## Transformation GroupsThe series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. APart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. THe works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. IN addition, it can serve as a guide for lectures and seminars on a graduate level. THe series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics.WHile the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. IN times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. IN this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. PLease submit any book proposals to Niels Jacob. |

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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 Chapter II Algebraic Topology 1 Equivariant CWcomplexes | 95 |

Maps between complexes | 104 |

Obstruction theory | 111 |

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 action additive algebraic algebraic topology apply assume bijective bundle called canonical characteristic classes closed cohomology commutative compact Lie group complex components composition connected consider consists construction contained continuous corresponding covering cyclic defined definition denote diagram differentiable dimension direct element equivariant Euler exact exercise exists extension fibre finite fixed point function functor G-complex G-homotopy G-map G-space G-vector bundle given group G Hausdorff hence holds homology homomorphism homotopy homotopy equivalence implies inclusion induced injective invariant isomorphism Let G Let H locally manifold module morphism multiplication natural neighbourhood normal numerable objects obtain orbit pair point set prime projective Proof proper properties Proposition relation relative representation restriction ring satisfies sequence Show space structure subgroup Suppose theorem theory topology trivial unique universal yields

### Popular passages

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