Morón, Manuel A. and Romero Ruiz del Portal, Francisco
(1997)
*Shape as a Cantor completion process.*
Mathematische Zeitschrift, 225
(1).
pp. 67-86.
ISSN 0025-5874

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## Abstract

Let X and Y be metric compacta, Y embedded in the Hilbert cube Q. For two maps f,g:X→Q the authors define F(f,g):=inf{ε>0:f is homotopic to g in the ε-neighborhood of Y}, and a sequence of maps fk :X→Q, k∈N, is said to be a Cauchy sequence provided for every ε>0 there is a k0∈N such that F(fk,fk′)<ε whenever k,k′≥k 0. Such sequences coincide with the approximative maps of K. Borsuk [Theory of shape, PWN, Warsaw, 1975] and represent shape morphisms from X to Y. The function F is not a pseudometric, but defining d(α,β):=lim k F(fk,gk), where the shape morphisms α,β∈Sh(X,Y) are represented by Cauchy sequences (fk),(gk), the authors prove that (Sh(X,Y),d) becomes a complete zero-dimensional ultrametric space, homeomorphic to a closed subset of the irrationals. Among other things, the authors prove that if two compacta X and Y are of the same shape, then for every compactum Z, the spaces Sh (X,Z) and Sh (Y,Z) are uniformly homeomorphic. In the last section, the authors show, for example, that for X compact and Y∈FANR , the space Sh (X,Y) is countable, give several characterizations of various kinds of movability, and translate their results to Z-sets in Q and sequences of proper maps between their complements.

Item Type: | Article |
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Uncontrolled Keywords: | Geometric finiteness theorems; controlled topology; homotopy types; spaces |

Subjects: | Sciences > Mathematics > Topology |

ID Code: | 15606 |

References: | Borsuk, K.: Theory of shape. Monografie Matematyczne 59, Polish Scientific Publishers, Warszawa (1975) Borsuk, K.: On some metrization of the hyperspace of compact set. Fund. Math. 41, 168-202 (1954) Borsuk, K.: On a metrization of the hyperspace of a metric space. Fund. Math. 94, 191-207 (1977) Borsuk, K., Oledzki, J.: Remark on the shape domination. Bull. Acad. Polon. Sci. 28, 67-70 (1980) Bogatyi, S.: Approximate and fundamental retracts. Math. USSR Sbornik 22, 91-103 (1974) Čerin, Z.: Homotopy properties of locally compact spaces at infinity – calmness and smoothness. Pacific Journal of Math. 79, 69-91 (1978) Chapman, T.A.: On some applications of infinite dimensional manifolds to the theory of shape. Fund. Math. 76, 181-193 (1972) Chapman, T.A.: Shape of finite-dimensional compacta. Fund. Math. 76, 261-276 (1972) Chapman, T.A.: Lectures on Hilbert cube manifolds. Regional Conferences Series in Math., Amer. Math. Soc. 28 (1976) Clapp, M.H.: On a generalization of absolute neighborhood retracts. Fund. Math. 70, 117-130 (1971) Dranisnikov, A.N., Ferry, S.C.: Cell like images of topological manifolds and limits of manifolds in Gromov-Hausdorff space. Preprint Dydak, J., Segal, J.: Shape Theory: an introduction. Lecture Notes in Math. 688, Springer-Verlag, Berlin (1978) Ferry, S.C.: Topological finiteness theorems for manifolds in Gromov-Hausdorff space. Preprint Gromov, M., Lafontaine, J., Pansu, P.: Structures métriques pour les varieties riemanniennes. Cedic/Fernand Nathan (1981) Gromov, M.: Groups of polynomials growth and expanding maps. Inst. Hautes Etude Sci. Publ. Math. 53, 53-78 (1981) Grove, K., Petersen, P.: Bounding homotopy types by geometry. Annals of Math. 128, 195-200 (1988) Grove, K., Petersen, P., Wu, J.: Geometric finiteness theorems via controlled topology. Inventiones Mathematicae 99, 205-213 (1990). Correction in Invent. Math. 104, 221-222 (1991) Laguna, V.F., Morón, M.A., To Nhu, N., Sanjurjo, J.M.R.: Movability and limits of polyhedra. Fund. Math. 143, 191-201 (1993) Laguna, V.F., Sanjurjo, J.M.R.: Spaces of approximative maps. Math. Japonica 31 (4) 623-633 (1986) Laguna, V.F., Sanjurjo, J.M.R.: Spaces of approximative maps. II, Pub. Mat. U. A. B. 30, 115-126 (1986) Laguna, V.F., Sanjurjo, J.M.R.: Shape morphisms and spaces of approximative maps. Fund. Math. 133, 225-235 (1989) Morón, M.A., Ruiz del Portal, F.R.: Counting shape and homotopy types among FANR's: an elementary approach. Manuscripta Math. 79, 411-414 (1993) Mardesic, S., Segal, J.: Shape theory. North-Holland (1982) Mardesic, S., Segal, J.: Shapes of compacta and ANR systems. Fund. Math. 72, 41-59 (1971) Noguchi, H.: A generalization of absolute neighbourhood retracts. Kodai Math. Sem. Rep. 1, 20-22 (1953) Nowak, S.: Some properties of fundamental dimension. Fund. Math. 85, 211-227 (1974) Petersen, P.: A finiteness theorem for metric spaces. Journal of Diff. Geometry 31, 387-395 (1990) Petersen, P.: Gromov Hausdorff convergence of metric spaces. Proc. of Symposia in Pure Math. 54, Part 3, 489-504 (1993) Schikhof, W.H.: Ultrametric calculus. An introduction to p-adic analysis, Cambridge Univ. Press (1984) Spiez, S.: Movability and uniform movability. Bull. Acad. Polon. Sci. 22, 43-45 (1974) Van Rooij, A.: Non-Archimedean functional analysis. Marcel Dekker Inc. New York (1978) Yamaguchi, T.: Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds. Proc. Amer. Math. Soc. 102, 660-666 (1988) |

Deposited On: | 13 Jun 2012 07:41 |

Last Modified: | 06 Feb 2014 10:28 |

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