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The KO-assembly map and positive scalar curvature

  • Geometry Of Manifolds
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Book cover Algebraic Topology Poznań 1989

Part of the book series: Lecture Notes in Mathematics ((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.

Partially supported by NSF Grants DMS-8400900 and DMS-8700551. This paper is in final form and is not merely an announcement of work to appear elsewhere.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, Maryland, U.S.A.

    Jonathan Rosenberg

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  1. Jonathan Rosenberg
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Stefan Jackowski Bob Oliver Krzystof Pawałowski

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© 1991 Springer-Verlag

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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. https://doi.org/10.1007/BFb0084745

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  • DOI: https://doi.org/10.1007/BFb0084745

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54098-4

  • Online ISBN: 978-3-540-47403-6

  • eBook Packages: Springer Book Archive

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