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**Homotopie des espaces de sections.**
*(English)*
Zbl 0535.55001

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.

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 |