Homotopy equivalence between diagrams of spaces.

*(English)*Zbl 0612.55017Any discrete group G can be considered as a small category having a single object and having G itself as the set of endomorphisms (in fact automorphisms) of that single object. An action of G on a space X can then be thought of as a functor from G to Top which takes X as its value on the single object of G. G. E. Bredon [Equivariant cohomology theories (Lect. Notes Math. 34) (1967; Zbl 0162.272)] proved that if \(f: X\to Y\) is a G-map between G-CW complexes, then it is a G-homology equivalence if and only if for each subgroup H of G the induced map \(f^ H: X^ H\to Y^ H\) on fixed point sets, is a homotopy equivalence.

By a careful analysis of Bredon’s result, the authors extend it to apply to diagrams of spaces indexed by an arbitrary small category, D. This requires them to generalize the concepts of orbits and fixed point set to the case of a general diagram. These concepts are then applied to give the generalization of Bredon’s result together with an application to diagrams of simplicial complexes.

By a careful analysis of Bredon’s result, the authors extend it to apply to diagrams of spaces indexed by an arbitrary small category, D. This requires them to generalize the concepts of orbits and fixed point set to the case of a general diagram. These concepts are then applied to give the generalization of Bredon’s result together with an application to diagrams of simplicial complexes.

Reviewer: T.Porter

##### MSC:

55P99 | Homotopy theory |

55P10 | Homotopy equivalences in algebraic topology |

55P91 | Equivariant homotopy theory in algebraic topology |

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\textit{E. Dror Farjoun} and \textit{A. Zabrodsky}, J. Pure Appl. Algebra 41, 169--182 (1986; Zbl 0612.55017)

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##### References:

[1] | Bredon, G.E., Equivariant cohomology theories, () · Zbl 0162.27202 |

[2] | Dror Farjoun, E., Weakly equivalent diagrams of spaces, (1984), to appear in Proc. AMS |

[3] | E. Dror Farjoun, Homotopy and homology theory for diagrams of spaces, Proc. of Seattle Topology Year, Lecture Notes in Math. (Springer, Berlin, to appear). · Zbl 0659.55011 |

[4] | Dwyer, W.; Kan, D., Singular functors and realization functors, (1983), to appear · Zbl 0555.55019 |

[5] | May, P., Simplicial objects in algebraic topology, Van nostrand mathematical studies, 11, (1967) · Zbl 0165.26004 |

[6] | Zabrodsky, A., Maps between classifying spaces, (1984), to appear · Zbl 0735.55011 |

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