Coarse Obstructions to Positive Scalar Curvature Metrics in Noncompact Arithmetic Manifolds

Details Coarse Obstructions to Positive Scalar Curvature Metrics in Noncompact Arithmetic Manifolds Coarse Obstructions to Positive Scalar Curvature Metrics in Noncompact Arithmetic Manifolds

2000-05-11

arxiv.org/abs/math/0005115v1

Mathematics

Differential Geometry

13 pages

Scientific paper

Block and Weinberger show that an arithmetic manifold can be endowed with a positive scalar curvature metric if and only if its $\rationals$-rank exceeds 2. We show in this article that these metrics are never in the same coarse class as the natural metric inherited from the base Lie group. Furthering the coarse $C^\ast$-algebraic methods of Roe, we find a nonzero Dirac obstruction in the $K$-theory of a particular operator algebra which encodes information about the quasi-isometry type of the manifold as well as its local geometry.

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