Topological Library

The following sections are included:

• TOPOLOGICAL LIBRARY

• Contents

• Foreword

The following sections are included:

• Introduction

• Chapter I. The notion of spectral sequence
  • Spectral sequence of a differential group with increasing filtration
  • The case of graded group
  • Transgression and suspension
  • Exact sequence
  • The cohomology spectral sequence
  • Spectral sequence of universal covering

• Chapter II. Singular homology and cohomology of fiber spaces
  • Singular cubic homology
  • Fiber spaces. Definitions and simple properties
  • Local family composed by homology of fiber
  • Filtration of singular complex of the space E
  • Calculation of the term E₁
  • Calculation of the term E₂
  • Properties of the homology spectral sequences
  • Cohomology spectral sequence
  • Properties of the cohomology spectral sequence
  • Transformation of second terms of homology and cohomology spectral sequences
  • Proof of Lemma 4
  • Proof of Lemma 5
  • Proof of Lemma 3

• Chapter III. Applications of spectral sequences of fiber spaces
  • First application
  • Euler–Poincaré characteristic of fiber spaces
  • Fibrations of Euclidean spaces
  • Exact sequence
  • Gysin exact sequence
  • Wang exact sequence
  • Leray–Hirsch theorem

• Chapter IV. Loop spaces
  • Loop spaces
  • Hopf theorem
  • Simplicity of H-spaces
  • The loop spaces fibrations
  • Fibration of a path space with fixed origin
  • Some general results on homology of loop spaces
  • Applications to variations calculus (Morse theory)
  • Applications to variations calculus: geodesics transversal to two sub-manifolds
  • The homology and cohomology of loop space on a sphere

• Chapter V. Homotopy groups
  • General method
  • First results
  • Finiteness of homotopy groups of odd-dimensional spheres
  • Auxiliary calculations
  • The first nontrivial modulo p homotopy group of an odd-dimensional sphere
  • Stiefel manifolds and even-dimensional spheres

• Chapter VI. Groups of Eilenberg–MacLane
  • Introduction
  • General results
  • The Hopf theorem

• Appendix. On homology of some coverings

• References

The following sections are included:

• Introduction

• Chapter I. The notion of a class
  • Definition of classes
  • C-notions
  • Torsion product
  • Two axioms on classes
  • A new axiom
  • Examples of classes satisfying axioms (II_A) and (III)
  • Examples of classes satisfying the axioms (II_B) and (III)

• Chapter II. Fiber spaces
  • Relative fiber spaces
  • The homology spectral sequence of a relative fiber space
  • The cohomology spectral sequence of a relative fibration
  • The main theorems
  • Applications
  • The loop spaces and the Eilenberg–MacLane groups

• Chapter III. The theorems of Hurewicz and J. H. C. Whitehead
  • Hurewicz theorem
  • Hurewicz theorem: the second proof
  • Relative Hurewicz theorem
  • Theorem of J. H. C. Whitehead
  • Criterions of applicability of J. H. C. Whitehead's theorem

• Chapter IV. Homotopy groups of spheres
  • Some endomorphisms
  • Manifold of vectors tangent to an even-dimensional sphere
  • Iterated suspension
  • Homotopy groups of even-dimensional spheres
  • The 3-dimensional sphere
  • Homotopy groups of spheres
  • Proof of the lemma 2

• Chapter V. Complements
  • Preliminary results
  • Maps of a polyhedron to an odd-dimensional sphere
  • Lie groups and products of spheres
  • Prime numbers which are regular for a given Lie group
  • Classical groups

• References

The following sections are included:

• Introduction

• Preliminaries
  • Notations
  • Steenrod's i-squares
  • Iterated i-squares
  • The Eilenberg–MacLane complexes
  • Elementary properties of the spaces K(Π, q)
  • Fibrations of the spaces K(Π, q)⁴

• Determination of the algebra H*(Π; q, Z₂)
  • Theorem of A. Borel
  • Determination of the algebra H*(Z₂; q, Z₂)
  • Examples
  • Determination of the algebra H*(Z; q, Z₂)
  • Determination of the algebra H*(Z_m; q, Z₂) for m = 2^h, h ≥ 2
  • Determination of the algebra H*(Π; q, Z₂) for an abelian group of finite type
  • Relations between different algebras H*(Π; q, Z₂)
  • Stable groups. The cohomology case
  • Stable groups. The homology case

