Complutense University Library

C0-coarse geometry of complements of Z-sets in the Hilbert cube

Cuchillo Ibáñez, Eduardo and Dydak, J. and Koyama, A. and Morón, Manuel A. (2008) C0-coarse geometry of complements of Z-sets in the Hilbert cube. Transactions of the American Mathematical Society, 360 (10). pp. 5229-5246. ISSN 0002-9947

[img] PDF
Restricted to Repository staff only until 31 December 2020.

253kB

Official URL: http://www.ams.org/journals/tran/2008-360-10/S0002-9947-08-04603-5/S0002-9947-08-04603-5.pdf

View download statistics for this eprint

==>>> Export to other formats

Abstract

Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube Q and the C0-coarse category of their complements. The C0-coarse morphisms are, in this particular case, intrinsically related to uniformly continuous proper maps. Using that fact we are able to relate in a natural way some of the topological invariants of Z-sets to the geometry of their complements.


Item Type:Article
Uncontrolled Keywords:C0 coarse geometry; covering dimension; asymptotic dimension; C0 coarse structure; ANR-space; C0 coarse morphism; uniformly continuous map; compact Z-set; Higson-Roe compactification and corona
Subjects:Sciences > Mathematics > Topology
ID Code:15176
References:

M. Alonso-Morón and A. González Gómez, The Hausdorff metric and classifications of compacta, Bull. London Math. Soc. 38 (2006), 314-322. MR2215924 (2006m:54019)

K. Borsuk, Theory of retracts, PWN-Polish Scientific Publishers, Warsaw, 1967. MR0216473 (35:7306)

T.A. Chapman, On some applications of infinite-dimensional manifolds to the theory of shape, Fund. Math. 76 (1972), 181–193. MR0320997 (47:9530)

T.A. Chapman, Lectures on Hilbert cube manifolds, American Mathematical Society, 1976. MR0423357 (54:11336)

T.A. Chapman and L. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976), 171–208. MR0425973 (54:13922)

A. Dranishnikov, Asymptotic Topology, Russian Math. Surveys 55 (2000), no. 6, 71-116. MR1840358 (2002j:55002)

B. Grave, Asymptotic dimension of coarse spaces, available at http://www.uni-math.gwdg.de/bgr/veroeffentl/, 2004.

M. Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture Notes 182 (1993). MR1253544 (95m:20041)

S-T. Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR0181977 (31:6202)

W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, NJ, 1941. MR0006493 (3:312b)

P. Mrozik, Hereditary shape equivalences and complement theorems, Topology Appl. 22 (1986), 131–137. MR836320 (88a:57030)

J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Memoirs of the American Mathematical Society 497, 1993. MR1147350 (94a:58193)

J. Roe, Index Theory, Coarse Geometry, and Topology of Manifolds, volume 90 of CBMS Conference Proceedings. American Mathematical Society, Providence, R.I., 1996. MR1399087 (97h:58155)

J. Roe, Lectures on Coarse Geometry, American Mathematical Society, 2003. MR2007488 (2004g:53050)

N. Wright, C0 coarse geometry, Ph.D. thesis, Penn State, 2002.

N. Wright, C0 coarse geometry and scalar curvature, J. Funct. Anal. 197 (2003), 469–488. MR1960422 (2003m:58034)

N. Wright, The coarse Baum-Connes conjecture via C0 coarse geometry, J. Funct. Anal. 220 (2005), 265–303. MR2119281 (2006d:58023)

Deposited On:10 May 2012 08:26
Last Modified:06 Feb 2014 10:18

Repository Staff Only: item control page