Math 203: Topology of manifolds
Course number: 22M:203:001: Topology of manifolds.
Course meets: TuTh 1:05--2:20pm Fall 2009 in 221 MacLean Hall.
Instructor: Julianna Tymoczko. Office hours and contact information are here.
Course webpage: http://www.divms.uiowa.edu/~tymoczko/teaching/203
Course description: Homology assigns a collection of groups to a topological space using natural topological constructions. Cohomology is an even better algebraic gadget: using very similar constructions to homology, it assigns a graded ring to a topological space. This course will introduce several cohomology theories, including singular cohomology, intersection cohomology, and equivariant cohomology.
We will start by quickly reviewing homology theory. Next we will introduce cohomology, discussing basic properties, products, the Universal Coefficient Theorem, the Kunneth Theorem, and Poincare duality. We will then describe intersection cohomology, which for manifolds can be thought of as a geometric/topological way to construct a cohomology ring product. We will finish with equivariant cohomology, a richer cohomology ring that exists for suitably nice topological spaces with group actions. We will have several guest lectures and will highlight topics of current research.
Goals: There are three course goals:
- To be able to calculate cohomology rings and compute in the ring.
- To be able to prove results using homological algebra.
- To develop professional skills, particularly LaTeX and mathematical writing.
Prerequisites: Math 201 or permission of instructor.
Note to students: If you feel that you may need an accomodation for any sort of disability, please make an appointment to see me.