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Equivariant stable homotopy theory. (English) Zbl 1476.55027

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 699-756 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: Equivariant stable homotopy theory considers spaces and spectra endowed with the action of a fixed group \(G\). There are many wonderful references for much of the foundational material in equivariant stable homotopy theory. The Cartesian product and function spaces with conjugation action give a closed symmetric monoidal structure on \(\operatorname{Top}^G\). There are several different conceptual approaches to stabilization in \(G\)-spectra, and somewhat surprisingly, these lead to the same results. There are two dominant themes: one geometric and one algebraic. Boardman’s stable homotopy category was defined as an extension of the ordinary Spanier-Whitehead category under colimits. Many of the standard arguments apply without change here. The equivariant Spanier-Whitehead category is additive: finite wedges and products exist and agree and the morphism sets are naturally abelian group valued. The composition and symmetric monoidal products induce bilinear maps on morphism sets.
For the entire collection see [Zbl 1468.55001].

MSC:

55P42 Stable homotopy theory, spectra
55P91 Equivariant homotopy theory in algebraic topology
18N40 Homotopical algebra, Quillen model categories, derivators
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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