Rational models for automorphisms of fiber bundles

TL;DR

This paper develops a differential graded Lie algebra model to understand the classifying space of automorphisms of fiber bundles, linking algebraic structures with topological automorphism groups.

Contribution

It introduces a novel algebraic model for the classifying space of fiber bundle automorphisms, bridging differential graded Lie algebras with topological automorphism spaces.

Findings

Constructed a dg Lie algebra model for the classifying space

Established a correspondence between algebraic and topological automorphisms

Provides tools for studying automorphisms via algebraic models

Abstract

Given a fiber bundle, we construct a differential graded Lie algebra model for the classifying space of the monoid of homotopy equivalences of the base covered by a fiberwise isomorphism of the total space.

Equations

$$E\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\operatorname{Hom}^{\tau}(\mathcal{C}L,\Pi)\langle 0\rangle\rtimes_{\tau_{*}}\big{(}\operatorname{Der}L\ltimes_{\operatorname{ad}}sL\big{)}\langle 1\rangle.$$

$$\big{(}H^{*}(X;\mathbb{Q})\otimes\pi_{*}(\Omega BU(n))\big{)}\langle 0\rangle\rtimes_{\rho_{*}}\big{(}\operatorname{Der}L\ltimes_{\operatorname{ad}}sL\big{)}\langle 1\rangle$$

$$\operatorname{Hom}^{\tau}(\mathcal{C}L,\Pi)\langle 0\rangle\to\operatorname{Hom}^{\tau}(\mathcal{C}L,\Pi)\langle 0\rangle\rtimes_{\tau_{*}}\big{(}\operatorname{Der}L\ltimes_{\operatorname{ad}}sL\big{)}\langle 1\rangle\to\big{(}\operatorname{Der}L\ltimes_{\operatorname{ad}}sL\big{)}\langle 1\rangle$$

$$Bau{t}_X(p)\to Bau{t}_{\circ }(p)\to Bau{t}_{\circ }(X),$$

$$Fib(X,\mathcal{F})=B\big{(}map(X,B_{\infty}),aut(X),*\big{)},$$

$$A\to Fib(X,\mathcal{F})$$

$${\pi }_{0}Fib(X,\mathcal{F}),$$

$$E\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\pi }_{0}Fib(X,\mathcal{F})\cong [X,B_{\infty }]/{\pi }_{0}aut(X).$$

$$Fib(X,\mathcal{F})\sim \coprod _{[p]}Bau{t}^{\mathcal{F}}(p),$$

$$aut(X)\to map(X,B_{\infty })\to Fib(X,\mathcal{F})\to Baut(X).$$

$$au{t}^{\mathcal{F}}(p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Baut^{\mathcal{F}}(p)\sim B\big{(}map(X,B_{\infty})_{\nu},aut(X)_{[\nu]},*\big{)},$$

$$Baut_{\circ}^{\mathcal{F}}(p)\sim B\big{(}map(X,B_{\infty})_{\nu},aut_{\circ}(X),*\big{)},$$

$$B\big{(}map(X_{\mathbb{Q}},Z_{\mathbb{Q}})_{\nu_{\mathbb{Q}}},aut_{\circ}(X_{\mathbb{Q}}),*\big{)}.$$

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

$$au{t}_{\circ }(r)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$map(X,Z{)}_{\nu }\stackrel{q_{\ast }}{\to }map(X,Z_{\mathbb{Q}}{)}_{q\nu }\stackrel{r^{\ast }}{\leftarrow }map(X_{\mathbb{Q}},Z_{\mathbb{Q}}{)}_{{\nu }_{\mathbb{Q}}}.$$

$$\textstyle{B\big{(}map(X,Z)_{\nu},aut_{\circ}(r),*\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$

$$\mathfrak{g}\langle n\rangle_{i}=\left\{\begin{array}[]{ll}\mathfrak{g}_{i},&i>n,\\ \ker(\mathfrak{g}_{n}\xrightarrow{d}\mathfrak{g}_{n-1}),&i=n,\\ 0,&i<n.\end{array}\right.$$

$$\mathfrak{g}={\Gamma }^1\mathfrak{g}\supseteq {\Gamma }^2\mathfrak{g}\supseteq \cdots$$

$$[x\otimes \alpha ,y\otimes \beta ]=(-1{)}^{|\alpha ||y|}[x,y]\otimes \alpha \beta$$

$$(\mathfrak{g}\otimes \Omega {)}_0=({\mathfrak{g}}_0\otimes {\Omega }^0)\oplus ({\mathfrak{g}}_1\otimes {\Omega }^1)\oplus \cdots \oplus ({\mathfrak{g}}_n\otimes {\Omega }^n).$$

$${\exp }_{\bullet }(\mathfrak{g})=\exp Z_0(\mathfrak{g}\otimes {\Omega }_{\bullet }).$$

$$d(\tau )+\frac{1}{2}[\tau ,\tau ]=0.$$

$${\mathsf{MC}}_{\bullet }(\mathfrak{g})=\mathsf{MC}(\mathfrak{g}\otimes {\Omega }_{\bullet }).$$

$$|\mathfrak{g}|=|{\mathsf{MC}}_{\bullet }(\mathfrak{g})|.$$

$$\Delta :C\to C{\otimes }_{\Omega }C,ϵ:C\to \Omega ,$$

$$\mathcal{G}(C)$$

Full text

1. Introduction

Consider a fiber bundle $p\colon E\to X$ with structure group $G$ over a simply connected CW-complex $X$ and let $aut_{\circ}(p)$ denote the topological monoid consisting of commutative diagrams

[TABLE]

such that $f$ is homotopic to the identity map of $X$ and $\varphi$ is a fiberwise isomorphism. The goal of this paper is to construct a differential graded Lie algebra model for the classifying space $Baut_{\circ}(p)$ in the sense of Quillen’s rational homotopy theory. We assume that $BG$ is a nilpotent space, i.e., that the group $\pi_{0}(G)$ is nilpotent and acts nilpotently on $\pi_{k}(G)$ for all $k\geq 1$.

