On a DGL-map between derivations of Sullivan minimal models

Abstract

For a map $f:X\rightarrow Y$, there is the relative model $M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)\simeq M(X)$ by Sullivan model theory (Félix et al., Rational homotopy theory, graduate texts in mathematics, Springer, Berlin, 2007). Let $\mathrm{Baut}_1X$ 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 $\mathrm{Der}M(X)$ of the Sullivan minimal model M(X) of X. Then we consider the condition that the restriction $b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d)$ is a DGL-map and the related topics.

1 Introduction

Let X (and also Y) be a connected and simply connected finite CW complex with $\dim \pi _*(X)_{{\mathbb {Q}}}<\infty$ ($G_{{\mathbb {Q}}}=G\otimes {\mathbb {Q}}$) and $\mathrm{Baut}_1X$ be the Dold–Lashof classifying space of orientable fibrations [5]. Here $\mathrm{aut}_1X=\mathrm{map}(X,X;id_X)$ is the identity component of the space $\mathrm{aut}X$ of self-equivalences of X. Then any orientable fibration $\xi$ with fibre X over a base space B is the pull-back of a universal fibration $X\rightarrow E_{\infty }^X\rightarrow \mathrm{Baut}_1X$ by a map $h:B\rightarrow \mathrm{Baut}_1X$ and equivalence classes of $\xi$ 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 $(\mathrm{Baut}_1X)_0\simeq \mathrm{Baut}_1(X_0)$ [17]. The differential graded Lie algebra (DGL) $\mathrm{Der}M(X)$, the negative derivations of M(X) (see §2), gives rise to the DGL model for $\mathrm{Baut}_1X$ due to Sullivan [24] (cf.[10, 25]), i.e., the spatial realization $||\mathrm{Der}M(X)||$ is $(\mathrm{Baut}_1X)_0$. Therefore, we obtain a map $(\mathrm{Baut}_1X)_0\rightarrow (\mathrm{Baut}_1Y)_0$ if there is a DGL-map $\mathrm{Der} M(X)\rightarrow \mathrm{Der}M(Y)$. However, it does not exist in general.

Let $f:X\rightarrow Y$ be a map whose homotopy fibration $\xi _f: F_f\rightarrow X\rightarrow Y$ is given by the relative model (Koszul–Sullivan extension)

$$\begin{aligned} M(Y)=(\Lambda V,d)\overset{i}{\hookrightarrow } (\Lambda V\otimes \Lambda W,D)\simeq M(X) \end{aligned}$$

for a certain differential D with $D\mid _{\Lambda V}=d$, where $M(F_f)\cong ( \Lambda W,\overline{D})$ for the homotopy fiber $F_f$ of f [7]. In this paper, we propose

Question 1.1

When is the restriction map given by $b_f (\sigma )=\mathrm{proj}_V\circ \sigma \circ i$

$$\begin{aligned} b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d) \end{aligned}$$

a DGL-map ?

Here $\mathrm{proj}_V:\Lambda V\otimes \Lambda W\rightarrow \Lambda V$ is the algebra map with $\mathrm{proj}_V(w)=0$ for $w\in W$ and $\mathrm{proj}_V\mid _{\Lambda V}=id_{\Lambda V}$.

Definition 1.2

We say that a ${\mathbb {Q}}$-w.t. map $f:X\rightarrow Y$ strictly induces the map

$$\begin{aligned} a_f: (\mathrm{Baut}_1X)_0\rightarrow (\mathrm{Baut}_1Y)_0 \end{aligned}$$

if its DGL model is given by the DGL-map $b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d)$ with $||b_f||= a_f$.

Let $\min \pi _*(S)_{{\mathbb {Q}}}{:=}\min \{ i>0\mid \pi _i(S)_{{\mathbb {Q}}}\ne 0\}$ and $\max \pi _*(S)_{{\mathbb {Q}}}{:=}\max \{ i\ge 0\mid \pi _i(S)_{{\mathbb {Q}}}\ne 0\}$ for a space S. In particular, $\min \pi _*(S)_{{\mathbb {Q}}}{:=}\infty$ when S is the one point space.

Definition 1.3

A fibration $\xi _f :F_f\rightarrow X\overset{f}{\rightarrow } Y$ or a map $f:X\rightarrow Y$ with homotopy fiber $F_f$ is said to be $\pi _{{\mathbb {Q}}}$-separable if $\min \pi _*(F_f)_{{\mathbb {Q}}}\ge \max \pi _*(Y)_{{\mathbb {Q}}}.$

We say a map is rationally weakly trivial (abbr., ${\mathbb {Q}}$-w.t.) if $\xi _f$ is rationally weakly trivial, i.e., $\pi _*(X)_{{\mathbb {Q}}}=\pi _*(F_f)_{{\mathbb {Q}}}\oplus \pi _*(Y)_{{\mathbb {Q}}}$. Then $(\Lambda V\otimes \Lambda W,D)$ is just the minimal model M(X) of X. If a map $f:X\rightarrow Y$ is $\pi _{{\mathbb {Q}}}$-separable, it is ${\mathbb {Q}}$-w.t. The condition to be $\pi _{{\mathbb {Q}}}$-separable is equivalent to the condition that $\min W=\min \{ i>0\mid W^i\ne 0\}\ge \max V=\max \{ i>0\mid V^i\ne 0\}$ in the relative minimal model $M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)$ of $\xi _f$.

Proposition 1.4

For a ${\mathbb {Q}}$-w.t. map $f:X\rightarrow Y$ with relative model $M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)$, the restriction $b_f: \mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d)$ is a DGL-map if and only if f is $\pi _{{\mathbb {Q}}}$-separable.

That means

Theorem 1.5

A ${\mathbb {Q}}$-w.t. map $f:X\rightarrow Y$ strictly induces $a_f: (\mathrm{Baut}_1X)_0\rightarrow (\mathrm{Baut}_1Y)_0$ if and only if f is $\pi _{{\mathbb {Q}}}$-separable.

In §2, we give the proofs under some preparations of models of [7] and [25]. In this paper, we consider only ${\mathbb {Q}}$-w.t. maps. For example, we do not consider the inclusion map $i_X:X\rightarrow X\times Y$, which is not ${\mathbb {Q}}$-w.t. However $i_X$ induces the monoid map $\psi :\mathrm{aut}_1X\rightarrow \mathrm{aut}_1(X\times Y)$ by $\psi (g)=g\times 1_Y$ and, therefore, there exists the induced map $B\psi :\mathrm{Baut}_1X\rightarrow B \mathrm{aut}_1(X\times Y)$ without rationalization. The DGL model is given by the natural inclusion $\mathrm{Der} M(X)\rightarrow \mathrm{Der} (M(X)\otimes M(Y))$, which is a DGL-map.

In §3, we give some conditions that the strictly induced map $a_f: (\mathrm{Baut}_1X)_0\rightarrow (\mathrm{Baut}_1Y)_0$ admits a section. Some results are obtained by Proposition 3.1 induced from [25, VI .1.(3) Proposition] that the DGL-model of the homotopy fibration $\chi _f: \ F_{a_f}\overset{}{\rightarrow } (\mathrm{Baut}_1X)_0\overset{a_f}{\rightarrow } (\mathrm{Baut}_1Y)_0$ is given by

$$\begin{aligned} \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)\rightarrow \mathrm{Der} (\Lambda V\otimes \Lambda W)\overset{b_f}{\rightarrow }\mathrm{Der} (\Lambda V). \end{aligned}$$

Let $\mathrm{aut}_1f$ be the identity component of the space of all fibre-homotopy self-equivalences of f, i.e., $\{ g:X\rightarrow X\mid f\circ g=f \ \}$ and $\mathrm{Baut}_1f$ be the classifying space of this topological monoid. It is known that $\mathrm{Baut}_1f\simeq \widetilde{\mathrm{map}}(Y,\mathrm{Baut}_1(F_f);h)$, where $h:Y\rightarrow \mathrm{Baut}_1(F_f)$ is the classifying map of the fibration $F_f\rightarrow X\overset{f}{\rightarrow } Y$ and $\widetilde{\mathrm{map}}$ denotes the universal cover of the function space [3]. Notice that

$$\begin{aligned} \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)=\mathrm{Der}_{\Lambda V} ( \Lambda V\otimes \Lambda W), \end{aligned}$$

where $\mathrm{Der}_{\Lambda V} ( \Lambda V\otimes \Lambda W)$ is the sub DGL of $\mathrm{Der} ( \Lambda V\otimes \Lambda W)$ sending the elements of $\Lambda V$ to zero and it is a DGL model of $\mathrm{Baut}_1f$ when Y and $F_f$ are finite [4, Theorem 1], [8]. Thus, we note

Theorem 1.6

If the homotopy fiber $F_f$ is finite for a $\pi _{{\mathbb {Q}}}$-separable map f, the fiber of $\chi _f$ has the rational homotopy type of $\mathrm{Baut}_1f$.

