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Lie models of homotopy automorphism monoids and classifying fibrations

June 2022
Advances in Mathematics 402:108359

June 2022
402:108359

DOI:10.1016/j.aim.2022.108359

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Authors:

Mario Fuentes Rumí

University of Malaga

Aniceto Murillo

University of Malaga

Given X a finite nilpotent simplicial set, consider the classifying fibrationsX→BautG⁎(X)→BautG(X)andX→Z→Bautπ⁎(X) where G and π denote, respectively, subgroups of the free and pointed homotopy classes of free and pointed self homotopy equivalences of X which act nilpotently on H⁎(X) and π⁎(X). We give algebraic models, in terms of complete differential graded Lie algebras (cdgl's), of the rational homotopy type of these fibrations. Explicitly, if L is a cdgl model of X, there are connected sub cdgl's DerGL and DerΠL of the Lie algebra of derivations of L such that the geometrical realizations of the sequences of cdgl morphismsL→adDerGL→DerGL×˜sLandL→L×˜DerΠL→DerΠL have the rational homotopy type of the above classifying fibrations. Among the consequences we also describe in cdgl terms the Malcev Q-completion of G and π together with the rational homotopy type of the classifying spaces BG and Bπ.

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Advances in Mathematics 402 (2022) 108359

Contents lists available at ScienceDirect

Advances in Mathematics

www.elsevier.com/locate/aim

Lie models of homotopy automorphism monoids

and classifying ﬁbrations

Yves Félix a, Mario Fuentes b, Aniceto Murillo b,∗,1

aInstitut de Mathématiques et Physique, Université Catholique de Louvain,

Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

bDepartamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29080

Málaga, Spain

a r t i c l e i n f o a b s t r a c t

Article history:

Received 11 March 2021

Received in revised form 5 March

2022

Accepted 12 March 2022

Available online xxxx

Communicated by Henning Krause

MSC:

primary 17B70, 55P62, 55R15,

55R35

secondary 17B55, 18N40

Keywords:

Lie models

Classifying spaces and ﬁbrations

Homotopy automorphisms

Rational homotopoy theory

Given Xa ﬁnite nilpotent simplicial set, consider the

classifying ﬁbrations

X→Baut∗

G(X)→BautG(X)andX→Z→Baut∗

π(X)

where Gand πdenote, respectively, subgroups of the free and

pointed homotopy classes of free and pointed self homotopy

equivalences of Xwhich act nilpotently on H∗(X)and π∗(X).

We give algebraic models, in terms of complete diﬀerential

graded Lie algebras (cdgl’s), of the rational homotopy type of

these ﬁbrations. Explicitly, if Lis a cdgl model of X, there are

connected sub cdgl’s DerGLand DerΠLof the Lie algebra of

derivations of Lsuch that the geometrical realizations of the

sequences of cdgl morphisms

Lad

→DerGL→DerGL

×sL and L→L

×DerΠL→DerΠL

have the rational homotopy type of the above classifying

ﬁbrations. Among the consequences we also describe in cdgl

*Corresponding author.

E-mail addresses: yves.felix@uclouvain.be (Y. Félix), m_fuentes@uma.es (M. Fuentes),

aniceto@uma.es (A. Murillo).

1The authors have been partially supported by the MICINN grant PID2020-118753GB-I00 of the Spanish

Government.

https://doi.org/10.1016/j.aim.2022.108359

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

2Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

terms the Malcev Q-completion of Gand πtogether with the

rational homotopy type of the classifying spaces BG and Bπ.

open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

A foundational result of Stasheﬀ [33] asserts that the classifying space Baut(X)of

the topological monoid aut(X)of self homotopy equivalences of a given ﬁnite complex

Xclassiﬁes ﬁbrations with ﬁber Xby means of a universal ﬁbration over Baut(X).

Furthermore, see [26], certain submonoids of aut(X) produce also classifying ﬁbrations

for distinguished families of ﬁbrations. The universal cover of Baut(X)has the homo-

topy type of the classifying space Baut1(X)of the monoid of self homotopy equivalences

homotopic to the identity on X. Whenever Xis simply connected (see [31,35], [2]for a

remarkable extension to the ﬁberwise setting or [3]for the relative case) the rational ho-

motopy type of Baut1(X)is well understood and described in terms of classical algebraic

models of X. However, the homotopy type of Baut(X)is as unmanageable, even for X

simply connected, as its fundamental group E(X)of homotopy classes of self homotopy

equivalences. For instance, as we will see, the homotopy type of Baut(Sn

Q) cannot be

obtained as the geometrical realization of any known algebraic structure shaping the

rational homotopy type of spaces.

Nevertheless, in this paper we are able to algebraically describe the rational homo-

topy type of the classifying space of certain non-connected distinguished submonoids of

aut(X). Furthermore, we model the corresponding universal classifying ﬁbrations. As a

consequence we also read in terms of these models the structure of nilpotent subgroups,

and their classifying spaces, of free and pointed homotopy classes of self homotopy equiv-

alences.

