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Bounded rigidity of manifolds and asymptotic dimension growth

Published online by Cambridge University Press:  07 January 2008

Stanley S. Chang ,
Steven Ferry  and
Guoliang Yu
Stanley S. Chang
Affiliation:
schang@wellesley.eduDepartment of Mathematics, Wellesley College, MA 02481, USA
Steven Ferry
Affiliation:
sferry@math.rutgers.eduDepartment of Mathematics, Rutgers University, Piscataway NJ 08854, USA
Guoliang Yu
Affiliation:
gyu@math.vanderbilt.eduDepartment of Mathematics, Vanderbilt University, Nashville TN 37240, USA
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Abstract

We provide a bounded rigidity result for uniformly contractible manifolds with bounded geometry and sufficiently slow asymptotic dimension growth. This notion of asymptotic growth is a generalization of Gromov's definition of asymptotic dimension. In particular for these manifolds we prove that the bounded assembly map is an isomorphism. Our result is inspired by the coarse Baum-Connes results of Yu and the development of squeezing structures.

Keywords

Bounded rigiditynoncompact manifoldsasymptotic dimension
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 1 , February 2008 , pp. 129 - 144
Copyright
Copyright © ISOPP 2008

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References

1.Bartels, A., Squeezing and higher algebraic K-theory, K-Theory 28 (2003), no. 1, 19–37CrossRefGoogle Scholar
2.Carlsson, G. and Goldfarb, B., The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension, Invent. Math. 157 (2004), no. 2, 405–418CrossRefGoogle Scholar
3.Chang, S. and Weinberger, S., On Novikov-type Conjectures, to appear in volumes of the Clay Institute, 2005Google Scholar
4.Chapman, T., Approximation results in topological manifolds, Mem. Amer. Math. Soc. 34 (1981), no. 251Google Scholar
5.Dranishnikov, A., On some approximation problems in topology, Ten mathematical essays on approximation in analysis and topology, 61–94, Elsevier B.V., Amsterdam, 2005CrossRefGoogle Scholar
6.Dranishnikov, A., Ferry, S. and Weinberger, S., Large Riemannian manifolds which are flexible, Ann. of Math. (2) 157 (2003), no. 3, 919–938CrossRefGoogle Scholar
7.Ferry, S., Epsilon-delta surgery over ℤ, preprintGoogle Scholar
8.Ferry, S. and Pedersen, E., Epsilon surgery theory, Novikov conjectures, index theorems and rigidity, vol. 2 (Oberwolfach, 1993), 167–226, London Math. Soc. Lecture Note Ser., 227, Cambridge Univ. Press, Cambridge, 1995CrossRefGoogle Scholar
9.Ferry, S. and Weinberger, S., A coarse approach to the Novikov conjecture, Novikov conjectures, index theorems and rigidity, vol. 1 (Oberwolfach, 1993), 147–163, London Math. Soc. Lecture Note Ser., 226, Cambridge Univ. Press, Cambridge, 1995CrossRefGoogle Scholar
10.Gromov, M., Asymptotic invariants for infinite groups, vol. 2, Proc. 1991 Sussex Conference on Geometric Group Theory, LMS Lecture Notes Series 182 (1993), 1–295Google Scholar
11.Gromov, M., Random walk in random groups Geom. Funct. Anal. 13 (2003), no. 1, 73–146CrossRefGoogle Scholar
12.Higson, N. and Roe, J., On the coarse Baum-Connes conjecture, Novikov conjectures, index theorems and rigidity, vol. 2 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser. 227, 1995, pp. 227–254CrossRefGoogle Scholar
13.Pedersen, E. and Ranicki, A., Projective surgery theory, Topology 19 (1980), no. 3, 239–254CrossRefGoogle Scholar
14.Pedersen, E. and Weibel, C., K-theory homology of spaces, Algebraic topology (Arcata, CA, 1986), 346–361, Lecture Notes in Math., 1370, Springer, Berlin, 1989CrossRefGoogle Scholar
15.Quinn, F., Ends of maps I, Ann. of Math.(2) 110 (1979), 275–311CrossRefGoogle Scholar
16.Ranicki, A., Lower K- and L-theory, London Mathematical Society Lecture Note Series, 178. Cambridge University Press, Cambridge, 1992Google Scholar
17.Ranicki, A. and Yamasaki, M., Controlled L-theory, arXiv:math.GT/0402217 v1, February 13, 2004Google Scholar
18.Roe, J., Coarse cohomology and index theory for complete Riemannian manifolds, Memoirs AMS no. 104 (1993)CrossRefGoogle Scholar
19.Shaneson, J., Wall's surgery obstruction groups for G×ℤ, Ann. of Math. (2) 90 (1969), 296–334CrossRefGoogle Scholar
20.Weinberger, S., The topological classification of stratified spaces, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1994Google Scholar
21.Weiss, M. and Williams, B., Automorphisms of manifolds and algebraic K-theory I, K-Theory 1 (1988), no. 6, 575–626CrossRefGoogle Scholar
22.Yu, G., The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), no. 2, 325–355CrossRefGoogle Scholar
23.Yu, G., The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (2000), no. 1, 201–240CrossRefGoogle Scholar