On rational equivariant \(H\)-spaces (Q1312357)

The author proves a result on the existence of an equivariant splitting of a rational \(H\)-space \(X\) in the case in which \(X\) has a finite group action compatible with the \(H\)-structure. If \(G\) is a finite group, \(G\)- spaces, \(G\)-maps and \(G\)-homotopies are considered. All \(G\)-spaces are \(G\)-connected (this means that the fixed-point spaces \(X^ H\) are connected for all subgroups \(H\) of \(G\)) and are \(G\)-homotopy-equivalent to \(G\)-complexes. A Hopf \(G\)-space is an \(H\)-space on which \(G\) acts in such a way that the multiplication \(m: X\times X \to X\) is a \(G\)- equivariant map, and the composition \(X \vee X \subset X \times X @>m>> X\) is \(G\)-homotopic to the folding map. A rational \(G\)-space is a \(G\)- nilpotent \(\phi\)-local \(G\)-space in the sense of [\textit{J. P. May, J. McClure, G. Triantafillou}, Bull. Lond. Math. Soc. 14, 223-230 (1982; Zbl 0492.57012)], where \(\phi\) denotes the empty set (for primes). The main result is the following Theorem A: Let \(X\) be a rational \(G\)-connected Hopf \(G\)-space such that the edge homomorphism \(\eta: H^ n_ G(X,\underline{\pi}_ n(X)) \to \text{Hom}( \underline{H}_ n(X),\underline{\pi}_ n(X))\) of the Bredon spectral sequence is an epimorphism for each \(n \geq 1\). Then \(X\) is \(G\)- homotopy equivalent to the weak product \(\prod_ n K(\underline{\pi}_ n(X),n)\). The author denotes by \(\underline{\pi}_ n(X)\) and \(\underline{H}_ n(X)\) the coefficient systems for \(G\) defined respectively by \(\underline{\pi}_ n(X)(G/_ H) = \pi_ n(X^ H)\) and \(\underline{H}_ n(X)(G/H) = H_ n(X^ H)\), \(n \geq 1\), where \(H_ n( )\) denotes the singular homology with \(\mathbb{Z}\)-coefficients.