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Manifolds of Positive Scalar Curvature

  • Jonathan Rosenberg
  • Stephan Stolz
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 27)

Abstract

We survey the status of the problem of determining which differentiable manifolds (without boundary) have Riemannian metrics of positive scalar curvature. Of course, if the manifold is non-compact, one requires the metric to be complete.

Key words and phrases

positive scalar curvature real K-theory spin manifold spin cobordism Dirac operator assembly map 

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References

  1. [ABP1]
    D.W. Anderson, E.H. Brown, Jr., and F.P. Peterson, The structure of the spin cobordism ring, Ann. of Math. 86 (1967), 271–298.CrossRefMATHMathSciNetGoogle Scholar
  2. [ABP2]
    D.W. Anderson, E.H. Brown, Jr., and F.P. Peterson, Pin cobordism and related topics, Comment. Math. Heiv. 44 (1969), 462–468.CrossRefMATHMathSciNetGoogle Scholar
  3. [AP]
    J.F. Adams and S. Priddy, Uniqueness of BSO, Math. Proc. Cambridge Philos. Soc. 80 (1978), 475–509.CrossRefMathSciNetGoogle Scholar
  4. [Au]
    T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Springer-Verlag, New York, 1982.CrossRefMATHGoogle Scholar
  5. [BB]
    L. Berard Bergery, Scalar curvature and isometry group, Spectra of Riemannian manifolds, ed. by M. Berger, S. Murakami and T. Ochiai, Proc. Franco-Japanese Seminar on Riemannian Geometry, Kyoto, 1981, Kagai, Tokyo, 1983, pp. 9–28.Google Scholar
  6. [Be]
    A.L. Besse, Einstein manifolds, Spinger Verlag, Berlin and New York, 1986.Google Scholar
  7. [Bo]
    J.M. Boardman, Stable homotopy theory, mimeographed notes, Warwick (1966).Google Scholar
  8. [Br]
    R. Brooks, The A-genus of complex hypersurfaces and complete intersections, Proc. Amer. Math. Soc. 87 (1983), 528–532.MATHMathSciNetGoogle Scholar
  9. [CGM]
    A. Connes, M. Gromov and H. Moscovici, Conjecture de Novikov et fibrés presques plats, C.R. Acad. Sci. Paris, Sér. I Math. 310 (1990), 273–277.MATHMathSciNetGoogle Scholar
  10. [CM]
    A. Connes and H. Moscovici, Conjecture de Novikov et groupes hyperboliques, C.R. Acad. Sci. Paris, Sér. I Math. 307 (1988), 475–480.MATHMathSciNetGoogle Scholar
  11. [G]
    V. Giambalvo, Pin and Pin’ cobordism, Proc. Amer. Math. Soc. 39 (1973), 395–401.MATHMathSciNetGoogle Scholar
  12. [Gr]
    M. Gromov, Stable mappings of foliations into manifolds, Math. USSR—Izv. 3 (1969), 671–694.CrossRefMATHGoogle Scholar
  13. [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.CrossRefMATHMathSciNetGoogle Scholar
  14. [GL2]
    M. Gromov The classification of simply connected manifolds of positive scalar cur-vature,Ann. of Math. 111 (1980), 423–434.Google Scholar
  15. [GL3]
    M. Gromov, Positive scalar curvature and the Dirac operator on complete Riemann-ian manifolds,Publ. Math.I.H.E.S. (1983), no. no. 58, 83–196.Google Scholar
  16. [Hi]
    N. Hitchin, Harmonic spinors, Adv. Math. 14 (1974), 1–55.MATHMathSciNetGoogle Scholar
  17. [K]
    G.G. Kasparov, Equivariant KK-theory and the Novikov Conjecture, Invent. Math. 91 (1988), 147–201.MATHMathSciNetGoogle Scholar
  18. [KaSk]
    G.G. Kasparov and G. Skandalis, Groups acting on buildings, operator K-theory, and Novikov’s conjecture, preprint, 1989.Google Scholar
  19. [Kaz]
    J.L. Kazdan, Prescribing the curvature of a Riemannian manifold, Conf. Board of the Math. Sciences, American Mathematical Society, Providence, R.I., 1985.Google Scholar
  20. [KW1]
    J.L. Kazdan and F.W. Warner, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann of Math. 101 (1975), 317–331.CrossRefMATHMathSciNetGoogle Scholar
  21. [KW2]
    J.L. Kazdan,Scalar curvature and conformal deformation of Riemannian stricture,J. Diff. Geom. 10 (1975), 113–134.Google Scholar
  22. [Kr]
    M. Kreck, Duality and surgery,preprint.Google Scholar
  23. [KrSt]
    M. Kreck and S. Stolz, IHIIP 2 -bundles and elliptic homology,preprint.Google Scholar
  24. [KwSc]
    S. Kwasik and R. Schultz, Positive scalar curvature and periodic fundamental groups, Comment. Math. Helvetici 65 (1990), 271–286.CrossRefMATHMathSciNetGoogle Scholar
  25. [LM]
    H.B. Lawson, Jr. and M.-L. Michelson, Spin Geometry, Princeton Math. Series, no. 38, Princeton Univ. Press, Princeton, N.J., 1989.Google Scholar
  26. [LY]
    H.B. Lawson, Jr. and S.-T. Yau, Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helvetici 49 (1974), 232–244.CrossRefMATHMathSciNetGoogle Scholar
  27. [Le]
    M. Lewkowicz, Positive scalar curvature and local actions of nonabelian Lie groups, J. Differential Geometry 31 (1990), 29–45.MathSciNetGoogle Scholar
  28. [Li]
