Resumen: We apply a Cantor completion process to construct a complete, non-Archimedean metric on the set of shape morphisms between pointed compacta. In the case of shape groups we obtain a canonical norm producing a complete, both left and right invariant ultrametric. On the other hand, we give a new characterization of movability and we use these spaces of shape morphisms and
uniformly continuous maps between them, to prove an infinite-dimensional theorem from which we can show, in a short and elementary way, some known Whitehead type theorems in shape theory.
Palabras clave: Pointed shape theory; Whitehead theorem; shape morphism; Cantor completion process; invariant ultrametric; shape theory
Resumen: We work over an o-minimal expansion of a real closed field. The o-minimal homotopy groups of a definable set are defined naturally using definable continuous maps. We prove that any two semialgebraic maps which are definably homotopic are also semialgebraically homotopic. This result together with known results on semialgebraic homotopy allows us to develop an o-minimal homotopy
theory. In particular, we obtain o-minimal versions of the Hurewicz theorems and the Whitehead theorem.
Resumen: In [E. Baro, M. Otero, On o-minimal homotopy, Quart. J. Math. (2009) 15pp, in press (doi:10.1093/qmath/hap011)] o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors.
We also study the concept of connectedness in V-definable groups - which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.