International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 27626, 14 pages
doi:10.1155/2007/27626
Research Article
On Relative Homotopy Groups of Modules
Department of Mathematics and Computer Science, Providence College, Providence 02918, RI, USA
Received 10 May 2007; Accepted 16 August 2007
Academic Editor: Mónica Clapp
Copyright © 2007 C. Joanna Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In his book “Homotopy Theory and Duality,” Peter Hilton described the concepts of relative homotopy theory in module theory. We study in this paper the possibility of parallel concepts of fibration and cofibration in module theory, analogous to the existing theorems in algebraic topology. First, we discover that one can study relative homotopy groups, of modules, from a viewpoint which is closer to that of (absolute) homotopy groups. Then, through the study of various cases, we learn that the classic fibration/cofibration relation does not come automatically. Nonetheless, the ability to see the relative homotopy groups as absolute homotopy groups, in a stronger sense, promises to justify our ultimate search.