56) FOUR MANIFOLDS AND KIRBY CALCULUS
Lecturer: Andras I. Stipsicz
No. of Credits: 3 and no. of ECTS credits 6
Prerequisites: Course Level:intermediate PhD
Brief introduction to the course:
The course introduces modern techniques of differential topology through handle calculus, and
pays special attention to the description of 4-dimensional manifolds. We also show how to
manipulate diagrams representing 4-manifolds. Smooth invariants of 3- and 4-manifolds
(Heegaard Floer invariants and Seiberg-Witten invariants) will be also discussed.
The goals of the course:
The aim is to get a working knowledge of all basic (algebraic) topologic notions such as homology,
cohomology theory, the theory of knots and handlebodies and some aspects of differential
geometry through the rich theory of 4-manifolds. This discussion quickly leads to some important
and unsolved questions in the field.
The learning outcomes of the course:
By the end of the course, students are enabled to do independent study and research in fields
touching on the topics of the course, and how to use these methods to solve specific problems. In
addition, they develop some special expertise in the topics covered, which they can use efficiently
in other mathematical fields, and in applications, as well. They also learn how the topic of the
course is interconnected to various other fields in mathematics, and in science, in general.
More detailed display of contents:
Week 1: Knots in the 3-space
Week 2: Invariants of knots, the Alexander and the Jones polynomial
Week 3: Morse theory, handle decompositions
Week 4: Decompositions of 3-manifolds: Heegaard diagrams
Week 5: Decompositions of 4-manifolds: Kirby diagrams
Week 6: Knots in 3-manifolds and Heegaard diagrams
Week 7: Surfaces in 4-manifolds; Freedman’s theorem
Week 8: New invariants of knots: grid homology
Week 9: Heegaard Floer invariants of 3-manifolds (combinatorial approach)
Week 10: Further structures on Heegaard Floer groups
Week 11: Seiberg-Witten invariants
Week 12: Basic properties of Seiberg-Witten invariants
References:
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2.
3.
4.
Milnor: Morse theory
Milnor: The h-cobordism theorem
Gompf-Stipsicz: 4-manifolds and Kirby calculus
Lickorish: An introduction to knot theory