Theorem 1.1**.**

Let $L$ be the minimal Quillen model for $X$ and let $\Pi$ be a dg Lie algebra model for $BG$. Furthermore, let $\tau\colon\mathcal{C}L\to\Pi$ be a twisting function that models the classifying map of the bundle $\nu\colon X\to BG$. Then the classifying space $Baut_{\circ}(p)$ is rationally homotopy equivalent to the geometric realization of the dg Lie algebra

[TABLE]

Here, $\mathcal{C}L$ is the Chevalley-Eilenberg complex of $L$, we use $\langle n\rangle$ to indicate the $n$-connected cover, and the decorations $\tau$ and $\tau_{*}$ indicate that we take a twisted semi-direct product (see §3.5 and §3.6).

In many cases of interest, there is an explicit formula for $\tau$ that yields a simplification of the model. For example, we have the following result in the case of complex vector bundles.

Theorem 1.2**.**

Let $\xi$ be an $n$-dimensional complex vector bundle over a simply connected finite CW-complex $X$ and let $L$ be the minimal Quillen model for $X$. Then $Baut_{\circ}(\xi)$ is rationally homotopy equivalent to the geometric realization of the dg Lie algebra

[TABLE]

where $\rho=\sum_{i=1}^{n}c_{i}(\xi)\otimes\gamma_{i}$, and $\gamma_{i}$ is the generator for $\pi_{2i-1}(\Omega BU(n))\otimes\mathbb{Q}=\pi_{2i}(BU(n))\otimes\mathbb{Q}$ dual to the universal Chern class $c_{i}\in H^{2i}(BU(n);\mathbb{Q})$.

Similar simplifications are possible whenever $G$ is a compact connected Lie group or, more generally, when $H^{*}(BG;\mathbb{Q})$ is a free graded commutative algebra, see §4.

Remark 1.3**.**

The fibration sequence of dg Lie algebras

[TABLE]

associated to the twisted semi-direct product (1) is a model for the homotopy fibration sequence

[TABLE]

where $aut_{\circ}(X)$ is the monoid of self-maps of $X$ homotopic to the identity, and $aut_{X}(p)\subseteq aut_{\circ}(p)$ is the submonoid of $aut_{\circ}(p)$ where $f$ is equal to the identity on $X$. In particular, $\operatorname{Hom}^{\tau}(\mathcal{C}L,\Pi)\langle 0\rangle$ is a model for $Baut_{X}(p)$. We should remark that rational models for $Baut_{X}(p)$ have been studied earlier, see e.g. [4].

2. Moduli spaces of $\mathcal{F}$-fibrations

We will utilize the general framework for classification of fibrations provided by May [10]. Let $(\mathcal{F},F)$ be a category of fibers in the sense of [10, Definition 4.1] and assume it satisfies the hypotheses of the classification theorem [10, Theorem 9.2]. Also recall the notions of $\mathcal{F}$-spaces and $\mathcal{F}$-maps from [3]. Let $G$ denote the group-like topological monoid $\mathcal{F}(F,F)$, to be thought of as the structure group for $\mathcal{F}$-fibrations.

In the special case when $\mathcal{F}$ is the category of all spaces weakly equivalent to a given CW-complex $X$, with morphisms all weak equivalences between such spaces, the ‘structure group’ $G=aut(X)$ is the monoid of homotopy automorphisms of $X$, and an $\mathcal{F}$-fibration is the same thing as a fibration with fiber weakly homotopy equivalent to $X$. We will refer to such fibrations as $X$-fibrations.

Returning to the general situation, let $p_{\infty}\colon E_{\infty}\to B_{\infty}$ denote the universal $\mathcal{F}$-fibration, the existence of which is ensured by May’s classification theorem, and define

[TABLE]

where the right hand side denotes the geometric bar construction of the group-like monoid $aut(X)$ acting on the space $map(X,B_{\infty})$ from the right by precomposition. It is a consequence of May’s ‘Classification of $Y$-structures’ [10, §11] that $Fib(X,\mathcal{F})$ may be thought of as a moduli space of $\mathcal{F}$-fibrations with base weakly equivalent to $X$. More precisely, we have the following:

Theorem 2.1**.**

For a CW-complex $A$, there is a bijective correspondence between homotopy classes of maps

[TABLE]

and equivalence classes of $X$-fibrations $p\colon E\to A$ with a $B_{\infty}$-structure $\theta\colon E\to B_{\infty}$.

Proof.

This follows readily from [10, Theorem 11.1]. ∎

In particular, since an $X$-fibration over a point is just a space weakly equivalent to $X$, we see that the set of path components,

[TABLE]

is in bijective correspondence with the set of equivalence classes of $\mathcal{F}$-fibrations with base weakly homotopy equivalent to $X$.

Definition 2.2**.**

Given an $\mathcal{F}$-fibration $p\colon E\to B$, let $aut^{\mathcal{F}}(p)$ denote the space of $\mathcal{F}$-self equivalences of $p$, i.e., the topological monoid consisting of commutative diagrams

[TABLE]

such that $f$ is a weak homotopy equivalence and $\varphi$ is a fiberwise $\mathcal{F}$-map, topologized as a subset of $map(B,B)\times map(E,E)$. Let $aut_{\circ}^{\mathcal{F}}(p)\subseteq aut^{\mathcal{F}}(p)$ denote the submonoid consisting of those pairs $(f,\varphi)$ such that $f$ is homotopic to the identity map on $X$. If $D\subseteq C\subseteq X$ are subsets, then let $aut_{C}^{\mathcal{F}}(p)$ denote the submonoid consisting of pairs as above such that $f$ restricts to the identity map on $C$, and write $aut_{C}^{D,\mathcal{F}}(p)$, or simply $aut_{C}^{D}(p)$, for the submonoid of $aut_{C}^{\mathcal{F}}(p)$ where $\varphi$ restricts to the identity isomorphism on the fibers over points in $D$. Finally, let $aut_{C,\circ}^{D}(p)$ denote $aut_{C}^{D}(p)\cap aut_{\circ}^{\mathcal{F}}(p)$.

By using standard properties of the geometric bar construction, we can obtain information about the homotopy types of the components of $Fib(X,\mathcal{F})$.

Theorem 2.3**.**

1. (1)

There is a bijection

[TABLE] 2. (2)

There is a weak equivalence of spaces over $Baut(X)$,

[TABLE]

where the union is over all equivalence classes of $\mathcal{F}$-fibrations $p\colon E\to B$, with $B$ weakly equivalent to $X$.

Proof.

As follows from [10, Proposition 7.9], there is a homotopy fiber sequence

[TABLE]

The first statement follows by looking at the induced long exact sequence of homotopy groups.

