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**Etale homotopy of simplicial schemes.**
*(English)*
Zbl 0538.55001

Annals of Mathematics Studies, 104. Princeton, New Jersey: Princeton University Press and University of Tokyo Press. VII, 190 p. $ 34.50; $ 15.00 (1982).

This book attempts to provide a coherent account of the current state of étale homotopy theory. Since its introduction by M. Artin and B. Mazur [Étale homotopy, Lect. Notes Math. 100 (1969; Zbl 0182.260)] étale homotopy theory has been refined - and thereby rendered prohibitively technical - in the course of the pursuit and development of the ideas of Quillen and Sullivan. With a view to making the subject accessible the author gives a fairly thorough, basic introduction to the étale homotopy type and cohomology related to the étale site of a simplicial scheme. Applications to such topics as the Adams conjecture and self-maps of classifying spaces are given. The generalised cohomology properties of the étale homotopy type are developed a little - treating such topics as Poincaré duality, tubular neighbourhoods, fibrations and function spaces.

The successes of étale homotopy theory have been in its application of results from algebraic geometry to solve problems in topology. Regrettably, the use of topological methods in conjunction with the étale topology to solve problems in K-theory are not touched on in this book. This is a shame. It came too early, I suppose, to include the major advances of R. Thomason and A. Suslin in algebraic K-theory. However, this volume would have been a suitable place for a discussion, for example, of characteristic classes in the generalised cohomology of the étale homotopy types of schemes - an inclusion which K-theorists would have welcomed.

The successes of étale homotopy theory have been in its application of results from algebraic geometry to solve problems in topology. Regrettably, the use of topological methods in conjunction with the étale topology to solve problems in K-theory are not touched on in this book. This is a shame. It came too early, I suppose, to include the major advances of R. Thomason and A. Suslin in algebraic K-theory. However, this volume would have been a suitable place for a discussion, for example, of characteristic classes in the generalised cohomology of the étale homotopy types of schemes - an inclusion which K-theorists would have welcomed.

Reviewer: V.Snaith

### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

55P99 | Homotopy theory |

14F25 | Classical real and complex (co)homology in algebraic geometry |

14F35 | Homotopy theory and fundamental groups in algebraic geometry |

55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |

55N15 | Topological \(K\)-theory |