preface to the third edition.

preface to the second edition

preface to the first edition

chapter 1　preliminaries on homotopy theory

1. category theory and homotopy theory

2. complexes

3. the spaces map (x,y) and map0 (x,y)

4. homotopy groups of spaces

5. fibre maps

part i　the general theory of fibre bundles

chapter 2　generalities on bundles

1. definition of bundles and cross sections

2. examples of bundles and cross sections

3. morphisms of bundles

4. products and fibre products

5. restrictions of bundles and induced bundles

6. local properties of bundles

7. prolongation of cross sections

exercises

chapter 3 vector bundles

.1. definition and examples of vector bundles

2. morphisms of vector bundles

3. induced vector bundles

4. homotopy properties of vector bundles

5. construction of gauss maps

6. homotopies of gauss maps

7. functorial description of the homotopy classification of vector bundles

8. kernel, image, and cokernel of morphisms with constant rank

9. riemannian and hermitian metrics on vector bundles

exercises

chapter 4　general fibre bundles

1. bundles defined by transformation groups

2. definition and examples of principal bundles

3. categories of principal bundles

4. induced bundles of principal bundles

5. definition of fibre bundles

6. functorial properties of fibre bundles

7. trivial and locally trivial fibre bundles

8. description of cross sections of a fibre bundle

9. numerable principal bundles over b x [0, 1]

10. the cofunctor k

11. the milnor construction

12. homotopy classification of numerable principal g-bundles

13. homotopy classification of principal g-bundles over

c w-complexes

exercises

chapter 5　local coordinate description of fibre bundles

1. automorphisms of trivial fibre bundles

2. charts and transition functions

3. construction of bundles with given transition functions

4. transition functions and induced bundles

5. local representation of vector bundle morphisms

6. operations on vector bundles

7. transition functions for bundles with metrics exercises

chapter 6　change of structure group in fibre bundles

1. fibre bundles with homogeneous spaces as fibres 2. prolongation and restriction of principal bundles

3. restriction and prolongation of structure group for fibre bundles

4. local coordinate description.of change of structure group

5. classifying spaces and the reduction of structure group exercises

chapter 7

the gauge group of a principal bundle

1. definition of the gauge group

2. the universal standard principal bundle of the gauge group

3. the standard principal bundle as a universal bundle

4. abelian gauge groups and the kiinneth formula

chpter 8

calculations involving the classical groups

1. stiefel varieties and the classical groups

2. grassmann manifolds and the classical groups

3. local triviality of projections from stiefel varieties

4. stability of the homotopy groups of the classical groups

5. vanishing of lower homotopy groups of stiefel varieties

6. universal bundles and classifying spaces for the classical groups

7. universal vector bundles

8. description of all locally trivial fibre bundles over suspensions

9. characteristic map of the tangent bundle over sn

10. homotopy properties of characteristic maps

11. homotopy groups of stiefel varieties

12. some of the homotopy groups of the classical groups

exercises

part ii

elements of k-theory

chapter 9

stability properties of vector bundles

1. trivial summands of vector bundles

2. homotopy classification and whitney sums

3. the k cofunctors

4. corepresentations of kf

5. homotopy groups of classical groups and kf(si)

exercises

chapter 10

relative k-theory

1. collapsing of trivialized bundles

2. exact sequences in relative k-theory

3. products in k-theory

4. the cofunctor l(x,a)

5. the difference morphism

6. products in l(x, a)

7. the clutching construction

8. the cofunctor ln(x. a)

9. half-exact cofunctors

exercises

chapter 11

bott periodicity in the complex case

1. k-theory interpretation of the periodicity result

2. complex vector bundles over x x s2

3. analysis of polynomial clutching maps

4. analysis of linear clutching maps

5. the inverse to the periodicity isomorphism..

chapter 12

clifford algebras

1. unit tangent vector fields on spheres: i

2. orthogonal multiplications

3. generalities on quadratic forms

4. clifford algebra of a quadratic form

5. calculations of clifford algebras

6, clifford modules

7. tensor products of clifford modules

8. unit tangent vector fields on spheres: ii

9. the group spin(k)

exercises

chapter 13

the adams operations and representations

1. λ-rings

2. the adams ψ-operations in λ-ring

3. the γi operations

4. generalities on g-modules

5. the representation ring of a group g and vector bundles

6. semisimplicity of g-modules over compact groups.

