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The KO-assembly map and positive scalar curvature

  • Jonathan Rosenberg
Geometry Of Manifolds
Part of the Lecture Notes in Mathematics book series (LNM, volume 1474)

Abstract

We state a geometrically appealing conjecture about when a closed manifold with finite fundamental group π admits a Riemannian metric with positive scalar curvature: this should happen exactly when there are no KO*-valued obstructions coming from Dirac operators. When the universal cover does not have a spin structure, the conjecture says there should always be a metric of positive scalar curvature, and we prove this if the dimension is ≥5 and if all Sylow subgroups of π are cyclic. In the spin case, the conjecture is closely tied to the structure of the assembly map KO*(Bπ) → KO*(Rπ), and we compute this map explicitly for all finite groups π. Finally, we give some evidence for the conjecture in the case of spin manifolds with π = Z/2.

Keywords

Fundamental Group Dirac Operator Sylow Subgroup Spin Manifold Oriented Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Jonathan Rosenberg
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkU.S.A.

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