Given an \(H\)-space \(E\) acting on a pointed nilpotent CW complex \(X\), the authors show that the rationalization of the map \(w : E \to X\) corresponding to the action on the basepoint factors through a rational homotopy monomorphism \(\Gamma_{w} : S_{w} \to X_{\mathbb{Q}}\) where \(S_{w}\) is a product of odd-dimensional spheres. The authors deduce a variety of interesting consequences of this basic result.
They prove that if \(X\) has nonvanishing Euler characterstic then the map induced by \(w\) on rational homology vanishes, complementing an old result of [D. H. Gottlieb, Mich. Math. J. 19, 289–297 (1972; Zbl 0232.55016)]. They also prove that the map induced by \(w\) is rationally essential if and only if \(w\) induces a nontrivial map on rational homotopy groups. Specializing to the case where \(E\) is the function space \(\text{Map}(X, X; 1)\) with the action on \(X\) obtained by evaluation at the basepoint, the map \(w\) corresponds to the evaluation map \(\omega : \text{Map}(X, X; 1) \to X.\) In this case, the image of \(\omega\) on homotopy groups is the Gottlieb group \(G_{*}(X)\).
The authors’ main result implies that the rationalized Gottlieb group \(G_{*}(X) \otimes \mathbb{Q}\) corresponds to the image of \(\Gamma_{\omega}\) on homotopy groups recovering, in particular, the result of [Y. Félix and S. Halperin, Trans. Am. Math. Soc. 273, No. 1, 1–38 (1982; Zbl 0508.55004)] on the vanishing of the even-dimensional rational Gottlieb groups. The main result also allows for the complete description of the space of rational cyclic maps as studied in [G. Lupton and S. B. Smith, Math. Z. 249, No. 1, 113–124 (2005; Zbl 1077.55008)].