then tori. We then discuss Dehn surgery along knots, as a specific construction.
As a reformulation of Morse theory, we will give a compact description of 4-manifolds called Kirby diagrams, and discuss Kirby calculus.
If time permits we will briefly discuss recent methods to study 3- and 4-manifolds.

Some of the lectures will be reserved to discuss solutions of exercises and for student presentations.

Essential Prerequisites:
>Point-Set Topology: basic notions of topological spaces, continuity, compactness, connectedness, fundamental concepts like quotient topology and metric spaces
>Basic Algebraic Topology: Fundamental group and covering spaces, Homology and cohomology
>Smooth manifolds, tangent spaces, smooth maps
>Basic Group Theory: fundamental concepts like group actions, free groups, and presentations.

Helpful but Not Strictly Necessary
>Some Exposure to 3-Manifolds: Basic examples (e.g., 3-sphere, torus, lens spaces).
>Basic Notions from Homotopy Theory: CW complexes, homotopy equivalence, cellular homology
>Basic Differential Topology: Sard’s theorem, transversality, vector bundles

Assessment and permitted materials

Regularly solve exercises and Present an additional topic. Here are some possible topics, but you can choose anything that I agree to:
>Alexander and Jones Polynomials
L. Kauffman, "On Knots," Princeton University Press, 1987.
C. Livingston, "Knot Theory," MAA, 1993.
>Hyperbolic 3-Manifolds and the Geometrization Conjecture
W. Thurston, "Three-Dimensional Geometry and Topology," Vol. 1, Princeton University Press, 1997.
J. Anderson, "Hyperbolic Geometry," Springer, 2005.
> Fibered Knots and Monodromy
J. Birman, "Braids, Links, and Mapping Class Groups," Princeton University Press, 1975.
W. Thurston, "The Geometry and Topology of 3-Manifolds," (Lecture Notes, freely available online).
> Knot Floer Homology
P. Ozsváth and Z. Szabó, "Knot Floer Homology," in Bulletin of the AMS, 2010 (survey paper, accessible introduction).
> Branched Coverings and 3-Manifolds
D. Rolfsen, "Knots and Links," Publish or Perish, 1976.
> Casson Invariant and Rohlin’s Theorem
S. Donaldson, "The Orientation of Yang-Mills Moduli Spaces and Rohlin’s Theorem," Journal of Differential Geometry, 1987
> 4-Manifold Intersection Forms and Freedman’s Theorem
R. Kirby, "The Topology of 4-Manifolds," Springer, 1989
> Trisections of 4-Manifolds
D. Gay and R. Kirby, "Trisecting 4-Manifolds," Geometry & Topology, 2016
> Khovanov Homology
M. Khovanov, "A Categorification of the Jones Polynomial," Duke Mathematical Journal, 2000 (first introduction to Khovanov homology).
> Legendrian and Transverse Knots
J. Etnyre, "Legendrian and Transverse Knots," Handbook of Knot Theory, 2005 (accessible introduction).

Minimum requirements and assessment criteria

Examination topics

Reading list

Differential Topology:
>J. Milnor: Morse Theory
>Morris W. Hirsch: Differential Topology
knots:
>G. Burde, M. Heusener and H. Zieschang: Knots
3-manifolds:
>Allen Hatcher: Notes on Basic 3-Manifold Topology
https://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html
>Bruno Martelli: An introduction to Geometric Topology
https://people.dm.unipi.it/martelli/Geometric_topology.pdf
>Dale Rolfsen: Knots and Links
4-manifolds:
>R. Gompf und A. Stipsicz: 4-Manifolds and Kirby Calculus

Association in the course directory

MGEV, MGET

Last modified: Mo 30.06.2025 11:06