Geometry & Topology

Manifolds with singularities accepting a metric of positive scalar curvature

Boris Botvinnik

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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