## EQUIVARIANT ORIENTATION THEORY

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### BibTeX

@MISC{Costenoble_equivariantorientation,

author = {S. R. Costenoble and J. P. May and S. Waner},

title = {EQUIVARIANT ORIENTATION THEORY},

year = {}

}

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### Abstract

Abstract. We give a long overdue theory of orientations of G-vector bundles, topological G-bundles, and spherical G-fibrations, where G is a compact Lie group. The notion of equivariant orientability is clear and unambiguous, but it is surprisingly difficult to obtain a satisfactory notion of an equivariant orientation such that every orientable G-vector bundle admits an orientation. Our focus here is on the geometric and homotopical aspects, rather than the cohomological aspects, of orientation theory. Orientations are described in terms of functors defined on equivariant fundamental groupoids of base G-spaces, and the essence of the theory is to construct an appropriate universal target category of G-vector bundles over orbit spaces G/H. The theory requires new categorical concepts and constructions that should be of interest in other subjects where analogous structures arise.

### Citations

68 | Equivariant Homotopy and Cohomology Theory. Volume 91 - May - 1996 |

26 |
Systems of fixed point sets
- Elmendorf
- 1983
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Citation Context ...-map Y ⊗D X −→ Z. We think of Y as a D-shaped diagram and call it a D-space. We think of X as a fixed functor used to induce a G-action on a coalescence of Y to a single space. For example, Elmendorf =-=[12]-=- applied this construction to construct a G-space ΨT associated to an OG-space T . Here ΨT = B(T, OG, ι), where ι : OG −→ GU is the evident functor that sends an orbit G/H, regarded as an object of th... |

19 | Transformation groups and representation theory - Dieck - 1979 |

7 | Fixed set systems of equivariant infinite loop spaces - Costenoble, Waner - 1991 |

7 | T.: “Transformation groups - Dieck - 1987 |

7 | Equivariant surgery theories and their periodicity properties - Dovermann, Schultz - 1990 |

3 |
Equivariant Poincaré duality
- Costenoble, Waner
- 1992
(Show Context)
Citation Context ...ok up the problem soon after. This paper is a revision of an undistributed 1989 preprint, and in the meantime a number of papers have appeared that are explicitly or implicitly based on that preprint =-=[1, 3, 4, 5, 6, 7, 25]-=-. The answer to the question is necessarily complicated, and our present categorical framework is a significant improvement on our original one. We give the idea by reviewing one approach to classical... |

3 | Equivariant algebraic K-theory
- Fiedorowicz, Hauschild, et al.
- 1983
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Citation Context ...⊂U (∐ n BOG(n, V ⊕ U)), where the colimit runs over the G-spaces V ⊂ U and the system of maps BOG(n, V ⊕ U) −→ BOG(n + |W − V |, W ⊕ U) given by addition of the plane W − V for V ⊂ W . It is standard =-=[14, 18]-=- that BOG(n, U) classifies n-dimensional G-vector bundles and that BOG(U) classifies virtual G-vector bundles over finite G-CW complexes X. For the latter, this means that KOG(X) = [X, BOG(U)]G. A dir... |

2 |
Equivariant orientations and G-bordism theory
- Costenoble, Waner
- 1989
(Show Context)
Citation Context ...ok up the problem soon after. This paper is a revision of an undistributed 1989 preprint, and in the meantime a number of papers have appeared that are explicitly or implicitly based on that preprint =-=[1, 3, 4, 5, 6, 7, 25]-=-. The answer to the question is necessarily complicated, and our present categorical framework is a significant improvement on our original one. We give the idea by reviewing one approach to classical... |

2 | The equivariant Spivak normal bundle and equivariant surgery
- Costenoble, Waner
- 1992
(Show Context)
Citation Context ...ok up the problem soon after. This paper is a revision of an undistributed 1989 preprint, and in the meantime a number of papers have appeared that are explicitly or implicitly based on that preprint =-=[1, 3, 4, 5, 6, 7, 25]-=-. The answer to the question is necessarily complicated, and our present categorical framework is a significant improvement on our original one. We give the idea by reviewing one approach to classical... |

1 |
The equivariant Thom isomorphism theorem
- Costenoble, Waner
- 1992
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1 |
Equivariant simple Poincaré duality
- Costenoble, Waner
- 1993
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1 |
The equivariant Euler Characteristic
- Costenoble, Waner, et al.
- 1991
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1 | The irreducibility of the space of curves of given genus. Institut des Hautes Études Scientifiques publications mathématiques 36(1969 - Deligne, Mumford |