A fundamental problem of algebraic topology is the classification of homotopy types and homotopy classes of maps. In this work the author extends results of rational homotopy theory to a subring of the rationale. The methods of proof employ classical commutator calculus of nilpotent group and Lie algebra theory and rely on an extensive and systematic study of the algebraic properties of the classical homotopy operations (composition and addition of maps, smash products, Whitehead products and higher order James-Hopi invariants). The account is essentially self-contained and should be accessible to non-specialists and graduate students with some background in algebraic topology and homotopy theory.
Author(s): Hans Joachim Baues
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1981
Language: English
Pages: 166
CONTENTS......Page 5
Introduction to Part B......Page 7
Introduction to Part A......Page 15
1 The exponential function and the Zassenhaus formula......Page 20
2 The exponential commutator......Page 28
3 A presentation for the exponential group......Page 32
4 The general type of Zassenhaus terms and its characterization modulo a prime......Page 35
II. Distributivity laws in homotopy theory......Page 41
1 Whitehead products and cup products......Page 42
2 Hopf invariants......Page 48
3 The Whitehead product of composition elements......Page 55
4 Proof of I (1. 13) and I (2. 6)......Page 63
5 Decomposition of suspensions and groups of homotopy classes......Page 67
1 Spherical Whitehead products and commutators......Page 74
2 Spherical Hopf invariants......Page 77
3 Deviation from commutativity of spherical cup products......Page 81
4 Cup products of spherical Hopf invariants......Page 84
5 Hopf invariants of a Hopf invariant, of a sum and of a cup product......Page 89
6 Hopf invariant of a composition element......Page 91
1 Fxamples of higher order Hopf invariants on spheres......Page 97
2 Proof of theorem (1. 3)......Page 100
3 Zassenhaus terms for an odd prime......Page 104
0 Notation......Page 107
1 The homotopy Lie algebra and the spherical cohomotopy algebra......Page 109
2 Homotopy groups of spheres and homotopy coefficients......Page 114
3 The Hurewicz and the degree map......Page 119
1 Nilpotent rational groups of homotopy classes......Page 128
2 The exponential group......Page 130
3 Groups of homotopy classes......Page 132
4 H-maps and Co-H-maps......Page 137
1 The category of coefficients......Page 141
2 Extension of algebras by homotopy coefficients......Page 145
3 The extension of Lie algebras by homotopy coefficients......Page 150
4 The Hilton-Milnor theorem and its dual......Page 156
Literature......Page 162
Index......Page 165
