Proper cohomologies and the proper classification problem

Resumen

We attack the classification problem for proper maps using proper cohomologies with coefficients either in a tower of abelian groups or in a morphism of a tower of abelian groups into an abelian group. To compute these proper cohomology groups we have considered a universal coefficient formula which works in a sligtly different way that the clasic one. In order to apply this formula we have done an analysis of the projective objects in the category of tower of abelian groups tow-Ab and in the category (Ab, tow-Ab) whose objects are morphisms from a tower of abelian groups to an abelian group.As a consequence of this study we prove that the group of proper homotopy classes of a finite dimension σ-compact simplicial complex K into the euclidean plane R2 is isomorphic to colim Hom (H1(Pi), Z) where {clousure of K-Pi} is cofinal in the set of compact subsets of K. In particular, we compute the group of proper homotopy classes of an open connected surface into the euclidean plane R2.