On a DGL-map between derivations of Sullivan minimal models

<jats:p>For a map <jats:inline-formula><jats:alternatives><jats:tex-math>$$f:X\rightarrow Y$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula>, there is the relative model <jats:inline-formula><jats:alternatives><jats:tex-math>$$M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)\simeq M(X)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm{Baut}_1X$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> be the Dold–Lashof classifying space of orientable fibrations with fiber <jats:italic>X</jats:italic> (Dold and Lashof, Ill J Math 3:285–305, 1959]). Its DGL (differential graded Lie algebra)-model is given by the derivations <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm{Der}M(X)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> of the Sullivan minimal model <jats:italic>M</jats:italic>(<jats:italic>X</jats:italic>) of <jats:italic>X</jats:italic>. Then we consider the condition that the restriction <jats:inline-formula><jats:alternatives><jats:tex-math>$$b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d) $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
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</mml:math></jats:alternatives></jats:inline-formula> is a DGL-map and the related topics.</jats:p>

Here proj V : V ⊗ W → V is the algebra map with proj V (w) = 0 for w ∈ W and proj V | V = id V . Definition 1. 2 We say that a Q-w.t. map f : X → Y strictly induces the map a f : (Baut 1 X ) 0 → (Baut 1 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 | π i (S) Q = 0} and max π * (S) Q := max{i ≥ 0 | π i (S) Q = 0} for a space S. In particular, min π * (S) Q :=∞ when S is the one point space.
We say a map is rationally weakly trivial (abbr., Q-w.t.) if ξ f is rationally weakly trivial, i.e., π * (X ) Q = π * (F f ) Q ⊕ π * (Y ) Q . Then ( V ⊗ W, D) is just the minimal model M(X ) of X . If a map f : X → Y is π Q -separable, it is Q-w.t. The condition to be π Q -separable is equivalent to the condition that min W = In §2, we give the proofs under some preparations of models of [7] and [25]. In this paper, we consider only Q-w.t. maps. For example, we do not consider the inclusion map i X : X → X × Y , which is not Q-w.t. However i X induces the monoid map ψ : aut 1 X → aut 1 (X × Y ) by ψ(g) = g × 1 Y and, therefore, there exists the induced map Bψ : Baut 1 X → Baut 1 (X × Y ) without rationalization. The DGL model is given by the natural inclusion Der M(X ) → Der(M(X ) ⊗ M(Y )), which is a DGL-map.
In §3, we give some conditions that the strictly induced map a f : (Baut 1 X ) 0 → (Baut 1 Y ) 0 admits a section.
Let aut 1 f be the identity component of the space of all fibre-homotopy self-equivalences of f , i.e., {g : X → X | f • g = f } and Baut 1 f be the classifying space of this topological monoid. It is known that Baut 1 f map(Y, Baut 1 (F f ); h), where h : Y → Baut 1 (F f ) is the classifying map of the fibration → Y and map denotes the universal cover of the function space [3]. Notice that A space X is said to be elliptic if the dimensions of the rational cohomology algebra and homotopy group are both finite [7]. An elliptic space X is said to be pure if d M(X ) even = 0 and d M(X ) odd ⊂ M(X ) rmeven . A pure space is said to be an F 0 -space (or positively elliptic) if dim π even (X ) ⊗ Q = dim π odd (X ) ⊗ Q and H odd (X ; Q) = 0. Then it is equivalent to H * (X ; Q) ∼ = Q[x 1 , . . . , x n ]/( f 1 , . . . , f n ), in which |x i |, the degree of x i , is even and f 1 , . . . , f n forms a regular sequence in the Q-polynomial algebra Q[x 1 , . . . , x n ], where M(X ) = (Q[x 1 , . . . , x n ] ⊗ (y 1 , . . . , y n ), d) with dx i = 0 and dy i = f i . In 1976, S. Halperin [12] conjectured that the Serre spectral sequences of all fibrations X → E → B of simply connected CW complexes collapse at the E 2 -terms for any F 0 -space X [7]. For compact connected Lie groups G and H where H is a subgroup of G, when rank G = rank H , the homogeneous space G/H satisfies the Halperin conjecture [21]. Also the Halperin conjecture is true when n ≤ 3 [16]. Finally, we note some relations with the Halperin conjecture [7, §39] due to Meier [18] as Theorem 1.7 Let Y be an F 0 -space. Then the fibration χ f is fibre-trivial for any π Q -separable map f : X → Y if and only if Y satisfies the Halperin's conjecture.
In §4, we observe the cellular obstruction for the liftingh for a map h : B → (Baut 1 Y ) 0 for a simply connected CW complex B of finite type: be a commutative diagram. Then, from Proposition 1.6, we define an obstruction class by derivations in Theorem 4.1 so that In §5, we consider an application to lifting actions. Let G be a topological group and acts on a CW complex Y . Recall the problem of lifting (up to homotopy) of Gottlieb [11]: Problem 1.9 When is a fibration F f → X f → Y fibre homotopy equivalent to a G-fibration? i.e., when is there a fibration f : X → Y such that f is fibre homotopy equivalent to f and there is a G-action on X such that f is equivariant?
Suppose that G is a compact connected Lie group. Since H * (BG; Q) is evenly graded, the obstruction classes of Theorem 1.8 are contained in π odd (Baut 1 f ) Q when B = BG. If π odd (Baut 1 f ) Q = 0, they vanish and there exists a lift h : BG → (Baut 1 X ) 0 . Then from Theorem 2.6 in the case that B = B = BG and g = (id BG ) 0 , we obtain using Theorem 5.1 of D. H. Gottlieb. Theorem 1.10 Let f : X → Y be a π Q -separable map with Y and F f finite. Suppose that a compact Lie group G acts on Y . If π odd (Baut 1 f ) Q = 0, the action on Y is rationally lifted to X , i.e., f is rationally fibre homotopy equivalent to a G-equivariant map f : X → Y for a G-space X .
Due to Theorem 1.7 and the result of Shiga and Tezuka [21], we have Corollary 1.11 Let f : X → Y be a π Q -separable map such that Y is a homogeneous space G/H with rank G = rank H . Then any group action on Y is rationally lifted to X . In particular, the natural G-action on Y is rationally lifted to X .
Furthermore, we apply the obstruction argument to a rational homotopical invariant: let r 0 (X ) be the rational toral rank of a simply connected complex X of dim H * (X ; Q) < ∞, i.e., the largest integer r such that an r -torus T r = S 1 ×· · ·× S 1 (r -factors) can act continuously on a CW-complex X in the rational homotopy type of X with all its isotropy subgroups finite (almost free action) [1,9,13]. It is very difficult to calculate r 0 ( ) in general. From the definition, we have the inequality r 0 (X × Y ) ≥ r 0 (X ) + r 0 (Y ). Notice that it may sometimes be a strict inequality since there is an example that r 0 (X × S 12 ) > 0 even though r 0 (X ) = r 0 (S 12 ) = 0 [15,Example 3.3]. For a map f : X → Y , we see r 0 (Y ) ≤ r 0 (X ) when X = F × Y for any space F and f is the projection F × Y → Y . In general, when does a map f : X → Y induce r 0 (Y ) ≤ r 0 (X ) ? Corollary 1.12 Let f : X → Y be a π Q -separable map with Y and F f finite. If π odd (Baut 1 f ) Q = 0, we have r 0 (Y ) ≤ r 0 (X ).

Sullivan models and derivations
Let M(X ) = ( V, d) be the Sullivan minimal model of simply connected CW complex X of finite type [24]. It is a free Q-commutative differential graded algebra (DGA) with a Q-graded vector space Here + V is the ideal of V generated by elements of positive degree. The degree of a homogeneous element x of a graded algebra is denoted as |x|. Then x y = (−1) |x||y| yx and d(x y) = d(x)y +(−1) |x| xd(y). Note that M(X ) determines the rational homotopy type of X , namely the spatial realization is given as ||M(X )|| X 0 . In particular, Here the second is an isomorphism as graded algebras. Refer to [7] for detail. Let Note that H * (DerM) = H * (Der N ) when free DGAs M and N are quasi-isomorphic [20]. Furthermore, recall the definition of Tanré [25, p. 25]: let (L , ∂) be a DGL of finite type with positive degree. Then where s −1 z; sx = (−1) |z| z; x and L is the dual space of L.
Two fibrations ξ f 1 and ξ f 2 are fibre homotopy equivalent if there is a diagram:

Then its Sullivan model is given as
where the left square is DGA-commutative and the right square is DGA-homotopy commutative.

Lemma 2.2
Suppose that two maps f 1 and f 2 strictly induce a f 1 and a f 2 , respectively. If ξ f 1 and ξ f 2 are fibre homotopy equivalent, there is a DGL-isomorphism φ : ., a f 1 and a f 2 are fibre homotopy equivalent as fibrations.

Theorem 2.4 If a π Q -separable map f admits a section, there is a commutative diagram:
in the following homotopy commutative diagram: That is the pointed homotopy classes of maps S n → aut 1 X = map(X, X ; id X ) are in bijection with the homotopy classes of those maps X × S n → X that composed with the inclusion i X : X → X ×  [26] since it admits a section. (However, c f is not a DGL-map in general.) Notice that The following is obvious from the definition of b f and useful: be a commutative diagram. Then there exists a map between total spaces k : E → E in the diagram: Here p : E → B 0 and p : E → B 0 are induced by the rationalized classifying maps h and h , respectively.
For a π Q -separable map f : X → Y , there exists a DGA-inclusion map ψ such that is commutative from the universality. Indeed, C * (Der( V ))⊗ V, D Y is a sub-DGA of C * (Der( V ⊗ W ))⊗ V ⊗ W, D X from Claim 2.5 and max V ≤ min W . Thus, there is a mapã f :=|ψ| :  (z, 1), respectively. Then the commutative diagram does not induce a map between total spaces f : E → E such that where |v| = n + 1 is homotopy commutative since h cannot be a DGA-map from Dh(z) = 0 but h D (z) = v.

When does a f admit a section?
Let f : X → Y be a π Q -separable map with homotopy fiber F f and Der( W, V ⊗ W ) the sub-DGL of Der( V ⊗ W ) restricted to derivations out of W .

Proposition 3.1 Let F a f be the homotopy fiber of a f . Then the DGL-model of the fibration
Proof Let L(F) = ⊕ i>0 L(F) i be the degree decomposition of a DGL-model of a space F.
Proof A chain-map ρ : The rational homotopy exact sequence of the strictly induced fibration χ f : is equivalent to the homology exact sequence: Then we have the following from an ordinary chain complex property: The connecting map δ f is given by Recall that the following implications hold for a general fibration χ : F → E p → B of simply connected spaces: Here δ : π * (B) → π * −1 (F) is the connecting map of the homotopy exact sequence for χ . The following may be a characteristic phenomenon in our fibration χ f .

Proposition 3.4 a f admits a section if and only if
is the model of χ f . From the assumption χ f is weakly equivalent, i.e., M(Baut 1 X ) ∼ = ( U ⊗ Z , D 2 ) as a minimal model. Let D = d 1 + d 2 as in §2. Notice that a linear component of any bracket representation of is the Lie bracket. That means Here I (S) is the ideal in C * (Der( V ⊗ W )) generated by a basis of a vector space S. Let σ be a non-exact ∂ X -cycle of Der( W, V ⊗ W ). Then for D = d 1 +d 2 as in §2. Since ρ 1 (D 1 ([s −1 σ * ])) and d 2 (s −1 σ * ) are D-cohomologous in C * (Der( V ⊗ W )), we can put from ( * * ), i.e., D 1 (Z ) ⊂ C * (Der V ) ⊗ + Z . By ρ 2 , D 2 (Z ) ⊂ U ⊗ + Z . Then we have done from [26].
(only if) It holds from the above implications ( * ).
Proof From the assumption and Claim 3.3, we have δ f = 0. Then it holds from Proposition 3.4.
Refer [19, page 292] for related topics. Conversely, when Y is an odd-sphere, Theorem 3.6 If a π Q -separable map f : X → Y = S 2n+1 is not rationally fibre-trivial, a f does not admit a section. Furthermore, a f ∼ * (the constant map).
Example 3.7 (1) Let S 5 × S 7 → X → Y = S 3 be a non-(fibre-)trivial π Q -separable fibration given by the model with |v 1 | = 3, |w 1 | = 5, |w 2 | = 7, Dw 1 = 0 and Dw 2 = v 1 w 1 . Then a f does not admit a section from Theorem 3.6. Indeed δ f : be a non-(fibre-)trivial π Q -separable fibration given by the model for any h i ∈ V even = Q[x 1 , . . . , x n ] with suitable degree and some θ ∈ Der( W, V ⊗ W ). Then we have δ f = 0 from Claim 3.3 since H even (DerM(Y )) = 0 [18] from the assumption. Then a f admits a section from Proposition 3.4. Furthermore, from Theorem 3.2, we can suppose that the Lie bracket decomposition of an element of Der( W, V ⊗ W ) does not have an element of Der( V ) as a factor since Der H * (Y ; Q) = 0 [18] again. Thus, we have (only if) Suppose that Y does not satisfies the Halperin conjecture, i.e., there is a non-zero element . , x n ], g j ∈ V and some m [18]. Let → Y be a fibration of the model: where |w 1 | = a is even and (Der( (w 1 , w 2 , w 3 ), V ⊗ (w 1 , w 2 , w 3 )) from Claim 3.3. In particular, χ f is not fibre-trivial.

Theorem 4.1 For a π
in Der( V ). Notice that the obstruction element Let Then h is a DGL-map since from (1) and (2). Furthermore, is commutative since b f (q) = 0. Thus, the (if)-part holds from the special realization of ( * ).
From Theorem 3.2, we have is commutative for all n and attaching α since O α (h n X , h α Y ) = 0. Indeed, π odd (Baut 1 f ) Q = 0 and L(BG) is oddly graded since H * (BG; Q) is evenly graded. Thus, we have the commutative diagram: From Theorem 2.6, there is a commutative diagram: for some space E. Let g : E → EG × G Y be the pull-back of g by the rationalization l 0 and f : X → Y be the pull-back of g by i Notice that the model is given by the DGA-commutative diagram: from Theorem 2.6. Here • is the one point space. We have dim H * (Ẽ; Q) < ∞ since dim H * (F g ; Q) < ∞ and dim H * (E; Q) = dim H * (E T r × T r Y ; Q) < ∞ for the fibration F g →Ẽ → E. Thus there is a free T r -action on X with X 0 X 0 andẼ (E T r × T r X ) 0 from Proposition 5.2. Thus, we have r 0 (X ) ≥ r .