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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


Exercises:

Exam from MA3H60

Exercises for Week 1

Exercises for Week 2

Exercises for Week 3

Exercises for Week 4

Exercises for Week 5

Exercises for Week 6

Exercises for Week 7

Exercises for Week 8

Exercises for Week 9

Exercises for Week 10