    A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris, Sér. A-B 257 (1963), 7–9.MATHMathSciNetGoogle Scholar
  29. [Lo]
    J.-L. Loday, K-théorie algébrique et représentations des groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), 309–377.MATHMathSciNetGoogle Scholar
  30. [Mah]
    M. Mahowald, The image of J in the EHP-sequence, Ann. of Math. 116 (1982), 65–112.CrossRefMATHMathSciNetGoogle Scholar
  31. [Man]
    C.A. Mann, Jr., A twenty-four dimensional spin manifold, Ph.D. Dissertation, Mass. Inst. of Technology, Cambridge, Mass., 1969.Google Scholar
  32. [Mar]
    H.R. Margolis, Eilenberg-MacLane spectra, Proc. Am. Math. Soc. 43 (1974), 409–415.CrossRefMATHMathSciNetGoogle Scholar
  33. [M]
    V. Mathai, Non-negative scalar curvature, preprint, University of Adelaide, 1990.Google Scholar
  34. [Mi1]
    J. Milnor, On the Stiefel-Whitney numbers of complex and spin manifolds, Topology 3 (1965), 223–230.CrossRefMATHMathSciNetGoogle Scholar
  35. [Mi2]
    Lectures on the h-cobordism theorem,Mathematical Notes, Princeton University Press, Princeton, 1965.Google Scholar
  36. [Miy1]
    T. Miyazaki, On the existence of positive scalar curvature metrics on nonsimply-connected manifolds, J. Fac. Sci. Univ. Tokyo, Sect. IA 30 (1984), 549–561.MATHMathSciNetGoogle Scholar
  37. [Miy2]
    Simply connected spin manifolds and positive scalar curvature,Proc. Amer. Math. Soc. 93 (1985), 730–734.Google Scholar
  38. [Mo]
    H. Moriyoshi, Positive scalar curvature and higher A-genus, J. Fac. Sci. Univ. Tokyo, Sect. IA 35 (1988), 199–224.MATHMathSciNetGoogle Scholar
  39. [Pe]
    D.J. Pengelley, H* (MO 8 ; Z/2) is an extended AZ-coalgebra,Proc. Am.Math. Soc. 87 (1983), 355–356.Google Scholar
  40. [Roe1]
    J. Roe, An index theorem on open manifolds, II, J. Differential Geom. 27 (1988), 115–136.MathSciNetGoogle Scholar
  41. [Roe2]
    J. Roe, Exotic cohomology and index theory,Bull. Amer. Math. Soc. 23 (1990), 447–453.Google Scholar
  42. [Roe3]
    J. Roe, Exotic cohomology and index theory on complete Riemannian mani-folds,preprint (1990).Google Scholar
  43. [R1]
    J. Rosenberg, CC*-algebras, positive scalar curvature, and the Novikov Conjecture, Publ. Math. I.H.E.S. (1983), no. no. 58, 197–212.Google Scholar
  44. [R2]
    J. Rosenberg, C*-algebras, positive scalar curvature, and the Novikov Conjecture, II, 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
  45. [R3]
    J. Rosenberg, C*-algebras, positive scalar curvature, and the Novikov Conjecture, III, Topology 25 (1986), 319–336.Google Scholar
  46. [R4]
    J. Rosenberg, The KO-assembly map and positive scalar curvature, Proc. Interna- tional Conf. on Algebraic Topology, Poznan, 1989, S. Jackowski, R. Oliver, and K. Pawalowski, eds., Lecture Notes in Math., Springer-Verlag, Berlin and New York (to appear).Google Scholar
  47. [Schn]
    R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (1984), 479–495.MATHMathSciNetGoogle Scholar
  48. [SY1]
    R. Schoen and S.-T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds of non-negative scalar curvature, Ann. of Math. 110 (1979), 127–142.CrossRefMATHMathSciNetGoogle Scholar
  49. [SY2]
    R. Schoen and S.-T. Yau, On the structure of manifolds with positive scalar curvature, Manuscrip-ta Math. 28 (1979), 159–183.Google Scholar
  50. [Schu]
    R. Schultz, private communication.Google Scholar
  51. [St1]
    S. Stolz, Simply connected manifolds of positive scalar curvature, Bull. Amer. Math. Soc. 23 (1990), 427–432.CrossRefMATHMathSciNetGoogle Scholar
  52. [St2]
    S. Stolz, Simply connected manifolds of positive scalar curvature,preprint.Google Scholar
  53. [St3]
    S. Stolz, Splitting MSpin-module spectra,preprint.Google Scholar
  54. [S]
    R.E. Stong, Notes on Cobordism Theory, Mathematical Notes, no. 7, Princeton Univ. Press, Princeton, N.J., 1968.Google Scholar
  55. [Sw]
    R.M. Switzer, Algebraic Topology — Homotopy and Homology, Spinxger Verlag, Berlin and New York, 1975.MATHGoogle Scholar
  56. [Ws]
    C.T. C. Wall, Determination of the cobordism ring, Ann. of Math. 72 (1969), 292–311.CrossRefGoogle Scholar
  57. [W]
    J.A. Wolf, Essential self-adjointness for the Dirac operator and its square, In-diana Univ. Math.J. 22 (1973), 611–640.MATHGoogle Scholar
  58. [Yau]
    S.-T. Yau, Minimal surfaces and their role in differential geometry, Global Riemannian Geometry, T.J. Willmore and N.J. Hitchin, eds., Ellis Horwood and Halsted Press, Chichester, England, and New York, 1984, pp. 99–103.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Jonathan Rosenberg
    • 1
    • 2
  • Stephan Stolz
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of Notre DameNotre DameUSA

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