Lectures on Homotopy Theory

Renzo A. Piccinini (Eds.)0444892389, 9780444892386, 9780080872827

The central idea of the lecture course which gave birth to this book was to define the homotopy groups of a space and then give all the machinery needed to prove in detail that the nth homotopy group of the sphere Sn , for n greater than or equal to 1 is isomorphic to the group of the integers, that the lower homotopy groups of Sn are trivial and that the third homotopy group of S2 is also isomorphic to the group of the integers. All this was achieved by discussing H-spaces and CoH-spaces, fibrations and cofibrations (rather thoroughly), simplicial structures and the homotopy groups of maps.
Later, the book was expanded to introduce CW-complexes and their homotopy groups, to construct a special class of CW-complexes (the Eilenberg-Mac Lane spaces) and to include a chapter devoted to the study of the action of the fundamental group on the higher homotopy groups and the study of fibrations in the context of a category in which the fibres are forced to live; the final material of that chapter is a comparison of various kinds of universal fibrations. Completing the book are two appendices on compactly generated spaces and the theory of colimits. The book does not require any prior knowledge of Algebraic Topology and only rudimentary concepts of Category Theory are necessary; however, the student is supposed to be well at ease with the main general theorems of Topology and have a reasonable mathematical maturity.

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Table of contents :
Content:
Edited by
Page iii

Copyright page
Page iv

Dedication
Page v

Preface
Pages vii-x
R. Piccinini

Chapter 1 Homotopy Groups
Pages 1-33

Chapter 2 Fibrations and Cofibrations
Pages 35-71

Chapter 3 Exact Homotopy Sequences
Pages 73-83

Chapter 4 Simplicial Complexes
Pages 85-116

Chapter 5 Relative Homotopy Groups
Pages 117-151

Chapter 6 Homotopy Theory of CW-complexes
Pages 153-213

Chapter 7 Fibrations revisited
Pages 215-265

Appendix A Colimits
Pages 267-276

Appendix B Compactly generated spaces
Pages 277-284

Bibliography
Pages 285-287

Index
Pages 289-293