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Stable Homotopy Around the Arf-Kervaire Invariant

Part of the Progress in Mathematics Book Series (volume 273)

DOI: 10.1007/978-3-7643-9904-7

ISBN: 978-3-7643-9903-0

Publisher: Birkhäuser Basel - Springer Journals

Jan 1, 2009

1 - 9 of 9 Chapters

LitStream

Chapter 1Algebraic Topology Background

Pages: 1 –32

[The objective of this chapter is to sketch the historical and technical stable homotopy background which we shall need in the course of this book. § 1 deals with the history of the calculations of stable homotopy groups of spheres (the so-called “stable stems”). §2 describes the framed manifold...
LitStream

Chapter 2The Arf-Kervaire Invariant via QX

Pages: 33 –47

[The objective of this chapter is to study πi(∑∞X) by identifying it with πi(QX) where QX is the space introduced in Chapter 1 § 1.5.8. Since we are interested in the two-primary part of the stable homotopy groups of spheres we would like to concentrate on the case when X = S0. However, for...
LitStream

Chapter 3The Upper Triangular Technology

Pages: 49 –64

[The object of this chapter is to establish the basic result which relates the upper triangular group to operations in connective K-theory. This result will identify a certain group of operations with the infinite upper triangular group with entries in the 2-adic integers. This identification...
LitStream

Chapter 4A Brief Glimpse of Algebraic K-theory

Pages: 65 –91

[The contents of this chapter began life as a short postgraduate course on the K-theory of Banach algebras given at the University of Southampton early in 2003. It seems appropriate for inclusion here because of the application to algebraic K-theory given in Chapter 3 § 3.3.3 which uses the...
LitStream

Chapter 5The Matrix Corresponding to 1 ∧ ψ3

Pages: 93 –118

[The objective of this chapter is to determine the conjugacy class of the map 1 ∧ψ3 in the upper triangular group U∞ℤ2 in the sense of Chapter 3, Theorem 3.1.2. §1 recapitulates the background and states the main result (Theorem 5.1.2). §2 contains the central calculations in which the effect of...
LitStream

Chapter 6Real Projective Space

Pages: 119 –142

[The objective of this chapter is to present the cohomological calculations (in MU*, KU* and -BP*) which will be needed in this and later chapters for the study of maps of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...
LitStream

Chapter 7Hurewicz Images, BP-theory and the Arf-Kervaire Invariant

Pages: 143 –153

[The objective of this chapter is to prove the conjecture of [30] in its original form, as stated in Chapter 1, Theorem 1.8.10. This result is reiterated in this chapter as Theorem 7.2.2. The conjecture states that an element of \documentclass[12pt]{minimal} \usepackage{amsmath}...
LitStream

Chapter 8Upper Triangular Technology and the Arf-Kervaire Invariant

Pages: 155 –197

[The objective of this chapter is to give applications of the upper triangular results of Chapter 3 and Chapter 5. § 1 gives some background concerning Adams operations, recapitulates the results and proves the main one (Theorem 8.1.2) assuming the results of § 4. § 2 contains technical...
LitStream

Chapter 9Futuristic and Contemporary Stable Homotopy

Pages: 199 –215

[The objective of this chapter is to give a brief overview of some current themes and developments in stable homotopy theory. In the 1990’s a new stable homotopy category was designed by Vladimir Voevodsky in order to construct MGL, a new type of spectrum which Voevodsky used to prove a famous...

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