Principal \(S^1\)-bundles and forgetful maps (Q2734874)

Given a principal \(G\)-bundle \(q: P\rightarrow B\), let \(\text{aut}_G(P)\) be the space of unbased \(G\)-equivariant self homotopy equivalences of \(P\), and aut\((P)\) the space of unbased self homotopy equivalences of \(P\). Then the forgetful map \(F: \pi_0(\text{aut}_G(P))\rightarrow \pi_0(\text{aut}(P)), F([f]_G)=[f]\), forgets the \(G\)-action on \(P\). NEWLINENEWLINENEWLINE\textit{Kouzou Tsukiyama} [Bull. Korean Math. Soc. 36, No. 4, 649-654 (1999; Zbl 0947.55009)] posed the question of when \(F\) is a monomorphism. \{The reviewer remarks that \textit{Pan Jianzhong} and \textit{Woo Mooha} [Phantom maps and injectivity of forgetful maps, to appear in J. Math. Soc. Japan] seem to have found an interesting relation between this problem and a generalization of phantom maps.\} NEWLINENEWLINENEWLINEIn the paper under review, the authors consider the above mentioned question for principal \(S^1\)-bundles \(q: P\rightarrow B\) with \(\pi_1(B)=0\). They give three negative examples: in each of them \(\text{Ker}(F)\) is nontrivial (finite, countable, and uncountable, respectively).NEWLINENEWLINEFor the entire collection see [Zbl 0960.00050].