Lectures on Homotopy Theory (eBook)

R.A. Piccinini (Autor)

eBook Download: PDF

1992 | 1. Auflage
292 Seiten
Elsevier Science (Verlag)
978-0-08-087282-7 (ISBN)

Infos

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 n^th 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.

Inhalt

Cover 1
Copyright Page 5
Dedication 6
Preface 8
TOC$Contents 12
CH$Chapter 1. Homotopy Groups 14
1.1 Function spaces 14
1.2 H-spaces and CoH-spaces 23
1.3 Homotopy groups 40
CH$Chapter 2. Fibrations and Cofibrations 48
2.1 Pullbacks and pushouts 48
2.2 Fibrations 54
2.3 Cofibrations 63
2.4 Applications of the mapping cylinder 76
CH$Chapter 3. Exact Homotopy Sequences 86
3.1 Exact sequence of a map: covariant case 86
3.2 Exact sequence of a map: contravariant case 92
CH$Chapter 4. Simplicial Complexes 98
4.1 Simplicial complexes 98
4.2 Simplical approximation theorem 108
4.3 Polyhedra 113
4.4 Fibrations and polyhedra 119
CH$Chapter 5. Relative Homotopy Groups 130
5.1 Homotopy groups of maps 130
5.2 Quasifibrations 154
5.3 Some homotopy groups of spheres 160
CH$Chapter 6. Homotopy Theory of CW-Complexes 166
6.1 CW-complexes 166
6.2 Homotopy theory of CW-complexes 187
6.3 Eilenberg-Mac Lane spaces 210
CH$Chapter 7. Fibrations Revisited 228
7.1 Sections of fibrations 228
7.2 F-Fibrations 249
7.3 Universal F-fibrations 268
Appendix A: Colimits 280
Appendix B: Compactly generated spaces 290
Bibliography 298
IDX$Index 302

Produkt-Details

• Erscheint lt. Verlag: 21.1.1992
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-087282-4 / 0080872824
• ISBN-13: 978-0-08-087282-7 / 9780080872827