Instructor Prof. Sergey Finashin

Homework II   selection from Milnor's book "Topology from differential viewpoint"

Homework III dvi   pdf   ps-zipped

Homework IV  dvi   pdf   ps-zipped

Homework V   dvi   pdf   ps-zipped    (Take-home exam I)

Homework VI   dvi  pdf  ps-zipped     (Take-home exam II)

Syllabus (2-13 are the questions for the theoretical part of the Final)

1. Computation of homology groups (review).
2. Homology of  fiber bundles. Transfer homomorphism. Gysin exact sequence.
3. Degree of a map. Hopf invariant.
4. Pontryagin-Thom construction. Calculation of $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$.
5. Cobordism groups.
6. Obstruction theory. Eilenberg-MacLane spaces.
7. Simplicial space associated to an open covering. Limit of the homology groups. Cech homology.
8. Sheaves as an abelian category. Cohomology in a sheaf (the approach with open coverings).
9. Cohomology with twisted coefficients.
10. Alexander polynomial for knots and links. Reidemeister torsion. Calculation for the Lens spaces.
11. Morse functions and handle decompositions of manifolds. The Morse inequalities.
12. Heegaard splitting of 3-manifolds.
13. Elements of the Kirby calculus.