Principal bundle maps via rational homotopy theory
The authors study the space \(\text{aut }P\) of self equivalences of a principal \(S^1\)-bundle \(P\) over \(B\), and the group \(\Aut P\) of homotopy classes of elements. For \(S^1\)-equivariant self-equivalences, they write \(\text{aut}_{S^1}P\) and \(\Aut_{S^1}P\). There is a natural forgetful map \[ \Aut_{S^1} P\to \Aut P. \] Finally, they write \(\text{aut}_{\text{Id}}B\) for the identity component of \(\text{aut}_B\). The main result is that when \(\pi_2(\text{aut}_{\text{Id}}B)\otimes \mathbb{Q}= 0\), then the kernel of the forgetful map is a finite group. The methods use the ideas of \textit{D. H. Gottlieb} [Trans. Am. Math. Soc. 171, 23-50 (1972; Zbl 0251.55018)] and the theory of minimal models. A corollary treats the case when the base space is a homogeneous space \(G/U\), with \(G\) and \(U\) of equal rank.
scientific article; zbMATH DE number 1917049