|
 -coarse geometry of complements of Z-sets in the Hilbert cube
Author(s):
E.
Cuchillo-Ibáñez;
J.
Dydak;
A.
Koyama;
M.
A.
Morón
Journal:
Trans. Amer. Math. Soc.
360
(2008),
5229-5246.
MSC (2000):
Primary 18B30, 54D35, 54E15;
Secondary 54C55, 54E35, 54F45
Posted:
May 20, 2008
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
Motivated by the Chapman Complement Theorem, we construct an isomorphism between the topological category of compact Z-sets in the Hilbert cube  and the  -coarse category of their complements. The  -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.
References:
-
- 1.
- M. Alonso-Morón and A. González Gómez, The Hausdorff metric and classifications of compacta, Bull. London Math. Soc. 38 (2006), 314-322. MR 2215924 (2006m:54019)
- 2.
- K. Borsuk, Theory of retracts, PWN-Polish Scientific Publishers, Warsaw, 1967.MR 0216473 (35:7306)
- 3.
- T.A. Chapman, On some applications of infinite-dimensional manifolds to the theory of shape, Fund. Math. 76 (1972), 181-193. MR 0320997 (47:9530)
- 4.
- T.A. Chapman, Lectures on Hilbert cube manifolds, American Mathematical Society, 1976. MR 0423357 (54:11336)
- 5.
- T.A. Chapman and L. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976), 171-208. MR 0425973 (54:13922)
- 6.
- A. Dranishnikov, Asymptotic Topology, Russian Math. Surveys 55 (2000), no. 6, 71-116. MR 1840358 (2002j:55002)
- 7.
- B. Grave, Asymptotic dimension of coarse spaces, available at http://www.uni-math. gwdg.de/bgr/veroeffentl/, 2004.
- 8.
- M. Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture Notes 182 (1993). MR 1253544 (95m:20041)
- 9.
- S-T. Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977 (31:6202)
- 10.
- W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, NJ, 1941. MR 0006493 (3:312b)
- 11.
- P. Mrozik, Hereditary shape equivalences and complement theorems, Topology Appl. 22 (1986), 131-137. MR 836320 (88a:57030)
- 12.
- J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Memoirs of the American Mathematical Society 497, 1993. MR 1147350 (94a:58193)
- 13.
- J. Roe, Index Theory, Coarse Geometry, and Topology of Manifolds, volume 90 of CBMS Conference Proceedings. American Mathematical Society, Providence, R.I., 1996. MR 1399087 (97h:58155)
- 14.
- J. Roe, Lectures on Coarse Geometry, American Mathematical Society, 2003. MR 2007488 (2004g:53050)
- 15.
- N. Wright,
 coarse geometry, Ph.D. thesis, Penn State, 2002. - 16.
- N. Wright,
 coarse geometry and scalar curvature, J. Funct. Anal. 197 (2003), 469-488. MR 1960422 (2003m:58034) - 17.
- N. Wright, The coarse Baum-Connes conjecture via
 coarse geometry, J. Funct. Anal. 220 (2005), 265-303. MR 2119281 (2006d:58023)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2000):
18B30, 54D35, 54E15,
54C55, 54E35, 54F45
Retrieve articles in all Journals with
MSC (2000):
18B30, 54D35, 54E15,
54C55, 54E35, 54F45
Additional Information:
E.
Cuchillo-Ibáñez
Affiliation:
Departamento Matemática Aplicada, E.T.S.I. Montes, Universidad Politécnica, 28040 Madrid, Spain
Email:
eduardo.cuchillo@upm.es
J.
Dydak
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email:
dydak@math.utk.edu
A.
Koyama
Affiliation:
Department of Mathematics, Shizuoka University, Shizuoka, Japan
Email:
sakoyam@ipc.shizuoka.ac.jp
M.
A.
Morón
Affiliation:
Departamento Geometría y Topología, Facultad de Cc.Matemáticas, Universidad Complutense, 28040 Madrid, Spain
Email:
ma\_moron@mat.ucm.es
DOI:
10.1090/S0002-9947-08-04603-5
PII:
S 0002-9947(08)04603-5
Keywords:
Covering dimension,
asymptotic dimension,
$C_{0}$-coarse structure,
ANR-space,
$C_{0}$-coarse morphism,
uniformly continuous map,
compact Z-set,
Higson-Roe compactification and corona.
Received by editor(s):
July 17, 2006
Posted:
May 20, 2008
Additional Notes:
The first and fourth named authors were supported by the MEC, MTM2006-0825.
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
