An induced map between rationalized classifying spaces for fibrations

TL;DR

This paper constructs a natural map between classifying spaces of fibrations induced by rationally weakly trivial maps, explores conditions for sections, and applies these to liftings of group actions and rational toral rank evaluations.

Contribution

It introduces a strictly induced map between classifying spaces for fibrations via Sullivan models and analyzes its properties and applications.

Findings

The induced map $a_f$ can admit a section under certain conditions.

Obstruction classes for lifting classifying maps are characterized.

Applications to liftings of $G$-actions and rational toral ranks are provided.

Abstract

Let $B{ aut}_1X$ be the Dold-Lashof classifying space of orientable fibrations with fiber $X$. For a rationally weakly trivial map $f:X\to Y$, our strictly induced map $a_f: (Baut_1X)_0\to (Baut_1Y)_0$ induces a natural map from a $X_0$-fibration to a $Y_0$-fibration. It is given by a map between the differential graded Lie algebras of derivations of Sullivan models. We note some conditions that the map $a_f$ admits a section and note some relations with the Halperin conjecture. Furthermore we give the obstruction class for a lifting of a classifying map $h: B\to (Baut_1Y)_0$ and apply it for liftings of $G$-actions on $Y$ for a compact connected Lie group $G$ as the case of $B=BG$ and evaluating of rational toral ranks as $r_0(Y)\leq r_0(X)$.

Equations

$$M(Y)=(\Lambda V,d)\to (\Lambda V\otimes \Lambda W,D)\simeq M(X)$$

$$a_f:(Bau{t}_{1}X{)}_0\to (Bau{t}_{1}Y{)}_0$$

$$b_f:Der(\Lambda V\otimes \Lambda W,D)\to Der(\Lambda V,d)$$

$$F_{f_1}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Lambda V,d\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$a_{g\circ f}=a_g\circ a_f:(Bau{t}_{1}X{)}_0\to (Bau{t}_{1}Z{)}_0$$

$$Der(\Lambda W,\Lambda V\otimes \Lambda W)\to Der(\Lambda V\otimes \Lambda W)\stackrel{b_f}{\to }Der(\Lambda V).$$

$$Der(\Lambda W,\Lambda V\otimes \Lambda W)=De{r}_{\Lambda V}(\Lambda V\otimes \Lambda W)$$

$$(Bau{t}_{1}X{)}_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$X_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

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

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

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

$$⟨d_1{s}^{-1}z;sx⟩=-⟨z;\partial x⟩\text{mbox}and⟨d_2{s}^{-1}z;s{x}_1,s{x}_2⟩=(-1{)}^{|x_1|}⟨z;[x_1,x_2]⟩,$$

$$[x,[y,z]]=[[x,y],z]+(-1{)}^{|x||y|}[y,[x,z]]$$

$$\partial [x,y]=[\partial x,y]+(-1{)}^{|x|}[x,\partial y].$$

$$\widetilde{H_n}(X;\mathbb{Q})\cong H_{n-1}(U,{\partial }_1)\text{mbox}and{\pi }_{\ast }(\Omega X{)}_{\mathbb{Q}}\cong H_{\ast }(\mathbb{L}U,\partial ),$$

$$Der(\Lambda V\otimes \Lambda W,D_1),{\partial }_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\partial }_2\varphi (\sigma )=D_2\psi \sigma {\psi }^{-1}-(-1{)}^i(\psi \sigma {\psi }^{-1})D_2=\psi D_1\sigma {\psi }^{-1}-(-1{)}^i\psi \sigma D_1{\psi }^{-1}$$

$$=\varphi (D_1\sigma -(-1{)}^i\sigma D_1)=\varphi {\partial }_1(\sigma )$$

$$Der(\Lambda V\otimes \Lambda W)=Der(\Lambda V)\oplus Der(\Lambda W,\Lambda V\otimes \Lambda W)$$

$$0\ne b_f(v,1)=b_f([(w,1),(v,w)])=[b_f(w,1),b_f(v,w)]=[0,0]=0$$

$$M(S^4)=(\Lambda (x,y),d)\to (\Lambda (x,y,z),D)\simeq M(S^7)$$

Full text

1. Introduction

Let $X$ (and also $Y$) be a connected and simply connected CW complex with $\dim\pi_{*}(X)_{\mathbb{Q}}<\infty$ ($G_{\mathbb{Q}}=G\otimes\mathbb{Q}$) and $B{aut}_{1}X$ 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 $autX$ 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 $X\to E_{\infty}^{X}\to B{aut}_{1}X$ by a map $h:B\to B{aut}_{1}X$ and equivalence classes of $\xi$ are classified by their homotopy classes [5], [25], [2] (So the map $h$ is often said as the classifying map for the fibration $\xi$). The Sullivan minimal model $M(X)$ ([26]) determines the rational homotopy type of $X$, the homotopy type of the rationalization $X_{0}$ [16] of $X$. The differential graded Lie algebra (DGL) $DerM(X)$, the negative derivations of $M(X)$ (see §2), gives rise to a Quillen model for $Baut_{1}X$ due to Sullivan [26] (cf.[27], [10]), i.e., the spatial realization $||DerM(X)||$ is $(Baut_{1}X)_{0}$. Therefore we obtain a map $(Baut_{1}X)_{0}\to(Baut_{1}Y)_{0}$ if there is a DGL-map $DerM(X)\to DerM(Y)$. However it does not exist in general.

Let $f:X\to Y$ be a map whose homotopy fibration $\xi_{f}:F_{f}\to X\to Y$ is given by the relative model (Koszul-Sullivan extension)

[TABLE]

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 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 $X$.

Definition 1.1**.**

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

[TABLE]

if its Quillen model is given by the DGL-map

[TABLE]

given by $b_{f}(\sigma)={\rm proj}_{V}\circ\sigma$ with $||b_{f}||=a_{f}$. Here ${\rm proj}_{V}:\Lambda V\otimes\Lambda W\to\Lambda V$ is the algebra map with ${\rm proj}_{V}(w)=0$ for $w\in W$ and ${\rm proj}_{V}\mid_{\Lambda V}=id_{\Lambda V}$.

Two fibration $\xi_{f_{1}}$ and $\xi_{f_{2}}$ are fibre homotopy equivalent if there is a diagram:

[TABLE]

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

[TABLE]

where the left square is DGA-commutative and the right square is DGA-homotopy commutative.

Lemma 1.2**.**

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 $\phi:Der(\Lambda V\otimes\Lambda W,D_{1})\cong Der(\Lambda V\otimes\Lambda W,D_{2})$ 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 $\phi:(Baut_{1}X_{1})_{0}\overset{\sim}{\to}(Baut_{1}X_{2})_{0}$ 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.

