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Infinite loop spaces and positive scalar curvature

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Abstract

We study the homotopy type of the space of metrics of positive scalar curvature on high-dimensional compact spin manifolds. Hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real K-theory spectrum. Our main results concern the nontriviality of this map. We prove that for \(2n \ge 6\), the natural KO-orientation from the infinite loop space of the Madsen–Tillmann–Weiss spectrum factors (up to homotopy) through the space of metrics of positive scalar curvature on any 2n-dimensional spin manifold. For manifolds of odd dimension \(2n+1 \ge 7\), we prove the existence of a similar factorisation. When combined with computational methods from homotopy theory, these results have strong implications. For example, the secondary index map is surjective on all rational homotopy groups. We also present more refined calculations concerning integral homotopy groups. To prove our results we use three major sets of technical tools and results. The first set of tools comes from Riemannian geometry: we use a parameterised version of the Gromov–Lawson surgery technique which allows us to apply homotopy-theoretic techniques to spaces of metrics of positive scalar curvature. Secondly, we relate Hitchin’s secondary index to several other index-theoretical results, such as the Atiyah–Singer family index theorem, the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem. Finally, we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories. The key new ingredient we use in this paper is the high-dimensional analogue of the Madsen–Weiss theorem, proven by Galatius and the third named author.

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Notes

  1. The notation g versus h carries no mathematical meaning, but we typically use g’s for metrics on a cobordism and h’s for metrics on the boundary of a cobordism.

  2. It is important here to adopt the correct convention for K-theory Thom classes of complex vector bundles: one should take the convention used in [35, Theorem C.8], which is characterised by the identity \((\lambda _L^{\mathbb {C}})^2 = (1-L) \cdot \lambda _L^{\mathbb {C}} \in K^0(\mathrm {Th}(L))\) when \(L \rightarrow \mathbb {C}\mathbb {P}^\infty \) is the universal line bundle.

References

  1. Adams, J.F.: On the groups \(J(X)\). II. Topology 3, 137–171 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, J.F.: On the groups \(J(X)\). IV. Topology 5, 21–71 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adams, J.F.: A variant of E. H. Brown’s representability theorem. Topology 10, 185–198 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atiyah, M.F.: \(K\)-theory and reality. Q. J. Math. Oxf. Ser. 2(17), 367–386 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  5. Atiyah, M.F., Bott, R., Shapiro, A.: Clifford modules. Topology 3(suppl. 1), 3–38 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  6. Atiyah, M.F., Segal, G.B.: Equivariant \(K\)-theory and completion. J. Differ. Geom. 3, 1–18 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  7. Atiyah, M.F., Singer, I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37, 5–26 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  8. Atiyah, M. F., Singer, I. M.: The index of elliptic operators. V. Ann. Math. (2) 93, 139–149 (1971)

  9. Bökstedt, M., Dupont, J., Svane, A.M.: Cobordism obstructions to independent vector fields. Q. J. Math. 66(1), 13–61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bruner, R.R., May, J.P., McClure, J.E., Steinberger, M.: \(H_\infty \) Ring Spectra and Their Applications. Lecture Notes in Mathematics, vol. 1176. Springer, Berlin (1986)

  11. Bunke, U.: A \(K\)-theoretic relative index theorem and Callias-type Dirac operators. Math. Ann. 303(2), 241–279 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chernysh, V.: On the homotopy type of the space \(\cal{R}^{+}({M})\). arXiv:math/0405235 (2004)

  13. Chernysh, V.: A quasifibration of spaces of positive scalar curvature metrics. Proc. Am. Math. Soc. 134(9), 2771–2777 (2006) (electronic)

  14. Crowley, D., Schick, T.: The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature. Geom. Topol. 17(3), 1773–1789 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dixmier, J., Douady, A.: Champs continus d’espaces hilbertiens et de \(C^{\ast } \)-algèbres. Bull. Soc. Math. Fr. 91, 227–284 (1963)

    Article  MATH  Google Scholar 

  16. Ebert, J.: A vanishing theorem for characteristic classes of odd-dimensional manifold bundles. J. Reine Angew. Math. 684, 1–29 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ebert, J.: The two definitions of the index difference. Trans. Am. Math. Soc. (2017). preprint arXiv:1308.4998 (to appear)

  18. Gajer, P.: Riemannian metrics of positive scalar curvature on compact manifolds with boundary. Ann. Glob. Anal. Geom. 5(3), 179–191 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Galatius, S., Randal-Williams, O.: Homological stability for moduli spaces of high dimensional manifolds. II. Ann. Math. (2017). preprint arXiv:1403.2334 (to appear)

  20. Galatius, S., Randal-Williams, O.: Stable moduli spaces of high-dimensional manifolds. Acta Math. 212(2), 257–377 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Galatius, S., Tillmann, U., Madsen, I., Weiss, M.: The homotopy type of the cobordism category. Acta Math. 202(2), 195–239 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Galatius, Søren, Randal-Williams, Oscar: Abelian quotients of mapping class groups of highly connected manifolds. Math. Ann. 365(1–2), 857–879 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gromov, M., Lawson, H.B.: The classification of simply connected manifolds of positive scalar curvature. Ann. Math. (2) 111(3), 423–434 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gromov, M., Lawson, H.B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58, 83–196 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hanke, B., Schick, T., Steimle, W.: The space of metrics of positive scalar curvature. Publ. Math. Inst. Hautes Études Sci. 120, 335–367 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  27. Hausmann, J.-C., Husemoller, D.: Acyclic maps. Enseign. Math. (2) 25(1–2), 53–75 (1979)

