Phantom elements and its applications (Q2734872)

This paper gives two characterizations of phantom elements in homotopy groups, together with an application to homotopy self-equivalences of bundles.NEWLINENEWLINENEWLINELet \(X\) be a CW complex, let \(Y\) be any space, and let \(\alpha\) be a member of a homotopy group \(\pi_j(\text{map}_*(X,Y))\) (with respect to any base point \(g: X\to Y\)), where \(\text{map}_*(X,Y)\) is the space of based maps. Then \(\alpha\) is called a phantom element if its restrictions to the skeleta of \(X\) are trivial. In the main result of this paper, \(X\) is nilpotent and of finite type, \(Y\) is simply connected, and the homotopy groups of \(Y\) have no nontrivial divisible subgroups. It is shown that \(\alpha\) is a phantom element if and only if it becomes zero in \(\pi_j(\text{map}_*(X,\hat Y))\), where \(\hat Y\) is the profinite completion of \(Y\). It is also shown that \(\alpha\) is a phantom element if and only if it becomes zero in \(\pi_j(\text{map}_*(X_\tau,Y))\), where \(X_\tau\) is the homotopy fibre of the rationalization map of \(X\).NEWLINENEWLINENEWLINEIn the application to bundles, let \(G\) be a group of the homotopy type of an Eilenberg-MacLane space, let \(P\) be the total space of a principal \(G\)-bundle, let \(\text{aut}^G(P)\) be the space of equivariant homotopy self-equivalences of \(P\), and let \(\text{aut}(P)\) be the space of all homotopy self-equivalences. It is shown that the forgetful map \(\pi_0\text{aut}^G(P)\to\pi_0\text{aut}(P)\) is in certain cases injective.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00050].