Homotopy and homology of diagrams of spaces. (English) Zbl 0659.55011

Dror Farjoun, Emmanuel

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 \(O\subset O_ D\).

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)

Keywords:

equivariant homotopy theory; homology of diagrams; homotopy theory of diagrams of spaces; simplicial sets; D-orbits; orbit category; free diagrams; homotopy limits; fixed points; Bredon homology and cohomology; spectral sequences