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Parametrized homotopy theory. (English) Zbl 1119.55001

Mathematical Surveys and Monographs 132. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3922-5/hbk). ix, 441 p. (2006).
In many situations it is important to study objects continuously parametrized by the points in a certain base space \(B\). This book aims at giving an in-depth discussion of the associated homotopy theory, especially in the stable equivariant situation.
A parametrized space, or ex-space, is a continuous map \(X\to B\) with a chosen section, thinking of \(B\) as a fixed parameter space. Ex-spaces are commonplace objects, but involve technicalities which are dealt with at the beginning of this book. Especially, the usual point-set topological issues related to mapping spaces become extra troublesome. The authors carefully study the effect of changing the base space. In particular, pulling back along the canonical map \(r\colon B\to *\) gives a functor \(r^*\) from ordinary homotopy theory to parametrized homotopy theory. For many applications the left adjoint \(r_!\) (as opposed to the right adjoint \(r_*\)) plays a prominent role, and the general theory is built around the pair \((r_!,r^*)\) with a more hands-on approach on questions needing attention to \(r_*\). Many important results can be formulated as relations between \(r_!\) and \(r_*\).
The main focus in this book is on the stable situation, providing foundations and results in parametrized equivariant stable homotopy theory. In particular, the authors develop a framework for duality statements and start a study of parametrized (co)homology theories and generalizations of Thom spectra.
The book is divided into five parts. Part I is concerned with point-set topological issues. Part II develops the relevant model structures for parametrized spaces and part III does the same for parametrized stable equivariant homotopy theory. The preferred model is that of orthogonal \(G\)-spectra (interpreted as enriched functors into the category of spaces parametrized over \(B\)) with sphere spectrum \(S_B\) given by sending a finite dimensional \(G\)-inner product space \(V\) to \(S^V\times B\).
In part IV duality is developed. Duality comes in two flavors. One – “fiberwise duality” – allows the construction and analysis of transfer maps, and another – “Costenoble-Waner duality” – is the appropriate analog of Spanier-Whitehead duality. Parametrized sphere spectra are fiberwise dualizable, but not generally Costenoble-Waner dualizable. The converse is true for parametrized finite cell spectra. The authors develop a fiberwise Costenoble-Waner duality theory of which the classical Wirthmüller and Adams isomorphisms are special instances.
The last part of the book considers parametrized homology and cohomology theories and various themes that more or less naturally follow from the developments in the first parts of the book. Most prominently, Poincaré duality is treated and axioms for parametrized (co)homology theories are discussed (representability for homology theories is less transparent than in the ordinary case). A chapter is devoted to twisted theories and spectral sequences. Thom spectra and their iterations as \(r_!\) of the bar-construction of commutative objects are discussed.

MSC:

55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
55P42 Stable homotopy theory, spectra
55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology
19-02 Research exposition (monographs, survey articles) pertaining to \(K\)-theory
19D99 Higher algebraic \(K\)-theory
18G55 Nonabelian homotopical algebra (MSC2010)
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