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

Definition 1.3**.**

A fibration $\xi_{f}:F_{f}\to X\overset{f}{\to}Y$ or a map $f:X\to Y$ with homotopy fiber $F_{f}$ is said to be $\pi_{\mathbb{Q}}$-separable if $\min\pi_{*}(F_{f})_{\mathbb{Q}}\geq\max\pi_{*}(Y)_{\mathbb{Q}}.$

If a map $f:X\to Y$ 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 $\min W=\min\{i>0\mid W^{i}\neq 0\}\geq\max V=\max\{i>0\mid V^{i}\neq 0\}$ in the relative minimal model $M(Y)=(\Lambda V,d)\to(\Lambda V\otimes\Lambda W,D)$ of $\xi_{f}$.

Remark 1.4**.**

Recall a question related to Gottlieb [11, §5]: Which map $f:X\to Y$ can be extended to a map between fibrations over a fixed base space $B$, that is, for any fibration $\xi:X\to E\to B$, does there exist a fibration $\eta:Y\to E^{\prime}\to B$ and a map $f^{\prime}:E\to E^{\prime}$ in the diagram:

[TABLE]

where $f^{\prime}\circ i\simeq i^{\prime}\circ f$ and $p=p^{\prime}\circ f^{\prime}$ ? [29, Example 3.8]. If a map $f:X\to Y$ is $\pi_{\mathbb{Q}}$-separable, then due to Sullivan minimal model theory, it is obvious that there are a fibration $\eta$ after rationalization and a map $f^{\prime}:E_{0}\to E^{\prime}$ in the diagram:

[TABLE]

where $f^{\prime}\circ i_{0}\simeq i^{\prime}\circ f_{0}$ and $p_{0}=p^{\prime}\circ f^{\prime}$. In particular, let $f:X\to X(n)$ be the rationalized Postnikov $n$-stage map of $X$, where $\pi_{>n}(X(n))=0$. Then there exists the rationally fibre-trivial fibration $\eta$ such that

[TABLE]

homotopically commutes when the rationalized classifying map $B\to(Baut_{1}X)_{0}\overset{a_{f}}{\to}(Baut_{1}X(n))_{0}$ is homotopic to the constant map for a sufficiently small $n$.

Proposition 1.5**.**

*A $\mathbb{Q}$-w.t. map $f:X\to Y$ strictly induces $a_{f}:(Baut_{1}X)_{0}\to(Baut_{1}Y)_{0}$ if and only if $f$ is $\pi_{\mathbb{Q}}$-separable. *

Theorem 1.6**.**

For a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$, let

[TABLE]

be a commutative diagram. Then there exists a map between total spaces $k:E\to E^{\prime}$ in the diagram:

[TABLE]

where $k\circ i\simeq i^{\prime}\circ f_{0}$ and $g\circ p=p^{\prime}\circ k$. Here $p:E\to B_{0}$ and $p^{\prime}:E^{\prime}\to B^{\prime}_{0}$ are induced by the rationalized classifying maps $h$ and $h^{\prime}$, respectively.

Let $f:X\to Y$ be a map with a section $s$, i.e., there is a map $s:Y\to X$ with $f\circ s\simeq id_{Y}$. Then there is a map $\psi_{f}:aut_{1}X\to aut_{1}Y$ with $\psi_{f}(g):=f\circ g\circ s$ for $g\in aut_{1}X$. In general, this does not preserve the monoid structures.

Theorem 1.7**.**

If a $\pi_{\mathbb{Q}}$-separable map $f$ admits a section, $\Omega a_{f}\simeq(\psi_{f})_{0}$.

In §2, we give the proofs under some preparations of models of [7] and [27]. In this paper, we consider only $\mathbb{Q}$-w.t. maps. For example, we do not consider the inclusion map $i_{X}:X\to X\times Y$, which is not $\mathbb{Q}$-w.t. However $i_{X}$ induces the monoid map $\psi:aut_{1}X\to aut_{1}(X\times Y)$ by $\psi(g)=g\times 1_{Y}$ and therefore there exists the induced map $B\psi:Baut_{1}X\to Baut_{1}(X\times Y)$ without rationalization. The DGL model is given by the natural inclusion $DerM(X)\to Der(M(X)\otimes M(Y))$, which is a DGL-map.

Let ${\rm Sep}_{\mathbb{Q}}$ be the category that the objects are simply connected CW-complexes of finite dimensional rational homotopy groups and morphisms are $\mathbb{Q}$-separable maps. When $f:X\to Y$ and $g:Y\to Z$ are $\pi_{\mathbb{Q}}$-separable maps, $g\circ f:X\to Z$ is also a $\pi_{\mathbb{Q}}$-separable map. Then

[TABLE]

by our construction. In particular $f={\rm id}_{X}:X\overset{=}{\to}X$ is $\pi_{\mathbb{Q}}$-separable and then $a_{f}$ is of course the identity map of $(Baut_{1}X)_{0}$. Namely, $(Baut_{1})_{0}$ is a functor from ${\rm Sep}_{\mathbb{Q}}$ to $ho(\mathbb{Q}$-$CW_{1})$ by $(Baut_{1})_{0}(f):=a_{f}$. Here $ho(\mathbb{Q}$-$CW_{1})$ is the homotopy category of rational simply connected CW-complexes of finite dimensional rational homotopy groups. This functor is not essentially surjective on objects, i.e., there are rational spaces that cannot be realized as $(Baut_{1}X)_{0}$ for any $X$ due to Lupton-Smith [19]([24]). Also we can easily find in Example 2.5 there exists a map $(Baut_{1}X)_{0}\to(Baut_{1}Y)_{0}$ that cannot be realized as $a_{f}$ for any $\pi_{\mathbb{Q}}$-separable map $f$.

In §3, we give some such conditions for

Question 1.8**.**

When does (is) the strictly induced map $a_{f}:(Baut_{1}X)_{0}\to(Baut_{1}Y)_{0}$ admit a section (fibre-trivial as a fibration) ?

Some results for this question are obtained by Proposition 3.1 induced from [27, VI .1.(3) Proposition] that the DGL-model of the homotopy fibration $\chi_{f}:\ F_{a_{f}}\overset{}{\to}(Baut_{1}X)_{0}\overset{a_{f}}{\to}(Baut_{1}Y)_{0}$ is given by

[TABLE]

Let $aut_{1}f$ be the identity component of the space of all fibre-homotopy self-equivalences of $f$, i.e., $\{g:X\to X\mid f\circ g=f\ \}$ and $Baut_{1}f$ be the classifying space of this topological monoid. It is known that $Baut_{1}f\simeq\widetilde{map}(Y,Baut_{1}(F_{f});h)$, where $h:Y\to Baut_{1}(F_{f})$ is the classifying map of the fibration $F_{f}\to X\overset{f}{\to}Y$ and $\widetilde{map}$ denotes the universal cover of the function space [3]. Notice that

[TABLE]

where $Der_{\Lambda V}(\Lambda V\otimes\Lambda W)$ is the sub DGL of $Der(\Lambda V\otimes\Lambda W)$ sending the elements of $\Lambda V$ to zero and it is a Quillen model of $Baut_{1}f$ when $Y$ and $F_{f}$ are finite [4, Theorem 1] ([8]). Thus we have

Proposition 1.9**.**

When $Y$ and the homotopy fiber $F_{f}$ are finite for $f:X\to Y$, the homotopy fiber $F_{a_{f}}$ of $a_{f}$ has the rational homotopy type of $Baut_{1}f$.

