On the rational homotopical nilpotency index of principal bundles
Let \(G\) be a compact connected topological group. For a principal \(G\)-bundle \(p : E \to X\) of simply connected CW complexes with \(E\) finite, let \(\mathrm{Aut}(X)\) denote the space of all self-fibre homotopy equivalences and let \(\mathrm{Aut}(p)_{0}\) be its rationalization. If the commutative differential graded \(\mathbb Q\)-algebra \((A, d)\) denotes a Sullivan model of \(X\), then it is well known that the minimal model of \(G\) is necessarily of the form \((\Lambda V, D)\) with \(V\) finite dimensional and concentrated in odd degrees. Then, the bundle \(p\) is modeled by an inclusion \((A, d) \to (A \otimes \Lambda V, D)\) for which \(DV \subset A\). A set \(\{v_{i} \}_{i\in I}\) generates the relative Sullivan algebra \((A \otimes \Lambda V, D)\), if \(v_{i} \notin A\), \(D(v_{i})\in A\) for each \(i\) and \(A\) and the linear space \(W\) spanned by \(\{v_{i} \}_{i\in I}\) generates multiplicatively \((A \otimes \Lambda V, D)\). Thus we can define a linear map of degree 1, \[ f:W\to H^{*}(A,d)\qquad f(v)=[Dv]. \] Let \(k\) be the number of dimensions \(m\) such that \(f:W^{m}\to H^{m+1}(A,d)\) is injective and let \(N(p)\) be the minimum of the different values \(k\) resulting from different choices of generators of \((A \otimes \Lambda V, D)\). The aim of the paper under review is to prove the following result \[ n-N(p)\leq \mathrm{Hnil}_{\mathbb Q}(\mathrm{Aut}(p)_{0})\leq n \] where \(n\) is the number of non-trivial rational homotopy groups of the space \(G\). Here Hnil\((\mathrm{Aut}(p)_{0})\) denotes the homotopical nilpotency of the space \(\mathrm{Aut}(p)_{0}\) defined as the least integer \(n\) such that the \((n+1)^{th}\)-commutator is nullhomotopic.
scientific article; zbMATH DE number 6835197