Geometry & Topology

Manifolds with singularities accepting a metric of positive scalar curvature

Boris Botvinnik

Full-text: Open access

Abstract

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.

Article information

Source
Geom. Topol., Volume 5, Number 2 (2001), 683-718.

Dates
Received: 2 November 1999
Revised: 28 August 2001
Accepted: 26 September 2001
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883041

Digital Object Identifier
doi:10.2140/gt.2001.5.683

Mathematical Reviews number (MathSciNet)
MR1857524

Zentralblatt MATH identifier
1002.57055

Subjects
Primary: 57R15: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Secondary: 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 55T15: Adams spectral sequences 57R90: Other types of cobordism [See also 55N22]

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

Citation

Botvinnik, Boris. Manifolds with singularities accepting a metric of positive scalar curvature. Geom. Topol. 5 (2001), no. 2, 683--718. doi:10.2140/gt.2001.5.683. https://projecteuclid.org/euclid.gt/1513883041


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References

  • D W Anderson, E H Brown, F P Peterson, The structure of $Spin$ cobordism ring, Ann. of Math. 86 (1967) 271–298
  • N Baas, On bordism theory of manifolds with singularities, Math. Scand. 33 (1973) 279–302
  • B Botvinnik, Manifold with singularities and the Adam–Novikov spectral sequence, Cambridge University Press (1992)
  • B Botvinnik, P Gilkey, S Stolz, The Gromov–Lawson–Rosenberg conjecture for groups with periodic cohomology, J. Diff. Geom. 46 (1997) 374–405
  • D S Freed, $\Z/k$–Manifolds and families of Dirac operators, Invent. Math. 92 (1988) 243–254
  • D S Freed, Two index theorems in odd dimensions, Comm. Anal. Geom. 6 (1998) 317–329
  • D S Freed, R B Melrose, A mod $k$ index theorem, Invent. Math. 107 (1992) 283–299
  • P Gajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom. 5 (1987) 179–191
  • M Gromov,H B Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. Math. 11 (1980) 423–434
  • M Gromov, H B Lawson, Positive scalar curvature and the Dirac operator on complete manifolds, Publ. Math. I.H.E.S. no. 58 (1983) 83–196
  • N Higson, An approach to $\Z/k$–index theory, Int. J. of math. Vol. 1, No. 2 (1990) 189–210
  • N Hitchin, Harmonic spinors, Advances in Math. 14 (1974) 1–55
  • M J Hopkins, M A Hovey, Spin cobordism determines real $K$–theory, Math. Z. 210 (1992) 181–196
  • J Kaminker, K P Wojciechowski, Index theory of ${\Z}/k$ manifolds and the Grassmannian, from: “Operator algebras and topology (Craiova, 1989)”, Pitman Res. Notes Math. Ser. 270, Longman Sci. Tech. Harlow (1992) 82–92
  • M Kreck, S Stolz, ${H}{\rm P}\sp 2$–bundles and elliptic homology, Acta Math. 171 (1993) 231–261
  • D Joyce, Compact $8$–manifolds with holonomy ${\rm Spin}(7)$, Invent. Math. 123 (1996) 507–552
  • M Mahowald, The image of $J$ in the EHP sequence, Ann. of Math. 116 (1982) 65–112
  • M Mahowald, J Milgram, Operations which detect $Sq^4$ in the connective $K$–theory and their applications, Quart. J. Math. 27 (1976) 415–432
  • O K Mironov, Existence of multiplicative structure in the cobordism theory with singularities, Math. USSR Izv. 9 (1975) 1007–1034
  • A Hassell, R Mazeo, R B Melrose, A signature formula for manifolds with corners of codimension two, Preprint, MIT (1996)
  • J W Morgan, D Sullivan, The transversality characteristic classes and linking cycles in surgery theory, Ann. of Math. 99 (1974) 461–544
  • J Rosenberg, Groupoid $C^*$–algebras and index theory on manifolds with singularities.
  • 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. Providence, RI (1995) 405–418,
  • J Rosenberg, S Stolz, Manifolds of positive scalar curvature, from: “Algebraic topology and its applications”, Math. Sci. Res. Inst. Publ. 27, Springer, New York (1994) 241–267,
  • J Rosenberg, S Stolz, Metrics of positive scalar curvature and connections with surgery, to appear in “Surveys on Surgery Theory”, vol. 2, Ann. of Math. Studies, vol. 149
  • T Schick, A counterexample to the (unstable) Gromov–Lawson–Rosenberg conjecture, Topology 37 (1998) 1165–1168
  • R Schoen, S T Yau, On the structure of manifolds with positive scalar curvature, Manuscripta Math. 28 (1979) 159–183
  • D Sullivan, Triangulating and smoothing homotopy equivalence and homeomorphisms, Geometric topology seminar notes, Princeton University Press (1967)
  • S Stolz, Simply connected manifolds of positive scalar curvature, Ann. of Math. 136 (1992) 511–540
  • S Stolz, Splitting of certain $MSpin$–module spectra, Topology, 133 (1994) 159–180
  • S Stolz, Concordance classes of positive scalar curvature metrics, to appear
  • R E Stong, Notes on Cobordism Theory, Princeton University Press (1968)
  • U Würgler, On the products in a family of cohomology theories, associated to the invariant prime ideal of $\pi_*(BP)$, Comment. Math. Helv. 52 (1977) 457–481
  • W Zhang, A proof of the mod $2$ index theorem of Atiyah and Singer, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993) 277–280
  • W Zhang, On the mod $k$ index theorem of Freed and Melrose, J. Differential Geom. 43 (1996) 198–206