Manifolds with singularities accepting a metric of positive scalar curvature

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 $S p i n$ 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

Forward citations

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

Volume 5, issue 2 (2001)

Boris Botvinnik