# MATH 7590 Geometric Topology

## Course Information

 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.

## Course Description

A hyperplane arrangement is a finite collection of (n-1)-dimensional subspaces in an n-dimensional vector space, such as lines in a plane, planes in 3-space, etc. Arrangements arise in a variety of mathematical contexts, and in applications ranging from mathematical physics to robotics.

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.

Department of Mathematics
Louisiana State University
Baton Rouge, LA, 70803
Office: 372 Lockett
Phone: (225) 578-1576
Email: cohen@math.lsu.edu