Equivariant Action of the Generalized Homotopy Group
by
J. Remedios
Dpto. Matematica Fundamental. Universidad de La Laguna. SPAIN.
Coauthors: F.J. Diaz , S. Rodriguez-Machin

In many theories, homotopy groups and exact sequences of them can be defined using different objects and morphisms as base point and zero morphism, respectively. So, the Brown homotopy groups can be defined using a sequence of points as base point, and the Steenrod homotopy groups use spaces based on a ray. H.J. Baues also uses trees as base point in proper homotopy theory.

J. Remedios and S. Rodriguez-Machin give at the International Category Theory Meeting CT99 (Coimbra, July 19-24, 1999) under the title ``Generalized Homotopy Groups in categories with a Natural Cylinder'' an abstract homotopy theory that generalizes the concept of point in the sense above described. Relative homotopy groupoids are built to define generalized homotopy groups. Exact homotopy sequences of these groups are also given.

The action of the first homotopy group on the homotopy groups is a property that most of the algebraic homotopy theories have. Here, an action of the m-th generalized homotopy group is defined on the groups of higher or equal order and the higher groups of a pair. In this way, higher generalized homotopy groups are proved to be abelian. The equivariant action of the first generalized homotopy group on the exact homotopy sequence is also shown. Finally the second generalized homotopy group of a pair has a structure of a crossed module under this action.

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-20.