Rational homotopy type of the classifying space for fibrewise self-equivalences (Q2838974)

Given a fibration of simply connected finite type CW-complexes \(F\to E\buildrel{p}\over{\to}B\) in which \(F\) and \(B\) are finite, the paper under review analyzes the rational homotopy type of \(Baut_1(p)\). This is the classifying space of the monoid \(aut_1(p)\) of fibrewise self equivalences of \(p\) which are homotopic to the identity. The main result describes an interesting and explicit Lie model (in the sense of Quillen) of \(Baut_1(p)\) as a result of regarding this space as the universal cover of the path component of the mapping space \(map\bigl(B,Baut_1(F)\bigr) \) containing the classifying map \(B\to aut_1(F)\) for the fibration \(p\). This model is constructed as follows:NEWLINENEWLINELet \(A\to A\otimes \Lambda V\) a cofibrant model of the fibration \(p: E\to B\). Consider the graded vector space \(Der_A(A\otimes\Lambda V)\) of \(A\)-derivations of \(A\otimes \Lambda V\), endowed with the usual differential and the Lie bracket given by commutators. Then, truncate this differential graded Lie algebra to get NEWLINE\[NEWLINE {\mathcal D}er_A(A\otimes\Lambda V)=\oplus_{n\geq 2} Der^n_A(A\otimes \Lambda V)\oplus Z_1, NEWLINE\]NEWLINE where \(Z_1\) is the space of cycles of \(Der^1_A(A\otimes \Lambda V)\). The authors prove that \({\mathcal D}er_A(A\otimes\Lambda V)\) is a differential graded Lie algebra modeling \(Baut_1(p)\).NEWLINENEWLINEInteresting applications on the classification problem of classifying \(G\) bundles are given, whenever \(G\) is a compact simply connected Lie group, or more generally, a simply connected topological group which is a finite CW-complex.