The KO-assembly map and positive scalar curvature

  • Jonathan Rosenberg
Geometry Of Manifolds

DOI: 10.1007/BFb0084745

Part of the Lecture Notes in Mathematics book series (LNM, volume 1474)
Cite this paper as:
Rosenberg J. (1991) The KO-assembly map and positive scalar curvature. In: Jackowski S., Oliver B., Pawałowski K. (eds) Algebraic Topology Poznań 1989. Lecture Notes in Mathematics, vol 1474. Springer, Berlin, Heidelberg

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.

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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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