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 $dM(X)^\mathrm{even}=0$ and $dM(X)^\mathrm{odd}\subset M(X)^{'rm even}$. A pure space is said to be an $F_0$-space (or positively elliptic) if $\dim \pi _\mathrm{even}(X)\otimes {\mathbb {Q}}=\dim \pi _\mathrm{odd}(X)\otimes {\mathbb {Q}}$ and $H^\mathrm{odd}(X;{\mathbb {Q}})=0$. Then it is equivalent to $H^*(X;{\mathbb {Q}})\cong {\mathbb {Q}}[x_1,\ldots ,x_n]/(f_1,\ldots ,f_n),$ in which $|x_i|$, the degree of $x_i$, is even and $f_1,\ldots ,f_n$ forms a regular sequence in the ${\mathbb {Q}}$-polynomial algebra ${\mathbb {Q}}[x_1,\ldots ,x_n]$, where $M(X)=( {\mathbb {Q}}[x_1,\ldots ,x_n]\otimes \Lambda (y_1, \ldots ,y_n),d)$ with $dx_i=0$ and $dy_i=f_i$. In 1976, S. Halperin [12] conjectured that the Serre spectral sequences of all fibrations $X\rightarrow E\rightarrow B$ 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 $\mathrm{rank}\ G=\mathrm{rank}\ H$, the homogeneous space G/H satisfies the Halperin conjecture [21]. Also the Halperin conjecture is true when $n\le 3$ [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 $\chi _f$ is fibre-trivial for any $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$ if and only if Y satisfies the Halperin’s conjecture.

In §4, we observe the cellular obstruction for the lifting ${\tilde{h}}$ for a map $h:B\rightarrow (\mathrm{Baut}_1Y)_0$ for a simply connected CW complex B of finite type:

for a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$. Of course, it is sufficient to define as $\tilde{h}=s\circ h$ if $a_f$ admits a section s. Specifically, for a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$, let

be a commutative diagram. Then, from Proposition 1.6, we define an obstruction class by derivations in Theorem 4.1 so that

Theorem 1.8

Let $f:X\rightarrow Y$ be a $\pi _{{\mathbb {Q}}}$-separable map with Y and $F_f$ finite. There is a lift h such that

is commutative if and only if ${\mathcal O}_{\alpha }(h_X, h_Y)=0$ in $\pi _{N-1}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}$.

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 $F_f\rightarrow X\overset{f}{\rightarrow } Y$ fibre homotopy equivalent to a G-fibration? i.e., when is there a fibration $f':X'\rightarrow Y$ 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 $H^*(BG;{\mathbb {Q}})$ is evenly graded, the obstruction classes of Theorem 1.8 are contained in $\pi _\mathrm{odd}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}$ when $B=BG$. If $\pi _\mathrm{odd}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}=0$, they vanish and there exists a lift $h:BG\rightarrow (\mathrm{Baut}_1X)_0$. 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 $f:X\rightarrow Y$ be a $\pi _{{\mathbb {Q}}}$-separable map with Y and $F_f$ finite. Suppose that a compact Lie group G acts on Y. If $\pi _\mathrm{odd}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}=0$, the action on Y is rationally lifted to X, i.e., f is rationally fibre homotopy equivalent to a G-equivariant map $f':X'\rightarrow Y$ for a G-space $X'$.

Due to Theorem 1.7 and the result of Shiga and Tezuka [21], we have

Corollary 1.11

Let $f:X\rightarrow Y$ be a $\pi _{{\mathbb {Q}}}$-separable map such that Y is a homogeneous space G/H with ${\mathrm{rank}}\ G={\mathrm{rank}}\ H$. 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 $\dim H^*(X;{\mathbb {Q}})<\infty$, i.e., the largest integer r such that an r-torus $T^r=S^1 \times \dots \times S^1$(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 $r_0(X\times Y)\ge r_0(X)+r_0(Y)$. Notice that it may sometimes be a strict inequality since there is an example that $r_0(X\times S^{12})>0$ even though $r_0(X)=r_0(S^{12})=0$ [15, Example 3.3]. For a map $f:X\rightarrow Y$, we see $r_0(Y)\le r_0(X)$ when $X= F\times Y$ for any space F and f is the projection $F\times Y\rightarrow Y$. In general, when does a map $f:X\rightarrow Y$ induce $r_0(Y)\le r_0(X)$ ?

Corollary 1.12

Let $f:X\rightarrow Y$ be a $\pi _{{\mathbb {Q}}}$-separable map with Y and $F_f$ finite. If $\pi _\mathrm{odd}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}=0$, we have $r_0(Y)\le r_0(X)$.

2 Sullivan models and derivations

Let $M(X)=(\Lambda {V},d)$ be the Sullivan minimal model of simply connected CW complex X of finite type [24]. It is a free ${\mathbb {Q}}$-commutative differential graded algebra (DGA) with a ${\mathbb {Q}}$-graded vector space $V=\bigoplus _{i\ge 1}V^i$ where $\dim V^i<\infty$ and a decomposable differential, i.e., $d(V^i) \subset (\Lambda ^+{V} \cdot \Lambda ^+{V})^{i+1}$ and $d \circ d=0$. Here $\Lambda ^+{V}$ is the ideal of $\Lambda {V}$ generated by elements of positive degree. The degree of a homogeneous element x of a graded algebra is denoted as |x|. Then $xy=(-1)^{|{x}||{y}|}yx$ and $d(xy)=d(x)y+(-1)^{|{x}|}xd(y)$. Note that M(X) determines the rational homotopy type of X, namely the spatial realization is given as $||M(X)||\simeq X_0$. In particular,

$$\begin{aligned} V^n\cong \mathrm{Hom}(\pi _n(X),{\mathbb {Q}}) \text{ and } H^*(\Lambda {V},d)\cong H^*(X;{\mathbb {Q}}). \end{aligned}$$

Here the second is an isomorphism as graded algebras. Refer to [7] for detail.

Let $\mathrm{Der}_i M(X)$ be the set of ${\mathbb {Q}}$-derivations of M(X) decreasing the degree by i with $\sigma (xy)=\sigma (x)y+(-1)^{i\cdot |x|}x\sigma (y)$ for $x,y\in M(X)$. The boundary operator $\partial : \mathrm{Der}_i M(X)\rightarrow \mathrm{Der}_{i-1} M(X)$ is defined by

$$\begin{aligned} \partial (\sigma )=d\circ \sigma -(-1)^i\sigma \circ d \end{aligned}$$

for $\sigma \in \mathrm{Der}_iM(X)$. We denote $\oplus _{i>0} \mathrm{Der}_iM(X)$ by $\mathrm{Der}M(X)$ in which $\mathrm{Der}_1M(X)$ is $\partial$-cycles. Then $\mathrm{Der}M(X)$ is a (non-free) DGL by the Lie bracket

$$\begin{aligned} {[}\sigma ,\tau ]{:=}\sigma \circ \tau -(-1)^{|\sigma ||\tau |} \tau \circ \sigma . \end{aligned}$$

Note that $H_*(\mathrm{Der}M)=H_*(\mathrm{Der} N)$ when free DGAs M and N are quasi-isomorphic [20]. Furthermore, recall the definition of Tanré [25, p. 25]: let $(L,\partial )$ be a DGL of finite type with positive degree. Then $C^*(L,\partial )=(\Lambda s^{-1}\sharp L, D=d_1+d_2)$ with

$$\begin{aligned} \langle d_1s^{-1}z; sx\rangle =-\langle z;\partial x\rangle \text{ and } \langle d_2s^{-1}z; sx_1,sx_2\rangle =(-1)^{|x_1|}\langle z;[x_1,x_2]\rangle , \end{aligned}$$

where $\langle s^{-1}z;sx\rangle =(-1)^{|z|}\langle z;x\rangle$ and $\sharp L$ is the dual space of L.

