Characteristic cohomology classes, deﬁned in modulo 2 coeﬃcients by Stiefel  and Whitney  and with integral coeﬃcients by Pontrjagin , make up the primary source of ﬁrst-order invariants of smooth manifolds. When their utility was ﬁrst recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques.
This book grew out of courses which I taught at Cornell University and
the University of Warwick during 1969 and 1970. I wrote it because of a
strong belief that there should be readily available a semi-historical and geometrically
motivated exposition of J. H. C. Whitehead's beautiful theory of
simple-homotopy types; that the best way to understand this theory is to
know how and why it was built.
Copyright c 2000 by Allen Hatcher
Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author. All other rights reserved.
.Table of Contents
Chapter 0. Some Underlying Geometric Notions
Homotopy and Homotopy Type 1. Cell Complexes 5.
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Operations on Spaces 8. Two Criteria for Homotopy Equivalence 11. The Homotopy Extension Property 14.
Chapter 1. The Fundamental Group
1. Basic Constructions
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