To do so we strongly rely on the homotopy theory developed in the category cdgl

of complete diﬀerential graded Lie algebras (cdgl’s henceforth) by means of the Quillen

pair

sset cdgl

·

(1)

given by the global model and realization functor (see next section for a quick review or

[11]for a detailed presentation).

From now on, and throughout the paper, we will often not distinguish a simplicial set

Xfrom its realization as a CW-complex. Such an object will always be pointed and we

denote by aut∗(X)the submonoid of aut(X)of pointed self homotopy equivalences.

Also, by a ﬁbration sequence F→E→Bwe understand its more general meaning:

the composition of the two maps is homotopic to a constant band the induced map from

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 3

Fto the homotopy ﬁber of E→Bover bis a weak homotopy equivalence. This also

applies to the category cdgl.

Let Xbe a ﬁnite connected complex, let G ⊂E(X)be a subgroup, and consider

autG(X) ⊂aut(X)the submonoid of self homotopy equivalences whose homotopy classes

are elements in G. Then, evaluating each equivalence at the base point induces a ﬁbration

sequence

X−→ Baut∗

G(X)−→ BautG(X)

which classiﬁes ﬁbrations X→E→Bwhere the image of the holonomy action

π1(X) →E(X) lies in G. Here, aut∗

G(X) ⊂aut∗(X)is the submonoid of pointed homo-

topy equivalences whose (free) homotopy classes lie in G.

Analogously, given π⊂E∗(X)a subgroup of pointed homotopy classes of pointed self

equivalences, there is a ﬁbration sequence of pointed spaces endowed with a homotopy

section (pointed ﬁbration from now on)

X−→ Z−→ Baut∗

π(X)

which classiﬁes pointed ﬁbrations X→E→Bwhere the image of the holonomy action,

this time understood as π1(X) →E∗(X), lies in π.

We now assume that Xis nilpotent and G ⊂E(X)acts nilpotently on H∗(X). Take , in

cdgl, the minimal model Lof X. Then, there is a subgroup Gof homotopy classes of cdgl

automorphisms of Ltogether with an action of H0(L), endowed with the group structure

given by the Baker-Campbell-Hausdorﬀ (BCH) product, such that G/H0(L)is isomorphic

to GQ, the rationalization of G. Consider DerGLthe connected sub diﬀerential graded

Lie algebra of DerL, the derivations of L, formed by the G-derivations of L. These are all

derivations of positive degrees and, in degree 0, those derivations θwhose exponential

eθis an automorphism of Lin G. Then, we prove (see §7for a thorough and precise

preamble of this result):

Theorem 0.1. The rational homotopy type of the classifying ﬁbration sequence

X−→ Baut∗

G(X)−→ BautG(X)

is modeled by the cdgl ﬁbration sequence

Lad

−→ DerGL−→ DerGL

×sL.

In fact, see Corollary 7.14, this cdgl sequence is part of a larger one

Hom(C(L),L)→Hom(C(L),L)→Lad

→DerGL→DerGL

×sL,

whose suitable restriction to certain components of Hom(C(L), L)and Hom(C(L), L)

models the ﬁbration sequence

4Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

aut∗

GQ(XQ)→autGQ(XQ)ev

→XQ→Baut∗

GQ(XQ)→BautGQ(XQ)

Whenever Xis simply connected, choosing universal covers in Theorem 0.1 recovers

the classical result mentioned above as the pair (1) extends, in the homotopy categories,

the classical Quillen model and realization functors [28]in the simply connected setting.

On the other hand, let π⊂E∗(X)acting nilpotently on π∗(X). Then, there is a

subgroup Πof homotopy classes of automorphisms of Lnaturally isomorphic to πQ. As

before, let DerΠLbe the connected dgl of Π-derivations of L. Then (see again §7for a

detailed statement):

Theorem 0.2. The rational homotopy type of the pointed classifying ﬁbration sequence

X−→ Z−→ Baut∗

π(X)

is modeled by the cdgl ﬁbration sequence

L−→ L

×DerΠL−→ DerΠL.

Under the assumptions and notation of the above results, immediate consequences

are:

Corollary 0.3. πQ∼

=H0(DerΠL)and GQ∼

=H0(DerGL)/Im H0(ad).

In both cases H0is considered as a group with the BCH product.

Corollary 0.4. BG and Bπ have the rational homotopy type of the realization of the cdgl’s

DerG

0L ⊕Rand DerΠ

0L ⊕Srespectively.

Here, Rand Sdenote a complement of the 1-cycles of DerGL

×sL and DerΠLrespec-

tively.

