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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


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:
\begin{theorem1}Every metric space $X$\space with bounded geometry whose asympt... ... embedding into the product of $n+1$\space locally finite trees. \end{theorem1}

\begin{theorem2}Every metric space $X$\space with bounded geometry whose asympt... ...ding into a non-positively curved manifold of dimension $2n+2$ . \end{theorem2}
The Corollary is used in the proof of the following.
\begin{theorem3}For every uniformly contractible manifold $X$ whose asymptotic d... ...\mathbf{R}^{n}$\space is integrally hyperspherical for some $n$ . \end{theorem3}

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 $X$ of bounded geometry is large scale uniformly embeddable into a Hilbert space, then $X$ is stably integrally hyperspherical.

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  • [Dr] A.N. Dranishnikov, Asymptotic topology, Uspekhi Mat. Nauk 55 (2000) 71-116, Russian Math. Surveys 55:6 (2000), 1085-1129.
  • [D-F-W] 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), 108-122. 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), 1-213. MR 98d:53052
  • [G4] M. Gromov, Spaces and questions, Geom. Funct. Anal., Special Volume, Part 1, 2000, pp. 118-161. MR 2002e:53056
  • [G-L] 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), 83-196. MR 85g:58082
  • [H-R1] N. Higson and J. Roe, On the coarse Baum-Connes conjecture, London Math. Soc. Lecture Notes 227 (1995), 227-254. MR 97f:58127
  • [H-R2] N. Higson and J. Roe, Amenable group actions and the Novikov conjecture, J. Reine Angew. Math. 519 (2000), 143-153. 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, 325-355. MR 99k:57072
  • [Yu2] G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Inventiones Mathematicae 139:1 (2000), 201-240. 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 32611-8105

PII: S 0002-9947(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