The KO-assembly map and positive scalar curvature
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.
KeywordsFundamental Group Dirac Operator Sylow Subgroup Spin Manifold Oriented Manifold
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- [BB]L. Berard Bergery, Scalar curvature and isometry group, in “Proc. Franco-Japanese Seminar on Riemannian Geometry, Kyoto, 1981,” to appear.Google Scholar
- [GL3]______, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. I.H.E.S. no. 58 (1983), 83–196.Google Scholar
- [KS1]S. Kwasik and R. Schultz, Positive scalar curvature and periodic fundamental groups, Math. Annalen (to appear).Google Scholar
- [KS2]______, Positive scalar curvature and spherical spaceforms, preprint.Google Scholar
- [MR]I. Madsen and J. Rosenberg, The universal coefficient theorem for equivariant K-theory of real and complex C*-algebras, in “Index Theory of Elliptic Operators, Foliations, and Operator Algebras,” J. Kaminker, K. Millett, and C. Schochet, eds., Contemp. Math., no. 70, Amer. Math. Soc., Providence, pp. 145–173.Google Scholar
- [R1]J. Rosenberg, C*-algebras, positive scalar curvature, and the Novikov Conjecture, Publ. Math. I.H.E.S. no. 58 (1983), 197–212.Google Scholar
- [R2]_____, C*-algebras, positive scalar curvature, and the Novikov Conjecture, II, in “Geometric Methods in Operator Algebras,” H. Araki and E. G. Effros, eds., Pitman Research Notes in Math., no. 123, Longman/Wiley, Harlow, Essex, England and New York, 1986, pp. 341–374.Google Scholar
- [Se]G. Segal, The representation ring of a compact Lie group, Publ. Math. I.H.E.S. no. 34 (1968), 113–128.Google Scholar
- [Sz]S. Stolz, Simply connected manifolds of positive scalar curvature, preprint.Google Scholar