## The Topology of Stiefel ManifoldsStiefel manifolds are an interesting family of spaces much studied by algebraic topologists. These notes, which originated in a course given at Harvard University, describe the state of knowledge of the subject, as well as the outstanding problems. The emphasis throughout is on applications (within the subject) rather than on theory. However, such theory as is required is summarized and references to the literature are given, thus making the book accessible to non-specialists and particularly graduate students. Many examples are given and further problems suggested. |

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### Contents

algebra versus topology | 1 |

The Stiefel manifolds | 13 |

The auxiliary spaces | 21 |

Retractible fibrations | 27 |

Thom spaces | 33 |

Homotopy equivariance | 40 |

Crosssections and the Stype | 45 |

Relative Stiefel manifolds | 52 |

Samelson products | 94 |

The Hopf construction | 100 |

The Bott suspension | 109 |

The intrinsic join again | 116 |

Homotopycommutativity | 123 |

The triviality problem | 129 |

When is P neutral? | 133 |

When is V neutral? | 139 |

Cannibalistic characteristic classes | 57 |

Exponential characteristic classes | 62 |

The main theorem of Jtheory | 71 |

The fibre suspension | 78 |

Canonical automorphisms | 83 |

The iterated suspension | 89 |

n | 146 |

Further results and problems | 150 |

155 | |

167 | |

### Common terms and phrases

Adams admits a cross-section algebra asserted automorphism basepoint Bott suspension changes the sign coefficients cofibration commutative defined denotes dimension element exists fibration fibre homotopy type fibre space finite following diagram follows at once given H-space Hence homomorphism homotopy class homotopy equivalence homotopy exact sequence homotopy groups Hopf construction Hopf line bundle inclusion integer intrinsic join intrinsic map isomorphism J-order J-trivial J. H. C. Whitehead J'-space k-frame Lemma line bundle m+n,k map f map h mapping cone mod 2 cohomology multiple natural projection neutral numbers orthonormal pair Pk(C proof prove Proposition prove Theorem Recall relative Samelson product retractible S-neutral S(nL self-map sense of fibre shown shows Sn_1 stable Steenrod Stiefel manifold Stiefel-Whitney classes stunted projective spaces subspace Suppose Thom space topological group trivial vector bundle Whitehead product

### Popular passages

Page 155 - WD Barcus and MG Barratt, On the homotopy classification of the extensions of a fixed map, Trans. Amer. Math. Soc. 88 ( 1958), 57-74.