The space of $\mathcal{F}$-maps $map^{\mathcal{F}}(p,p_{\infty})$ is weakly contractible for every $\mathcal{F}$-fibration $p\colon E\to X$ by [3, Proposition 3.1]. Consider the diagram

[TABLE]

According to [10, Proposition 7.9] the rows are quasifibration sequences. The leftmost square is homotopy cartesian. It follows that the third vertical map from the left induces a weak equivalence between the connected components containing $\nu$. Since $map^{\mathcal{F}}(p,p_{\infty})$ is weakly contractible, the rightmost map in the top row is a weak homotopy equivalence. The rightmost square yields a zig-zag of weak homotopy equivalences showing Baut^{\mathcal{F}}(p)\sim B\big{(}map(X,B_{\infty}),aut(X),*\big{)}_{\nu} as spaces over $Baut(X)$, where $\nu$ indicates the component containing (the class of) $\nu$. ∎

Corollary 2.4**.**

There are weak homotopy equivalences

[TABLE]

[TABLE]

where $aut(X)_{[\nu]}$ denotes the monoid of homotopy equivalences $\varphi\colon X\to X$ such that $\nu\circ\varphi\simeq\nu$ and $map(X,B_{\infty})_{\nu}$ denotes the component of $\nu$.

Proof.

We have just seen that Baut^{\mathcal{F}}(p)\sim B\big{(}map(X,B_{\infty}),aut(X),*\big{)}_{\nu}. The latter is easily seen to be weakly equivalent to B\big{(}map(X,B_{\infty})_{\nu},aut(X)_{[\nu]},*\big{)}. The second claim is proved similarly. ∎

3. Rational models

This section contains the proof of the main theorem. We begin by examining the effect of $\mathbb{Q}$-localization on the geometric bar construction. Then we will construct a dg Lie model for the $\mathbb{Q}$-localized bar construction, by combining Schlessinger-Stasheff’s [12] and Tanré’s [13] theory of fibrations of dg Lie algebras with Quillen’s theory of principal dg coalgebra bundles [11].

3.1. Rationalization

Lemma 3.1**.**

Let $X$ be a connected nilpotent finite CW-complex, let $Z$ be a connected nilpotent space, and fix a map $\nu\colon X\to Z$. Then B\big{(}map(X,Z)_{\nu},aut_{\circ}(X),*\big{)} is rationally homotopy equivalent to

[TABLE]

Proof.

By using a functorial $\mathbb{Q}$-localization for nilpotent spaces, e.g., the Bousfield-Kan $\mathbb{Q}$-completion, we can construct a commutative diagram

[TABLE]

where the vertical maps are $\mathbb{Q}$-localizations. We may also assume that $r$ is a cofibration. Define the monoid $aut_{\circ}(r)$ as the pullback

[TABLE]

Thus, the monoid $aut_{\circ}(r)$ consists of pairs $(f,g)$ where $f$ and $g$ are self-maps homotopic to the identity of $X$ and $X_{\mathbb{Q}}$, respectively, such that $r\circ f=g\circ r$. Since $r$ is a cofibration, the map $r^{*}$ is a fibration. It is also a weak equivalence by standard properties of $\mathbb{Q}$-localization. The map $r_{*}$ is a rational homotopy equivalence by [7, Theorem II.3.11]. It follows that the projections from $aut_{\circ}(r)$ to $aut_{\circ}(X)$ and $aut_{\circ}(X_{\mathbb{Q}})$ are a weak equivalence and a rational homotopy equivalence, respectively.

There are right actions of the monoid $aut_{\circ}(r)$ on $map(X,Z)$ and $map(X_{\mathbb{Q}},Z_{\mathbb{Q}})$ through the projections to $aut_{\circ}(X)$ and $aut_{\circ}(X_{\mathbb{Q}})$, respectively. We get a zig-zag of rational homotopy equivalences of right $aut_{\circ}(r)$-spaces

[TABLE]

This accounts for the top horizontal zig-zag in the following diagram, where we write $\bullet$ instead of B\big{(}map(X,Z_{\mathbb{Q}})_{q\nu},aut_{\circ}(r),*\big{)} to save space,

[TABLE]

∎

3.2. Geometric realization of dg Lie algebras

Let $\mathfrak{g}$ be a dg Lie algebra over $\mathbb{Q}$, possibly unbounded as a chain complex. For $n\geq 0$, the $n$-connected cover is the dg Lie subalgebra $\mathfrak{g}\langle n\rangle\subseteq\mathfrak{g}$ defined by

[TABLE]

We call $\mathfrak{g}$ connected if $\mathfrak{g}=\mathfrak{g}\langle 0\rangle$ and simply connected if $\mathfrak{g}=\mathfrak{g}\langle 1\rangle$.

The lower central series of $\mathfrak{g}$ is the descending filtration

[TABLE]

characterized by $\Gamma^{1}\mathfrak{g}=\mathfrak{g}$ and $[\Gamma^{k}\mathfrak{g},\mathfrak{g}]=\Gamma^{k+1}\mathfrak{g}$. We call $\mathfrak{g}$ nilpotent if the lower central series terminates degree-wise, meaning that for every $n$, there is a $k$ such that $(\Gamma^{k}\mathfrak{g})_{n}=0$. This definition of nilpotence mirrors the notion of nilpotence for topological spaces. Indeed, a connected dg Lie algebra $\mathfrak{g}$ is nilpotent if and only if the Lie algebra $\mathfrak{g}_{0}$ is nilpotent and the action of $\mathfrak{g}_{0}$ on $\mathfrak{g}_{n}$ is nilpotent for all $n$. And, clearly, every simply connected dg Lie algebra is nilpotent.