It is suitable since $F^{-1}(id_{Y})=aut_{1}f$ if there exists a map $F:aut_{1}X\to aut_{1}Y$ such that $f\circ g=F(g)\circ f$ for $g\in aut_{1}X$.

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 $dM(X)^{even}=0$ and $dM(X)^{odd}\subset M(X)^{even}$. Furthermore a pure space is said to be an $F_{0}$-space (or positively elliptic) if $\dim\pi_{\rm even}(X)\otimes\mathbb{Q}=\dim\pi_{\rm odd}(X)\otimes\mathbb{Q}$ and $H^{\rm odd}(X;\mathbb{Q})=0$. Then it is equivalent to $H^{*}(X;\mathbb{Q})\cong\mathbb{Q}[x_{1},\cdots,x_{n}]/(f_{1},\cdots,f_{n}),$ in which $|x_{i}|$, the degree of $x_{i}$, are even and $f_{1},\cdots,f_{n}$ forms a regular sequence in the $\mathbb{Q}$-polynomial algebra $\mathbb{Q}[x_{1},\cdots,x_{n}]$, where $M(X)=(\mathbb{Q}[x_{1},\cdots,x_{n}]\otimes\Lambda(y_{1},\cdots,y_{n}),d)$ with $dx_{i}=0$ and $dy_{i}=f_{i}$. In 1976, S. Halperin [14] conjectured that the Serre spectral sequences of all fibrations $X\to E\to B$ 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 ${\rm rank}\ G={\rm rank}\ H$, the homogeneous space $G/H$ satisfies the Halperin conjecture [23]. Also the Halperin conjecture is true when $n\leq 3$ [18]. In this paper we note some relations with the Halperin conjecture [7, §39] due to W. Meier [20] as

Theorem 1.10**.**

Let $Y$ be an $F_{0}$-space. Then the fibration $\chi_{f}$ is fibre-trivial for any $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$ 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 $h:B\to(Baut_{1}Y)_{0}$:

[TABLE]

for a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$. Of course, it is sufficient to define as $\tilde{h}=s\circ h$ if $a_{f}$ admits a section $s$. Thus it is a general approach for Question 1.8, which is the case of $B=Baut_{1}Y$. Specifically, for a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$, let

[TABLE]

be a commutative diagram. Then, from Proposition 1.9, we define an obstruction class by derivations in Theorem 4.1 so that

Theorem 1.11**.**

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map with $Y$ and $F_{f}$ finite. There is a lift $h$ such that

[TABLE]

is commutative if and only if ${\mathcal{O}}_{\alpha}(h_{X},h_{Y})=0$ in $\pi_{N-1}(Baut_{1}f)_{\mathbb{Q}}$.

From Theorem 1.6 and Theorem 1.11, we obtain

Corollary 1.12**.**

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map with $Y$ and $F_{f}$ finite. Suppose that there is a fibration $Y\to E\to B$. If $H^{n+1}(B,\pi_{n}(Baut_{1}f)_{\mathbb{Q}})=Hom(H_{n+1}(B),\pi_{n}(Baut_{1}f)_{\mathbb{Q}})=0$ for all $n$, there exist a fibration $X_{0}\to\tilde{E}\to B_{0}$ and a map between the fibrations:

[TABLE]

Remark 1.13**.**

Let $g:Y\to Z$ and $f:X\to Y$ be $\pi_{\mathbb{Q}}$-separable maps given by the models $M(Z)=\Lambda U\to\Lambda(U\oplus V)=M(Y)$ and $M(Y)=\Lambda(U\oplus V)\to\Lambda(U\oplus V\oplus W)=M(X)$, respectively. Then $\pi_{*}(Baut_{1}g\circ f)_{\mathbb{Q}}=H_{*-1}(Der_{\Lambda U}(\Lambda(U\oplus V\oplus W))=H_{*-1}(Der_{\Lambda U}(\Lambda(U\oplus V))\oplus H_{*-1}(Der_{\Lambda(U\oplus V)}(\Lambda(U\oplus V\oplus W))=\pi_{*}(Baut_{1}g)_{\mathbb{Q}}\oplus\pi_{*}(Baut_{1}f)_{\mathbb{Q}}$.

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 D. H. Gottlieb [13]:

Problem 1.14**.**

When is a fibration $F_{f}\to X\overset{f}{\to}Y$ fibre homotopy equivalent to a $G$-fibration ? i.e., when is there a fibartion $f^{\prime}:X^{\prime}\to Y$ such that $f^{\prime}$ is fibre homotopy equivalent to $f$ and there is a $G$-action on $X^{\prime}$ such that $f^{\prime}$ is equivariant ?

Suppose that $G$ is a compact connected Lie group. Since $H^{*}(BG;\mathbb{Q})$ is evenly graded, the obstruction classes of Theorem 1.11 are contained in $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}$ when $B=BG$. If $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}=0$, they vanish and there exists a lift $h:BG\to(Baut_{1}X)_{0}$. Then, from Theorem 1.6 in the case that $B=B^{\prime}=BG$ and $g=(id_{BG})_{0}$, we obtain by using Theorem 5.1 of D. H. Gottlieb

Theorem 1.15**.**

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map with $Y$ and $F_{f}$ finite. Suppose that a compact Lie group $G$ acts on $Y$. If $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}=0$, the action on $Y$ is rationally lifted to $X$, i.e., $f$ is rationally fibre homotopy equivalent to a $G$-equivariant map $f^{\prime}:X^{\prime}\to Y$ for a $G$-space $X^{\prime}$.

Due to Theorem 1.10 and the result of H. Shiga - M. Tezuka [23], we have

Corollary 1.16**.**

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map such that $Y$ is a homogeneous space $G/H$ with ${\rm rank}\ G={\rm rank}\ H$. 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 $X^{\prime}$ in the rational homotopy type of $X$ with all its isotropy subgroups finite (almost free action) [1], [9], [15]. It is very difficult to calculate $r_{0}(\ \ )$ in general. From the definition, we have the inequality $r_{0}(X\times Y)\geq r_{0}(X)+r_{0}(Y)$. Notice that it may sometimes be a strict inequality since there is an example that $r_{0}(X\times S^{12})>0$ even though $r_{0}(X)=r_{0}(S^{12})=0$ [17, Example 3.3]. For a map $f:X\to Y$, we see $r_{0}(Y)\leq r_{0}(X)$ when $X=F\times Y$ for any space $F$ and $f$ is the projection $F\times Y\to Y$. In general, when does a map $f:X\to Y$ induce $r_{0}(Y)\leq r_{0}(X)$ ?

