Topics/Goals
- Spaces: Manifolds, CW-complexes
- Homotopy and Coverings Spaces
- Integration on Manifolds, Stokes Theorem (with chains)
- Homology and Cohomology Theories: de Rham, simplicial, singular,
cellular, Cech
- Applications: Brouwer Fixed Point, Jordan Curve, Hairy Ball Theorem,
Euler Characteristics, Lefschetz Fixed Point
Prerequisites
This course will require a higher level of mathematical maturity than an
undergraduate course. First it requires knowledge of, or willingness to
learn, material from a typical grad-school-prep undergrad curriculum. For
us, some basic point-set topology and algebra will be helpful. Second, you
will also need to have the ability to locate and learn some theorems on your
own. I will try and give you guys good references but sometimes what I
provide won't be perfect and you will need to cobble some things together.
Finally, as a fair warning, I'm trying to cut out a path in a ton of
material in one semester. A typical graduate Geometry-Topology sequence is
two semesters, and there is enough material here to easily take up
four semesters. I don't expect things to be in an optimal order my first
time teaching and I don't expect to finish everything.
Grading
- Grading will be based on homework assignments. They will typically be
due on Fridays (probably not every Friday).
- You may work with others and consult references but you must write
the homework using your own words and you must cite your sources
(including collaborators)!