## Transformation Groups, Volume 8“This book is a jewel – it explains important, useful and deep topics in Algebraic Topology that you won’t find elsewhere, carefully and in detail.” |

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### Contents

Foundations | 1 |

Basic notions | 2 |

General remarks Examples | 10 |

Elementary properties | 22 |

Functorial properties | 32 |

Differentiable manifolds Tubes and slices | 38 |

Families of subgroups | 46 |

Equivariant maps | 50 |

Localization | 177 |

Cohomology of some classifying spaces | 183 |

Localization | 190 |

Applications of localization | 197 |

BorelSmith functions | 210 |

Further results for cyclic groups Applications | 218 |

The Burnside Ring | 227 |

The Burnside ring | 240 |

Bundles | 54 |

Vector bundles | 67 |

Orbit categories fundamental groups and coverings | 72 |

Elementary algebra of transformation groups | 77 |

Algebraic Topology | 95 |

Maps between complexes | 104 |

Obstruction theory lll | 109 |

The classification theorem of Hopf | 122 |

Maps between complex representation spheres | 133 |

Stable homotopy Homology Cohomology | 139 |

Homology with families | 150 |

The Burnside ring and stable homotopy | 155 |

Bredon homology and Mackey functors | 160 |

Homotopy representations | 167 |

The space of subgroups | 248 |

Prime ideals | 251 |

Congruences | 256 |

Finiteness theorems | 260 |

Idempotent elements | 266 |

Induction categories | 271 |

Induction theory | 279 |

The Burnside ring and localization | 285 |

295 | |

Further reading | 306 |

307 | |

More symbols | 312 |

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### Common terms and phrases

abelian group action additive invariant algebraic automorphism bijective Burnside ring called canonical closed subgroup coefficient cohomology commutative compact Lie group complex representation congruences conjugacy classes conjugate consider cyclic defined degree denote diagram Dieck differentiable dimension function element equivariant Euler characteristic Euler class exact sequence exercise exists fibre finite group finite number finite orbit type fixed point set G acts G-action G-ENR G-homotopy G-manifold G-map G-sets G-space G-vector G)-bundle given H c G Hausdorff space hence homology theories homomorphism homotopy equivalence homotopy representation injective integer irreducible isotropy group Lemma Let G Let H Lie group G locally compact Mackey functor manifold module morphism multiplication natural transformation neighbourhood numerable obtain orbit space pair Proof properties Proposition quotient r-modules representation ring resp Show sphere structure subgroup of G subspace Suppose surjective topological group topology torus transformation groups vector bundles

### Popular passages

Page 304 - Fixed points of periodic transformations, Appendix B in S. Lefschetz, Algebraic topology, Amer.