If $\mathfrak{g}$ is an ordinary nilpotent Lie algebra, then $\exp(\mathfrak{g})$ denotes the nilpotent group whose underlying set is $\mathfrak{g}$ and where the group operation is given by the Campbell-Baker-Hausdorff formula, see e.g. [11]. The following generalizes this to dg Lie algebras. Let $\mathfrak{g}$ be a connected nilpotent dg Lie algebra. If $\Omega$ is a commutative cochain algebra, then the chain complex $\mathfrak{g}\otimes\Omega$ becomes a dg Lie algebra with

[TABLE]

for $x,y\in\mathfrak{g}$ and $\alpha,\beta\in\Omega$. If $\Omega^{k}=0$ unless $0\leq k\leq n$ for some $n$, then the degree [math] component of $\mathfrak{g}\otimes\Omega$ decomposes as

[TABLE]

From the fact that $[\mathfrak{g}_{i}\otimes\Omega^{i},\mathfrak{g}_{j}\otimes\Omega^{j}]\subseteq\mathfrak{g}_{i+j}\otimes\Omega^{i+j}$ and that $\mathfrak{g}_{0}$ acts nilpotently on $\mathfrak{g}_{k}$ for all $k$, one sees that $(\mathfrak{g}\otimes\Omega)_{0}$ is a nilpotent Lie algebra. Hence, so is the Lie subalgebra of zero-cycles $Z_{0}(\mathfrak{g}\otimes\Omega)$.

Let $\Omega_{\bullet}$ be the simplicial commutative differential graded algebra where $\Omega_{n}$ is the Sullivan-de Rham algebra of polynomial differential forms on the $n$-simplex, see [5]. Since $\Omega_{n}^{k}=0$ unless $0\leq k\leq n$, the above construction may be applied levelwise to the simplicial dg Lie algebra $\mathfrak{g}\otimes\Omega_{\bullet}$.

Definition 3.2**.**

Let $\mathfrak{g}$ be a connected nilpotent dg Lie algebra. We define $\exp_{\bullet}(\mathfrak{g})$ to be the simplicial nilpotent group

[TABLE]

Next, we recall the definition of the nerve $\mathsf{MC}_{\bullet}(\mathfrak{g})$ of a dg Lie algebra $\mathfrak{g}$. As we will see below, the nerve $\mathsf{MC}_{\bullet}(\mathfrak{g})$ is a delooping of the simplicial group $\exp_{\bullet}(\mathfrak{g})$.

Definition 3.3**.**

A Maurer-Cartan element in $\mathfrak{g}$ is an element $\tau$ of degree $-1$ such that

[TABLE]

The set of Maurer-Cartan elements is denoted $\mathsf{MC}(\mathfrak{g})$. The nerve of $\mathfrak{g}$ is the simplicial set

[TABLE]

Define the geometric realization of a dg Lie algebra to be the geometric realization of its nerve,

[TABLE]

3.3. Geometric realization of dg coalgebras

Let $\Omega$ be a commutative cochain algebra over $\mathbb{Q}$. A dg coalgebra over $\Omega$ is a coalgebra in the symmetric monoidal category of $\Omega$-modules, i.e., a dg $\Omega$-module $C$ together with a coproduct and a counit,

[TABLE]

such that the appropriate diagrams commute. We let $dgc(\Omega)$ denote the category of dg coalgebras over $\Omega$. If $C$ is a dg coalgebra over $\Omega$, we let

[TABLE]

denote the set of group-like elements, i.e., elements $\xi\in C$ of degree [math] such that

[TABLE]

Given a dg coalgebra $C$ over $\mathbb{Q}$, the free $\Omega$-module $C\otimes\Omega$ is a dg coalgebra over $\Omega$. Clearly,

[TABLE]

defines a functor from commutative cochain algebras to sets.

Definition 3.4**.**

Let $C$ be a dg coalgebra. We defined the spatial realization of $C$ to be the simplicial set

[TABLE]

A dg Lie algebra over $\Omega$ is a dg $\Omega$-module $L$ together with a Lie bracket $\ell\colon L\otimes_{\Omega}L\to L$ satisfying the usual anti-symmetry and Jacobi relations. Quillen’s generalization of the Chevalley-Eilenberg construction can be extended to dg Lie algebras over $\Omega$, yielding a functor

[TABLE]

The underlying coalgebra $\mathcal{C}_{\Omega}(L)$ is the symmetric coalgebra $S_{\Omega}(sL)$, where

[TABLE]

for an $\Omega$-module $V$. The differential is defined as usual, see e.g., [5, p.301]. If $L$ is a dg Lie algebra over $\mathbb{Q}$, then $L\otimes\Omega$ is a dg Lie algebra over $\Omega$ and there is a natural isomorphism of dg coalgebras over $\Omega$,

[TABLE]

Proposition 3.5**.**

Let $L$ be a connected dg Lie algebra and let $\Omega$ be a bounded commutative cochain algebra. There is a natural bijection

[TABLE]

[TABLE]

Proof.

The crucial observation is that this series converges since $\Omega$ is bounded and $L$ is connected. Say $\Omega^{k}=$ unless $0\leq k\leq n$. Then

[TABLE]

whence $\tau\in L\otimes\Omega^{+}$ for every element $\tau$ of degree $-1$. Since $(\Omega^{+})^{k}=0$ for $k>n$, this implies that

[TABLE]

whenever there are more than $n$ factors. Clearly, $\Delta(e(\tau))=e(\tau)\otimes e(\tau)$ and $\epsilon(e(\tau))=0$. As the reader may check, the equation $d(e(\tau))=0$ is equivalent to the Maurer-Cartan equation for $\tau$. ∎

Corollary 3.6**.**

There is a natural isomorphism of simplicial sets,

[TABLE]

for every connected dg Lie algebra $L$.

Proof.

Indeed, \mathsf{MC}_{\bullet}(L)=\mathsf{MC}(L\otimes\Omega_{\bullet})\cong\mathcal{G}\big{(}\mathcal{C}_{\Omega}(L\otimes\Omega_{\bullet})\big{)}\cong\mathcal{G}\big{(}\mathcal{C}(L)\otimes\Omega_{\bullet}\big{)}. ∎

Recall that for a commutative dg algebra $A$, the spatial realization is defined by

[TABLE]

see e.g., [1]. We use the same notation as for the coalgebra realization, but it should be clear from the context which one is used.

Proposition 3.7**.**

Let $A$ be a commutative cochain algebra of finite type with dual dg coalgebra $A^{\vee}$. Then there is a natural isomorphism

[TABLE]

Proof.