Theorem 2.1

[24, §11],[10, 25] For a Sullivan model M(X) of X, $\mathrm{Der} M(X)$ is a DGL-model of $\mathrm{Baut}_1X$. In particular, there is an isomorphism of graded Lie algebras $H_*(\mathrm{Der} M(X))\cong \pi _*(\Omega \mathrm{Baut}_1X)_{{\mathbb {Q}}}$ where the right hand has the Samelson bracket. Furthermore, $C^*(\mathrm{Der} M(X) )$ is a DGA-model of $\mathrm{Baut}_1X$.

Two fibrations $\xi _{f_1}$ and $\xi _{f_2}$ are fibre homotopy equivalent if there is a diagram:

where $\psi \circ i_2\simeq i_1\circ \overline{\psi }$ and $f_1\circ \psi =f_2$. 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 $\xi _{f_1}$ and $\xi _{f_2}$ are fibre homotopy equivalent, there is a DGL-isomorphism $\phi :\mathrm{Der} (\Lambda V\otimes \Lambda W,D_1)\cong \mathrm{Der} (\Lambda V\otimes \Lambda W,D_2)$ such that $\phi (\sigma )=\psi \circ \sigma \circ \psi ^{-1}$ and $b_{f_2}\circ \phi =b_{f_1}$. Thus, there is a homotopy equivalence map $\phi : (\mathrm{Baut}_1X_1)_0\overset{\sim }{\rightarrow } (\mathrm{Baut}_1X_2)_0$ such that $a_{f_2}\circ \phi =a_{f_1}$, i.e., $a_{f_1}$ and $a_{f_2}$ are fibre homotopy equivalent as fibrations.

Proof

Recall the DGA-diagram of §1. Then $D_2=\psi \circ D_1\circ \psi ^{-1}$. Therefore, there is a DGL-isomorphism $\phi$ given by $\phi (\sigma )=\psi \circ \sigma \circ \psi ^{-1}$ and

is DGL-commutative since $\psi |_{\Lambda V} =id_{\Lambda V}$. In particular, we can check $\partial _2\circ \phi =\phi \circ \partial _1$ by

$$\begin{aligned} \partial _2 \phi (\sigma )= & {} D_2\psi \sigma \psi ^{-1}-(-1)^i(\psi \sigma \psi ^{-1})D_2=\psi D_1\sigma \psi ^{-1}-(-1)^i\psi \sigma D_1\psi ^{-1} \\= & {} \phi (D_1\sigma -(-1)^i\sigma D_1)=\phi \partial _1(\sigma ) \end{aligned}$$

for $\sigma \in \mathrm{Der}_i (\Lambda V\otimes \Lambda W,D_1)$. Similarly, we have $\phi ([\sigma ,\tau ])=[\phi (\sigma ),\phi (\tau )]$. $\square$

Convention. For a DGA-model $(\Lambda V,d)$ the symbol (v, f) means the elementary derivation that takes a generator v of V to an element f of $\Lambda V$ and the other generators to 0. Note that $|(v,f)|=|v|-|f|$.

Proof of Proposition 1.4

Let $M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)$ be the model of f. Note that $b_f(\sigma )$ is a derivation on $\Lambda V$ for $\sigma \in \mathrm{Der} (\Lambda V\otimes \Lambda W)$ since

$$\begin{aligned} b_f(\sigma )(a\cdot b)= & {} (\mathrm{proj}_V\circ \sigma \circ i)(a\cdot b) =\mathrm{proj}_V( \sigma (a)\cdot b+(-1)^{|\sigma ||a|}a\cdot \sigma (b)) \\= & {} (\mathrm{proj}_V\circ \sigma \circ i )(a)\cdot b+(-1)^{|\sigma ||a|}a\cdot (\mathrm{proj}_V\circ \sigma \circ i )(b) =b_f(\sigma )(a)\cdot b+(-1)^{|\sigma ||a|}a\cdot b_f(\sigma )(b) \end{aligned}$$

for $a,b\in \Lambda V$. Thus, $b_f$ is well defined.

(if) When $\min W\ge \max V$, there is a decomposition of vector spaces

$$\begin{aligned} \mathrm{Der} (\Lambda V\otimes \Lambda W)=\mathrm{Der} (\Lambda V)\oplus \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W) \end{aligned}$$

from degree arguments. Then there is a DGL-map $b_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d)$ by $b_f(\sigma _1)=\sigma _1$ and $b_f (\sigma _2 )=0$ for $\sigma =\sigma _1+\sigma _2$ with $\sigma _1\in \mathrm{Der} (\Lambda V)$ and $\sigma _2\in \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$. Then $b_f$ is a Lie algebra map since

$$\begin{aligned} b_f([\sigma ,\tau ])= & {} b_f([\sigma _1+\sigma _2 ,\tau _1+\tau _2 ])=b_f([\sigma _1 ,\tau _1 ]) \\= & {} \mathrm{proj}_V\circ \sigma _1\circ \tau _1\circ i+(-1)^{|\sigma _1||\tau _1|} \mathrm{proj}_V\circ \tau _1 \circ \sigma _1\circ i \\= & {} (\mathrm{proj}_V\circ \sigma _1\circ i)\circ (\mathrm{proj}_V \circ \tau _1\circ i)+(-1)^{|\sigma _1||\tau _1|} (\mathrm{proj}_V\circ \tau _1\circ i)\circ (\mathrm{proj}_V\circ \sigma _1\circ i) \\= & {} [b_f(\sigma _1 ),b_f(\tau _1 )] =[b_f(\sigma ),b_f(\tau )] \end{aligned}$$

for $\sigma ,\tau \in \mathrm{Der} (\Lambda V\otimes \Lambda W)$. Furthermore, it preserves the differential since

$$\begin{aligned} (b_f\circ \partial _X)(\sigma )= & {} b_f(\partial _X(\sigma _1+\sigma _2) )=b_f(\partial _X(\sigma _1) )=b_f(\partial _Y(\sigma _1) +\tau ) \\= & {} b_f(\partial _Y(\sigma _1))=\partial _Y(\sigma _1)=\partial _Y(b_f(\sigma _1 ))=\partial _Y(b_f(\sigma _1 +\sigma _2))=( \partial _Y\circ b_f)(\sigma ) \end{aligned}$$

for $\sigma \in \mathrm{Der} (\Lambda V\otimes \Lambda W)$ with some $\tau \in \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$.