• The Poincaré series of the algebras H*(Π; q, Z₂)
  • Definition of the Poincaré series
  • The sereies ϑ(Z₂; q, t)
  • The series ϑ(Z; q, 1)
  • The series ϑ(Z₂, t) and ϑ(Z, t)
  • Examples
  • Convergence of the series ϑ(Π; q, t)
  • Increasing of the functions φ(Z₂; q, x)
  • Increasing of the function φ(Π; q, x) for a group II of finite type
  • Topological applications
  • Remarks

• Cohomology operations
  • Definition of cohomology operations
  • Examples
  • Characterization of the cohomology operations
  • First applications
  • Characterization of the i-squares
  • Cohomology operations in characteristic 2
  • Relations between the iterated i-squares
  • Method to derive relations between the iterated i-squares

• Applications to the homotopy groups of spheres
  • The method
  • The spaces (S₃, q)
  • Cohomology of the space (S₃, 4)
  • Cohomology of the space (S₃, 5)
  • Cohomology of the space (S₃, 6)
  • Cohomology of the space (S₃, 7)
  • The groups π_n+3(S_n)
  • Groups π_n+4(S_n)

• Remarks

• References

The following sections are included:

• Introduction

• Chapter I. Preliminaries
  • Algebraic notions
  • Fiber bundles
  • The Leray theory: cohomology of compact spaces
  • The Leray theory: the fiber bundles
  • Transgression

• Chapter II. Hopf theorem
  • Algebraic Hopf theorem
  • Topological consequences

• Chapter III. Cohomology of Stiefel–Whitney manifolds (elementary theory)
  • Remarks on spectral sequences of fiber bundles
  • Complex and quaternionic Stiefel manifolds
  • Real Stiefel manifolds

• Chapter IV. The main theorem
  • The notion of a relation
  • Auxiliary propositions
  • The main theorem
  • The first part of the proof
  • The second part of the proof
  • Appendix in the case of the characteristic 2
  • Appendix in the case of the characteristic different from 2

• Chapter V. Transgression in principal fiber bundles
  • Universal and classifying spaces
  • Cohomology of classifying spaces and transgression
  • Universally transgressive and primitive elements
  • Three homomorphisms associated to a subgroup
  • Two spectral sequences
  • Cohomology of classifying spaces of unimodular orthogonal groups

• Chapter VI. Real cohomology of principal fiber bundles and homogeneous spaces
  • Cohomology of compact principal fiber bundles
  • Cohomology of homogeneous spaces
  • Quotient of a compact group with respect to its subgroup of the same rank
  • Invariants of the group of H. Weyl
  • Interpretation of the homomorphism ϱ*

• Chapter VII. Integer cohomology and cohomology mod p of some homogeneous spaces
  • The quotient of a compact group by its maximal torus
  • Quotient of a group by its subgroup of the maximal rank
  • Studies of some particular cases

• Editorial notes

• References

The following sections are included:

• Introduction

• Universal spaces, classifying spaces

• Spectral sequence of a fiber space

• Auxiliary remarks

• Classifying spaces of the orthogonal groups; Stiefel manifolds
  • Cohomology of the space F_n
  • Cohomology of the space B_O(n); reduced characteristic classes
  • Duality formulae mod 2
  • The Steenrod squares of the reduced characteristic classes
  • Cohomologies of the space B_SO(n)
  • Steenrod squares in Stiefel manifolds

• Some homogeneous spaces
  • General remarks
  • Homogeneous spaces
  • Homogeneous spaces U(n)/Q(n) and U(n)/O(n)
  • Homogeneous spaces G₂/Q(3) and G₂/SO(4)

• References

The following sections are included:

• Summary

• Prerequisites: Sign conventions, Hopf algebras, the Steenrod algebra

• The homomorphism ψ*

• The homomorphism λ*

• The structure of the dual algebra ℐ_*

• A basis for ℐ^*

• The canonical anti-automorphism

• Miscellaneous remarks

• Appendix 1. The case p = 2

• Appendix 2. Sign conventions

• References

The following sections are included:

• Introduction

• Summary of Results and Methods

• The Spectral Sequence

• Multiplicative Properties of the Spectral Sequence

• The Structure of the Steenrod Algebra

• The Cohomology of the Steenrod Algebra

• References

The following sections are included:

• Introduction

• A cohomology theory derived from the unitary groups

• The spectral sequence

• The differentiable Riemann–Roch theorem and some applications

• The classifying space of a compact connected Lie group

• References

The goal of this work is the construction of the analogue to the Adams spectral sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of applications: to the problem of computing homotopy groups and the classical Adams spectral sequence, fixed points of transformations of period p, and others.

The following sections are included:

• Index