For a bounded commutative cochain algebra $\Omega$ and a finite type dg algebra $A$, there is a natural isomorphism of chain complexes

[TABLE]

Under this isomorphism, group-like elements in the dg coalgebra $A^{\vee}\otimes\Omega$ correspond to morphisms of dg algebras $A\to\Omega$. ∎

Note that the spatial realization of dg coalgebras preserves products, $\langle C\otimes D\rangle\cong\langle C\rangle\times\langle D\rangle$. In particular, since the universal enveloping algebra $U\mathfrak{g}$ of a dg Lie algebra $\mathfrak{g}$ is a dg Hopf algebra, i.e., a group object in the category of dg coalgebras, its spatial realization $\langle U\mathfrak{g}\rangle$ is a simplicial group. We also remark that for every commutative cochain algebra $\Omega$, the forgetful functor $dga(\Omega)\to dgl(\Omega)$ admits a left adjoint $U_{\Omega}\colon dgl(\Omega)\to dga(\Omega)$.

Proposition 3.8**.**

Let $\mathfrak{g}$ be a simply connected dg Lie algebra. There is a natural isomorphism of simplicial groups

[TABLE]

Proof.

Let $\Omega$ be a bounded commutative cochain algebra, say $\Omega^{k}=0$ unless $0\leq k\leq n$. Observe that there is a canonical isomorphism $U\mathfrak{g}\otimes\Omega\cong U_{\Omega}(\mathfrak{g}\otimes\Omega)$. The isomorphism is effected by the exponential map

[TABLE]

[TABLE]

where the product $x^{k}$ is taken in $U_{\Omega}(\mathfrak{g}\otimes\Omega)$. The crucial point is that the sum converges. Indeed, since $\mathfrak{g}$ is simply connected,

[TABLE]

so $x\in\mathfrak{g}\otimes\Omega^{+}$, whence $x^{k}=0$ for $k>n$, whenever $x$ is an element of degree [math]. The fact that $\exp$ respects the group structure is essentially by design of the Campbell-Baker-Hausdorff group structure. ∎

3.4. Principal dg coalgebra bundles

Next, recall Quillen’s theory of principal dg coalgebra bundles [11, Appendix B, §5]. In particular, recall that $\mathcal{C}(\mathfrak{g})$ serves as a classifying space for principal $\mathfrak{g}$-bundles. Quillen’s universal principal $\mathfrak{g}$-bundle may be identified with

[TABLE]

where $U\mathfrak{g}$ is the universal enveloping algebra of $\mathfrak{g}$ and $\mathcal{C}(U\mathfrak{g};\mathfrak{g})$ is the Chevalley-Eilenberg complex of $\mathfrak{g}$ with coefficients in the right $\mathfrak{g}$-module $U\mathfrak{g}$.

Theorem 3.9**.**

Let $\mathfrak{g}$ be a simply connected dg Lie algebra of finite type. The realization of the universal principal $\mathfrak{g}$-bundle,

[TABLE]

is a universal principal $\langle U\mathfrak{g}\rangle$-bundle.

Proof.

This is proved in [5, Chapter 25]. Indeed, when $\mathfrak{g}$ is simply connected and of finite type, the coalgebra realization of $U\mathfrak{g}$ is the same as the algebra realization of the dual dg algebra $U\mathfrak{g}^{\vee}$. ∎

Corollary 3.10**.**

Let $\mathfrak{g}$ be a simply connected dg Lie algebra of finite type. The nerve $\mathsf{MC}_{\bullet}(\mathfrak{g})$ is a delooping of the simplicial group $\exp_{\bullet}(\mathfrak{g})$.

Proof.

We have the isomorphisms $\exp_{\bullet}(\mathfrak{g})\cong\langle U\mathfrak{g}\rangle$ and $\mathsf{MC}_{\bullet}(\mathfrak{g})\cong\langle\mathcal{C}(\mathfrak{g})\rangle$. ∎

Remark 3.11**.**

Since we work with coalgebras, the finite type hypothesis on $\mathfrak{g}$ can be dropped in Theorem 3.9 and Corollary 3.10. However, we will not repeat the lengthy argument here since $\mathfrak{g}$ will be of finite type in our applications.

3.5. Twisted semi-direct products and Borel constructions

We begin by recalling certain aspects of Tanré’s classification of fibrations in the category of dg Lie algebras [13, Chapitre VII].

Definition 3.12**.**

Let $\mathfrak{g}$ and $L$ be dg Lie algebras. An outer action of $\mathfrak{g}$ on $L$ consists of two maps

[TABLE]

satisfying the following conditions for all $x,y\in\mathfrak{g}$ and $a,b\in L$, where we write

[TABLE]

Firstly, the map $\alpha$ defines an action of $\mathfrak{g}$ on $L$ by derivations, i.e.,

[TABLE]

[TABLE]

Secondly, the map $\xi$ is a chain map of degree $-1$ and a derivation, i.e.,

[TABLE]

[TABLE]

Finally, the action and $\xi$ are connected by the equation

[TABLE]

Definition 3.13**.**

Given an outer action of $\mathfrak{g}$ on $L$, the twisted semi-direct product $L\rtimes_{\xi}\mathfrak{g}$ is the dg Lie algebra whose underlying graded Lie algebra is the semi-direct product of $\mathfrak{g}$ acting on $L$,

[TABLE]

and whose differential is twisted by $\xi$ in the sense that

[TABLE]

The twisted semi-direct product is the total space in a short exact sequence (i.e.  fibration sequence) of dg Lie algebras,

[TABLE]

The section $\mathfrak{g}\to L\rtimes_{\xi}\mathfrak{g}$, $x\mapsto(0,x)$, is a morphism of graded Lie algebras, but it commutes with differentials if and only if $\xi=0$.

Outer actions on $L$ are classified by the dg Lie algebra

[TABLE]

whose underlying graded Lie algebra is the semi-direct product of $\operatorname{Der}L$ acting on the abelian dg Lie algebra $sL$ from the left by

[TABLE]

and whose differential is given by

[TABLE]

where $\operatorname{ad}_{x}\in\operatorname{Der}L$ is given by $\operatorname{ad}_{x}(y)=[x,y]$.

Proposition 3.14**.**

Specifying an outer action of $\mathfrak{g}$ on $L$ is tantamount to specifying a morphism of dg Lie algebras

[TABLE]

The correspondence is given by

[TABLE]

where $\theta_{x}(a)=x\ldotp a$.

Proof.

The proof is a straightforward calculation. ∎

An outer action of $\mathfrak{g}$ on $L$ defines an action of $\mathfrak{g}$ on $\mathcal{C}(L)$ by coderivations by the following formula:

[TABLE]

Equivalently, $\mathcal{C}(L)$ becomes a $U\mathfrak{g}$-module coalgebra, i.e., a right $U\mathfrak{g}$-module such that the structure map $\mathcal{C}(L)\otimes U\mathfrak{g}\to\mathcal{C}(L)$ is a morphism of dg coalgebras.