(only if) Suppose that $\min W< \max V$. There are elements $w\in W$ and $v\in V$ with $|w|<|v|$. Then $b_f$ is not a DGL-map since $b_f([(w,1),(v,w)])=b_f(v,1)\ne 0$ but $[b_f(w,1),b_f(v,w)]=[0,0]=0$ from the definition of $b_f$. $\square$

Example 2.3

Consider the case that f is not $\pi _{{\mathbb {Q}}}$-separable (not ${\mathbb {Q}}$-w.t.). Let $f:S^7\rightarrow S^4$ be the Hopf map. Then the model is given by

$$\begin{aligned} M(S^4)=(\Lambda (x,y),d)\rightarrow (\Lambda (x,y,z),D)\simeq M(S^7) \end{aligned}$$

with $|x|=4$, $|y|=7$, $|z|=3$, $dx=Dx=0$, $Dy=dy=x^2$, $Dz=x$, $Dy=x^2$. Then the bases of derivations are given as

n	$\mathrm{Der}_n (\Lambda (x,y,z),D)$
7	(y, 1)
4	(x, 1) (y, z)
3	(y, x) (z, 1)
1	(x, z)

n	$\mathrm{Der}_n (\Lambda (x,y),d)$
7	(y, 1)
4	(x, 1)
3	(y, x)
1

, where $H_*(\mathrm{Der} (\Lambda (x,y))={\mathbb {Q}}\{ (y,1)\}$. By degree reason, any DGL-map

$$\begin{aligned} \psi :(\mathrm{Der} (\Lambda (x,y,z),D)\rightarrow (\mathrm{Der} (\Lambda (x,y),d) \end{aligned}$$

is given by $\psi (y,1)=a_1(y,1)$, $\psi (x,1)=a_2(x,1)$, $\psi (y,z)=a_3(x,1)$, $\psi (y,x)=a_4(y,x)$, $\psi (z,1)=a_5(y,x)$ and $\psi _f (x,z)=0$ for some $a_i\in {\mathbb {Q}}$.

From $(x,1)=[(z,1),(x,z)]$ and $(y,z)=[(x,z),(y,x)]$ we have $a_2=0$ and $a_3=0$, respectively. Then from $2(y,1)=[(z,1),(y,z)]+[(x,1),(y,x)]$, we obtain $a_1=0$. Thus, $||\psi ||$ is homotopic to the constant map.

Let $f:X\rightarrow Y$ be a map with a section s, i.e., there is a map $s:Y\rightarrow X$ with $f\circ s\sim id_Y$. Then there is a map $\psi _f :\mathrm{aut}_1X\rightarrow \mathrm{aut}_1Y$ with $\psi _f(g){:=}f\circ g\circ s$ for $g\in \mathrm{aut}_1X$. In general, this does not preserve the monoid structures.

Theorem 2.4

If a $\pi _{{\mathbb {Q}}}$-separable map f admits a section, there is a commutative diagram:

Proof

The map $\pi _n(\psi _f):\pi _n(\mathrm{aut}_1X)\rightarrow \pi _n(\mathrm{aut}_1Y)$ is given by $\pi _n(\psi _f)([\sigma ])=[\tau ]{:=}[f\circ \sigma \circ (s\times 1_{S^n})]$ in the following homotopy commutative diagram:

from adjointness. That is the pointed homotopy classes of maps $S^n\rightarrow \mathrm{aut}_1X=\mathrm{map}(X,X;id_X)$ are in bijection with the homotopy classes of those maps $X\times S^n\rightarrow X$ that composed with the inclusion $i_X:X\rightarrow X\times S^n$ yield the identity [20, p. 43–44]. Let $M(Y)=(\Lambda V,d)\overset{i}{ \rightarrow } (\Lambda V\otimes \Lambda W,D)\simeq M(X)$ be the model of f. We identify the rationalized map $\sigma _0:(X\times S^n)_0\rightarrow X_0$ to an element of $\mathrm{Der}_n(\Lambda V\otimes \Lambda W)$ [24, p. 313] (cf. [20, Proposition 11]). Then there is a chain map

$$\begin{aligned} c_f:\mathrm{Der} (\Lambda V\otimes \Lambda W,D)\rightarrow \mathrm{Der}(\Lambda V,d) \end{aligned}$$

given by $c_f (\sigma _0 )=\mathrm{proj}_V\circ \sigma _0 \circ i$. It is well defined, i.e., $\partial _Y\circ c_f=c_f\circ \partial _X$, from $DW\subset \Lambda V\otimes \Lambda ^+W$ [26] since it admits a section. (However, $c_f$ is not a DGL-map in general.) Notice that

$$\begin{aligned} \pi _n(\psi _f)_{{\mathbb {Q}}}([\sigma _0 ])\equiv H_n(c_f)([\sigma _0 ]). \end{aligned}$$

If f is a $\pi _{{\mathbb {Q}}}$-separable map, $c_f$ is the DGL-map $b_f$. $\square$

The following is obvious from the definition of $b_f$ and useful:

Claim 2.5

For any $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$, we have $b_f(C)=0$ and $b_f\mid _{\mathrm{Der}(\Lambda V)}=id_{\mathrm{Der}(\Lambda V)}$ for $\mathrm{Der} (\Lambda V\otimes \Lambda W)=C\oplus \mathrm{Der}(\Lambda V)$.

Theorem 2.6

For a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$, let

be a commutative diagram. Then there exists a map between total spaces $k :E\rightarrow E'$ in the diagram:

where $k\circ i\simeq i'\circ f_0$ and $g\circ p=p'\circ k$. Here $p:E\rightarrow B_0$ and $p':E'\rightarrow B'_0$ are induced by the rationalized classifying maps h and $h'$, respectively.

Proof

Let $X\rightarrow E_{\infty }^X\overset{p_{\infty }^X}{\rightarrow } \mathrm{Baut}_1X$ and $Y\rightarrow E_{\infty }^Y\overset{p_{\infty }^Y}{\rightarrow } \mathrm{Baut}_1Y$ be the universal fibrations of X and Y, respectively. Let $C^*(\mathrm{Der} (\Lambda V))\otimes \Lambda V,D_Y$ be the DGA-model of $E_{\infty }^Y$ and $C^*(\mathrm{Der} (\Lambda V\otimes \Lambda W)) \otimes \Lambda V\otimes \Lambda W,D_X$ be the DGA-model of $E_{\infty }^X$. For a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$, there exists a DGA-inclusion map $\psi$ such that

is commutative from the universality. Indeed, $C^*(\mathrm{Der} (\Lambda V))\otimes \Lambda V,D_Y$ is a sub-DGA of $C^*(\mathrm{Der} (\Lambda V\otimes \Lambda W)) \otimes \Lambda V\otimes \Lambda W,D_X$ from Claim 2.5 and $\max V\le \min W$. Thus, there is a map $\tilde{a}_f{:=}|\psi |: (E_{\infty }^X)_0 \rightarrow (E_{\infty }^Y)_0$ such that $(p_{\infty }^Y)_0\circ \tilde{a}_f=a_f\circ (p_{\infty }^X)_0$. Since $p'$ is the pull-back of $(p_{\infty }^Y)_0$ by $h'$, there exists a map $k:E\rightarrow E'$ such that

is commutative from the universality since $h'\circ g\circ p=a_f\circ h\circ p=a_f\circ (p_{\infty }^X)_0\circ \tilde{h}=(p_{\infty }^Y)_0\circ \tilde{a}_f\circ \tilde{h}$. $\square$

Example 2.7

Let $X=K({\mathbb {Q}},n)\times K({\mathbb {Q}}, 2n)$ and $Y=K({\mathbb {Q}}, n)$ for some even integer n. Then $M(X)=\Lambda (x,y),0$ and $M(Y)=\Lambda (z),0$ with $|x|=|z|=n$ and $|y|=2n$. Let a map $f:X\rightarrow Y$ be given by $M(f):\Lambda (z)\rightarrow \Lambda (x,y)$ with $M(f)(z)=x$. The homotopy fibration of any $\pi _{{\mathbb {Q}}}$-separable map is given by $\Lambda (z),0\rightarrow \Lambda (z,y),0\cong \Lambda (x,y),0$ from the degree reason. Therefore, the DGL-map $\psi :\mathrm{Der} \Lambda (x,y)\rightarrow \mathrm{Der} \Lambda (z)$ such that $\psi ((y,x))=\psi ((x,1))=(z,1)$ is not DGL-homotopic to $b_f$ from Claim 2.5.

