Fibrewise Homotopy Theory

Topology occupies a central position in the mathematics of today. One of the most useful ideas to be introduced in the past sixty years is the concept of fibre bundle, which provides an appropriate framework for studying differential geometry and much else. Fibre bundles are examples of the kind of structures studied in fibrewise topology. Just as homotopy theory arises from topology, so fibrewise homotopy the ory arises from fibrewise topology. In this monograph we provide an overview of fibrewise homotopy theory as it stands at present. It is hoped that this may stimulate further research. The literature on the subject is already quite extensive but clearly there is a great deal more to be done. Efforts have been made to develop general theories of which ordinary homotopy theory, equivariant homotopy theory, fibrewise homotopy theory and so forth will be special cases. For example, Baues [7] and, more recently, Dwyer and Spalinski [53], have presented such general theories, derived from an earlier theory of Quillen, but none of these seem to provide quite the right framework for our purposes. We have preferred, in this monograph, to develop fibre wise homotopy theory more or less ab initio, assuming only a basic knowledge of ordinary homotopy theory, at least in the early sections, but our aim has been to keep the exposition reasonably self-contained.

Ioan James is an Internationally well-known expert in this field and an experienced author Fibrewise Homotopy Theory is a very large subject that has attracted a good deal of research in recent years The authors provide an overview of fibrewise homotopy theory as it stands at present; a kind of expository survey Ioan James has been interested in this subject for a number of years, and in addition to the expository text, a good deal of the material is completely new The main advantage of this monograph is that one does not have to be an expert in classical stable theory to understand and enjoy it

Contenu
I. A Survey of Fibrewise Homotopy Theory.- 1: An Introduction to Fibrewise Homotopy Theory.- 2: The Pointed Theory.- 3: Applications.- II. An Introduction to Fibrewise Stable Homotopy Theory.- 1: Foundations.- 2: Fixed-point Methods.- 3: Manifold Theory.- 4: Homology Theory.- References.