Sull'omotopia uniforme

Abstract

Following in the classical theory's footsteps, it is possible to construct a « rational homotopy theory ». To this purpose, we take all the uniform maps f: I Q n →S (where IQ is the closed unit interval of the rational line equipped with the standard metric uniformity) as paths of a uniform space S. This brings us to the definition of the rational homotopy groups Qn(S, UQ, x) and enables us to consider the related exact sequences. All these objects are uniform invariants. The main problem which arises now is to find a suitable space S* in order that its classical homotopy groups Πn(S*,x) are isomorphic to the classical ones of S. We are able to reach an answer to this question if S is a metrizable uniform space. Since considering the completion Ŝ of S serves no useful purpose (as it is shown by a simple example), we prove that the required space is the « rational path completion » S* of S, with S⊆S*⊆Ŝ. We finally recall that the rational uniform homotopy is a special case of regular homotopy, which has been defined and widely investigated in[1].