Let $h:S^{n+1}_0\rightarrow (\mathrm{Baut}_1X)_0$ and $h':S^{n+1}_0\rightarrow (\mathrm{Baut}_1Y)_0$ be given by $L(h): {\mathbb {L}}(u)\rightarrow \mathrm{Der} (\Lambda (x,y))$ with $|u|=n$, $L(h)(u)=(y,x)$ and $L(h'): {\mathbb {L}}(u)\rightarrow \mathrm{Der} (\Lambda (z))$ with $L(h')(u)=(z,1)$, respectively. Then the commutative diagram

does not induce a map between total spaces $f':E\rightarrow E'$ such that

is homotopy commutative. Indeed, there does not exist a DGA-map $h:\Lambda (v,z),D'\rightarrow \Lambda (v,x,y),D$ with $D'z=v$, $Dy=vx$ and $Dx=0$ such that

where $|v|=n+1$ is homotopy commutative since h cannot be a DGA-map from $Dh(z)=0$ but $hD'(z)=v$.

3 When does $a_f$ admit a section?

Let $f:X\rightarrow Y$ be a $\pi _{{\mathbb {Q}}}$-separable map with homotopy fiber $F_f$ and $\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$ the sub-DGL of $\mathrm{Der} (\Lambda V\otimes \Lambda W)$ restricted to derivations out of $\Lambda W$.

Proposition 3.1

Let $F_{a_f}$ be the homotopy fiber of $a_f$. Then the DGL-model of the fibration $\chi _f :F_{a_f}\overset{j}{\rightarrow } (\mathrm{Baut}_1X)_0\overset{a_f}{\rightarrow } (\mathrm{Baut}_1Y)_0$ is given by

$$\begin{aligned} \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)\overset{\mathrm{incl.}}{\rightarrow } \mathrm{Der} (\Lambda V\otimes \Lambda W)\overset{b_f}{\rightarrow }\mathrm{Der} (\Lambda V). \end{aligned}$$

Proof

Since $b_f$ is surjective and $\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$ is $\mathrm{Ker}\ b_f$, it follows from [25, VI .1.(3) Proposition]. $\square$

Let $L(F)=\oplus _{i>0}L(F)_i$ be the degree decomposition of a DGL-model of a space F.

Theorem 3.2

$L(F_{a_f})_n\cong \oplus _{i-j=n}\mathrm{Der}_i ( \Lambda W)\otimes H^j(\Lambda V)$.

Proof

A chain-map $\rho :\mathrm{Der}_i ( \Lambda W)\otimes H^j(\Lambda V)\rightarrow \mathrm{Der}_i (\Lambda W, \Lambda W\otimes (\Lambda V)^j)$ is given by $\rho (\sigma \otimes [f])(w){:=}(-1)^{|w|j}\sigma (w)\cdot f$ induced by an inclusion $H^j(\Lambda V)\rightarrow (\Lambda V)^j$. It is quasi-isomorphic, i.e., there is a decomposition $\mathrm{Der} (\Lambda W, \Lambda W\otimes \Lambda V)=(\mathrm{Der} ( \Lambda W)\otimes H^*(\Lambda V))\oplus C$ for a complex C of derivations with ${H}_*(C)=0$. $\square$

The rational homotopy exact sequence of the strictly induced fibration $\chi _f$:

$$\begin{aligned}&\ \ \ \ \ \cdots \overset{j_{\sharp }}{\rightarrow } \pi _{n+2}(\mathrm{Baut}_1X)_{{\mathbb {Q}}}\overset{{a_f}_{\sharp }}{\rightarrow }\pi _{n+2}(\mathrm{Baut}_1Y)_{{\mathbb {Q}}} \overset{\delta _f}{\rightarrow } \\&\pi _{n+1}(F_{a_f})_{{\mathbb {Q}}}\overset{j_{\sharp }}{\rightarrow } \pi _{n+1}(\mathrm{Baut}_1X)_{{\mathbb {Q}}}\overset{{a_f}_{\sharp }}{\rightarrow }\pi _{n+1}(\mathrm{Baut}_1Y)_{{\mathbb {Q}}} \overset{\delta _f}{\rightarrow } \cdots \end{aligned}$$

is equivalent to the homology exact sequence:

$$\begin{aligned}&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdots \rightarrow H_{n+1}(\mathrm{Der} (\Lambda V\otimes \Lambda W))\overset{{b_f}_*}{\rightarrow }H_{n+1}(\mathrm{Der} (\Lambda V)) \overset{\delta _f}{\rightarrow } \\&H_n(\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W))\rightarrow H_n(\mathrm{Der} (\Lambda V\otimes \Lambda W))\overset{{b_f}_*}{\rightarrow }H_n(\mathrm{Der} (\Lambda V)) \overset{\delta _f}{\rightarrow } \cdots \end{aligned}$$

Then we have the following from an ordinary chain complex property:

Claim 3.3

The connecting map $\delta _f$ is given by $\delta _f ([\sigma ])=[\tau ]$ when $\partial _X (\sigma )=\tau$ for a $\partial _Y$-cycle $\sigma$ of $\mathrm{Der} (\Lambda V)$ and a $\partial _X$-cycle $\tau$ of $\mathrm{Der} (\Lambda W, \Lambda V\otimes \Lambda W)$.

Recall that the following implications hold for a general fibration $\chi : F\rightarrow E\overset{p}{\rightarrow } B$ of simply connected spaces:

$$\begin{aligned} \chi \text{ is } \text{ fibre-trivial } \Rightarrow \ p \text{ admits } \text{ a } \text{ section } \Rightarrow \ \chi \text{ is } \text{ weakly } \text{ trivial } \Leftrightarrow \ \delta =0 .\ \ \ \ \ (*) \end{aligned}$$

Here $\delta :\pi _*(B)\rightarrow \pi _{*-1}(F)$ is the connecting map of the homotopy exact sequence for $\chi$. The following may be a characteristic phenomenon in our fibration $\chi _f$.

Proposition 3.4

$a_f$ admits a section if and only if $\delta _f =0$.

Proof

(if) Let the DGA-model of the fibration $\chi _f:F_{a_f}\overset{j}{\rightarrow } (\mathrm{Baut}_1X)_0\overset{a_f}{\rightarrow } (\mathrm{Baut}_1Y)_0$ be given as the commutative diagram:

where $M(\mathrm{Baut}_1Y)\cong (\Lambda U,d)$ with $U^{n+1}= H_n(\mathrm{Der} (\Lambda V))$ and $M(F_{a_f})\cong ( \Lambda Z,\overline{D}_2)$ with $Z^{n+1}= H_n(\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W))$. Here

$$\begin{aligned} ({\Lambda U,d})\rightarrow (\Lambda U\otimes \Lambda Z,D_2)\rightarrow (\Lambda Z,\overline{D}_2) \end{aligned}$$

is the model of $\chi _f$. From the assumption $\chi _f$ is weakly equivalent, i.e., $M(\mathrm{Baut}_1X)\cong (\Lambda U\otimes \Lambda Z,D_2)$ as a minimal model. Let $D=d_1+d_2$ as in §2. Notice that a linear component of any bracket representation of $\sigma \in \mathrm{Der}( \Lambda W, \Lambda V\otimes \Lambda W)$ is not contained in $[\mathrm{Der} (\Lambda V),\mathrm{Der} (\Lambda V)]$, where $[\ ,\ ]$ is the Lie bracket. That means

$$\begin{aligned} d_2(s^{-1}\sigma ^*)\in I( C^*(\mathrm{Der}( \Lambda W, \Lambda V\otimes \Lambda W))) \ \ \ \ \ \ \ \ \ (**). \end{aligned}$$

Here I(S) is the ideal in $C^*(\mathrm{Der}(\Lambda V\otimes \Lambda W))$ generated by a basis of a vector space S. Let $\sigma$ be a non-exact $\partial _X$-cycle of $\mathrm{Der}( \Lambda W, \Lambda V\otimes \Lambda W)$. Then

$$\begin{aligned} H^*( C^*(\mathrm{Der}( \Lambda W, \Lambda V\otimes \Lambda W)),d_1')/H^*\cdot H^*\cong Z\ni [s^{-1}\sigma ^* ] \end{aligned}$$

for $D'=d_1'+d_2'$ as in §2. Since $\rho _1(D_1([s^{-1}\sigma ^* ]))$ and $d_2(s^{-1}\sigma ^*)$ are D-cohomologous in $C^*(\mathrm{Der}(\Lambda V\otimes \Lambda W))$, we can put

$$\begin{aligned} D_1([s^{-1}\sigma ^* ])\in I(Z) \end{aligned}$$

from $(**)$, i.e., $D_1(Z)\subset C^*(\mathrm{Der}\Lambda V)\otimes \Lambda ^+ Z$. By $\rho _2$, $D_2(Z)\subset \Lambda U\otimes \Lambda ^+ Z$. Then we have done from [26].

