Date: 30 Sep 2006

The KO-assembly map and positive scalar curvature

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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.