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Positive scalar curvature for manifolds with elementary abelian fundamental group


Authors: Boris Botvinnik and Jonathan Rosenberg
Journal: Proc. Amer. Math. Soc. 133 (2005), 545-556
MSC (2000): Primary 53C20; Secondary 53C21, 55S30, 55N22, 55U25, 57R75
DOI: https://doi.org/10.1090/S0002-9939-04-07762-7
Published electronically: September 16, 2004
MathSciNet review: 2093079
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Abstract: The statement often called the Gromov-Lawson-Rosenberg Conjecture asserts that a manifold with finite fundamental group should admit a metric of positive scalar curvature except when the $KO_*$-valued index of some Dirac operator with coefficients in a flat bundle is non-zero. We prove spin and oriented non-spin versions of this statement for manifolds (of dimension $\ge5$) with elementary abelian fundamental groups $\pi$, except for ``toral'' classes, and thus our results are automatically applicable once the dimension of the manifold exceeds the rank of $\pi$. The proofs involve the detailed structure of $BP_*(B\pi)$, as computed by Johnson and Wilson.


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Additional Information

Boris Botvinnik
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
Email: botvinn@poincare.uoregon.edu

Jonathan Rosenberg
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
Email: jmr@math.umd.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07762-7
Received by editor(s): June 21, 2002
Published electronically: September 16, 2004
Additional Notes: We thank Sergey Novikov for helping to make this collaboration possible
This work was partially supported by NSF grant DMS-0103647
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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