Course: | MATH 7590 Geometric Topology - Arrangements |
Time/Place: | Tuesday & Thursday, 12:10 - 1:30 pm, in 135 Lockett |
Instructor: | Dan Cohen |
Office Hours: | tentatively Monday, Wednesday, Friday, 9:00 - 10:00 am, in 372 Lockett, and by appointment |
Prerequisites: | The exposure to the fundamental group and covering
spaces provided by MATH 7512 Topology II |
Grade: |
Based on an in-class presentation. Potential presentation topics
will be posted
here: http://www.math.lsu.edu/~cohen/courses/FALL05/M7590/M7590hw.html. I will probably give, but not grade, homework problems occasionally. |
Text: |
There will be no formal textbook for this course.
Potential sources for material covered in this course are listed here: http://www.math.lsu.edu/~cohen/courses/FALL05/M7590/M7590ref.html. |
The complement of an arrangement, what is left of space after the hyperplanes have been removed, is an object of fundamental interest in the topological study of arrangements. For instance, a collection of lines cuts the real plane into pieces, and understanding the topology of the complement amounts to counting the pieces. This depends on combinatorial aspects of the collection of lines such as parallelism, how many of the lines intersect at a given point, and so on. This example illustrates a central theme in the subject, the relationship between combinatorial and topological aspects of arrangements.
A complex hyperplane does not disconnect space, and the complement of an arrangement in a complex vector space has rich topological structure. The focus of this course will be on the (low-dimensional) topology of these spaces. Specific topics include braid groups, both as primary objects of study and as tools, configuration spaces, the Fox calculus, and Alexander polynomials and generalizations. In particular, we will develop general algorithms for computing these latter invariants, and investigate the extent to which these invariants of arrangements are combinatorially determined.
The topological aspects of the course will assume familiarity with the fundamental group, while other (combinatorial) aspects will be self-contained.