On hypersphericity of manifolds with finite asymptotic dimension

Author:
A. N. Dranishnikov

Journal:
Trans. Amer. Math. Soc. **355** (2003), 155-167

MSC (2000):
Primary 53C23

Published electronically:
September 6, 2002

MathSciNet review:
1928082

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following embedding theorems in the coarse geometry:

The Corollary is used in the proof of the following.

Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature.

We also prove that if a uniformly contractible manifold of bounded geometry is large scale uniformly embeddable into a Hilbert space, then is stably integrally hyperspherical.

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Additional Information

**A. N. Dranishnikov**

Affiliation:
Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville Florida 32611-8105

Email:
dranish@math.ufl.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03115-X

Keywords:
Hyperspherical manifold,
uniform embedding,
asymptotic dimension,
scalar curvature,
Gromov-Lawson conjecture

Received by editor(s):
January 23, 2001

Received by editor(s) in revised form:
May 20, 2002

Published electronically:
September 6, 2002

Additional Notes:
The author was partially supported by NSF grant DMS-9971709

Article copyright:
© Copyright 2002
American Mathematical Society