Rational equivalence and phantom map out of a loop space (Q5939020)

The author proves the following two generalizations of a result of \textit{C. A. McGibbon} and \textit{C. W. Wilkerson} [Proc. Am. Math. Soc. 96, 698-702 (1986; Zbl 0594.55006)]. Let \(X\) be a simply connected, rationally elliptic, finite complex. Then there exists a rational equivalence \(\Omega X\to\prod_\alpha S^{2n_\alpha-1} \times\prod_\beta\Omega S^{2n_\beta-1}\). Let \(p\) be a prime and \(k\geq 1\). If \(X\) is a (non-necessarily 1-connected) pseudo-finite CW-complex then there exists a \(p\)-local map \(\Omega^k_0 X\to\prod_\alpha S^{2n_\alpha-1}\times\prod_\beta\Omega S^{2n_\beta-1}\) which is a rational homotopy equivalence. \((\Omega_0^kX\) is the base point component of \(\Omega^kX\).) The author applies these results to study phantom maps from \(\Omega^kX\) to a finite type target.