Geometry & Topology, Vol. 5 (2001) Paper no. 22, pages 683--718.

Manifolds with singularities accepting a metric of positive scalar curvature

Boris Botvinnik

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.

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, A(1)-modules.

AMS subject classification. Primary: 57R15. Secondary: 53C21, 55T15, 57R90.

DOI: 10.2140/gt.2001.5.683

E-print: arXiv:math.DG/9910177

Submitted to GT on 2 November 1999. (Revised 28 August 2001.) Paper accepted 26 September 2001. Paper published 26 September 2001.

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Boris Botvinnik
Department of Mathematics, University of Oregon
Eugene, OR 97403, USA

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