Topics and Exercises
Week |
Topics |
1 | Introduction to cohomology, Universal coefficient theorem for cohomology |
2 | Cohomology of spaces, applications of Universal coefficient theorem |
3 | Cellular cohomology, cup product, cohomology rings of RP^2 and T^2 |
4 | Graded commutativity of cup product, tensor product of graded rings, Kunneth theorem |
5 | Proof of Kunneth theorem, cohomology of real projective space |
6 | Cohomology of projective spaces, applications, Manifolds and orientations |
7 | Fundamental class of an oriented closed manifold, cap product |
8 | Relative cap product, cohomology with compact supports, direct limits, PD map |
9 | Proof of Poincare duality, application to Euler characteristic |
10 |
More applications of PD: recognizing manifolds, middle-dimensional bilinear forms, Lefschetz duality |