The action of $\operatorname{Der}L\ltimes_{\operatorname{ad}}sL$ by coderivations on $\mathcal{C}(L)$, derived from the tautological outer action on $L$, gives rise to a morphism of dg Lie algebras that we will denote

[TABLE]

Theorem 3.15**.**

Let $\mathfrak{g}$ be a simply connected dg Lie algebra of finite type with an outer action on a connected dg Lie algebra $L$. There is a right action of the simplicial group $G=\exp_{\bullet}(\mathfrak{g})$ on $\mathsf{MC}_{\bullet}(L)$ and a weak equivalence of simplicial sets over $\mathsf{MC}_{\bullet}(\mathfrak{g})$,

[TABLE]

Proof.

The action of $\mathfrak{g}$ on $\mathcal{C}(L)$ makes $\mathcal{C}(L)$ into a right $U\mathfrak{g}$-module coalgebra. This yields a right action of $\exp_{\bullet}(\mathfrak{g})\cong\langle U\mathfrak{g}\rangle$ on $\mathsf{MC}_{\bullet}(L)\cong\langle\mathcal{C}(L)\rangle$. The key observation, which may be checked by hand, is that there is an isomorphism of dg coalgebras

[TABLE]

Secondly, we have the standard isomorphism

[TABLE]

By combining these isomorphisms and taking realizations, we get isomorphisms of simplicial sets

[TABLE]

By Theorem 3.9, the simplicial set $\langle\mathcal{C}(U\mathfrak{g};\mathfrak{g})\rangle$ is a model for $EG$. This finishes the proof. ∎

Let $L$ be a simply connected cofibrant dg Lie algebra of finite type with geometric realization

[TABLE]

and consider the simply connected dg Lie algebra

[TABLE]

with associated topological group

[TABLE]

There is an evident outer action of $\mathfrak{g}$ on $L$, whence an action of the simplicial group $\exp_{\bullet}(\mathfrak{g})$ on the nerve $\mathsf{MC}_{\bullet}(L)$, cf. Theorem 3.15, whence an action of $G$ on $X$. Since $\mathfrak{g}$ is simply connected, the simplicial group $\exp_{\bullet}(\mathfrak{g})$ is reduced, i.e., has only one vertex. In particular, the topological group $G$ is connected. Therefore, the action yields a map of grouplike monoids

[TABLE]

This map is a weak homotopy equivalence, as follows from, e.g., Tanré’s theory [13, Chapitre VII].

3.6. Twisting functions and mapping spaces

Let $C$ be a dg coalgebra with coproduct $\Delta\colon C\to C\otimes C$ and let $L$ a dg Lie algebra with Lie bracket $\ell\colon L\otimes L\to L$. Recall that a twisting function $\tau\colon C\to L$ is a Maurer-Cartan element in the dg Lie algebra $\operatorname{Hom}(C,L)$, whose differential and Lie bracket are given by

[TABLE]

[TABLE]

If $\tau$ is a twisting function, then $\operatorname{Hom}^{\tau}(C,L)$ denotes the dg Lie algebra with the same underlying graded Lie algebra but twisted differential

[TABLE]

Furthermore, there is an outer action of $\operatorname{Coder}C$ on $\operatorname{Hom}^{\tau}(C,L)$ given by

[TABLE]

for $f\in\operatorname{Hom}^{\tau}(C,L)$ and $\theta\in\operatorname{Coder}C$. We note for future reference that we may make the identification

[TABLE]

for every twisting function $\tau\colon C\to L$.

Theorem 3.16**.**

Let $L$ and $\Pi$ be connected dg Lie algebras and suppose $\Pi$ is nilpotent and of finite type. There is a natural weak homotopy equivalence of simplicial sets

[TABLE]

Proof.

Let $\Omega$ be a bounded commutative cochain algebra. We define a natural map

[TABLE]

as follows. First, make the identifications

[TABLE]

the second of which is justified by Proposition 3.5, and then define

[TABLE]

simply by evaluation,

[TABLE]

We need to verify that $\tau(\xi)$ satisfies the Maurer-Cartan equation. Since $\tau$ is a twisting function, it satisfies the equation

[TABLE]

Evaluating both sides at the group-like element $\xi$ yields

[TABLE]

showing that $\tau(\xi)$ satisfies the Maurer-Cartan equation.

The map is clearly natural in $\Omega$ and yields a simplicial map

[TABLE]

The map in the theorem is defined to be the adjoint of this map.

To show it is a weak homotopy equivalence, one argues as in [1, Theorem 6.6] by induction on a suitable complete filtration of $\Pi$. The proof is entirely analogous so we omit the details. ∎

Remark 3.17**.**

The dg Lie algebra $\operatorname{Hom}(\mathcal{C}L,\Pi)$ with the descending filtration

[TABLE]

is a complete dg Lie algebra in the sense of [1, Definition 5.1]. By [1, Theorem 6.3] (see also Definition 5.3 and Remark 6.4 in loc.cit.), the Kan complex

[TABLE]

is homotopy equivalent to $map(\mathsf{MC}_{\bullet}(L),\mathsf{MC}_{\bullet}(\Pi))$. We would like to remark how this relates to the statement in Theorem 3.16.

Since $\Pi/\Pi\langle r\rangle$ is finite dimensional for all $r$, we have

[TABLE]

Upon taking the inverse limit, we get an isomorphism of simplicial sets

[TABLE]

Thus, Theorem 3.16 and Theorem 6.3 in [1] say the same thing. The advantage of Theorem 3.16 is that the explicit formula for the map gives us control over equivariance properties, as we will see next.