(only if) It holds from the above implications $(*)$. $\square$

Theorem 3.5

If a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$ is rationally fibre-trivial (i.e., $X_0\sim (F_f)_0\times Y_0$), $a_f$ admits a section.

Proof

From the assumption and Claim 3.3, we have $\delta _f=0$. Then it holds from Proposition 3.4. $\square$

Refer [19, page 292] for related topics. Conversely, when Y is an odd-sphere,

Theorem 3.6

If a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y=S^{2n+1}$ is not rationally fibre-trivial, $a_f$ does not admit a section. Furthermore, ${a_f}\sim *$(the constant map).

Proof

Let $M(S^{2n+1})=(\Lambda v,0)$. Since there exists an element $w\in W$ such that $Dw\in \Lambda v\otimes \Lambda ^+W$ from the assumption, $\partial _X(v,1)=\pm (w,\partial Dw/\partial v)+\cdots \ne 0$ in $\mathrm{Der} (\Lambda W)$. From Claim 3.3$\delta _f$ is injective since $\delta _f([(v,1)])=[\pm (w,\partial Dw/\partial v)+\cdots ]\ne 0$. Then the former holds from Proposition 3.4. Furthermore, from the homotopy exact sequence, we have ${a_f}_{\sharp }=0$. Thus, the latter holds. $\square$

Example 3.7

• (1) Let $S^5\times S^7\rightarrow X\rightarrow Y=S^3$ be a non-(fibre-)trivial $\pi _{{\mathbb {Q}}}$-separable fibration given by the model

  $$\begin{aligned} (\Lambda (v_1),0)\rightarrow (\Lambda (v_1, w_1,w_2),D) \rightarrow (\Lambda (w_1,w_2),0) \end{aligned}$$

  with $|v_1|=3$, $|w_1|=5$, $|w_2|=7$, $Dw_1=0$ and $Dw_2=v_1w_1$. Then $a_f$ does not admit a section from Theorem 3.6. Indeed $\delta _f:H_{3}(\mathrm{Der} (\Lambda v')) \overset{}{\rightarrow } H_2(\mathrm{Der}(\Lambda (w_1,w_2), \Lambda (v_1,w_1,w_2)))$ is non-trivial from $\delta _f ([(v_1,1)])=[(w_2,w_1)]\ne 0$.

• (2) Let $S^5\times S^7\rightarrow X'\rightarrow Y'$ be a non-(fibre-)trivial $\pi _{{\mathbb {Q}}}$-separable fibration given by the model

  $$\begin{aligned} (\Lambda (v_1,v_2,v_3),d_{Y'})\rightarrow (\Lambda (v_1,v_2,v_3, w_1,w_2),D') \rightarrow (\Lambda (w_1,w_2),0) \end{aligned}$$

  with $|v_1|=|v_2|=3$, $|v_3|=5$, $|w_1|=7$, $|w_2|=9$, $d_{Y'}(v_1)=d_{Y'}(v_2)=0$, $d_{Y'}(v_3)=v_1v_2$, $D'w_1=0$ and $D'w_2=v_1w_1$. Then $a_f$ admits a section from Proposition 3.4 since $\delta _f ([(v_3,1)])= 0$ for $H_*(\mathrm{Der}(\Lambda (v_1,v_2,v_3)))={\mathbb {Q}}\{ [(v_3,1)]\}$. However, $\chi _f$ is not trivial from $[(v_3,1),(w_2,v_2v_3)]=(w_2,v_2)$. Indeed, then

  $$\begin{aligned} {\mathcal {D}}(s^{-1}(w_2,v_2)^*)=d_2(s^{-1}(w_2,v_2)^*)=s^{-1}(v_3,1)^*\cdot s^{-1}(w_2,v_2v_3)^* \end{aligned}$$

  for $(C^*(\mathrm{Der}(\Lambda (v_1,v_2,v_3, w_1,w_2)),{\mathcal D})$ with ${\mathcal {D}}=d_1+d_2$. Refer the proof of Proposition 3.4.

Proof of Theorem 1.7

Let $M(Y)=(\Lambda V,d)=( {\mathbb {Q}}[x_1,\ldots ,x_n]\otimes \Lambda (y_1, \ldots ,y_n),d)$ with $dx_i=0$ and $dy_i=f_i$ for $i=1,..,n$.

(if) Let $M(Y)=(\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda W,D)$ be the model of f. From the regularity of $f_1,\ldots ,f_n$, $Im D\subset {\mathbb {Q}}[x_1,\ldots ,x_n]\otimes \Lambda W$. Thus,

$$\begin{aligned} \partial _X(x_i,h_i)=\sum _{j=1}^n(y_j,(\partial f_j/\partial x_i)\cdot h_i)+\theta \ \ \text{ and } \ \ \partial _X(y_i,h_i)=0\ \ \ \ (i=1,\ldots ,n) \end{aligned}$$

for any $h_i\in \Lambda V^\mathrm{even}={\mathbb {Q}}[x_1,\ldots ,x_n]$ with suitable degree and some $\theta \in \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$. Then we have $\delta _f=0$ from Claim 3.3 since $H_\mathrm{even}(\mathrm{Der}M(Y))=0$ [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 $\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$ does not have an element of $\mathrm{Der} (\Lambda V)$ as a factor since $\mathrm{Der} H^*(Y;{\mathbb {Q}})=0$ [18] again. Thus, we have

$$\begin{aligned} D_2=d\otimes 1\pm 1\otimes \overline{D}_2 \end{aligned}$$

for the Sullivan minimal model $(\Lambda U,d)\rightarrow (\Lambda U\otimes \Lambda Z,D_2)\rightarrow (\Lambda Z,\overline{D}_2)$ of $\chi _f$ (in the proof of Proposition 3.4).

(only if) Suppose that Y does not satisfies the Halperin conjecture, i.e., there is a non-zero element $[\sum _{i}(x_i,h_i)+\sum _{j}(y_j,g_j) ]\in H_{2m}(\mathrm{Der}M(Y))$ for $h_i\in {\mathbb {Q}}[x_1,\ldots ,x_n]$, $g_j\in \Lambda V$ and some m [18]. Let $(S^a\times S^b\simeq _0 ) F\rightarrow X\overset{f}{\rightarrow } Y$ be a fibration of the model:

$$\begin{aligned} (\Lambda V,d)\rightarrow (\Lambda V\otimes \Lambda (w_1,w_2,w_3),D)\rightarrow (\Lambda (w_1,w_2,w_3),d_F), \end{aligned}$$

where $|w_1|=a$ is even and $|w_2|=b$ is odd with $b-a=|x_k|-1$ for some k, $d_Fw_1=d_Fw_2=0$, $d_Fw_3=w_1^2$, $Dw_1=0$ and $Dw_2=x_kw_1$. When $h_k$ is not $d_Y$-exact, the element $h_kw_1$ is not D-exact. Then

$$\begin{aligned} \delta _f\left( \left[ \sum _{i}(x_i,h_i)+\sum _{j}(y_j,g_j) \right] \right) =[(w_2,h_kw_1)]\ne 0 \end{aligned}$$

for $\delta _f:H_{2m}(\mathrm{Der} \Lambda V)\rightarrow H_{2m-1}(\mathrm{Der} (\Lambda (w_1,w_2,w_3), \Lambda V\otimes \Lambda (w_1,w_2,w_3))$ from Claim 3.3. In particular, $\chi _f$ is not fibre-trivial. $\square$