  28. Higson, N., Roe, J.: Analytic \(K\)-Homology. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  29. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Joyce, D.D.: Compact \(8\)-manifolds with holonomy Spin(7). Invent. Math. 123(3), 507–552 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Karoubi, M.: Espaces classifiants en \(K\)-théorie. Trans. Am. Math. Soc. 147, 75–115 (1970)

    MATH  Google Scholar 

  32. Kervaire, M.A., Milnor, J.W.: Groups of homotopy spheres. I. Ann. Math. 2(77), 504–537 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kochman, S.O.: Homology of the classical groups over the Dyer–Lashof algebra. Trans. Am. Math. Soc. 185, 83–136 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kreck, M.: Surgery and duality. Ann. Math. (2) 149(3), 707–754 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1989)

    MATH  Google Scholar 

  36. Lichnerowicz, A.: Spineurs harmoniques. C. R. Acad. Sci. Paris 257, 7–9 (1963)

    MathSciNet  MATH  Google Scholar 

  37. Madsen, I., Schlichtkrull, C.: The circle transfer and \(K\)-theory. In: Geometry and Topology: Aarhus (1998). Contemporary Mathematics, vol. 258, pp. 307–328. Am. Math. Soc., Providence, RI (2000)

  38. May, J.P.: Classifying spaces and fibrations. Mem. Am. Math. Soc. 1(1, 155), xiii+98 (1975)

  39. Milnor, J.W., Stasheff, J.D.: Characteristic Classes. Princeton University Press, Princeton. Annals of Mathematics Studies, no. 76 (1974)

  40. Nagata, M.: On the uniqueness of Dyer–Lashof operations on the Bott periodicity spaces. Publ. Res. Inst. Math. Sci. 16(2), 499–511 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Palais, R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  42. Priddy, S.: Dyer–Lashof operations for the classifying spaces of certain matrix groups. Q. J. Math. Oxf. Ser. (2) 26(102), 179–193 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  43. Robbin, J., Salamon, D.: The spectral flow and the Maslov index. Bull. Lond. Math. Soc. 27(1), 1–33 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rosenberg. J.: Algebraic \(K\)-Theory and Its Applications. Graduate Texts in Mathematics, vol. 147. Springer, New York (1994)

  45. Rosenberg, J.: Manifolds of positive scalar curvature: a progress report. In: Surveys in Differential Geometry. Surveys in Differential Geometry, vol. XI, pp. 259–294. International Press, Somerville (2007)

  46. Schick, T.: The topology of positive scalar curvature. Proc, ICM (2014). arXiv:1405.4220

  47. Schrödinger, E.: Diracsches Elektron im Schwerefeld. I. Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1932, 105–128 (1932)

  48. Stolz, S.: Simply connected manifolds of positive scalar curvature. Ann. Math. (2), 136(3), 511–540 (1992)

  49. Stong, R.E.: Notes on Cobordism Theory. Mathematical Notes. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1968)

  50. Sullivan, D.: Genetics of homotopy theory and the Adams conjecture. Ann. Math. 2(100), 1–79 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  51. Toda, H.: Composition Methods in Homotopy Groups of Spheres. Annals of Mathematics Studies, no. 49. Princeton University Press, Princeton (1962)

  52. Tuschmann, W., Wraith, D.: Moduli spaces of Riemannian Metrics. Oberwolfach Seminars 46. Birkhäuser (2015)

  53. Wall, C.T.C.: Surgery on Compact Manifolds. London Mathematical Society Monographs, vol. 1. Academic Press, London (1970)

  54. Wall, C.T.C.: Geometrical connectivity. I. J. Lond. Math. Soc. 2(3), 597–604 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  55. Walsh, M.: Metrics of positive scalar curvature and generalised Morse functions, Part I. Mem. Am. Math. Soc. 209(983), xviii+80 (2011)

  56. Walsh, M.: Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics. Proc. Am. Math. Soc. 141(7), 2475–2484 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  57. Walsh, M.: \(H\)-spaces, loop spaces and the space of positive scalar curvature metrics on the sphere. Geom. Topol. 18(4), 2189–2243 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  58. Walsh, M.: The space of positive scalar curvature metrics on a manifold with boundary. arXiv:1411.2423 (2014)

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Acknowledgements

The authors would like to thank Mark Walsh for helpful discussions concerning surgery results for psc metrics. An early draft of the paper had intended to contain a detailed appendix written by Walsh on this subject; later we found a way to prove the main results without using these delicate geometric arguments. The appendix grew instead into a separate paper, [58], which is of independent interest. The authors would also like to thank Don Zagier and Benjamin Young for their interest and advice regarding the numbers A(m, 2) arising in Theorem E.

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Boris Botvinnik

  2. Mathematisches Institut, Westfälische Wilhelms-Universität, Einsteinstr. 62, 48149, Münster, Germany

    Johannes Ebert

  3. Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, UK

    Oscar Randal-Williams

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  1. Boris Botvinnik
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  2. Johannes Ebert
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  3. Oscar Randal-Williams
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Corresponding author

Correspondence to Johannes Ebert.

Additional information

Johannes Ebert was partially supported by the SFB 878.

Oscar Randal-Williams acknowledges Herchel Smith Fellowship support.

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Botvinnik, B., Ebert, J. & Randal-Williams, O. Infinite loop spaces and positive scalar curvature. Invent. math. 209, 749–835 (2017). https://doi.org/10.1007/s00222-017-0719-3

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