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Topics and Exercises


Week
Topics
1 Definition of cohomology, Universal coefficient theorem for cohomology
2 Universal coefficient Theorem contd., parallels between homology and cohomology
3 Definition of cup product, properties and elementary examples
4 Cross product and the cohomology of the n-torus Notes available here.
5 Cup product structure of torus and projective spaces
6 Tensor products, a Kunneth formula, division algebras over the reals
7 Kunneth for pairs, Poincare duality for surfaces, orientation of manifolds
8 Construction of the fundamental class, definition and properties of cap product
9 Direct limits, cohomology with compact supports, proof of Poincare duality
10
Applications of Poincare duality to Euler characteristic and to cup product structure of manifolds


Exercises: Link to PDF file