Corollary 1.17**.**

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map with $Y$ and $F_{f}$ finite. If $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}=0$, we have $r_{0}(Y)\leq r_{0}(X)$.

2. Sullivan models, derivations and Quillen models

Let $M(X)=(\Lambda{V},d)$ be the Sullivan minimal model of simply connected CW complex $X$ of finite type [26]. It is a free $\mathbb{Q}$-commutative differential graded algebra (DGA) with a $\mathbb{Q}$-graded vector space $V=\bigoplus_{i\geq 1}V^{i}$ 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,

[TABLE]

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

Let $Der_{i}M(X)$ 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 $\partial:Der_{i}M(X)\to Der_{i-1}M(X)$ is defined by

[TABLE]

for $\sigma\in Der_{i}M(X)$. We denote $\oplus_{i>0}Der_{i}M(X)$ by $DerM(X)$ in which $Der_{1}M(X)$ is $\partial$-cycles. Then ${Der}M(X)$ is a (non-free) DGL by the Lie bracket

[TABLE]

Note that $H_{*}(DerM)=H_{*}(DerN)$ when free DGAs $M$ and $N$ are quasi-isomorphic [22]. Furthermore, recall the definition of D.Tanré [27, 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

[TABLE]

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**.**

[26, §11]**,[27],[10] For a Sullivan model $M(X)$ of $X$, $DerM(X)$ is a DGL-model of $Baut_{1}X$. In particular, there is an isomorphism of graded Lie algebras $H_{*}(DerM(X))\cong\pi_{*}(\Omega Baut_{1}X)_{\mathbb{Q}}$ where the right-hand has the Samelson bracket. Furthermore $C^{*}(DerM(X))$ is a DGA-model of $Baut_{1}X$.

Let $L(X)=(\mathbb{L}U,\partial)$ be the Quillen model of $X$ [27, III.3.], [7, §24]. It is a free $\mathbb{Q}$-commutative differential graded Lie algebra (DGL) with a $\mathbb{Q}$-graded vector space $U=\bigoplus_{i\geq 1}U_{i}$ where $\dim U_{i}<\infty$ and $\partial(U_{i})\subset(\mathbb{L}U)_{i-1}$, which is the space of elements of $\mathbb{L}U$ with degree $i-1$. Note that $[x,y]=-(-1)^{|x||y|}[y,x]$ and Jacobi identity:

[TABLE]

for $x,y,z\in\mathbb{L}U$ and Leibniz rule:

[TABLE]

Note that $L(X)$ determines the rational homotopy type of $X$, namely $||L(X)||\simeq X_{0}$. In particular, there are isomorphisms

[TABLE]

where $\partial_{1}:U\to U$ is the linear part of $\partial$. Here the second is an isomorphism as graded Lie algebras. Refer to [7] for detail.

Proof of Lemma 1.2. 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

[TABLE]

is DGL-commutative since $\psi|_{\Lambda V}=id_{\Lambda V}$. In particular, we can check $\partial_{2}\circ\phi=\phi\circ\partial_{1}$ by

[TABLE]

[TABLE]

for $\sigma\in Der_{i}(\Lambda V\otimes\Lambda W,D_{1})$. Similarly we have $\phi([\sigma,\tau])=[\phi(\sigma),\phi(\tau)]$. ∎

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 [math]. Note that $|(v,f)|=|v|-|f|$.

Proof of Proposition 1.5. Let $M(Y)=(\Lambda V,d)\to(\Lambda V\otimes\Lambda W,D)$ be the model of $f$.

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

[TABLE]

from degree arguments. Then there is a DGL-map $b_{f}:Der(\Lambda V\otimes\Lambda W,D)\to Der(\Lambda V,d)$ by $b_{f}(\sigma_{1})=\sigma_{1}$ and $b_{f}(\sigma_{2})=0$ for $\sigma=\sigma_{1}+\sigma_{2}$ with $\sigma_{1}\in Der(\Lambda V)$ and $\sigma_{2}\in Der(\Lambda W,\Lambda V\otimes\Lambda W)$. In particular, it preserves the differential since $b_{f}(\tau(\sigma_{1}))=0$ when $\partial_{X}(\sigma_{1})=\partial_{Y}(\sigma_{1})+\tau(\sigma_{1})$ for $\tau(\sigma_{1})\in Der(\Lambda W,\Lambda V\otimes\Lambda W)$.

(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

[TABLE]

from the definition of $b_{f}$. ∎

Example 2.2**.**

Let $f:S^{2n}\to K(\mathbb{Z},2n)$ be the natural inclusion. Then the homotopy fibre is $S^{4n-1}$ and therefore $f$ is $\pi_{\mathbb{Q}}$-separable. Thus $b_{f}:Der(\Lambda(x,y),D)\to Der(\Lambda(x),0)$ with $|x|=2n$, $|y|=4n-1$, $Dx=0$ and $Dy=x^{2}$ is given by $b_{f}((x,1))=(x,1)$ and $b_{f}((y,1))=b_{f}((y,x))=0$. Refer Theorem 3.9.

Example 2.3**.**

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

[TABLE]

*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

[TABLE]

[TABLE]

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

[TABLE]

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. *

Proof of Theorem 1.7. The map $\pi_{n}(\psi_{f}):\pi_{n}(aut_{1}X)\to\pi_{n}(aut_{1}Y)$ is given by $\pi_{n}(\psi_{f})([\sigma])=[\tau]:=[f\circ\sigma\circ(s\times 1_{S^{n}})]$ in the following homotopy commutative diagram:

[TABLE]

from adjointness. That is the pointed homotopy classes of maps $S^{n}\to aut_{1}X=map(X,X;id_{X})$ are in bijection with the homotopy classes of those maps $X\times S^{n}\to X$ that composed with the inclusion $i_{X}:X\to X\times S^{n}$ yield the identity [22, p.43-44]. Let $M(Y)=(\Lambda V,d)\to(\Lambda V\otimes\Lambda W,D)$ be the model of $f$. There is a chain map

