On hypersphericity of manifolds with finite asymptotic dimension
Author:
A. N. Dranishnikov
Journal:
Trans. Amer. Math. Soc. 355 (2003), 155167
MSC (2000):
Primary 53C23
Published electronically:
September 6, 2002
MathSciNet review:
1928082
Fulltext PDF Free Access
Abstract 
References 
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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 GromovLawson 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.
 [Dr]
A.N. Dranishnikov, Asymptotic topology, Uspekhi Mat. Nauk 55 (2000) 71116, Russian Math. Surveys 55:6 (2000), 10851129.
 [DFW]
A.N. Dranishnikov, S. Ferry and S. Weinberger, Large Riemannian manifolds which are flexible, Preprint (1994).
 [DZ]
A.N. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Preprint (2002).
 [G1]
M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory, vol. 2, London Math. Soc. Lecture Notes 182, Cambridge University Press, 1993. MR 95m:20041
 [G2]
M. Gromov, Large Riemannian manifolds, Lecture Notes in Math. 1201 (1986), 108122. MR 87k:53091
 [G3]
M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional Analysis on the eve of the 21st century, Vol. 2, Progr. Math. 132 (1996), 1213. MR 98d:53052
 [G4]
M. Gromov, Spaces and questions, Geom. Funct. Anal., Special Volume, Part 1, 2000, pp. 118161. MR 2002e:53056
 [GL]
M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. 58 (1983), 83196. MR 85g:58082
 [HR1]
N. Higson and J. Roe, On the coarse BaumConnes conjecture, London Math. Soc. Lecture Notes 227 (1995), 227254. MR 97f:58127
 [HR2]
N. Higson and J. Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000), 143153. MR 2001h:57043
 [Ro1]
J. Roe, Coarse cohomology and index theory for complete Riemannian manifolds, Memoirs Amer. Math. Soc. No. 497, Providence, RI, 1993. MR 94a:58193
 [Ro2]
J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, Number 90 (1996). MR 97h:58155
 [Yu1]
G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math 147 (1998), no. 2, 325355. MR 99k:57072
 [Yu2]
G. Yu, The coarse BaumConnes conjecture for spaces which admit a uniform embedding into Hilbert space, Inventiones Mathematicae 139:1 (2000), 201240. MR 2000j:19005
 [Dr]
 A.N. Dranishnikov, Asymptotic topology, Uspekhi Mat. Nauk 55 (2000) 71116, Russian Math. Surveys 55:6 (2000), 10851129.
 [DFW]
 A.N. Dranishnikov, S. Ferry and S. Weinberger, Large Riemannian manifolds which are flexible, Preprint (1994).
 [DZ]
 A.N. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Preprint (2002).
 [G1]
 M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory, vol. 2, London Math. Soc. Lecture Notes 182, Cambridge University Press, 1993. MR 95m:20041
 [G2]
 M. Gromov, Large Riemannian manifolds, Lecture Notes in Math. 1201 (1986), 108122. MR 87k:53091
 [G3]
 M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional Analysis on the eve of the 21st century, Vol. 2, Progr. Math. 132 (1996), 1213. MR 98d:53052
 [G4]
 M. Gromov, Spaces and questions, Geom. Funct. Anal., Special Volume, Part 1, 2000, pp. 118161. MR 2002e:53056
 [GL]
 M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Etudes Sci. Publ. Math. 58 (1983), 83196. MR 85g:58082
 [HR1]
 N. Higson and J. Roe, On the coarse BaumConnes conjecture, London Math. Soc. Lecture Notes 227 (1995), 227254. MR 97f:58127
 [HR2]
 N. Higson and J. Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000), 143153. MR 2001h:57043
 [Ro1]
 J. Roe, Coarse cohomology and index theory for complete Riemannian manifolds, Memoirs Amer. Math. Soc. No. 497, Providence, RI, 1993. MR 94a:58193
 [Ro2]
 J. Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, Number 90 (1996). MR 97h:58155
 [Yu1]
 G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math 147 (1998), no. 2, 325355. MR 99k:57072
 [Yu2]
 G. Yu, The coarse BaumConnes conjecture for spaces which admit a uniform embedding into Hilbert space, Inventiones Mathematicae 139:1 (2000), 201240. MR 2000j:19005
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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 326118105
Email:
dranish@math.ufl.edu
DOI:
http://dx.doi.org/10.1090/S000299470203115X
PII:
S 00029947(02)03115X
Keywords:
Hyperspherical manifold,
uniform embedding,
asymptotic dimension,
scalar curvature,
GromovLawson 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 DMS9971709
Article copyright:
© Copyright 2002
American Mathematical Society
