K-theory

Michael Atiyah

• Publisher: CRC Press
• Year: 2018
• eBook ISBN: 9780429973178
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

These notes are based on the course of lectures I gave at Harvard in the fall of 1964. They constitute a self-contained account of vector bundles and K-theory assuming only the rudiments of point-set topology and linear algebra. One of the features of the treatment is that no use is made of ordinary homology or cohomology theory. In fact, rational cohomology is defined in terms of K-theory.The theory is taken as far as the solution of the Hopf invariant problem and a start is mode on the J-homomorphism. In addition to the lecture notes proper, two papers of mine published since 1964 have been reproduced at the end. The first, dealing with operations, is a natural supplement to the material in Chapter III. It provides an alternative approach to operations which is less slick but more fundamental than the Grothendieck method of Chapter III, and it relates operations and filtration. Actually, the lectures deal with compact spaces, not cell-complexes, and so the skeleton-filtration does not figure in the notes. The second paper provides a new approach to K-theory and so fills an obvious gap in the lecture notes.

CHAPTER I. Vector Bundles

§1.1 Basic definitions. We shall develop the theory of complex vector bundles only, though much of the elementary theory is the same for real and symplectic bundles. Therefore, by vector space, we shall always understand complex vector space unless otherwise specified.

Let X be a topological space, A family of vector spaces over X is a topological space E, together with:

(i)	a continuous map p : E → X
(ii)	a finite dimensional vector space structure on each Ex = p^−1(x) for x ∈ X, compatible with the topology on Ex induced from E.

The map p is called the projection map, the space E is called the total space of the family, the space X is called the base space of the family, and if x ∈ X, E_x is called the fiber over x.

A section of a family p : E → X is a continuous map s : X ∈ E such that ps(x) = x for all x ∈ X.

A homomorphism from one family p : E → X to another family q : F → X is a continuous map φ : E → F such that:

(i)	qφ = p
(ii)	for each x G X, φ: EX → FX is a linear map of vector spaces.

We say that φ is an isomorphism if φ is bijective and φ⁻¹ is continuous. If there exists an isomorphism between E and F, we say that they are isomorphic.

Example 1. Let V be a vector space, and let E = X × V, p : E → X be the projection onto the first factor. E is called the product family with fiber V. If F is any family which is isomorphic to some product family, F is said to be a trivial family.

If Y is a subspace of X, and if E is a family of vector spaces over X with projection p, p : p⁻¹(Y) → Y is clearly a family over Y. We call it the restriction of E to Y, and denote it by E|Y. More generally, if Y is any space, and f : Y → X is a continuous map, then we define the induced family f^*(p) : f^*(E) → Y as follows:

f^*(E) is the subspace of Y × E consisting of all points (y, e) such that f(y) = p(e), together with the obvious projection maps and vector space structures on the fibers. If g : Z → Y, then there is a natural isomorphism g_*f^* ≅ (fg)^* (E) given by sending each point of the form (z,e) into the point (z, g(z), e), where z ∈ Z, e ∈ E. If f : Y φ X is an inclusion map, clearly there is an isomorphism E|Y ≅ f_^*(E) given by sending each e ∈ E into the corresponding (p(e), e).

A family E of vector spaces over X is said to be locally trivial if every x ∈ X posesses a neighborhood U such that E|U is trivial, A locally trivial family will also be called a vector bundle. A trivial family will be called a trivial bundle. If f : Y → X, and if E is a vector bundle over X, it is easy to see that f^*(E) is a vector bundle over Y. We shall call f^*(E) the induced bundle in this case.

Example 2. Let V be a vector space, and let X be its associated projective space. We define E ⊂ X × V to be the set of all (x, v) such that x ∈ X, v ∈ V, and v lies in the line determining x. We leave it to the reader to show that E is actually a vector bundle.

Notice that if E is a vector bundle over X, then dim(E_x) is a locally constant function on X, and hence is a constant on each connected component of X. If dim(E_x) is a constant on the whole of X, then E is said to have a dimension, and the dimension of...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• CHAPTER I Vector Bundles
• CHAPTER II K-Theory
• CHAPTER III Operations
• APPENDIX
• REFERENCES
• REPRINTS