Rational equivariant cohomology theories with toral support (Q312379)

For a compact Lie group \(G\), the category of \(G\)-equivariant cohomology theories is the homotopy category of the category of rational \(G\)-spectra. This category breaks up into parts and among them the toral part plays a special role. In the paper under review, the author provides an effective method for calculating with toral \(G\)-spectra. He constructs an abelian category \({\mathcal A}(G, {\mathrm{toral}})\) and a homology functor \(\pi_{*}^{{\mathcal A}(G)}\) from the category of \(G\)-spectra to \({\mathcal A}(G, {\mathrm{toral}})\) so that there is a convergent Adams spectral sequence, \({\mathrm{Ext}}^{*,*}_{{\mathcal A}(G, {\mathrm{toral}})} (\pi_{*}^{{\mathcal A}(G)}(X),\pi_{*}^{{\mathcal A}(G)}(Y)) \Rightarrow [X,Y]_{*}^{G}\).NEWLINENEWLINEThe special cases when \(G\) is a torus, \({\mathrm O}(2)\) or \({\mathrm{SO}}(3)\) have been studied in previous works of the author. In the general case of a compact Lie group \(G\), the model is assembled from data at individual subgroups \(K\) of the maximal torus of \(G\). The contribution from \(K\) is captured by a module over \(H^*(BW^e_{G}(K))\) with an action of \(\pi_{0}(W_{G}(K))\), where \(W^e_{G}(K)\) is the identity component of the Weyl group \(W_{G}(K)=N_{G}(K)/K\).