Self-dual sheaves and L²-cohomology of locally symmetric spaces. Conference on Spectral Analysis on Noncompact Manifolds . Hausdorff Center for Mathematics, Bonn, Germany. June 24, 2010
Invited Lectures ; Goresky and MacPherson were motivated to introduce intersection cohomology in part to recover a generalized form of Poincare duality for singular spaces: for each pair of dual perversities, such as the lower middle and the upper middle, there is a nondegenerate pairing between the corresponding intersection cohomology groups. However for singular spaces with even codimension strata, or more generally for Witt spaces, the upper middle and the lower middle theories coincide, yielding a nondegenerate pairing on what is simply called middle perversity intersection cohomology. This enabled Goresky and MacPherson to define an L-class for Witt spaces. For non-Witt spaces, Banagl has shown there exist a well-defined L-class provided there exists a self-dual sheaf that interpolates the lower middle and the upper middle intersection cohomology sheaves. In the case of the reductive Borel-Serre compactification of a Hilbert modular surface, Banagl and Kulkarni show that such a self-dual sheaf exists. In this talk I will address the existence of such self-dual sheaves on the reductive Borel-Serre compactifications of general locally symmetric spaces, a question raised by Banagl and Kulkarni. Note that a completely independent analytic approach to restoring Poincare duality and producing characteristic classes was developed by Cheeger using L²-cohomology. I will also relate the existence of these self-dual sheaves to L²-cohomology.
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Conference on Spectral Analysis on Noncompact Manifolds
Hausdorff Center for Mathematics, Bonn, Germany