Homotopy and homology of diagrams of spaces.

*(English)*Zbl 0659.55011
Algebraic topology, Proc. Workshop, Seattle/Wash. 1985, Lect. Notes Math. 1286, 93-134 (1987).

[For the entire collection see Zbl 0621.00017.]

The present notes, which arose from a lecture series given by the author in 1984-85 at the University of Heidelberg and the University of Washington at Seattle, give an excellent introduction to some of the basic ideas about the homotopy theory of diagrams of spaces (D-spaces, i.e. functors from a small category D to simplicial sets or topological spaces) developed among others by W. G. Dwyer, D. M. Kan, A. D. Elmendorf, A. Zabrodsky and the author. The following subjects are covered among other things: the category \(O_ D\) of D-orbits generalizing the orbit category \(O_ G\), where G is a group whose objects are the G-sets G/H for all subgroups H of G; free diagrams; homotopy limits; spaces of ‘fixed points’ for diagrams of spaces; Bredon homology and cohomology; applications to spectral sequences. One of the main results is the following Proposition: Let X, Y be D-spaces. Then there exist free O-spaces \(X^ O\), \(Y^ O\) and a weak homotopy equivalence \(\hom_ D(X,Y)\approx \hom_ O(X^ O,Y^ O)\) of function complexes for some small orbit category \(O\subset O_ D\).

The present notes, which arose from a lecture series given by the author in 1984-85 at the University of Heidelberg and the University of Washington at Seattle, give an excellent introduction to some of the basic ideas about the homotopy theory of diagrams of spaces (D-spaces, i.e. functors from a small category D to simplicial sets or topological spaces) developed among others by W. G. Dwyer, D. M. Kan, A. D. Elmendorf, A. Zabrodsky and the author. The following subjects are covered among other things: the category \(O_ D\) of D-orbits generalizing the orbit category \(O_ G\), where G is a group whose objects are the G-sets G/H for all subgroups H of G; free diagrams; homotopy limits; spaces of ‘fixed points’ for diagrams of spaces; Bredon homology and cohomology; applications to spectral sequences. One of the main results is the following Proposition: Let X, Y be D-spaces. Then there exist free O-spaces \(X^ O\), \(Y^ O\) and a weak homotopy equivalence \(\hom_ D(X,Y)\approx \hom_ O(X^ O,Y^ O)\) of function complexes for some small orbit category \(O\subset O_ D\).

Reviewer: K.H.Kamps

##### MSC:

55P91 | Equivariant homotopy theory in algebraic topology |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

55P65 | Homotopy functors in algebraic topology |

55P10 | Homotopy equivalences in algebraic topology |

18F99 | Categories in geometry and topology |

18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |