On a DGL-map between derivations of Sullivan minimal models
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Arabian Journal of Mathematics
Toshihiro Yamaguchi
On a DGL-map between derivations of Sullivan minimal models
Received: 6 May 2020 / Accepted: 4 August 2020 © The Author(s) 2020
Abstract 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 3:285–305, 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 the restriction b f : Der(V ⊗ W, D) → Der(V, d) is a DGL-map and the related topics. Mathematics Subject Classification
55P62 · 55R15
1 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 Baut1 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 oriX → Baut X entable fibration ξ with fibre X over a base space B is the pull-back of a universal fibration X → E ∞ 1 by a map h : B → Baut1 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 (Baut1 X )0 Baut1 (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 (Baut1 X )0 . Therefore, we obtain a map (Baut1 X )0 → (Baut1 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) i
M(Y ) = (V, d) → (V ⊗ W, D) M(X ) for a certain differential D with D |V = d, where M(F f ) ∼ = (W, 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 ) = projV ◦ σ ◦ i b f : Der(V ⊗ W, D) → Der(V, d) a DGL-map ? T. Yamaguchi Kochi University, 2-5-1, Kochi 780-8520, Japan E-mail: [email protected]
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Here projV : V ⊗ W → V is the algebra map with projV (w) = 0 for w ∈ W and projV |V = idV . Definition 1.2 We say that a Q-w.t. map f : X → Y strictly induces the map a f : (Baut1 X )0 → (Baut1 Y )0 if its DGL model is given by the DGL-map b f : Der(V ⊗ W, D) → Der(V, d) with ||b f || = a f . Let min π∗ (S)Q := min{i > 0 | πi (S)Q = 0} and max π∗ (S)Q := max{i ≥ 0 | πi (S)Q = 0} for a space S. In particular, min π∗ (S)Q :=∞ when S is the one point space. f
Definition 1.3 A fibration ξ f : F f → X → Y or a map f : X → Y with homotopy fiber F f is said to be πQ -separable if min π∗ (
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