Splittings in the Burnside ring and in \(SF_G\) (Q2462758)

Let \(BO\) and \(BO_\otimes\) be the classifying space for the group of virtual real vector bundles of virtual dimension 0 under direct sum and the classifying space for the group of virtual real vector bundles of virtual dimension 1 under tensor product, respectively, and likewise let \(B\operatorname{Spin}\) and \(B\operatorname{Spin}_\otimes\) be the analogous classifying spaces for virtual bundles with Spin structure. Then there are two maps \(\psi^k-1 : BO \to B\operatorname{Spin}\) and \(\psi^k/1 :BO_\otimes \to B\operatorname{Spin}_\otimes\) determined by the \(k\)th Adams operation \(\psi^k\). So let \(J\) and \(J_\otimes\) denote the fibres of these two maps, respectively, and also by \(J_{(p)}\) and \(J_{\otimes (p)}\) denote their \(p\)-localizations, where \(p\) is a prime not dividing \(k\). \textit{J. P. May} in [``\(E_\infty\) ring spaces and \(E_\infty\) ring spectra. With contributions by Frank Quinn, Nigel Ray, and Jorgen Tornehave'', Berlin-Heidelberg-New York: Springer (1977; Zbl 0345.55007)] constructed a map from \(J\) to \(J_{\otimes (p)}\) which induces a weak equivalence from \(J_{(p)}\) to \(J_{\otimes (p)}\), and proved that it can be factored as \[ J@>\alpha^k>> SF_{(p)}@>\varepsilon^k>> J_{\otimes (p)} \] through the \(p\)-localization of the space \(SF\) of stable degree 1 self-maps of spheres. Thus a splitting of \(SF_{(p)}\) is obtained. The author previously in [Homology Homotopy Appl. 5, 161--212 (2003; Zbl 1032.55016)] considered the equivariant analogue of this splitting. In order to do this the author tried to generalize all the arguments presented above to the equivariant setting, and succeeded in obtaining an equivariant splitting of the \(p\)-completion of the \(G\)-connected cover of \(SF_G\), the space of degree 1 self-maps of equivariant spheres, where \(G\) is a finite \(p\)-group, \(p\neq 2\). (By the \(G\)-connected cover of a based \(G\)-space \(X\) one means a \(G\)-map \(\iota : X_0 \to X\) such that for each \(H \leqslant G\), \(\iota^H : X_0^H \to X^H\) is, up to homotopy, the inclusion of the basepoint component.) But it is quite natural to expect that there will be a splitting of \((SF_G)_p\), the \(p\)-completion of \(SF_G\), much more similar to the above one of \(SF_{(p)}\). In the present paper the author refines the method of his work [op. cit.] and actually obtains a finer splitting of \((SF_G)_p\). Using the same notation, write \[ J_G@>\alpha^k>> (SF_G)_p@>\varepsilon^k>>(J_{\otimes G})_p \] for the maps corresponding to \(\alpha^k\) and \(\epsilon^k\) above. Then the main theorem of this paper states that for each \(H \leqslant G\), the composite \(\varepsilon^k\circ\alpha^k\) induces a \(p\)-completion from \(\pi_0(J_G^H)\) to \(\pi_0((J_{\otimes G}^H)_p)\) (Corollary 5.5). It is clear that this gives a natural splitting of the \(p\)-completions \(\pi_0((SF_G^H)_p)\), \(H \leqslant G\), of the component groups of fixed point subspaces, which is just the splitting desired here. The proof consists in constructing \(\alpha^k\) and \(\varepsilon^k\). However \(\varepsilon^k\) in fact can be found in the process of devising \(\alpha^k\). The method is based on the techniques of \textit{D. Quillen} [Topology 10, 67--80 (1971; Zbl 0219.55013)] and [\textit{T. tom Dieck}, ``Transformation groups and representation theory'', Berlin-Heidelberg-New York: Springer (1979; Zbl 0445.57023)] together with the previous results of the author [op. cit.]. The paper ends with an announcement that an equivariant splitting of \(SF_G\) for a more general group \(G\) has been studied in a separate paper.