Martínez Pérez , Álvaro and Morón, Manuel A.
(2010)
*Inverse sequences, rooted trees and their end spaces.*
Topology and its Applications, 157
(16).
pp. 2480-2494.
ISSN 0166-8641

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Official URL: http://www.sciencedirect.com/science/article/pii/S0166864110002634

## Abstract

In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inverse sequences in terms of the metrically proper homotopy type of the corresponding tree and its maximal geodesically complete subtree. We also obtain some consequences in shape theory. In particular we describe some new representations of shape morphisms related to infinite branches in trees.

Item Type: | Article |
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Uncontrolled Keywords: | Tree; Inverse sequence; End space; Coarse map; Mittag-Leffler property; Shape theory |

Subjects: | Sciences > Mathematics > Topology Sciences > Mathematics > Algebra |

ID Code: | 14996 |

References: | [1] H.J. Baues, A. Quintero, Infinite Homotopy Theory, Kluwer Academic Publishers, Boston, 2001. [2] M. Bestvina, R-trees in topology, geometry and group theory, in: Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, pp. 55–91. [3] K. Borsuk, Theory of Retracts, Polish Scientific Publishers, Warszawa, 1967. [4] K. Borsuk, Theory of Shape, Monogr. Mat., vol. 59, Polish Scientific Publishers, Warszawa, 1975. [5] J.-M. Cordier, T. Porter, Shape Theory. Categorical Methods of Approximation, Math. Appl., Ellis Horwood Ltd., Halsted Press, John Wiley and Sons, Inc., Chichester, New York, 1989. [6] J. Dydak, J. Segal, Shape Theory. An Introduction, Lecture Notes in Math., vol. 688, Springer-Verlag, Berlin, Heidelberg, New York, 1978. [7] A. Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique II, in: Séminaire Bourbaki, 12 ème année, 1959–1960, exposé pp. 190–195. [8] B. Hughes, Trees and ultrametric spaces: a categorial equivalence, Adv. Math. 189 (2004) 148–191. [9] B. Hughes, Trees, ultrametrics, and noncommutative geometry, arXiv:math/0605131v2[math.OA]. [10] S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, New York, 1971. [11] S. Mardešic, J. Segal, Shapes of compacta and ANR systems, Fund. Math. 72 (1971) 41–59. [12] S. Mardešic, J. Segal, Shape Theory, North-Holland, 1982. [13] Á. Martínez-Pérez, M.A. Morón, Uniformly continuous maps between ends of R-trees, Math. Z. 263 (3) (2009) 583–606. [14] J.W. Morgan, _-trees and their applications, Bull. Amer. Math. Soc. 26 (1) (1992) 87–112. [15] M.A. Morón, F.R. Ruiz del Portal, Shape as a Cantor completion process, Math. Z. 225 (1997) 67–86. [16] A.M. Robert, A Course in p-Adic Analysis, Grad. Texts in Math., vol. 198, Springer, 2000. [17] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., 2003. [18] J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 497 (1993). [19] J.P. Serre, Trees, Springer-Verlag, New York, 1980. |

Deposited On: | 25 Apr 2012 08:47 |

Last Modified: | 06 Feb 2014 10:13 |

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