[TABLE]

given by $c_{f}(\sigma)={\rm proj}_{V}\circ\sigma$. It is well-defined, i.e., $\partial_{Y}\circ c_{f}=c_{f}\circ\partial_{X}$, from $DW\subset\Lambda V\otimes\Lambda^{+}W$ [28] since it admits a section. Notice that $\pi_{n}(\psi_{f})_{\mathbb{Q}}([\sigma_{0}])=H_{n}(c_{f})([\sigma])=[\tau]$ by identifying the maps $\sigma$ and $\tau$ to their induced derivations of $\Lambda V\otimes\Lambda W$ and $\Lambda V$, respectively [26, p.313](cf.[22, Proposition 11]). (However $c_{f}$ is not a DGL-map in general.) If $f$ is a $\pi_{\mathbb{Q}}$-separable map, $c_{f}=b_{f}$ as DGL-maps and then $\pi_{n}(\Omega a_{f})$ is identified with

[TABLE]

for any $n$. Thus we obtain $\Omega a_{f}\simeq(\psi_{f})_{0}$. ∎

The following is obvious from the definition of $b_{f}$ and useful:

Claim 2.4**.**

For any $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$, we have $b_{f}(C)=0$ and $b_{f}\mid_{Der(\Lambda V)}=id_{Der(\Lambda V)}$ for $Der(\Lambda V\otimes\Lambda W)=C\oplus Der(\Lambda V)$.

Proof of Theorem 1.6. Let $X\to E_{\infty}^{X}\overset{p_{\infty}^{X}}{\to}Baut_{1}X$ and $Y\to E_{\infty}^{Y}\overset{p_{\infty}^{Y}}{\to}Baut_{1}Y$ be the universal fibrations of $X$ and $Y$, respectively. Let $C^{*}(Der(\Lambda V))\otimes\Lambda V,D_{Y}$ be the DGA-model of $E_{\infty}^{Y}$ and $C^{*}(Der(\Lambda V\otimes\Lambda W))\otimes\Lambda V\otimes\Lambda W,D_{X}$ be the DGA-model of $E_{\infty}^{X}$. For a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$, there exists a DGA-inclusion map $\psi$ such that

[TABLE]

is commutative from the universality. Indeed, $C^{*}(Der(\Lambda V))\otimes\Lambda V,D_{Y}$ is a sub-DGA of $C^{*}(Der(\Lambda V\otimes\Lambda W))\otimes\Lambda V\otimes\Lambda W,D_{X}$ from Claim 2.4 and $\max V\leq\min W$. Thus there is a map $\tilde{a}_{f}:=|\psi|:(E_{\infty}^{X})_{0}\to(E_{\infty}^{Y})_{0}$ such that $(p_{\infty}^{Y})_{0}\circ\tilde{a}_{f}=a_{f}\circ(p_{\infty}^{X})_{0}$. Since $p^{\prime}$ is the pull-back of $(p_{\infty}^{Y})_{0}$ by $h^{\prime}$, there exists a map $k:E\to E^{\prime}$ such that

[TABLE]

is commutative from the universality since $h^{\prime}\circ g\circ p=a_{f}\circ h\circ p=a_{f}\circ(p_{\infty}^{X})_{0}\circ\tilde{h}=(p_{\infty}^{Y})_{0}\circ\tilde{a}_{f}\circ\tilde{h}$. ∎

Example 2.5**.**

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 $f:X\to Y$ be given by $M(f):\Lambda(z)\to\Lambda(x,y)$ with $M(f)(z)=x$. The homotopy fibration of any $\pi_{\mathbb{Q}}$-separable map is given by $\Lambda(z),0\to\Lambda(z,y),0\cong\Lambda(x,y),0$ from the degree reason. Therefore the DGL-map $\psi:Der\Lambda(x,y)\to Der\Lambda(z)$ such that $\psi((y,x))=\psi((x,1))=(z,1)$ is not DGL-homotopic to $b_{f}$ from Claim 2.4.

Let $h:S^{n+1}_{0}\to(Baut_{1}X)_{0}$ and $h^{\prime}:S^{n+1}_{0}\to(Baut_{1}Y)_{0}$ be given by $L(h):\mathbb{L}(u)\to Der(\Lambda(x,y))$ with $|u|=n$, $L(h)(u)=(y,x)$ and $L(h^{\prime}):\mathbb{L}(u)\to Der(\Lambda(z))$ with $L(h^{\prime})(u)=(z,1)$, respectively. Then the commutative diagram

[TABLE]

does not induce a map between total spaces $f^{\prime}:E\to E^{\prime}$ such that

[TABLE]

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

[TABLE]

where $|v|=n+1$ is homotopy commutative since $h$ can not be a DGA-map from $Dh(z)=0$ but $hD^{\prime}(z)=v$.

Remark 2.6**.**

Recall that a map $f:X\to Y$ is said to be a Gottlieb map [29] if its homotopy group map induces the map between their Gottlieb groups [12] $f_{\sharp}:G_{*}(X)\to G_{*}(Y)$. For example, if a map $f$ admits a section, it is a Gottlieb map. Since elements of the rational Gottlieb group $G_{*}(X)_{\mathbb{Q}}$ is described by certain derivations of $M(X)$ [6], we see that if a map $f:X\to Y$ is $\pi_{\mathbb{Q}}$-separable, it is a rational Gottlieb map.

3. When does $a_{f}$ admit a section ?

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map with homotopy fiber $F_{f}$ and $Der(\Lambda W,\Lambda V\otimes\Lambda W)$ the sub-DGL of $Der(\Lambda V\otimes\Lambda W)$ 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 $\chi_{f}:F_{a_{f}}\overset{j}{\to}(Baut_{1}X)_{0}\overset{a_{f}}{\to}(Baut_{1}Y)_{0}$ is given by

[TABLE]

Proof.

Since $b_{f}$ is surjective and $Der(\Lambda W,\Lambda V\otimes\Lambda W)$ is ${\rm Ker}\ b_{f}$, it follows from [27, VI .1.(3) Proposition]. ∎

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

Theorem 3.2**.**

$L(F_{a_{f}})_{n}=\oplus_{i-j=n}Der_{i}(\Lambda W)\otimes H^{j}(\Lambda V)$*. *

Proof.

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

The rational homotopy exact sequence of the strictly induced fibration $\chi_{f}$:

[TABLE]

[TABLE]

is equivalent to the homology exact sequence:

[TABLE]

[TABLE]

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 $Der(\Lambda V)$ and a $\partial_{X}$-cycle $\tau$ of $Der(\Lambda W,\Lambda V\otimes\Lambda W)$.

