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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Manifolds with singularities accepting a metric of positive scalar curvature

Boris Botvinnik

Geometry & Topology 5 (2001) 683–718

arXiv: math.DG/9910177

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, $\mathcal{A}(1)$–modules
Mathematical Subject Classification 2000
Primary: 57R15
Secondary: 53C21, 55T15, 57R90
References
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Publication
Received: 2 November 1999
Revised: 28 August 2001
Accepted: 26 September 2001
Published: 26 September 2001
Proposed: Ralph Cohen
Seconded: Haynes Miller, Steven Ferry
Authors
Boris Botvinnik
Department of Mathematics
University of Oregon
Eugene
Oregon 97403
USA