Math 541 - Topics in Topology:
L^{2} Invariants in Topology and Geometry | Spring 2019

Instructor

Professor Shelly Harvey
Herman Brown 446
Phone: x3659
shelly at rice dot edu

Course Information

Class meets: MWF 10-10:50pm in HB 423
Office Hours: Monday 3-3:30pm, Wednesday 3-3:30pm, Thursday 3-4pm, or by appointment
Webpage: http://math.rice.edu/~shelly/541s19/

In algebraic topology, one considers invariants of finite CW-complexes like Euler characteristic, signature, and Betti numbers. However, often this doesn't tell you much about the space. Instead of studying the space X, one can study the its universal cover X' and consider the algebraic invariants of X'.
If the fundamental group of X, G=π_{1}(X), is finite, then X' is still a finite CW-complex and everything works as usual. However, it is often the case that G is infinite in which these invariants don't make sense (or are infinite). For example, let X=S^{1} ∨ S^{2}.
Then H_{2}(X') ≅ Z^{∞} and H_{p}(X') = 0. Since G≅Z is isomorphic to the group of deck translation of
X', there is a left Z=&t> action on X' making H_{p}(X') into a left Z[t,t^{-1}]-module and we can easily verify that H_{2}(X')≅ Z[t,t^{-1}] is a free module of rank 1. More generally, if X is a finite CW-complex with G=Z^{m} then H_p(X') is a finitely generated module over the Laurent polynomial ring in m variables for all p, hence it has a well-defined and finite rank. We could define the L^{2} p^{th} Betti number of this space to be the rank of this module. For general G, H_p(X') is a left module over the group ring Z[G]. In most cases, this ring is quite difficult to work with. For example, it is not Noetherian and finitely generated modules over it don't have a nice well-defined notion of "rank." The remedy to this is to pass to the L^{2} completion to obtain Hilbert spaces and use powerful tools from functional analysis to define L^{2}-invariants. These invariants have many connections to group theory, differential geometry, ergodic theory, K-theory, and low-dimensional topology.

We will start by discussing the basics of group von Neumann algebras of countable discrete groups including canonical trace, the Fuglede-Kadison determinant, and the Ore localization. We will uses these to to define L^{2} algebraic topological invariants including L^{2}-Betti and L^{2}-torsion, and discuss their applications to geometry and topology. If we have time, we will define L^{2}-signatures and discuss their applications.

Prerequisites

You should have completed Math 444/539 (Geometric Topology), 445/540 (Algebraic Topology), and 463/563 (Algebra II). I will assume that each student is familiar with fundamental groups, covering spaces, finitely presented groups, homology, cohomology, basic homological algebra, Poincare duality, and modules over general rings. I will assume no knowledge of functional analysis and cover the necessary material in class. Depending on your background, you may need to do some extra reading outside of class to fill in details.

Course objectives and learning outcomes

By the end of the course, the student will have a thorough understanding of the material presented in class. In particular, they will have a good working knowledge of group von Neumann algebras, L^{2} Betti numbers, and L^{2}-torsion.

Homework and in class participation

This is advanced topics course primarily intended for graduate students in mathematics interested in topology and geometry.
I will assign homework problems for you to do to understand the material. You are encouraged to do the problems but are not required to turn them in. In addition, we will have in class activities. You are required to participate in these as much as you can with your mathematical background (there may some homework problems or activities that first year students may not have the background to do). There will be no exams. Attendance is required.

Exams

There will be no exams.

Grades

Students are required to attend every lecture and participate in the in class activities (unless they have permission from the instructor to miss). Your grade is completely determined by attendance. In order to get an A, you must miss fewer than 4 lectures. If you miss 4-6 lectures, you get a B. If you miss 7-9 lectures, you get a C. If you miss 10-12 lectures, you get a D. If you miss more than 12 lectures, you get an F.

Disability Support

Any student with a documented disability seeking academic adjustments or accommodations is requested to speak with me during the first two weeks of class. All discussions will remain as confidential as possible. Students with disabilities will also need to contact the Disability Support Services Office in the Ley Student Center.