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

[TABLE]

Here $\delta:\pi_{*}(B)\to\pi_{*-1}(F)$ 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 $\chi_{f}$ be given as the commutative diagram:

[TABLE]

where $M(Baut_{1}Y)\cong(\Lambda U,d)$ with $U^{n+1}=H_{n}(Der(\Lambda V))$ and $M(F_{a_{f}})\cong(\Lambda Z,\overline{D}_{2})$ with $Z^{n+1}=H_{n}(Der(\Lambda W,\Lambda V\otimes\Lambda W))$. From the assumption $\chi_{f}$ is weakly equivalent, i.e., $M(Baut_{1}X)\cong(\Lambda U\otimes\Lambda Z,D_{2})$. Let $D=d_{1}+d_{2}$ in §2. Then

[TABLE]

Notice that $(w,h)\not\in[Der(\Lambda V),Der(\Lambda V)]$ for any $w\in W$ and $h\in\Lambda V\otimes\Lambda W$, where $[\ ,\ ]$ is the Lie bracket. That means

[TABLE]

Here $I(S)$ is the ideal generated by a basis of a vector space $S$. Let $\sigma$ be a $\partial_{X}$-cycle of $Der(\Lambda W,\Lambda V\otimes\Lambda W)$. Then $[s^{-1}\sigma^{*}]\in H_{*}(C^{*}(Der(\Lambda W,\Lambda V\otimes\Lambda W)),d_{1}^{\prime})\cong Z$ for $D^{\prime}=d_{1}^{\prime}+d_{2}^{\prime}$ in §2. Since

[TABLE]

we have $D_{1}([s^{-1}\sigma^{*}])\in I(Z)$ by $(**)$, i.e., $D_{1}(Z)\subset C^{*}(Der\Lambda V)\otimes\Lambda^{+}Z$. By $\rho_{2}$, $D_{2}(Z)\subset\Lambda U\otimes\Lambda^{+}Z$. Then we have done from [28].

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

Theorem 3.5**.**

If a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$ 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. ∎

Refer [21, page 292] for related topics. Conversely, when $Y$ is an odd-sphere,

Theorem 3.6**.**

If a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y=S^{2n+1}$ 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, $\partial_{X}(v,1)=\pm(w,\partial Dw/\partial v)+\cdots\neq 0$ in $Der(\Lambda W)$. From Claim 3.3 $\delta_{f}$ is injective since $\delta_{f}([(v,1)])=[\pm(w,\partial Dw/\partial v)+\cdots]\neq 0$. Then the former holds from Proposition 3.4. Furthermore, from the homotopy exact sequence, we have ${a_{f}}_{\sharp}=0$. Thus the latter holds. ∎

Example 3.7**.**

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

[TABLE]

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 $\delta_{f}:H_{3}(Der(\Lambda v^{\prime}))\overset{}{\to}H_{2}(Der(\Lambda(w_{1},w_{2}),\Lambda(v_{1},w_{1},w_{2})))$ is non-trivial from $\delta_{f}([(v_{1},1)])=[(w_{2},w_{1})]\neq 0$.

(2) Let $S^{5}\times S^{7}\to X^{\prime}\to Y^{\prime}$ be a non-(fibre-)trivial $\pi_{\mathbb{Q}}$-separable fibration given by the model

[TABLE]

with $|v_{1}|=|v_{2}|=3$, $|v_{3}|=5$, $|w_{1}|=7$, $|w_{2}|=9$, $d_{Y^{\prime}}(v_{1})=d_{Y^{\prime}}(v_{2})=0$, $d_{Y^{\prime}}(v_{3})=v_{1}v_{2}$, $D^{\prime}w_{1}=0$ and $D^{\prime}w_{2}=v_{1}w_{1}$. Then $a_{f}$ admits a section from Proposition 3.4 since $\delta_{f}([(v_{3},1)])=0$ for $H_{*}(Der(\Lambda(v_{1},v_{2},v_{3})))=\mathbb{Q}\{[(v_{3},1)]\}$. However $\chi_{f}$ is not trivial from $[(v_{3},1),(w_{2},v_{2}v_{3})]=(w_{2},v_{2})$. Indeed, then

[TABLE]

for $(C^{*}(Der(\Lambda(v_{1},v_{2},v_{3},w_{1},w_{2})),{\mathcal{D}})$ with ${\mathcal{D}}=d_{1}+d_{2}$. Refer the proof of Proposition 3.4.

From Claim 2.4, we obtain

Lemma 3.8**.**

Let $X$ be a pure space with $M(X)=(\mathbb{Q}[x_{1},\cdots,x_{m}]\otimes\Lambda(y_{1},\cdots,y_{n}),d)$ with $\max\{|x_{1}|,\cdots,|x_{m}|\}<\min\{|y_{1}|,\cdots,|y_{n}|\}$. If $f:X\to\Pi_{i=1}^{m}K(\mathbb{Q},|x_{i}|)$ is the rational principal fibration given by $M(f):(\mathbb{Q}[x_{1},\cdots,x_{m}],0)\to M(X)$ with $M(f)(x_{i})=x_{i}$ for all $i$. Then $a_{f}\sim*$ if and only if $S=0$ for $H_{*}(DerM(X))=S\oplus T$ with $S\subset Der(\mathbb{Q}[x_{1},\cdots,x_{m}])$ and $T\cap Der(\mathbb{Q}[x_{1},\cdots,x_{m}])=0$.

Theorem 3.9**.**

Let $X$ be an $F_{0}$-space with $\max\{|x_{1}|,\cdots,|x_{n}|\}<\min\{|y_{1}|,\cdots,|y_{n}|\}$. Then $a_{f}\sim*$ for the map $f:X\to Y:=\Pi_{i=1}^{n}K(\mathbb{Q},|x_{i}|)$ of above if and only if $X$ satisfies the Halperin’s conjecture.

Proof.

It follows from Lemma 3.8 since the Halperin’s conjecture is equivalent to that $DerH^{*}(X;\mathbb{Q})=Der(\mathbb{Q}[x_{1},\cdots,x_{n}]/(f_{1},\cdots,f_{n}))=0$ [20]. ∎

Proof of Theorem 1.10. Let $M(Y)=(\Lambda V,d)=(\mathbb{Q}[x_{1},\cdots,x_{n}]\otimes\Lambda(y_{1},\cdots,y_{n}),d)$ with $dx_{i}=0$ and $dy_{i}=f_{i}$ for $i=1,..,n$.

(if) Let $M(Y)=(\Lambda V,d)\to(\Lambda V\otimes\Lambda W,D)$ be the model of $f$. From the regularity of $f_{1},\cdots,f_{n}$, $ImD\subset\mathbb{Q}[x_{1},\cdots,x_{n}]\otimes\Lambda W$. Thus

[TABLE]

for any $h_{i}\in\mathbb{Q}[x_{1},\cdots,x_{n}]$ with suitable degree and some $\theta\in Der(\Lambda W,\Lambda V\otimes\Lambda W)$. Then we have $\delta_{f}=0$ from Claim 3.3 since $H_{even}(DerM(Y))=0$ [20] from the assumption. Then $a_{f}$ admits a section from Proposition 3.4. Furthermore the Lie bracket decomposition of an element of $Der(\Lambda W,\Lambda V\otimes\Lambda W)$ does not have an element of $Der(\Lambda V)$ as a factor from Theorem 3.2 since $DerH^{*}(Y;\mathbb{Q})=0$ [20] again. Thus we have $D_{2}=d\otimes 1\pm 1\otimes\overline{D}_{2}$ for the Sullivan minimal model $(\Lambda U,d)\to(\Lambda U\otimes\Lambda Z,D_{2})\to(\Lambda Z,\overline{D}_{2})$ of $\chi_{f}$ (in the proof of Proposition 3.4).

