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

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

Published electronically:
September 6, 2002

MathSciNet review:
1928082

Full-text PDF

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.

**[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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
53C23

Retrieve articles in all journals with MSC (2000): 53C23

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