7. characters and the structure of the group rf(g)

8. maximal tort

9. the representation ring of a torus

10. the o-operations on k(x) and ko(x)

11. the o-operations on k(sn)

chapter 14

representation rings of classical groups

1. symmetric functions

2. maximal tori in su(n) and u(n)

3. the representation rings of su(n) and u(n)

4. maximal toff in sp(n)

5. formal identities in polynomial rings

6. the representation ring of sp(n)

7. maximal tori and the weyl group of so(n)

8. maximal tori and the weyl group of spin(n)

9. special representations of so(n) and spin(n)

10. calculation of rso(n) and r spin(n)

11. relation between real and complex representation rings

12. examples of real and quaternionic representations

13. spinor representations and the k-groups of spheres

chapter 15

the hopf lnyariant

1. k-theory definition of the hopf invariant

2. algebraic properties of the hopf invariant

3. hopf invariant and bidegree

4. nonexistence of elements of hopf invariant 1

chapter 16

vector fields on the sphere

1. thorn spaces of vector bundles

2. s-category

3. s-duality and the atiyah duality theorem

4. fibre homotopy type

5. stable fibre homotopy equivalence

6. the groups j(sk) and ktop(sk)

7. thom spaces and fibre homotopy type

8. s-duality and s-reducibility

9. nonexistence of vector fields and reducibility

10. nonexistence of vector fields and coreducibility

11. nonexistence of vector fields and j(rpk)

12. real k-groups of real projective spaces

13. relation between ko(rpn) and j(rpn)

14. remarks on the adams conjecture

part iii

characteristic classes

chapter 17

chern classes and stiefei-whitney classes

1. the leray-hirsch theorem

2. definition of the stiefei-whitney classes and chern classes

3. axiomatic properties of the characteristic classes

4. stability properties and examples of characteristic classes

5. splitting maps and uniqueness of characteristic classes

6. existence of the characteristic classes

7. fundamental class of sphere bundles. gysin sequence

8. multiplicative property of the euler class

9. definition of stiefei-whitney classes using the squaring

operations of steenrod

10. the thom isomorphism

11. relations between real and complex vector bundles

12. orientability and stiefei-whitney classes

exercises

chapter 18

differentiable manifolds

1. generalities on manifolds

2. the tangent bundle to a manifold

3. orientation in euclidean spaces

4. orientation of manifolds

5. duality in manifolds

6. thorn class of the tangent bundle

7. euler characteristic and class of a manifold

8. wu's formula for the stiefei-whitney class of a manifold

9. stiefei-whitney numbers and cobordism

10. immersions and embeddings of manifolds

exercises

chapter 19

characteristic classes and connections

1. differential forms and de rham cohomology

2. connections on a vector bundle

3. invariant polynomials in the curvature of a connection

4. homotopy properties of connections and curvature

5. homotopy to the trivial connection and the chern-simons form

6. the levi-civita or riemannian connection

chapter 20

general theory of characteristic classes

1. the yoneda representation theorem

2. generalities on characteristic classes

3. complex characteristic classes in dimension n

4. complex characteristic classes

5. real characteristic classes mod 2

6. 2-divisible real characteristic classes in dimension n

7. oriented even-dimensional real characteristic classes

8. examples and applications

9. bott periodicity and integrality theorems

10. comparison of k-theory and cohomology definitions

of hopf invariant

11. the borel-hirzebruch description of characteristic classes

appendix 1

dold's theory of local properties of bundles

appendix 2

on the double suspension

1. h*(ωs(x)) as an algebraic functor of h(x)

2. connectivity of the pair (ω2s2n+1, s2n-1) localized at p

3. decomposition of suspensions of products and lis(x)

4. single suspension sequences

5. mod p hopf invariant

6. spaces where the pth power is zero

7. double suspension sequences

8. study of the boundary map △: ω3s2np+1→ωs2n-1

bibliography

index...