(only if) Suppose there is a non-zero element $[\sum_{i}(x_{i},h_{i})+\sum_{j}(y_{j},g_{j})]\in H_{2m}(DerM(Y))$ for $h_{i}\in\mathbb{Q}[x_{1},\cdots,x_{n}]$, $g_{j}\in\Lambda V$ and some $m$. Let $S^{a}\times S^{b}\to X\overset{f}{\to}Y$ be a rational fibration of the model:

[TABLE]

where $|w_{1}|=a$ and $|w_{2}|=b$ are odd with $b-a=|x_{k}|-1$ for some $k$, $h_{k}$ is not $d_{Y}$-exact, $Dw_{1}=0$ and $Dw_{2}=x_{k}w_{1}$. Then

[TABLE]

for $\delta_{f}:H_{2m}(Der\Lambda V)\to H_{2m-1}(Der(\Lambda(w_{1},w_{2}),\Lambda V\otimes\Lambda(w_{1},w_{2}))$. In particular, $\chi_{f}$ is not fibre-trivial. ∎

Example 3.10**.**

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 ${\rm rank}\ SU(6)=5>4={\rm rank}\ (SU(3)\times SU(3))$. Let $\xi:S^{11}\times S^{23}\to X\overset{f}{\to}Y$ be a fibration whose relative model is given as

[TABLE]

where $|x_{1}|=4$, $|x_{2}|=6$, $|y_{1}|=7$, $|y_{2}|=9$, $|y_{3}|=11$, $|w_{1}|=11$, $|w_{2}|=23$, $d_{Y}y_{1}=x_{1}^{2}$, $d_{Y}y_{2}=x_{1}x_{2}$, $d_{Y}y_{3}=x_{2}^{2}$, $Dw_{1}=0$ and $Dw_{2}=(x_{1}y_{2}-x_{2}y_{1})w_{1}$. Then $\partial_{X}((y_{1},1))=(w_{2},x_{2}w_{1})$, i.e., $\delta_{f}[(y_{1},1)]=[(w_{2},x_{2}w_{1})]\neq 0$. In particular $\chi_{f}$ is not trivial. Refer [21, Example 1.14(2)] for the Sullivan minimal model of $Baut_{1}Y$.

In this section finally we mention about a heredity property for a pull-back:

Theorem 3.11**.**

Let $f^{\prime}:X^{\prime}\to Y^{\prime}$ be the pull-back of a map $f:X\to Y$ by a map $g:Y^{\prime}\to Y$ with a rational section. Suppose that both $f$ and $f^{\prime}$ are $\pi_{\mathbb{Q}}$-separable. If $a_{f}$ admits a section, then $a_{f^{\prime}}$ does so.

Proof.

From [7, Proposition 15.8], the Sullivan model of the pull-back diagram:

[TABLE]

is given as

[TABLE]

where $D^{\prime}W=DW\subset\Lambda V\otimes\Lambda W$. Then, from Claim 3.3, the following is commutative:

[TABLE]

where $c_{g}$ is same as $b_{g}$ as a chain map (see the proof of Theorem 1.7). Here $Der(g^{*}):Der(\Lambda W,\Lambda W\otimes H^{*}(Y;\mathbb{Q}))\to Der(\Lambda W,\Lambda W\otimes H^{*}(Y^{\prime};\mathbb{Q}))$ is given by $Der(g^{*})((w,w^{\prime}\otimes y))=(w,w^{\prime}\otimes g^{*}(y))$. Then $\delta_{f^{\prime}}=0$ if $\delta_{f}=0$. Thus it follows from Proposition 3.4. ∎

Remark that $\chi_{f^{\prime}}$ is not a pull-back of $\chi_{f}$. Moreover the converse of Theorem 3.11 is false in general. Indeed, we see in Example 3.7 that the fibration of (2) is the pull-back of (1) by a map $g:Y^{\prime}\to Y=S^{3}$ with $M(g)(v_{1})=v_{1}$.

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 $\partial_{\alpha}(u)\in L(B)$ [27, Proposition III.3.(6)].

Theorem 4.1**.**

For a $\pi_{\mathbb{Q}}$-separable map $f:X\to Y$, let

[TABLE]

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

[TABLE]

is commutative if and only if

[TABLE]

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

[TABLE]

with

$\bullet$* $\partial_{X}\mid_{Der(\Lambda V,\Lambda V\otimes\Lambda W)}=\partial_{Y}+\tau$ and $\partial_{X}\mid_{Der(\Lambda W,\Lambda V\otimes\Lambda W)}=\tau$ for some $\tau:Der_{*}(\Lambda V\otimes\Lambda W)\to Der_{*-1}(\Lambda W,\Lambda V\otimes\Lambda W)$ and*

$\bullet$* $h_{X}=h_{X}^{\prime}+h_{X}^{\prime\prime}$ where $h_{X}^{\prime}(b)\in Der(\Lambda V)$ and $h_{X}^{\prime\prime}(b)\in Der(\Lambda W,\Lambda V\otimes\Lambda W)$ for $b\in L(B)$.*

Proof.

Since $b_{f}\circ h_{X}=h_{Y}\circ i$ and $h_{Y}$ is a DGL-map,

[TABLE]

in $Der(\Lambda V)$. Notice that the obstruction element $\partial_{X}(h_{Y}(u))-h_{X}(\partial_{\alpha}(u))$ is a $\partial_{X}$-cycle in $Der(\Lambda V\otimes\Lambda W)$. Therefore $\tau(h_{Y}(u))-h^{\prime\prime}_{X}(\partial_{\alpha}(u))$ is a $\partial_{X}$-cycle in $Der(\Lambda W,\Lambda V\otimes\Lambda W)$ from (1).