Let $L$ be a simply connected cofibrant dg Lie algebra of finite type. Precomposition defines a right action of the dg Lie algebra $\operatorname{Coder}\mathcal{C}(L)$ on the complete dg Lie algebra $\operatorname{Hom}(\mathcal{C}(L),\Pi)$. By composing with (3), and restricting to the simply connected cover, we get an action of the dg Lie algebra

[TABLE]

on $\operatorname{Hom}(\mathcal{C}(L),\Pi)$. By Theorem 3.15, this induces an action of the simplicial group $\exp_{\bullet}(\mathfrak{g})$ on the simplicial set $\mathsf{MC}\operatorname{Hom}(\mathcal{C}(L),\Pi\otimes\Omega_{\bullet})\cong\widehat{\mathsf{MC}}_{\bullet}(\operatorname{Hom}(\mathcal{C}(L),\Pi))$. On the other hand, $\exp_{\bullet}(\mathfrak{g})$ acts on $\mathsf{MC}_{\bullet}(L)$ and hence also on $map(\mathsf{MC}_{\bullet}(L),\mathsf{MC}_{\bullet}(\Pi))$. The following is an important addendum to Theorem 3.16.

Proposition 3.18**.**

The weak equivalence of Theorem 3.16,

[TABLE]

is equivariant with respect to the action of the simplicial group $\exp_{\bullet}(\mathfrak{g})$.

Proof.

The proof boils down to the easily checked fact that the map $\epsilon$ in the proof of Theorem 3.16 satisfies

[TABLE]

for $\theta\in\mathcal{G}U_{\Omega_{\bullet}}(\mathfrak{g}\otimes\Omega_{\bullet})$, $f\in\mathsf{MC}\operatorname{Hom}_{\Omega}(\mathcal{C}_{\Omega}(L\otimes\Omega),\Pi\otimes\Omega)$ and \xi\in\mathcal{G}\big{(}\mathcal{C}_{\Omega}(L\otimes\Omega)\big{)}. ∎

Proposition 3.19**.**

Let $X_{\mathbb{Q}}$ and $Z_{\mathbb{Q}}$ be $\mathbb{Q}$-local connected nilpotent spaces of finite $\mathbb{Q}$-type. Let $L$ be a finite type cofibrant dg Lie algebra model for $X_{\mathbb{Q}}$ and let $\Pi$ be any dg Lie model for $Z_{\mathbb{Q}}$. The geometric bar construction,

[TABLE]

is weakly homotopy equivalent to the geometric realization of the dg Lie algebra

[TABLE]

Proof.

We may as well assume $X_{\mathbb{Q}}=\mathsf{MC}_{\bullet}(L)$ and $Z_{\mathbb{Q}}=\mathsf{MC}_{\bullet}(\Pi)$. By Theorem 3.15, there is a weak homotopy equivalence

[TABLE]

The weak equivalence $\exp_{\bullet}(\mathfrak{g})\to aut_{\circ}(X)$ of group-like simplicial monoids and the weak equivalence of $\exp_{\bullet}(\mathfrak{g})$-spaces of Proposition 3.18 combine to give a weak homotopy equivalence

[TABLE]

∎

3.7. Proof of the main result

Theorem 3.20**.**

Suppose that $\mathcal{F}$ is a category of fibers such that the classifying space $B_{\infty}$ is connected and nilpotent. Let $p\colon E\to X$ be an $\mathcal{F}$-fibration over a simply connected finite CW-complex $X$.

Let $L$ be a simply connected cofibrant dg Lie algebra model for $X$ and let $\Pi$ be a connected nilpotent dg Lie algebra model for $B_{\infty}$. Let $\tau\colon\mathcal{C}L\to\Pi$ be a twisting function that models the map $\nu\colon X\to B_{\infty}$ that classifies $p$.

Then the classifying space $Baut_{\circ}^{\mathcal{F}}(p)$ is rationally homotopy equivalent to the geometric realization of the dg Lie algebra

[TABLE]

Proof.

For notational convenience, let $Z=B_{\infty}$. As before, let

[TABLE]

That the dg Lie algebras $L$ and $\Pi$ are models for $X$ and $Z$ means that we may use their geometric realizations as models for the $\mathbb{Q}$-localizations of $X$ and $Z$;

[TABLE]

By Corollary 2.4 and Lemma 3.1 we have

[TABLE]

The latter space is weakly homotopy equivalent to the component

[TABLE]

By Proposition 3.19

[TABLE]

Let $\tau\colon\mathcal{C}L\to\Pi$ be a twisting function that corresponds to $\nu_{\mathbb{Q}}$. It follows from [1, Corollary 1.3] that the component

[TABLE]

Finally, as in (5) one checks that there is an isomorphism of dg Lie algebras

[TABLE]

This finishes the proof. ∎

Remark 3.21**.**

It is straightforward to derive the following variants of the main result: If $A\subseteq X$ is a subspace such that the inclusion of $A$ into $X$ is a cofibration, then we may consider the submonoid $aut_{\circ,A}^{\mathcal{F}}(p)\subseteq aut_{\circ}^{\mathcal{F}}(p)$ where the homotopy automorphism of the base restricts to the identity map on $A$. If

[TABLE]

is a dg Lie algebra morphism that models the map $Baut_{A,\circ}(X)\to Baut_{\circ}(X)$, then

[TABLE]

is a dg Lie algebra model for the space $Baut_{A,\circ}^{\mathcal{F}}(p)$. Similarly, if we pick a base-point $*\in A\subseteq X$, and let $aut_{A,\circ}^{*}(p)$ denote the submonoid of $aut_{A,\circ}^{\mathcal{F}}(p)$ where the automorphism of the total space restricts to the identity over the base-point, then one gets a model for $Baut_{A,\circ}^{*}(p)$ by replacing $\mathcal{C}(L)$ with the reduced Chevalley-Eilenberg chains.

4. Examples and applications

Many classifying spaces of interest in geometry have simple rational homotopy types:

• •

If $G$ is a compact connected Lie group, then $H^{*}(BG;\mathbb{Q})$ is a polynomial algebra on finitely many generators of even degree (see, e.g., [6, Theorem 1.81]).

• •

The stable classifying spaces $BO$, $BTop$, $BPL$ are infinite loop spaces and have rational cohomology rings of the form $\mathbb{Q}[p_{1},p_{2},\ldots]$, where $p_{i}$ is a generator of degree $4i$ (see, e.g., [9]).

• •

Halperin’s conjecture, which has been verified in many cases, asserts that $H^{*}(Baut_{\circ}(X);\mathbb{Q})$ is a polynomial algebra whenever $X$ is an elliptic space with non-zero Euler characteristic.

In this section, we will provide a simplification of the model arrived at in the previous section in the case when $H^{*}(B_{\infty};\mathbb{Q})$ is a polynomial algebra with finitely many generators in each degree.

