Let X be a simply connected space with finite-dimensional rational homotopy groups. Let \(p_\infty :UE \rightarrow B\mathrm {aut}_1(X)\) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map \( \omega :\mathrm {aut}_1(B\mathrm {aut}_1(X_\mathbb {Q})) \rightarrow B\mathrm {aut}_1(X_\mathbb {Q})\) expressed in terms of derivations of the relative Sullivan model of \(p_\infty \). We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space \(B\mathrm {aut}_1(X_\mathbb {Q})\) as a consequence. We also prove that \(\mathbb {C} P^n_\mathbb {Q}\) cannot be realized as \(B\mathrm {aut}_1(X_\mathbb {Q})\) for \(n \le 4\) and X with finite-dimensional rational homotopy groups.

中文翻译：

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有理同伦理论中纤维X的普遍成因

令X为具有有限维有理同伦群的简单连通空间。令\（p_ \ infty：UE \ rightarrow B \ mathrm {aut} _1（X）\）为带有纤维X的简单连通空间的通用纤维。我们为评估图给出了DG Lie代数模型\（\ omega：\ mathrm {aut} _1（B \ mathrm {aut} _1（X_ \ mathbb {Q}））\ rightarrow B \ mathrm {aut} _1（X_ \ mathbb {Q}）\）以\（p_ \ infty \）的相对Sullivan模型的导数表示。因此，我们推导了有理Gottlieb组和分类空间\（B \ mathrm {aut} _1（X_ \ mathbb {Q}）\）的评估子组的公式。我们还证明\（\ mathbb {C} P ^ n_ \ mathbb {Q} \）对于具有有限维有理同伦群的\（n \ le 4 \）和X，不能将其实现为\（B \ mathrm {aut} _1（X_ \ mathbb {Q}）\）。