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The rational homotopy type of the space of self-equivalences of a fibration. (English) Zbl 1214.55011

Let \(p: E\to B\) a fibration of simply connected CW-complexes, in which \(E\) is finite, and let Aut\((p)\) be the space of self-fibre-homotopy equivalences of \(p\). On the other hand, given a Sullivan model of \(p\), \(\Lambda V\to\Lambda V\otimes\Lambda W\), consider the Lie algebra Der\(_{\Lambda V}(\Lambda V\otimes\Lambda W)\) of derivations of the model of \(E\) which vanish on the model of \(B\). Then, it is proven that the homology of this Lie algebra is naturally isomorphic to the rational Samelson Lie algebra Aut\((p)\otimes\mathbb Q\).
Ideas and techniques used to prove this result are extended to the study of the components of Aut\((p)\): denote by \({\mathcal E}_\sharp(p)\) the subgroup of \(\pi_0\)Aut\((p)\) consisting of homotopy equivalence classes of maps \(E\to E\) over \(B\) inducing the identity on the image and cokernel of the connecting homomorphism in the long exact sequence on homotopy groups of the fibration. Then it is proven that the rationalization of this group \({\mathcal E}_\sharp(p)_{\mathbb Q}\) is isomorphic to its algebraic counterpart in terms of derivations, namely the group \(H_0\bigl(\)Der\(_\sharp(\Lambda V\otimes\Lambda W)\bigr)\).

MSC:

55P62 Rational homotopy theory
55Q15 Whitehead products and generalizations
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