## Algebraic Methods in Unstable Homotopy TheoryThe most modern and thorough treatment of unstable homotopy theory available. The focus is on those methods from algebraic topology which are needed in the presentation of results, proven by Cohen, Moore, and the author, on the exponents of homotopy groups. The author introduces various aspects of unstable homotopy theory, including: homotopy groups with coefficients; localization and completion; the Hopf invariants of Hilton, James, and Toda; Samelson products; homotopy Bockstein spectral sequences; graded Lie algebras; differential homological algebra; and the exponent theorems concerning the homotopy groups of spheres and Moore spaces. This book is suitable for a course in unstable homotopy theory, following a first course in homotopy theory. It is also a valuable reference for both experts and graduate students wishing to enter the field. |

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### Contents

11 | |

A general theory of localization | 35 |

Fibre extensions of squares and the PetersonStein formula | 94 |

HiltonHopf invariants and the EHP sequence | 107 |

JamesHopf invariants and TodaHopf invariants | 135 |

Samelson products | 158 |

Lie algebras and universal enveloping algebras | 251 |

Applications of graded Lie algebras | 283 |

Differential homological algebra | 313 |

Odd primary exponent theorems | 437 |

Differential homological algebra of classifying spaces | 489 |

545 | |

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### Common terms and phrases

abelian group acyclic Bockstein spectral sequence bundle chain coalgebra structure coefficients cofibration cofibration sequence cohomology commutative diagram comodules composition construction Corollary defined Definition diagonal differential coalgebras differential comodules differential Cotor differential graded dimension Eilenberg-Moore spectral sequence elements epimorphism Exercises factors fibration sequence filtration finite free graded Lie functor graded Lie algebra ground ring H-space Hence homology isomorphism homology suspension homomorphism homotopy commutative homotopy equivalence homotopy groups homotopy pullback homotopy theoretic fibre Hopf algebra Hurewicz map Hurewicz theorem induces Lemma localization long exact loop space map of differential mod p homology mod pr monomorphism Moore spaces multiplication nonzero null homotopic odd degree odd primary odd prime p-completion power map primitive Proof Proposition relative Samelson products S-local Serre spectral sequence short exact sequence Show simply connected spaces Suppose tensor algebra tensor product torsion total complex twisted tensor product twisting morphism universal enveloping algebra Z/fcZ Z/prZ Z/pZ