Example 3.8

Let Y be the homogeneous space $SU(6)/SU(3)\times SU(3)$. Then Y is a pure space but not an $F_0$-space since ${\mathrm{rank}}\ SU(6)=5>4={\mathrm{rank}}\ (SU(3)\times SU(3))$. Let $\xi :(S^{11}\times S^{23}\simeq _0 ) F\rightarrow X\overset{f}{\rightarrow } Y$ be a fibration whose relative model is given as

$$\begin{aligned} (\Lambda (x_1,x_2,y_1,y_2,y_3),d_Y)\rightarrow (\Lambda (x_1,x_2,y_1,y_2,y_3)\otimes \Lambda (w_1,w_2),D)\rightarrow (\Lambda (w_1,w_2),0), \end{aligned}$$

where $|x_1|=4$, $|x_2|=6$, $|y_1|=7$, $|y_2|=9$, $|y_3|=11$, $|w_1|=11$, $|w_2|=23$, $d_Yy_1=x_1^2$, $d_Yy_2=x_1x_2$, $d_Yy_3=x_2^2$, $Dw_1=0$ and $Dw_2=(x_1y_2-x_2y_1)w_1$. Then $\partial _X((y_1,1))=(w_2,x_2w_1)$, i.e., $\delta _f [(y_1,1)]=[(w_2,x_2w_1)]\ne 0$ from Claim 3.3. In particular, $\chi _f$ is not trivial. Refer [19, Example 1.14(2)] for the Sullivan minimal model of $\mathrm{Baut}_1Y$.

4 The obstruction class for a lifting

Let $L(B)=(L(B),\partial _B)$ be the Quillen model of a simply connected CW complex B of finite type. Then $L(B\cup _{\alpha } e^N)$ is given by $L(B)\coprod {\mathbb {L}}(u),\partial _{\alpha }$ where $|u|=N-1$, $\partial _{\alpha }\mid _{L (B)}=\partial _B$ and $\partial _{\alpha }(u)\in L(B)$ [25, Proposition III.3.(6)].

Theorem 4.1

For a $\pi _{{\mathbb {Q}}}$-separable map $f:X\rightarrow Y$, let

be a commutative diagram. Then there is a lift h such that

is commutative if and only if

$$\begin{aligned} {\mathcal O}_{\alpha }(h_X, h_Y){:=}[\tau (h_Y(u))-h''_X(\partial _{\alpha }(u)) ]=0 \end{aligned}$$

in $H_{N-2}(\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W))=\pi _{N-1}(F_{a_f})_{{\mathbb {Q}}}$ for the DGL-commutative diagram

with

• $\partial _X\mid _{ \mathrm{Der} (\Lambda V, \Lambda V\otimes \Lambda W)}=\partial _Y+\tau$ and $\partial _X \mid _{ \mathrm{Der} (\Lambda W, \Lambda V\otimes \Lambda W)}=\tau$ for some $\tau : \mathrm{Der}_*(\Lambda V\otimes \Lambda W)\rightarrow \mathrm{Der}_{*-1} (\Lambda W, \Lambda V\otimes \Lambda W)$ and

• $h_X=h_X'+h_X''$ where $h_X'(b)\in \mathrm{Der} (\Lambda V)$ and $h_X''(b)\in \mathrm{Der} (\Lambda W, \Lambda V\otimes \Lambda W)$ for $b\in L(B)$.

Proof

Since $b_f\circ h_X= h_Y\circ i$ and $h_Y$ is a DGL-map,

$$\begin{aligned} h'_X\partial _{\alpha }(u)= h_Y\partial _{\alpha }(u)= \partial _{Y}h_Y(u) \end{aligned}$$

(1)

in $\mathrm{Der} ( \Lambda V)$. Notice that the obstruction element $\partial _X (h_Y(u))-h_X(\partial _{\alpha }(u))$ is a $\partial _X$-cycle in $\mathrm{Der} ( \Lambda V\otimes \Lambda W)$. Therefore, $\tau (h_Y(u))-h''_X(\partial _{\alpha }(u))$ is a $\partial _X$-cycle in $\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$ from (1).

(if) Suppose that ${\mathcal O}_{\alpha }(h_X, h_Y)=0$. Then there is an element $q\in \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$ such that

$$\begin{aligned} \partial _X(q)= \tau (h_Y(u))-h''_X(\partial _{\alpha }(u)). \end{aligned}$$

(2)

Let

$$\begin{aligned} h\mid _{L(B)} {:=}h_X \ \text{ and } \ \ h(u){:=}h_Y(u)-q. \end{aligned}$$

Then h is a DGL-map since

$$\begin{aligned} \partial _{X}(h(u))= & {} \partial _X(h_Y(u))-\partial _X(q) \\= & {} (h'_X\partial _{\alpha }(u)+\tau h_Y(u))- (\tau h_Y(u)-h''_X\partial _{\alpha }(u)) \\= & {} h'_X\partial _{\alpha }(u)+h''_X\partial _{\alpha }(u) =h_X(\partial _{\alpha }(u))=h(\partial _{\alpha }(u)) \end{aligned}$$

from (1) and (2). Furthermore,

is commutative since $b_f(q)=0$. Thus, the (if)-part holds from the special realization of $(*)$.

(only if) Suppose that there exists a map h such that $(*)$ is commutative. Since h is a DGL-map,

$$\begin{aligned} \partial _X(h(u)) =h(\partial _{\alpha }(u)) \end{aligned}$$

(3)

in $\mathrm{Der} ( \Lambda V\otimes \Lambda W)$ and

$$\begin{aligned} h_X''\partial _{\alpha }(u)= \tau h (u) \end{aligned}$$

(4)

from (1) and (3). Furthermore,

$$\begin{aligned} \tau h (u) \sim \tau h_Y(u) \end{aligned}$$

(5)

in $\mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$. Here $\sim$ means “homologous”. Indeed, (5) follows since

$$\begin{aligned} h(u)=h_Y(u)+x \end{aligned}$$

for some element $x\in \mathrm{Der} ( \Lambda W, \Lambda V\otimes \Lambda W)$ from $b_f\circ h= h_Y$ and then since

$$\begin{aligned} \tau h(u)=\tau (h_Y(u)+x)=\tau h_Y(u)+\partial _X( x). \end{aligned}$$

Thus, we obtain that ${\mathcal O}_{\alpha }(h_X, h_Y)=[\tau (h_Y(u))-h''_X(\partial _{\alpha }(u)) ]=0$ from (4) and (5). $\square$

From Theorem 3.2, we have

Corollary 4.2

If $\pi _{\ge N-1}(\mathrm{Baut}_1F_f)_{{\mathbb {Q}}}=0$ for the homotopy fiber $F_f$ of f, there exists a lift h for the pair $(h_X,h_Y)$ of above.

Example 4.3

Let $B=S^2={\mathbb {C}}P^1$. Let $S^3\times S^5\rightarrow X\overset{f}{\rightarrow } Y=S^3$ be the fibration given by the model

$$\begin{aligned} (\Lambda (v),0)\rightarrow (\Lambda (v,w_1,w_2),D)\rightarrow (\Lambda (w_1,w_2),0) \end{aligned}$$

with $|v|=|w_1|=3$, $|w_2|=5$, $Dw_1=0$ and $Dw_2=vw_1$. Let $L({\mathbb {C}}P^2)=L (B\cup _{\alpha } e^4)=({\mathbb {L}}(u_1,u_2),\partial )$ with $|u_1|=1$, $|u_2|=3$, $\partial u_1=0$ and $\partial u_2=[u_1,u_1]$ [25]. Let

be a commutative diagram given by the DGL-model

by $h_X(u_1)=h_Y(u_1)=0$ and $h_Y(u_2)=(v,1)$. Then ${\mathcal O}_{\alpha }(h_X, h_Y)\ne 0$ in $H_2(\mathrm{Der} (\Lambda (w_1,w_2), \Lambda (v,w_1,w_2))$ since

$$\begin{aligned} \tau h_Y(u_2)=\partial _X(v,1)=(w_2,w_1)\not \sim 0=h_X''([u_1,u_1])=h_X''(\partial _{\alpha }(u_2)). \end{aligned}$$

Thus $h_Y:{\mathbb {C}}P^2\rightarrow (\mathrm{Baut}_1Y)_0$ cannot lift to $h:{\mathbb {C}}P^2\rightarrow (\mathrm{Baut}_1X)_0$. Note that $h_Y$ is extended to ${\mathbb {C}}P^{\infty }\rightarrow (\mathrm{Baut}_1Y)_0$. Since $BS^1={\mathbb {C}}P^{\infty }$, we obtain that any free $S^1$-action on Y cannot lift to X.

