Un ejemplo de teoría de homotopia en los grupos abelianos

Abstract

For a commutative unitary ring R, we have developed a new homotopy theory in the category of abelian groups. The homotopy category of this theory has the following property: if an abelian group A admits the structrure of an R-module, then A has the homotopy type of the zero abelian group. As a consequence of this fact, this theory is a useful tool to analyze the obstruction of an abelian group to be an R-module. This work contains a detailed study of the analogues of homotopy groups and the construction of homotopy sequences associated to a homomorphism of abelian groups. We have also analyzed the theories associated to some particular rings, for example, for the ring of rational numbers, Q, we have the following version of the Whitehead theorem: An abelian group A has the structure of a Q-module ( A is contractible) if and only if A has trivial homotopy groups.