Manifolds with singularities accepting a metric of positive scalar curvature

Botvinnik, Boris¹

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

Geometry & topology, Tome 5 (2001) no. 2, pp. 683-718.

Résumé

We study the question of existence of a Riemannian metric of positive scalar curvature metric on manifolds with the Sullivan–Baas singularities. The manifolds we consider are $Spin$ and simply connected. We prove an analogue of the Gromov–Lawson Conjecture for such manifolds in the case of particular type of singularities. We give an affirmative answer when such manifolds with singularities accept a metric of positive scalar curvature in terms of the index of the Dirac operator valued in the corresponding “$K$–theories with singularities”. The key ideas are based on the construction due to Stolz, some stable homotopy theory, and the index theory for the Dirac operator applied to the manifolds with singularities. As a side-product we compute homotopy types of the corresponding classifying spectra.

Détail

DOI : 10.2140/gt.2001.5.683

Keywords: Positive scalar curvature, Spin manifolds, manifolds with singularities, Spin cobordism, characteristic classes in $K$–theory, cobordism with singularities, Dirac operator, $K$–theory with singularities, Adams spectral sequence, $\mathcal{A}(1)$–modules

Bibliographie

[1] D W Anderson, E H Brown Jr., F P Peterson, The structure of the Spin cobordism ring, Ann. of Math. $(2)$ 86 (1967) 271

[2] N A Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973)

[3] B I Botvinnik, Manifolds with singularities and the Adams–Novikov spectral sequence, London Mathematical Society Lecture Note Series 170, Cambridge University Press (1992)

[4] B Botvinnik, P Gilkey, S Stolz, The Gromov–Lawson–Rosenberg conjecture for groups with periodic cohomology, J. Differential Geom. 46 (1997) 374

[5] D S Freed, $\mathbb{Z}/k$–manifolds and families of Dirac operators, Invent. Math. 92 (1988) 243

[6] D S Freed, Two index theorems in odd dimensions, Comm. Anal. Geom. 6 (1998) 317

[7] D S Freed, R B Melrose, A mod $k$ index theorem, Invent. Math. 107 (1992) 283

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

[9] M Gromov, H B Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. $(2)$ 111 (1980) 423

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

[11] N Higson, An approach to $\mathbb{Z}/k$–index theory, Internat. J. Math. 1 (1990) 189

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

[13] M J Hopkins, M A Hovey, Spin cobordism determines real $K$–theory, Math. Z. 210 (1992) 181

[14] J Kaminker, K P Wojciechowski, Index theory of $\mathbb{Z}/k$ manifolds and the Grassmannian, from: "Operator algebras and topology (Craiova, 1989)", Pitman Res. Notes Math. Ser. 270, Longman Sci. Tech. (1992) 82

[15] M Kreck, S Stolz, $\mathbb{H}\mathrm{P}^2$–bundles and elliptic homology, Acta Math. 171 (1993) 231

[16] D D Joyce, Compact 8–manifolds with holonomy $\mathrm{Spin}(7)$, Invent. Math. 123 (1996) 507

[17] M Mahowald, The image of $J$ in the $EHP$ sequence, Ann. of Math. $(2)$ 116 (1982) 65

[18] M Mahowald, R J Milgram, Operations which detect Sq4 in connective $K$–theory and their applications, Quart. J. Math. Oxford Ser. $(2)$ 27 (1976) 415

[19] O K Mironov, Existence of multiplicative structures in the theory of cobordism with singularities, Izv. Akad. Nauk SSR Ser. Mat. 39 (1975) 1065, 1219

[20] A Hassell, R Mazzeo, R B Melrose, A signature formula for manifolds with corners of codimension two, Topology 36 (1997) 1055

[21] J W Morgan, D P Sullivan, The transversality characteristic class and linking cycles in surgery theory, Ann. of Math. $(2)$ 99 (1974) 463

[22] J Rosenberg, Groupoid $C^*$–algebras and index theory on manifolds with singularities, Geom. Dedicata 100 (2003) 65

[23] J Rosenberg, S Stolz, A “stable” version of the Gromov–Lawson conjecture, from: "The Čech centennial (Boston, MA, 1993)", Contemp. Math. 181, Amer. Math. Soc. (1995) 405

[24] J Rosenberg, S Stolz, Manifolds of positive scalar curvature, from: "Algebraic topology and its applications", Math. Sci. Res. Inst. Publ. 27, Springer (1994) 241

[25] J Rosenberg, S Stolz, Metrics of positive scalar curvature and connections with surgery, from: "Surveys on surgery theory, Vol. 2", Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 353

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

[27] R Schoen, S T Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979) 159

[28] D P Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms. Geometric Topology Seminar Notes, from: "The Hauptvermutung book", $K$–Monogr. Math. 1, Kluwer Acad. Publ. (1996) 69

[29] S Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. $(2)$ 136 (1992) 511

[30] S Stolz, Splitting certain $M\mathrm{Spin}$–module spectra, Topology 33 (1994) 159

[31] S Stolz, Concordance classes of positive scalar curvature metrics, to appear

[32] R E Stong, Notes on cobordism theory, Mathematical notes, Princeton University Press (1968)

[33] U Würgler, On products in a family of cohomology theories associated to the invariant prime ideals of $\pi_{*}(\mathrm{BP})$, Comment. Math. Helv. 52 (1977) 457

[34] W Zhang, A proof of the mod 2 index theorem of Atiyah and Singer, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 277

[35] W Zhang, On the mod $k$ index theorem of Freed and Melrose, J. Differential Geom. 43 (1996) 198