5 An application to lifting actions

Let BG and EG be the classifying space and the universal space of a compact connected Lie group G of ${\mathrm{rank}}\ G=r$, respectively. If G acts on a space Y by $\mu :G\times Y\rightarrow Y$, there is the Borel fibration

$$\begin{aligned} Y \overset{i}{\rightarrow } EG \times _{G}^{\mu } Y \rightarrow BG, \end{aligned}$$

where the Borel space $EG \times _{G}^{\mu } Y$ (or simply $EG \times _{G} Y$) is the orbit space of the diagonal action $g(e,y)=(eg^{-1},gy)$ on the product $EG \times Y$. It is rationally given by the KS extension (model)

$$\begin{aligned} ({\mathbb {Q}}[t_1,\dots ,t_r],0) \rightarrow ({\mathbb {Q}}[t_1,\dots ,t_r] \otimes \Lambda {V},D_{\mu }) \rightarrow (\Lambda {V},d)=M(Y)\ \ \ \ (*), \end{aligned}$$

where $|{t_i}|$ are even for $i=1,\dots ,r$, $D_{\mu }(t_i)=0$ and $D_{\mu }(v) \equiv d(v)$ modulo the ideal $(t_1,\dots ,t_r)$ for $v\in V$.

Recall the lifting theorem of D. H. Gottlieb:

Theorem 5.1

[11, Theorem 1] Let a topological group G acts on a space Y. A fibration $X\overset{f}{\rightarrow } Y$ is fibre homotopy equivalent to a G-fibration if and only if it is fibre homotopy equivalent to the pull-back of a fibration over $EG\times _GY$ induced by the inclusion $i: Y\rightarrow EG\times _GY$.

Proof of Theorem 1.10

Let $h_Y: BG\rightarrow (\mathrm{Baut}_1Y)_0$ be the rationalization of the classifying map of the Borel fibration $Y\overset{i}{\rightarrow } EG \times _{G}^{\mu } Y \rightarrow BG$ of the action $\mu : G\times Y\rightarrow Y$. Let $B^n$ be the n-skeleton of a CW complex B. From Theorem 1.8, there is a lift $h_X^{\alpha }$ such that

is commutative for all n and attaching $\alpha$ since ${\mathcal O}_{\alpha }(h_X^n, h_Y^{\alpha })=0$. Indeed, $\pi _\mathrm{odd}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}=0$ and L(BG) is oddly graded since $H^*(BG;{\mathbb {Q}})$ is evenly graded. Thus, we have the commutative diagram:

From Theorem 2.6, there is a commutative diagram:

for some space E. Let $g':E'\rightarrow EG\times _GY$ be the pull-back of g by the rationalization $l_0$ and $f':X'\rightarrow Y$ be the pull-back of $g'$ by i

Notice that the model is given by the DGA-commutative diagram:

for $R{:=}H^*(BG;{\mathbb {Q}})={\mathbb {Q}}[t_1,\ldots ,t_r]$. Notice that the third square is given by the push-out [7, Proposition 15.8]. Thus, from Theorem 5.1, we obtain the commutative diagram

since $M(X')\cong \Lambda V\otimes \Lambda W=M(X)$. $\square$

If the r-torus $T^r$ acts on a space Y, $|t_1|=\cdots =|t_r|=2$ in $(*)$.

Proposition 5.2

[13, Proposition 4.2] Suppose that Y is a simply connected CW-complex with $\dim H^*(Y;{\mathbb {Q}})<\infty$. Put $M(Y)=(\Lambda V,d)$. Then $r_0(Y) \ge r$ if and only if there is a KS extension $(*)$ satisfying $\dim H^*({\mathbb {Q}}[t_1,\dots ,t_r] \otimes \wedge {V},D)<\infty$. Moreover, if $r_0(Y) \ge r$, then $T^r$ acts freely on a finite complex $Y'$ that has the same rational homotopy type as Y and $M(ET^r\times _{T^r}Y')\cong ({\mathbb {Q}}[t_1,\dots ,t_r] \otimes \wedge {V},D)$.

Proof of Corollary 1.12

Let $r_0(Y)=r$. Notice that $L(BT^r)$ is oddly generated since $H^*(BT^r;{\mathbb {Q}})={\mathbb {Q}}[t_1,\ldots ,t_r]$. Since $\pi _\mathrm{odd}(\mathrm{Baut}_1f)_{{\mathbb {Q}}}=0$, there exists a lift $(BT^r)_0\rightarrow (\mathrm{Baut}_1X)_0$ from Theorem 1.8. Then we have the homotopy commutative diagram:

from Theorem 2.6. Here $\bullet$ is the one point space. We have $\dim H^*(\tilde{E};{\mathbb {Q}})<\infty$ since $\dim H^*(F_g;{\mathbb {Q}})<\infty$ and $\dim H^*(E;{\mathbb {Q}})=\dim H^*(ET^r\times _{T^r}Y;{\mathbb {Q}})<\infty$ for the fibration $F_g\rightarrow \tilde{E}\rightarrow E$. Thus there is a free $T^r$-action on $X'$ with $X'_0\simeq X_0$ and $\tilde{E}\simeq (ET^r\times _{T^r}X')_0$ from Proposition 5.2. Thus, we have $r_0(X)\ge r$. $\square$

Example 5.3

Let $S^5\rightarrow X\overset{f}{\rightarrow } Y$ be a rationally non-trivial fibration given by the model

$$\begin{aligned} (\Lambda V,d_Y)=(\Lambda (v_1,v_2,v_3,v_4,v_5),d_Y)\rightarrow (\Lambda (v_1,v_2,v_3,v_4,v_5,w),D) \end{aligned}$$

with $|v_1|=|v_2|=2$, $|v_3|=|v_4|=|v_5|=|w|=5$, $d_Y(v_1)=d_Y(v_2)=0$, $d_Y(v_3)=v_1^3$, $d_Y(v_4)=v_1^2v_2$, $d_Y(v_5)=v_2^3$ and $D(w)=v_1v_2^2$. Then

$$\begin{aligned} \pi _\mathrm{odd} (\mathrm{Baut}_1f)_{{\mathbb {Q}}}\cong H_\mathrm{even}(\mathrm{Der}_{\Lambda V}(\Lambda V\otimes \Lambda (w)))=0 \end{aligned}$$

since there is no element of odd-degree $<5$ in $\Lambda V$. Therefore, $r_0(Y)\le r_0(X)$. Indeed, we can directly check that $r_0(Y)=1$ and $r_0(X)=2$.

On the other hand, let $S^5\rightarrow X\overset{f}{\rightarrow } Y=S^3\times S^3$ be a rationally non-trivial fibration. Then the model is given by

$$\begin{aligned} (\Lambda (v_1,v_2),0)\rightarrow (\Lambda (v_1,v_2,w),D) \end{aligned}$$

with $|v_1|=|v_2|=3$, $|w|=5$, and $D(w)=v_1v_2$. Then

$$\begin{aligned} \pi _{3} (\mathrm{Baut}_1f)_{{\mathbb {Q}}}\cong H_{2}(\mathrm{Der}_{\Lambda V}(\Lambda V\otimes \Lambda (w)))={\mathbb {Q}}\{ (w,v_1)\} \oplus {\mathbb {Q}}\{ (w,v_2)\} \ne 0 \end{aligned}$$

and $r_0(Y)=2>1= r_0(X)$.