# Mathematics Colloquia and Seminars

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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 |

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