Algebraic Topology Poznań 1989 pp 170-182 | Cite as

# The *KO*-assembly map and positive scalar curvature

## 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*_{*}(*B*π) → *KO*_{*}(**R**π), 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## Preview

Unable to display preview. Download preview PDF.

## References

- [ABP]D. W. Anderson, E. H. Brown, Jr., and F. P. Peterson,
*Pin cobordism and related topics*, Comment. Math. Helv.**44**(1969), 462–468.MathSciNetCrossRefzbMATHGoogle Scholar - [AS]M. F. Atiyah and G. B. Segal,
*Equivariant K-theory and completion*, J. Differential Geometry**3**(1969), 1–18.MathSciNetzbMATHGoogle Scholar - [BB]L. Berard Bergery,
*Scalar curvature and isometry group*, in “Proc. Franco-Japanese Seminar on Riemannian Geometry, Kyoto, 1981,” to appear.Google Scholar - [CE]H. Cartan and S. Eilenberg, “Homological Algebra,” Princeton Math. Ser., no. 19, Princeton Univ. Press, Princeton, N. J., 1956.zbMATHGoogle Scholar
- [G]V. Giambalvo,
*Pin and Pin' cobordism*, Proc. Amer. Math. Soc.**39**(1973), 395–401.MathSciNetzbMATHGoogle Scholar - [GL1]M. Gromov and H. B. Lawson, Jr.,
*Spin and scalar curvature in the presence of a fundamental group, I*, Ann. of Math.**111**(1980), 209–230.MathSciNetCrossRefzbMATHGoogle Scholar - [GL2]______,
*The classification of simply connected manifolds of positive scalar curvature*, Ann. of Math.**111**(1980), 423–434.MathSciNetCrossRefzbMATHGoogle 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 - [K]G. G. Kasparov,
*Equivariant KK-theory and the Novikov Conjecture*, Invent. Math.**91**(1988), 147–201.MathSciNetCrossRefzbMATHGoogle 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 - [L]J.-L. Loday,
*K-théorie algébrique et représentations des groupes*, Ann. Sci. École Norm. Sup. (4)**9**(1976), 309–377.MathSciNetzbMATHGoogle 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 - [Ra]A. Ranicki, “Exact Sequences in the Algebraic Theory of Surgery,” Mathematical Notes, no. 26, Princeton Univ. Press, Princeton, N. J., 1981.zbMATHGoogle 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 - [R3]_____,
*C*-algebras, positive scalar curvature, and the Novikov Conjecture, III*, Topology**25**(1986), 319–336.MathSciNetCrossRefzbMATHGoogle 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 - [S]R. E. Stong, “Notes on Cobordism Theory,” Mathematical Notes, no. 7, Princeton Univ. Press, Princeton, N. J., 1968.zbMATHGoogle Scholar
- [Y]Z. I. Yosimura,
*Universal coefficient sequences for cohomology theories of CW-spectra*, Osaka J. Math.**12**(1975), 305–323.MathSciNetzbMATHGoogle Scholar