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On Gromov's scalar curvature conjecture

Authors: Dmitry Bolotov and Alexander Dranishnikov
Journal: Proc. Amer. Math. Soc. 138 (2010), 1517-1524
MSC (2010): Primary 55M30; Secondary 53C23, 57N65, 55N15
Published electronically: December 8, 2009
MathSciNet review: 2578547
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Abstract: We prove the Gromov conjecture on the macroscopic dimension of the universal covering of a closed spin manifold with a positive scalar curvature under the following assumptions on the fundamental group.

$ 0.1$. Theorem.

Suppose that a discrete group $ \pi$ has the following properties:

$ 1$. The Strong Novikov Conjecture holds for $ \pi$.

$ 2$. The natural map $ per:ko_n(B\pi)\to KO_n(B\pi)$ is injective. Then the Gromov Macroscopic Dimension Conjecture holds true for spin $ n$-manifolds $ M$ with the fundamental groups $ \pi_1(M)$ that contain $ \pi$ as a finite index subgroup.

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Additional Information

Dmitry Bolotov
Affiliation: Verkin Institute of Low Temperature Physics, Lenina Avenue, 47, Kharkov, 631103, Ukraine

Alexander Dranishnikov
Affiliation: Department of Mathematics, 358 Little Hall, University of Florida, Gainesville, Florida 32611-8105

Keywords: Positive scalar curvature, macroscopic dimension, connective $K$-theory, Strong Novikov Conjecture
Received by editor(s): January 28, 2009
Received by editor(s) in revised form: September 11, 2009
Published electronically: December 8, 2009
Additional Notes: This work was supported by NSF grant DMS-0604494
Communicated by: Brooke Shipley
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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