Homotopy groups of the moduli space of metrics of positive scalar curvature

Botvinnik, Boris¹ ; Hanke, Bernhard² ; Schick, Thomas³ ; Walsh, Mark⁴

¹ Department of Mathematics, University of Oregon, Eugene OR 97403, USA
² Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
³ Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3, 37073 Göttingen, Germany
⁴ Mathematisches Institut, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany

Geometry & topology, Tome 14 (2010) no. 4, pp. 2047-2076.

Résumé

We show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov–Lawson to certain nonlinear smooth sphere bundles constructed by Hatcher.

Détail

DOI : 10.2140/gt.2010.14.2047

Keywords: metrics of positive scalar curvature, moduli space of positive scalar curvature metrics, classifying space of a diffeomorphism group, Gromov–Lawson surgery parametrized by a Morse function, rational homotopy type, Hatcher map

Bibliographie

[1] K Akutagawa, B Botvinnik, The relative Yamabe invariant, Comm. Anal. Geom. 10 (2002) 935

[2] M Belolipetsky, A Lubotzky, Finite groups and hyperbolic manifolds, Invent. Math. 162 (2005) 459 | DOI

[3] A L Besse, Einstein manifolds, 10, Springer (1987)

[4] M Bökstedt, The rational homotopy type of ΩWhDiff(∗), from: "Algebraic topology, Aarhus 1982 (Aarhus, 1982)", Lecture Notes in Math. 1051, Springer (1984) 25 | DOI

[5] B Botvinnik, P B Gilkey, The eta invariant and metrics of positive scalar curvature, Math. Ann. 302 (1995) 507 | DOI

[6] J P Bourguignon, Une stratification de l’espace des structures riemanniennes, Compositio Math. 30 (1975) 1

[7] G E Bredon, Sheaf theory, 170, Springer (1997)

[8] D G Ebin, The manifold of Riemannian metrics, from: "Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968)", Amer. Math. Soc. (1970) 11

[9] F T Farrell, W C Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, from: "Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1", Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc. (1978) 325

[10] P Gajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom. 5 (1987) 179 | DOI

[11] S Goette, Morse theory and higher torsion invariants I

[12] M Gromov, H B Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980) 423 | DOI

[13] M Gromov, H B Lawson Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. (1983)

[14] H Herrera, R Herrera, Higher –genera on certain non-spin S1–manifolds, Topology Appl. 157 (2010) 1658 | DOI

[15] N Hitchin, Harmonic spinors, Advances in Math. 14 (1974) 1 | DOI

[16] K Igusa, Higher Franz–Reidemeister torsion, 31, American Mathematical Society (2002)

[17] K Igusa, Higher complex torsion and the framing principle, Mem. Amer. Math. Soc. 177 (2005)

[18] K Igusa, Axioms for higher torsion invariants of smooth bundles, J. Topol. 1 (2008) 159 | DOI

[19] A Kriegl, P W Michor, The convenient setting of global analysis, 53, American Mathematical Society (1997)

[20] H B Lawson Jr., M L Michelsohn, Spin geometry, 38, Princeton University Press (1989)

[21] E Leichtnam, P Piazza, On higher eta-invariants and metrics of positive scalar curvature, –Theory 24 (2001) 341 | DOI

[22] J W Milnor, M A Kervaire, Bernoulli numbers, homotopy groups, and a theorem of Rohlin, from: "Proc Internat. Congress Math. 1958", Cambridge Univ. Press (1960) 454

[23] S B Myers, N E Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. 40 (1939) 400 | DOI

[24] P Piazza, T Schick, Bordism, rho-invariants and the Baum–Connes conjecture, J. Noncommut. Geom. 1 (2007) 27 | DOI

[25] J Rosenberg, C∗–algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. (1983)

[26] T Schick, A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture, Topology 37 (1998) 1165 | DOI

[27] S Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992) 511 | DOI

[28] M Walsh, Metrics of positive scalar curvature and generalised Morse functions, part 1, Mem. Amer. Math. Soc. (2010) | DOI

[29] M Walsh, Metrics of positive scalar curvature and generalised Morse functions, part 2