Middle East Technical University   Department of Mathematics     Spring 2009

 

 

Low-dimensional Topology
Math 559

Instructor  Prof. Sergey Finashin

Prerequisites:   Some elementary Topology (Math 420 or Math 422)

 

Description of the Course: The aim is to introduce the basic techniques to work with manifolds in dimension 3 and 4 (Kirby Calculus, branched cover presentation, Lefschetz fibrations and open books). Links and surfaces in the beginning of the course are considered as the tools to approach this main goal. Algebraic and Differential Topology are not considered as prerequisites, and so some technicalities in the proofs will be avoided (replaced by geometrically intuitive explanations). For that reason this course does not follow any single textbook, but some chapters from the reference books will be recommended instead.

 

Grading policy:  Each exam (two Midterms and Final, 30+30+40) will consists in two parts: I a set of questions to be answered in the class (easy questions examining general understanding), and II take-home problems (more advanced). The work in the class (answering questions of the lecturer, posing reasonable questions and giving good remarks) will be also taken into account and may influence to the final grade.

 

Reference books:

  1. N.D.Gilbert, T.Potter, Knots and surfaces
  2. L.Kauffman, On Knots
  3. N.Saveliev, Lectures on the Topology of 3-manifolds
  4. R.Kirby, The topology of 4-manifolds
  5. R.Gompf, A.Stipsicz, An introduction to 4-manifolds and Kirby Calculus
  6. B.Ozbagi, A.Stipsicz, Surgery on Contact 3-manifolds and Stein Surfaces

 

 


Syllabus


1. Knots and Links: Reidemeister moves, Braid presentation (Alexander and Markov theorems).

     Writhe and linking number. Crossing number, bridge number. Satellite knots: sum, cabling, doubling.

     Rational tangles and 2-bridge knots.

2. Seifert surfaces and genus of a knot (link). Slice knots, ribbon presentation, 4-genus. Plumbing surfaces.

3. Framing of knots. The writhe as the blackboard framing. Seifert matrices, Alexander polynomial,

    signature of a knot. Detecting of chirality and invertibility.

4. Links of singularities, and the Milnor fibration. Monodromy of an open book.

    Braid monodromy of Riemann surfaces. Positivity and quasipositivity.

5. Mapping class groups of a surface. Dehn twists, relations in the mapping class group.

    Connection with the braid groups. SL2(Z) as the mapping class group of a torus.

    Elliptic, parabolic and hyperbolic matrices, hyperbolic plane action of  SL2(Z).

6. Handlebody decompositions of 3-manifolds and Heegaard diagrams. Lens spaces.

 

   Optional topics: Orientability and class w1. Fundamental group, van Kampen theorem.

                              Skein relation for the Alexander and the Convey polynomials. Finite type invariants.

                              Chord diagrams. The Arf  invariant of a knot. The Casson invariant.

                              Handlebody decomposition of n-manifolds.

 

Midterm 1

 

7. Dehn surgery in 3-manifolds. Rational surgery and continued fractions.

    Handle decomposition of 4-manifolds and Kirby calculus. Handle slides, topological blowup.

8. Weighted graphs and plumbing 4-manifolds, their quadratic forms and Kirby diagrams. K3-surface.

9. Seifert manifolds and Brieskorn spheres. Rational homology balls. Branched covering presentation.

10. Resolution of singularities. Exceptional divisor, multiplicities of its components.

      Examples: the simple (A-D-E) singularities.

 

Optional topics: Intersection forms, their parity and class w2, signature and Rokhlin’s theorem.

                          Poincare conjecture, Exotic Milnor spheres, Morse theory and h-cobordism theorem.

                          Rational blow-down, logarithmic transform, and exotic 4-manifolds.

 

 

Midterm 2

 

11. Symplectic 4-manifolds, Darboux theorem. Lefschetz fibrations  and their monodromy.

       Fiber sum. Elliptic fibrations E(n).

12. Contact 3-manifolds, Legendrian and transverse curves. Legendrian lifting of a planar curve.

       Thurston-Bennequin invariant and the rotation number. Tight and overtwisted contact structures.

13. Contact boundary of symplectic manifolds: Stein and symplectic fillability. Eliashberg theorems.

14. Open book decomposition of 3-manifolds and contact topology (Giroux correspondence).

      Relation to braids via branched coverings.

 

Optional topics: The Chern class c1, Spinc-structures. Arnold conjecture.

                           Characteristic foliation, convexity, and dividing curve. Contact surgery.

 

 

Final