Call the generators $p_{1},p_{2},\ldots$ and let $d_{i}=|p_{i}|$. Let $X$ be a simply connected finite CW complex together with an $\mathcal{F}$-bundle classified by a map

[TABLE]

The characteristic classes of the bundle are defined by pulling back the universal classes $p_{i}$ along $\nu$;

[TABLE]

A dg Lie algebra model for $B_{\infty}$ is provided by the abelian dg Lie algebra with zero differential

[TABLE]

This graded vector space is spanned by classes $\pi_{i}\in\pi_{d_{i}-1}(\Omega B_{\infty})\otimes\mathbb{Q}=\pi_{d_{i}}(\Omega B_{\infty})\otimes\mathbb{Q}$ that are dual to $p_{i}$ under the Hurewicz pairing between cohomology and homotopy.

Let $L$ denote the minimal Quillen model of $X$. Recall that it has the form

[TABLE]

where $V=s^{-1}\widetilde{H}_{*}(X;\mathbb{Q})$ and the differential $\delta$ is decomposable in the sense that $\delta(L)\subseteq[L,L]$. Thus, the suspension of the space of indecomposables, $sL/[L,L]$, may be identified with $\widetilde{H}_{*}(X;\mathbb{Q})$.

We will work with based maps, so in this section we let $\overline{\mathcal{C}}(L)$ denote the reduced Chevalley-Eilenberg chains on a dg Lie algebra $L$. It is defined as the cokernel of the coaugmentation $\eta\colon\mathbb{Q}\to\mathcal{C}(L)$.

The restriction of (3) to $\operatorname{Der}L$ gives a morphism of dg Lie algebras

[TABLE]

In particular, $\overline{\mathcal{C}}(L)$ is a module over $\operatorname{Der}L$. Moreover, the universal twisting function $\tau_{L}\colon\overline{\mathcal{C}}(L)\to L$ is a morphism of $\operatorname{Der}L$-modules. Indeed, for $\theta\in\operatorname{Der}L$, the coderivation $\chi(\theta)\in\operatorname{Coder}\overline{\mathcal{C}}(L)$ is characterized by commutativity of the diagram

[TABLE]

which means that $\tau_{L}$ is a morphism of $\operatorname{Der}L$-modules.

Consider the composite of the universal twisting function and the abelianization,

[TABLE]

Since $a$ is a morphism of dg Lie algebras, this composite is a twisting function. But a twisting function with abelian target is the same thing as a chain map of degree $-1$. Thus, for every dg Lie algebra $L$, there is a canonical chain map

[TABLE]

Moreover, $g_{L}$ is a morphism of $\operatorname{Der}L$-modules. Indeed, we have just seen that $\tau_{L}$ is a morphism of $\operatorname{Der}L$-modules, and $a\colon L\to L/[L,L]$ is obviously a morphism of $\operatorname{Der}L$-modules.

Proposition 4.1**.**

If $L$ is a cofibrant dg Lie algebra, then the canonical map

[TABLE]

is a quasi-isomorphism.

Proof.

See, e.g., [5, Proposition 22.8] ∎

Next, let $L$ be a cofibrant minimal dg Lie algebra model for $X$ and let $\Pi$ denote the abelian graded Lie algebra $\pi_{*}(\Omega B_{\infty})\otimes\mathbb{Q}$. Consider the degree $-1$ map of graded vector spaces

[TABLE]

where $\langle-,-\rangle$ denotes the standard pairing between homology and cohomology (and $\langle p,e\rangle=0$ unless $e$ and $p$ have the same degree).

Proposition 4.2**.**

The composite map

[TABLE]

is a twisting function that models the map $\nu\colon X\to B_{\infty}$.

Proof.

The map

[TABLE]

is a dg Lie model for $\nu\colon X\to B_{\infty}$. Hence, composing with the universal twisting function we get a twisting function that models $\nu$:

[TABLE]

This composite map is the same as the map in the statement of the proposition. ∎

Theorem 4.3**.**

The classifying space $Baut_{*,\circ}^{*}(\nu)$ is rationally homotopy equivalent to the geometric realization of the dg Lie algebra

[TABLE]

Proof.

By Proposition 4.2, the map $\tau=\rho\circ g_{L}$ is a twisting function that models the map $\nu\colon X\to B_{\infty}$. By Theorem 3.20 the dg Lie algebra

[TABLE]

is a model for $Baut_{*,\circ}^{*}(\nu)$. Since $\Pi$ is abelian, twisting has no effect, i.e., we have \operatorname{Hom}^{\tau}\big{(}\overline{\mathcal{C}}L,\Pi\big{)}=\operatorname{Hom}\big{(}\overline{\mathcal{C}}L,\Pi\big{)}. Since $g_{L}^{*}(\rho)=\tau$ by definition of $\tau$, we see that the quasi-isomorphism of Proposition 4.1 induces a quasi-isomorphism of dg Lie algebras

[TABLE]

Finally note that $sL/[L,L]=\widetilde{H}_{*}(X;\mathbb{Q})$. ∎

Remark 4.4**.**

When $\widetilde{H}_{*}(X;\mathbb{Q})$ is finite dimensional, we can rewrite the result in terms of cohomology since

[TABLE]

The twisting function takes the form

[TABLE]

in this case.

Remark 4.5**.**

Again there are easily proved variants of this result. If $A\subseteq X$ is a subspace containing the base-point such that the inclusion of $A$ into $X$ is a cofibration, then we may consider the submonoid $aut_{A,\circ}^{*}(\nu)\subseteq aut_{*,\circ}^{*}(\nu)$ where the homotopy automorphism of the base restricts to the identity map on $A$. If $\mathfrak{g}\to\operatorname{Der}L\langle 1\rangle$ is a morphism of dg Lie algebras that models the map $Baut_{A}(X)\to Baut_{*}(X)$, then

[TABLE]

is a dg Lie algebra model for the space $Baut_{A,\circ}^{*}(\nu)$. In [2] we apply this result in the case $(X,A)=(M,\partial M)$ to construct a dg Lie algebra model for the block diffeomorphism group of a simply connected smooth $n$-manifold $M$ with boundary $\partial M=S^{n-1}$ ($n\geq 5$).

Acknowledgments

The author thanks Wojciech Chachólski for discussions about classification of fibrations and Ib Madsen for numerous discussions. This work was supported by the Swedish Research Council through grant no. 2015-03991.