Betti numbers of compact locally symmetric spaces | |
Student No.： | 50 |
Time： | Tue. /Thu. 13:00-14:50 |
Instructor： | MS Raghunathan [Indian Institute of Technology, Powai Mumbai, India] |
Place： | Conference Room 3, Floor 2, Jin Chun Yuan West Building |
Starting Date： | 2012-7-17 |
Ending Date： | 2012-8-14 |
Course description:
Let $G$ be a connected semi-simple Lie group with no compact factors and $\Gamma$ a torsion-free co-compact discrete subgroup of $G$. Let $K$ be a maximal compact subgroup of $G$ and $S = K\G$, the associated symmetric Riemannian space. This course is concerned with the topology of the locally symmetric compact manifold $X = K\G/\Gamma$, more specifically about the Betti numbers of this manifold; in fact we will be concerned principally with the first Betti number. We will begin with a very general theorem due to Matsushima which relates the Betti numbers of $X$ with the unitary representation of $G$ on $L^2 (G/ \Gamma)$. We will then establish the vanishing of the first Betti number in the case when $\Gamma$ is irreducible and $S$ has rank > 1 using the method of Kazdan (this is also based on representation theory but has a very different flavour from the ideas of Matsushima) . The rest of the course will concentrate on the case when $X$ has rank 1 (and hence $G$ is almost simple). Here again we will prove vanishing theorems for the first Betti number for a family of rank 1 spaces $S$. These results will show that non-vanishing of the first Betti number can happen only when $S$ is a space of constant curvature or is a hermitian symmetric domain of constant holomorphic curvature. From this point on we describe various non-vanishing results using different techniques: theorem of Kazdan for the case $G = SU(n,1)$ - this uses Matsushim's theorem and certain other deep results from the cohomology of unitary representations of $G$; Millson's theorem for the first Betti number in the case $G = SO(n,1)$ and a theorem due to Millson and Raghunathan for higher Betti numbers $SO(n,1)$ as well as certain other $G$ which use geometric techniques; an application of ideas arising in connection with the congruence subgroup problem to obtain non-vanishing results for the first Betti number. Time permitting I will speak a little bit on the recent solurion of vthe Thurston conjecture on hyperbolic manifolds.
Prerequisite:
I will assume knowledge of basic differential geometry and the theory of Lie groups. Familiarity with semi-simple Lie groups and symmetric Riemannian manifolds will be advantageous – I will from time to time recall facts about these and so advanced knowledge of these topics will not be essential. Similarly some knowledge about arithmetic groups while not essential will be advantageous.
Reference for the course:
1. S Helgason Differentai Geometry, Lie Groups and Symmetric Spaces, Academic Press 1978, (new edition: 1995).
2. M S Raghunathan Discrtete Subgroups of Lie Groups, Springer Velag, Ergebnisse der Mathematik 1973.
3. Y Matsushima, On Betti numbers of compact, locally sysmmetric Riemannian manifolds. Osaka Math. J. 14 1962 1–20.
4. A Borel and N Wallach Continuous cohomology,
5. Annals of Mathematics Studies, 94.iscrete subgroups, and representations of reductive groups. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.
6. Raghunathan, M. S.; Venkataramana, T. N. The first Betti number of arithmetic groups and the congruence subgroup problem. Linear algebraic groups and their representations (Los Angeles, CA, 1992), 95–107, Contemp. Math., 153, Amer. Math. Soc., Providence, RI, 1993.
7. Millson, John J.; Raghunathan, M. S. Geometric construction of cohomology for arithmetic groups. I. Proc. Indian Acad. Sci. Math. Sci. 90 (1981), no. 2, 103–123.
8. Millson, John J. On the first Betti number of a constant negatively curved manifold. Ann. of Math. (2) 104 (1976), no. 2, 235–247.