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

[TABLE]

Let

[TABLE]

Then $h$ is a DGL-map since

[TABLE]

[TABLE]

[TABLE]

from (1) and (2). Furthermore

[TABLE]

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,

[TABLE]

in $Der(\Lambda V\otimes\Lambda W)$ and

[TABLE]

from (1) and (3). Furthermore

[TABLE]

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

[TABLE]

for some element $x\in Der(\Lambda W,\Lambda V\otimes\Lambda W)$ from $b_{f}\circ h=h_{Y}$ and then since

[TABLE]

Thus we obtain that ${\mathcal{O}}_{\alpha}(h_{X},h_{Y})=[\tau(h_{Y}(u))-h^{\prime\prime}_{X}(\partial_{\alpha}(u))]=0$ from (4) and (5). ∎

From Theorem 3.2, we have

Corollary 4.2**.**

If $\pi_{\geq N-1}(Baut_{1}F_{f})_{\mathbb{Q}}=0$ 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 $S^{3}\times S^{5}\to X\overset{f}{\to}Y=S^{3}$ be the fibration given by the model

[TABLE]

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 $\partial u_{2}=[u_{1},u_{1}]$ [27]. Let

[TABLE]

be a commutative diagram given by the DGL-model

[TABLE]

by $h_{X}(u_{1})=h_{Y}(u_{1})=0$ and $h_{Y}(u_{2})=(v,1)$. Then ${\mathcal{O}}_{\alpha}(h_{X},h_{Y})\neq 0$ in $H_{2}(Der(\Lambda(w_{1},w_{2}),\Lambda(v,w_{1},w_{2}))$ since

[TABLE]

Thus $h_{Y}:\mathbb{C}P^{2}\to(Baut_{1}Y)_{0}$ cannot lift to $h:\mathbb{C}P^{2}\to(Baut_{1}X)_{0}$. Note that $h_{Y}$ is extended to $\mathbb{C}P^{\infty}\to(Baut_{1}Y)_{0}$. Since $BS^{1}=\mathbb{C}P^{\infty}$, we obtain that any free $S^{1}$-action on $Y$ cannot lift to $X$ (See §5).

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 ${\rm rank}\ G=r$, respectively. If $G$ acts on a space $Y$ by $\mu:G\times Y\to Y$, there is the Borel fibration

[TABLE]

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)

[TABLE]

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**.**

[13, Theorem 1]** Let a topological group $G$ acts on a space $Y$. A fibration $X\overset{f}{\to}Y$ 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 $EG\times_{G}Y$ induced by the inclusion $i:Y\to EG\times_{G}Y$.

Proof of Theorem 1.15. Let $h_{Y}:BG\to(Baut_{1}Y)_{0}$ be the rationalization of the classifying map of the Borel fibration $Y\overset{i}{\to}EG\times_{G}^{\mu}Y\to BG$ of the action $\mu:G\times Y\to Y$. Let $B^{n}$ be the n-skelton of a CW complex $B$. From Theorem 1.11 there is a lift $h_{X}^{\alpha}$ such that

[TABLE]

is commutative for all $n$ and attachings $\alpha$ since ${\mathcal{O}}_{\alpha}(h_{X}^{n},h_{Y}^{\alpha})=0$. Indeed, $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}=0$ and $L(BG)$ is oddly graded since $H^{*}(BG;\mathbb{Q})$ is evenly graded. Thus we have the commutative diagram:

[TABLE]

From Theorem 1.6 there is a commutative digram:

[TABLE]

for some space $E$. Let $g^{\prime}:E^{\prime}\to EG\times_{G}Y$ be the pull-back of $g$ by the rationalization $l_{0}$ and $f^{\prime}:X^{\prime}\to Y$ be the pull-back of $g^{\prime}$ by $i$:

[TABLE]

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

[TABLE]

for $R:=H^{*}(BG;\mathbb{Q})=\mathbb{Q}[t_{1},\cdots,t_{r}]$. 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

[TABLE]

since $M(X^{\prime})\cong\Lambda V\otimes\Lambda W=M(X)$. ∎

If the $r$-torus $T^{r}$ acts on a space $Y$, $|t_{1}|=\cdots=|t_{r}|=2$ in $(*)$.

Proposition 5.2**.**

[15, 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 $r_{0}(Y)\geq r$ 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 $r_{0}(Y)\geq r$, then $T^{r}$ acts freely on a finite complex $Y^{\prime}$ that has the same rational homotopy type as $Y$ and $M(ET^{r}\times_{T^{r}}Y^{\prime})\cong(\mathbb{Q}[t_{1},\dots,t_{r}]\otimes\wedge{V},D)$.

Proof of Corollary 1.17. Let $r_{0}(Y)=r$. Notice that $L(BT^{r})$ is oddly generated since $H^{*}(BT^{r};\mathbb{Q})=\mathbb{Q}[t_{1},\cdots,t_{r}]$. Since $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}=0$, there exists a lift $(BT^{r})_{0}\to(Baut_{1}X)_{0}$ from Theorem 1.11 (Corollary 1.12). Then we have the homotopy commutative diagram:

[TABLE]

from Theorem 1.6. 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 $F_{g}\to\tilde{E}\to E$. Thus there is a free $T^{r}$-action on $X^{\prime}$ with $X^{\prime}_{0}\simeq X_{0}$ and $\tilde{E}\simeq(ET^{r}\times_{T^{r}}X^{\prime})_{0}$ from Proposition 5.2. Thus we have $r_{0}(X)\geq r$. ∎

Example 5.3**.**

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

[TABLE]

with $|v_{1}|=|v_{2}|=2$, $|v_{3}|=|v_{4}|=|v_{5}|=|w|=5$, $d_{Y}(v_{1})=d_{Y}(v_{2})=0$, $d_{Y}(v_{3})=v_{1}^{3}$, $d_{Y}(v_{4})=v_{1}^{2}v_{2}$, $d_{Y}(v_{5})=v_{2}^{3}$ and $D(w)=v_{1}v_{2}^{2}$. Then

[TABLE]

since there is no element of odd-degree $<5$ in $\Lambda V$. Therefore $r_{0}(Y)\leq r_{0}(X)$. Indeed, we can directly check that $r_{0}(Y)=1$ and $r_{0}(X)=2$.

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

[TABLE]

with $|v_{1}|=|v_{2}|=3$, $|w|=5$, and $D(w)=v_{1}v_{2}$. Then

[TABLE]

and $r_{0}(Y)=2>1=r_{0}(X)$.

Finally recall [30]. For a map $f:X\to Y$, we say that the rational toral rank of $f$, denoted as $r_{0}(f)$, is $r$ when it is the largest integer such that there is a map $F$ between an $X_{0}$-fibration and a $Y_{0}$-fibration over $(BT^{r})_{0}$ as

[TABLE]

with $\dim H^{*}(E_{i};\mathbb{Q})<\infty$ for $i=1,2$. From the definition, $r_{0}(f)\leq\min\{r_{0}(X),r_{0}(Y)\}$ for any map $f:X\to Y$. From the proof of Corollary 1.17, we obtain

Corollary 5.4**.**

Let $f:X\to Y$ be a $\pi_{\mathbb{Q}}$-separable map with $Y$ and $F_{f}$ finite. If $\pi_{odd}(Baut_{1}f)_{\mathbb{Q}}=0$, we have $r_{0}(f)=r_{0}(Y)$.