Homotopie des espaces de sections. (English) Zbl 0535.55001

Legrand, André

Lecture Notes in Mathematics. 941. Berlin-Heidelberg-New York: Springer-Verlag. VII, 132 p. DM 19.80: $ 8.80 (1982).

The author generalizes the theory of simplicial fibrations to objects over a simplicial base set B, where a B-fibration is a map of simplicial sets over B which also is a fibration. Using the associated concept of a group over B, he studies principal B-fibrations and their classification, and considers B-cohomology and B-cohomology operations. B-fibrations with Eilenberg-MacLane complexes as fibres are classified under suitable assumptions. The author is mainly interested in fibrations \(\eta\) : \({\mathcal G}\to B\) which are also groups over B, and he sets up a theory of Postnikov systems for these and constructs a spectral sequence \(\{E_ r\}\) associated with \(\eta\), thereby generalizing the spectral sequence of W. Shih [Publ. Math., Inst. Hautes Étud. Sci. 13, 93-176 (1962; Zbl 0105.169)]. It is then demonstrated that \(\{E_ r\}\) is isomorphic to a second spectral sequence \(\{\) \(\bar E_ r\}\) and that the Serre spectral sequence of a fibration over B can essentially be considered as such a spectral sequence \(\{\) \(\bar E_ r\}\). This leads to a description of the differential \(d_ 2\) in the Serre spectral sequence as a ”richer” invariant than the classical obstruction, and to an explicit determination of \(d_ 2\) when \(\pi_ 1(B)\) is free, and also yields - for simply connected B - an old result of E. Fadell and W. Hurewicz [Ann. Math., II. Ser. 68, 314-347 (1958; Zbl 0084.385)]. Unfortunately, no concrete examples are given.
The individual chapters are as follows: I. Simplicial sets over B (pp. 1- 16). II. Principal B-fibrations (pp. 17-42). III. Group fibrations of fibre type \(K(\pi\),n) (pp. 43-83). IV. Homotopy of the space of sections of a group fibration (pp. 84-108). V. Differential of the Shih spectral sequence (pp. 109-127). The volume also contains a bibliography, a terminological index and a helpful index of notations.

Reviewer: S.Thomeier

MSC:

55-02	Research exposition (monographs, survey articles) pertaining to algebraic topology
55U10	Simplicial sets and complexes in algebraic topology
55S40	Sectioning fiber spaces and bundles in algebraic topology
55R05	Fiber spaces in algebraic topology
55T10	Serre spectral sequences
55R20	Spectral sequences and homology of fiber spaces in algebraic topology
55S45	Postnikov systems, \(k\)-invariants

Keywords:

simplicial fibrations; simplicial sets; principal B-fibrations; B- cohomology operations; Eilenberg-MacLane complexes; Postnikov systems; differential; Serre spectral sequence; Homotopy of the space of sections

Citations:

Zbl 0105.169; Zbl 0084.385