An Algebraic Model for Rational S1‐Equivariant Stable Homotopy Theory

By B. Shipley. 2001

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Greenlees dened an abelian categoryA whose derived category is equivalent to the rational S 1 -equivariant stable homotopy category whose objects represent rational S 1 - equivariant cohomology theories. We show that in fact the model category of dierential graded objects inA models the whole rational S 1 -equivariant stable homotopy theory. That is, we show that there is a Quillen equivalence between dgA and the model category of ra- tional S 1 -equivariant spectra, before the quasi-isomorphisms or stable equivalences have been inverted. This implies that all of the higher order structures such as mapping spaces, function spectra and homotopy (co)limits are reflected in the algebraic model. The construction of this equivalence involves calculations with Massey products. In an appendix we show that Toda brackets, and hence also Massey products, are determined by the derived category.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5dbf01afb3f3d017b4d999436444faf4fbdd434

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Stable model categories are categories of modules (2003, methodology): “This is carried out in [58] and applied in [61] [19].”