arXiv subjects
Publications and source records attributed to Boris Okun.
We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree.
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a cubical complex Σ_L on which W_L acts properly and cocompactly. Its two most salient features are that (1) the link of each vertex of Σ_L is L and (2) Σ_L is contractible. It follows that if L is a triangulation of S^{n-1}, then Σ_L is a contractible n-manifold. We describe a program for proving the Singer Conjecture (on the vanishing of the reduced L^2-homology except in the middle dimension) in the case of Σ_L where L is a triangulation of S^{n-1}. The program succeeds when n < 5. This implies the Charney-Davis Conjecture on flag triangulations of S^3. It also implies the following special case of the Hopf-Chern Conjecture: every closed 4-manifold with a nonpositively curved, piecewise Euclidean, cubical structure has nonnegative Euler characteristic. Our methods suggest the following generalization of the Singer Conjecture. Conjecture: If a discrete group G acts properly on a contractible n-manifold, then its L^2-Betti numbers b_i^{(2)} (G)$ vanish for i>n/2.
In this article, we generalize Eberlein's Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure. Any hyperbolic manifold, on the other hand, does admit such a structure.