Bordism, Stable Homotopy, and Adams Spectral Sequences
This book is a compilation of lecture notes that were prepared for the graduate course "Adams Spectral Sequences and Stable Homotopy Theory" given at The Fields Institute during the fall of 1995. The aim of this volume is to prepare students with a knowledge of elementary algebraic topology to study recent developments in stable homotopy theory, such as the nilpotence and periodicity theorems.
Suitable as a text for an intermediate course in algebraic topology, this book provides a direct exposition of the basic concepts of bordism, characteristic classes, Adams spectral sequences, Brown-Peterson spectra and the computation of stable stems. The key ideas are presented in complete detail without becoming encyclopedic. The approach to characteristic classes and some of the methods for computing stable stems have not been published previously.
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abelian group Adams resolution Adams spectral sequence Adams—Novikov spectral sequence Assume Atiyah—Hirzebruch spectral sequence BO(n bordism bordism ring boundary brackets canonical map cobar construction cohomology theory commutative ring coproduct Corollary CW—complex deﬁning system deﬁnition denote differential direct limit dual E—oriented element equals exact couple f)—structure ﬁber ﬁbration ﬁnd ﬁnite dimensional ﬁnite type ﬁrst following diagram commutes formal product functor given graded h0—tower homology homotopy equivalence homotopy groups Hopf algebra Hurewicz homomorphism identiﬁcation Image inclusion map induced inﬁnite cycle integer Kernel lambda algebra Lemma long exact sequence Massey products modulo decomposables monomials monomorphism multiplicative nonzero normal bundle numbers Observe oriented orthogonal subspaces pairing polynomial algebra power series prime reduced cohomology represented ring spectrum Section Serre spectral sequence short exact sequence smooth manifold Steenrod algebra structure maps subcomplex summand theorem Thom isomorphism Thom spectrum vector bundle wedge axiom zero