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On a DGL-map between derivations of Sullivan minimal models

Toshihiro Yamaguchi

2020
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Arabian Journal of Mathematics
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For a map f : X → Y , there is the relative model M(Y ) = ( V, d) → ( V ⊗ W, D) M(X ) by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let Baut 1 X be the Dold-Lashof classifying space of orientable fibrations with fiber X (Dold and Lashof, Ill J Math 3: 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations DerM(X ) of the Sullivan minimal model M(X ) of X . Then we consider the condition that
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... the restriction b f : Der( V ⊗ W, D) → Der( V, d) is a DGL-map and the related topics. Mathematics Subject Classification 55P62 · 55R15 Introduction Let X (and also Y ) be a connected and simply connected finite CW complex with dim π * (X ) Q < ∞ (G Q = G ⊗ Q) and Baut 1 X be the Dold-Lashof classifying space of orientable fibrations [5] . Here aut 1 X = map(X, X ; id X ) is the identity component of the space aut X of self-equivalences of X . Then any orientable fibration ξ with fibre X over a base space B is the pull-back of a universal fibration X → E X ∞ → Baut 1 X by a map h : B → Baut 1 X and equivalence classes of ξ are classified by their homotopy classes [2, 5, 23] . The Sullivan minimal model M(X ) [24] determines the rational homotopy type of X , the homotopy type of the rationalization X 0 [14] of X . Notice that (Baut 1 X ) 0 Baut 1 (X 0 ) [17]. The differential graded Lie algebra (DGL) DerM(X ), the negative derivations of M(X ) (see §2), gives rise to the DGL model for Baut 1 X due to Sullivan [24] (cf.[10,25]), i.e., the spatial realization ||DerM(X )|| is (Baut 1 X ) 0 . Therefore, we obtain a map (Baut 1 X ) 0 → (Baut 1 Y ) 0 if there is a DGL-map DerM(X ) → DerM(Y ). However, it does not exist in general. Let f : X → Y be a map whose homotopy fibration ξ f : F f → X → Y is given by the relative model (Koszul-Sullivan extension) D) for the homotopy fiber F f of f [7]. In this paper, we propose Question 1.1 When is the restriction map given by b f (σ ) = proj V • σ • i b f : Der( V ⊗ W, D) → Der( V, d) a DGL-map ?

doi:10.1007/s40065-020-00291-0
fatcat:taidsck7onfcrfiypubpluue7i