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The cohomology with local coefficients of compact hyperbolic manifolds
Geometry/Topology| Speaker: | John Millson, University of Maryland |
| Location: | 693 Kerr |
| Start time: | Wed, Jan 21 2004, 3:10PM |
Description
I will begin my talk by reviewing my old work on nonvanishing of
rational homology groups of certain (arithmetically defined) compact
hyperbolic n-manifolds:
1. The first Betti number - Annals of Math 104, (1976).
2. Higher Betti numbers - (with M.S. Raghunathan), Proc. Indian
Math. Soc. 90 (1981).
3. The first cohomology with coefficients in the harmonic powers of
the standard representation of SO(n,1) - Topology 24 (1985).
I will then explain a new result (to appear in the upcoming
issue of Tata Inst. Fund. Res Stud. Math. in honor of the sixtieth birthday
of M.S. Raghunathan) realizing all possible (i.e. consistent with the
vanishing theorem of Vogan and Zuckerman) nonvanishing results
for the cohomology of compact hyperbolic manifolds with coefficients
in an irreducible representation W of SO(n,1).
The statement of the main theorem is very simple. Nonzero cohomology with
coefficients in W can be realized (for a suitable cocompact lattice) for an
unbroken string of degrees beginning with degree = i(W) : = the number of
nonzero entries in the highest weight of W and ending at n - i(W). The proof
is also very simple depending on the existence and intersection pairings of
totally geodesic cycles with coefficients in W. The theorem and its proof
can be found at arXiv:math.GR/0306296