Title: L2 methods, topology of manifolds, and rational homotopy
Speaker: Grigori Avramidi (OSU)
Abstract: A theme connecting analysis and topology is that global analysis (elliptic operators, L^2-differential forms, and all that stuff) on the universal cover of a closed manifold tells us about the topology of the base manifold (e.g. about the Euler characteristic, signature and growth of Betti numbers in finite covers). If the universal cover is contractible, then the simplest possible analytic picture (suggested by Dodziuk and Singer) is that its L^2-de Rham cohomology (represented by the harmonic L^2 forms) appears only in the middle dimension. A combinatorial form of global analysis works on the universal cover of a finite complex (instead of a closed manifold). Viewed from this perspective, the Dodziuk-Singer picture would have striking topological consequences. It would imply for instance (by a result of Okun and Schreve), that if the universal cover of a finite complex is contractible and has a non-zero L^2 harmonic k-form, then that complex does not embed in R^{2k-1}, and is not even homotopy equivalent to a 2k-1 manifold. Topologically, such embedding and ``thickening'' obstructions are fragile, finite order things that tend to disappear rationally. In fact, one can build closed manifolds whose universal covers are ``rationally contractible'' but whose harmonic L^2 forms are not concentrated in the middle dimension.
This leads to the curious conclusion that the Dodziuk-Singer picture of closed aspherical manifolds, if true, cannot be proved via global analysis but is tied up with finite order phenomena in embedding theory. Alternatively, there may be a more interesting analytic picture (for the spectrum of the Laplacian on L^2-forms on the universal cover) that has yet to be discovered.