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Bordism, Stable Homotopy and Adams Spectral Sequences

Contents

Context

Stable Homotopy theory

Manifolds and cobordisms

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

This page collects material related to the book

  • Stanley Kochman,

    Bordism, Stable Homotopy and Adams Spectral Sequences,

    Fields Institute Monographs

    American Mathematical Society, 1996

on cobordism theory, stable homotopy theory, complex oriented cohomology, and the Adams spectral sequence.

The approach to stable homotopy presented in this book originated with graduate courses taken by the author at the University of Chicago from 1966 to 1970 given by Frank Adams, Arunas Liulevicius and Peter May. The content of the lectures by Adams have been published in [[Adams: Stable homotopy and generalised homology (1974)]]. However, the content of the courses given by Liulevicius on bordism and by May on characteristic classes and on the Adams spectral sequence have not been published. Most of the material in the first four chapters has been given by the author as graduate courses at Yale University, Purdue University and the University of Western Ontario. (Kochmann, p. 13)

Contents

1 Bordism

1.2 Manifolds

1.3 Classifying spaces

1.4 Manifolds with (B,f)(B,f)-structure

1.5 Pontrjagin-Thom Theorem

2 Characteristic Classes

2.1 Introduction

2.2 Serre spectral sequence

2.3 Cohomology of classifying spaces

2.4 Homology of Classifying spaces

2.5 Steenrod algebra

2.6 Homology of Thom spectra

3 Stable Category

3.2 Underlying theorems

3.3 Spectra

3.4 Generalized homology

3.5 Eilenberg-MacLane spectra

3.6 Adams spectral sequence

3.7 Homotopy of Thom spectra

4 Complex Bordism

4.2 Atiyah-Hirzebruch spectral sequences

4.3 MU characteristic classes

4.4 Formal products

4.5 MU operation

4.6 Brown-Peterson spectra

4.7 Adams-Novikov spectral sequence

5 Computing Stable Stems

5.2 Lambda algebras

5.3 May spectral sequences

5.4 Massey products

5.5 The Lambda Complex Γ *\Gamma^\ast

5.6 The Lambda Complex Γ BP *\Gamma^\ast_{BP}

5.7 Toda brackets and stable stems

Errata

  • p. 36, second line: on the right replace pp by p1p-1, i.e. replace π p:G nD p,np\pi_p \colon G^n \to D^{p,n-p} by π p1:G nD p1,np+1\pi_{p-1} \colon G^n \to D^{p-1,n-p+1} (see end of the proof on the same page)

  • p. 41, second diagram: the top horizontal moprhism is missing a superscript star.

  • p. 192, second displayed formula from below: The expression for CotorCotor on the left has a superfluous argument AA.

  • p. 199, the spectral sequence in prop. 5.3.1 converges not to the cobar complex, but to the cohomology of that complex, namely to Cotor 𝒜 *(/2,/2)Cotor_{\mathcal{A}^\ast}(\mathbb{Z}/2, \mathbb{Z}/2) (as shown correctly in the paragraph just before);

  • p. 199, very last line: E 0E_0 must be E 0E^0 (not the 0-page of the spectral sequence, but the associated graded module, as defined on p. 198)

category: reference

Last revised on December 10, 2018 at 04:59:35. See the history of this page for a list of all contributions to it.