Middle East Technical University
Department of Mathematics
Spring 2005
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Instructor Prof. Sergey Finashin
Syllabus
1. Knots and Links. Isotopy and homeomorphism. Knot diagrams, Reidemeister
moves. Crossing number, Gordion (unknotting) number. Examples (torus knots,
etc.). Reverse, obverse and inverse knots,
amphi-chiral knots. Characterization of the unknot.
2. Construction of Seifert surface. Seifert matrix. Alexander polynomial, signature of a knot. Genus of a knot. Slice knots, 4-genus and estimate of the unknotting number. Cobordism of knots and cobordism invariants.
3. Commutativity of the connected sum. Satellite knots. Linking number. Framing of knots. Blackboard framing. The writhe of a knot diagram.
4. Alexander polynomial and branched coverings. The case of fibered knots. Links of singularities (examples).
5. Skein relation (explanation) for the Alexander and the Convey polynomial. The Arf and the Casson invariants as the coefficients of Convey polynomial.
6. Finite type invariants and Chord diagrams.
7. Definition of the Arf invariant of Z/2-valued quadratic form and its properties. Definition of the Arf invariant of a knot.
8. Presentation of links by braids. Generators and relations of the braid groups. Braids and the discriminant hypersurface of polynomials.
9. Mapping class groups of a surface. Dehn twists, relations in the mapping class group
and their proof using their connection with the braid groups. Action in the homology and the Torelli group.
10. SL_2(Z) as the mapping class group of a torus.
Midterm 1 (take-home) 3.04 – 17.04 dvi
11. Morse functions and the proof of existence of a handlebody decomposition for 3-manifolds. Heegaard diagrams.
12. Dehn surgery in 3-manifolds. Examples (Lens spaces, etc.). Rational surgery and continued fractions.
13. Open book decomposition of 3-manifolds. Examples: branched coverings over 3-sphere ramified along braids.
14. Classification problems in 3- and 4-dimensional topology. Diff and Top categories. Poincare conjecture in dimension 3 and 4.
Examples of homology spheres.
15. Unimodular quadratic forms (lattices): odd/even, definite/indefinite, signature. Classification of indefinite forms.
Realizability of quadratic forms by 4-manifolds in categories Diff and Top: Donaldson’s, Freedman’s and Rokhlin’s theorems.
16. Handle decomposition of 4-manifolds and Kirby diagrams. Handle slides, topological blowup and anti-blowup.
17. Algebraic blowup and resolution of singularities of curves (examples). The complete and the proper image of a curve. Exceptional divisor,
multiplicities of its components.
18. Blowup of surface singularities (example: stabilization of a curve singulary). The simple (A-D-E) singularities.
19. Weighted graphs and the plumbing 4-manifolds, their quadratic forms and Kirby diagrams. Blowup and blowdown.
20. The Hodge diamond and its calculation for surfaces in CP3.
21. The arithmetic and the geometric genus of algebraic curves. Rational curves. Rational curves through (3d-1) points in CP2.
22. The Euler characteristic and the signature of the branched coverings.
23. The characteristic classes w_1 and w_2. Relation of w_2 to the quadratic form of 4-manifolds. Class c_1 for the complex line bundles and
for the complex surfaces. Its relation to w_2, to the signature and the Euler characteristic.
24. The adjunction formula. Calculation of c_1 and its square for the branched coverings.
Midterm II (take-home)
3.05 – 23.05 DVI ps
Final (Oral
exam) 25.05