Rational \(S^1\)-equivariant homotopy theory (Q2750941)

In his paper [Publ. Math., Inst. Hautes Étud. Sci. 47, 269-331 (1977; Zbl 0374.57002)], \textit{D. Sullivan} exhibited a passage between certain commutative differential graded algebras (DGAs) over the rationals to rational homotopy types of topological spaces. A CW-complex corresponds to a DGA of de Rham-like forms, its rational cohomology to the cohomology of that algebra, and its rational homotopy can be computed via the generators of its minimal model having a decomposable differential [cf. \textit{Y. Félix, S. Halperin} and \textit{J.-C. Thomas}, Rational homotopy theory, Grad. Texts Math. 205 (2001; Zbl 0961.55002) for a modern account]. \textit{G. V. Triantafillou} [Trans. Am. Math. Soc. 274, 509-532 (1982; Zbl 0516.55010)] generalized this approach to CW-complexes with actions of a finite group \(G\). Models take the form of a functor from the orbit category of \(G\) to the category of DGAs (satifsying extra conditions) and organising the DGAs corresponding to the fixed point sets of subgroups of \(G\). But the notion of minimality is more intricate in the equivariant case. The present author shows that the correspondence to an equivariant Postnikov tower cannot be fully captured by decomposability of the differential but has to be phrased using sequences of minimal extensions instead. More importantly, the author develops an extension of the equivariant model to \(T=S^1\)-spaces. In this case, one has to consider \(T\)-systems consisting of DGAs corresponding to the Borel constructions of the subgroups (with a \(\mathbb{Q}[c]=H^*(BT)\)-module structure) and to handle the \(T\)-fixed point set separately. As a result, one obtains a 1-1-correspondence between rational homotopy types of \(T\)-spaces and isomorphism classes of minimal \(T\)-systems. Moreover, a minimal \(T\)-system allows to compute the equivariant cohomology and encodes the \(T\)-equivariant Postnikov decomposition of the corresponding space.