Along the way to the above results we must translate to cdgl the needed geometric

phenomena in each situation, mostly arising from the non triviality of the fundamental

group. Distinguishing free and pointed homotopy classes from the cdgl point of view, or

describing evaluation ﬁbrations as cdgl’s morphisms of derivations, are examples of this

conveyance which enlarges the dictionary between the homotopy categories of simplicial

sets and cdgl’s. This is precisely and brieﬂy outlined in the following summary of the

content.

In §1we give a short digest of the main facts concerning the homotopy theory of

cdgl’s and its connection with that of simplicial sets by means of the pair (1). From this

point on, we refer to this part for general notation in this matter.

In the second section we describe the subtle but important diﬀerence between free and

pointed homotopy classes in cdgl: given cdgl’s Land L, the group H0(L) endowed with

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 5

the BCH product acts on the set of cdgl homotopy classes {L, L}by means of the expo-

nential map [x] •[ϕ] =eadx◦ϕ. Whenever Lis connected and Lis the minimal model

of the connected simplicial set X, {L, L}is in bijective correspondence with the set

of pointed homotopy classes {X, L}∗(Corollary 2.2) and the mentioned action corre-

sponds to that of π1(X)on {X, L}∗(Theorem 2.4). That is, {X, L}

∼

={L, L}/H0(L).

Again by the presence of non trivial fundamental groups, we need to analyze distin-

guished subgroups of degree 0elements of particular cdgl’s, as always endowed with the

BCH product. To this end we recall in Section 3the isomorphism between the category

of (ungraded) complete Lie algebras and (Malcev) complete groups and prove in Theo-

rem 3.3 a useful characterization of these groups. Also in this section we recall the formal

deﬁnition of the logarithm and exponential for a given (ungraded) complete Lie algebra.

Via the exponential we observe that, whenever Mis a certain sub Lie algebra of Der0L

for some cdgl L, then the group (M, ∗)can be naturally identiﬁed with a subgroup of

aut(L).

In §4we describe cdgl models of evaluation ﬁbrations map∗(X, Y) →map(X, Y) →Y

(Theorem 4.2) and any of its components (Proposition 4.6) by reﬁning existing Lie models

of mapping spaces. Later on, in Section 5we transform these models in terms of deriva-

tions of the models of the involved simplicial sets (Theorem 5.1 and Corollary 5.2). Of

special interest is the description of the long homotopy exact sequence of the evaluation

ﬁbration as the long homology sequence of a short exact sequence of chain complexes of

derivations (Theorem 5.3).

Although cdgl is complete and cocomplete from the categorical point of view, essential

manipulations of cdgl’s may lie outside of this category. The twisted product L

×Mof

cdgl’s, or even the dgl DerLof derivations of a given Lie algebra are examples of this

malfunction, as they fail to be complete even if Land Mare. In §6we ﬁx these issues

for distinguished sub dgl’s of DerL(Proposition 6.6) and the particular twisted products

we use (Propositions 6.9, 6.10 and 6.12).

In Section 7we present a detailed and precise statement of our main results, together

with their ﬁrst consequences.

Section 8contains the proofs of Theorems 0.1 and 0.2 and ﬁnally, in §9, we present

several consequences of these results, of which Corollary 0.3 is an immediate one. Also

(see Theorem 9.9 and Propositions 9.11, 9.12), we model the rational homotopy type of

the ﬁbration sequences,

autG(X)→G→Baut1(X)→BautG(X)→BG

and

aut∗

π(X)→π→Baut∗

1(X)→Baut∗

π(X)→Bπ,

obtaining in particular Corollary 0.4. We ﬁnish the section with two sets of examples

covering a wide range: every ﬁnitely generated rational nilpotent group is ﬁrst easily

6Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

realized as a subgroup of self homotopy equivalences acting nilpotently either in the

(rational) homology or homotopy groups of some suitable complexes. We then apply our

main results to ﬁnd explicit Lie models of derivations for the corresponding classifying

spaces.

Acknowledgment. We thank Alexander Berglund for helpful conversations. We are also

extremely grateful to the referee. His/her numerous suggestions and corrections have

prevented some ﬂaws in this article, while greatly improving it.

1. Homotopy theory of complete Lie algebras

In this section we recall the basics for complete diﬀerential graded Lie algebras, from

the homotopy point of view. For it, original references are [7–10]whose main results

are developed in the complete and detailed monograph [11]. Sometimes we will also use

classical facts from the Sullivan commutative approach to rational homotopy theory.

These are not included here and we refer to [14,15]as standard references.

For any category C, we denote by HomCits morphisms, except for that of (graded)

vector spaces whose morphisms will be denoted by the unadorned Hom.

All considered chain (or cochain) complexes, with possibly extra structures, are un-

bounded and have Qas ground ﬁeld. The suspension of such a complex Vis denoted by

sV where (sV )n=Vn−1, n ∈Zand dsv =−sdv. The desuspension complex is denoted

by s−1V.

We denote by sset the category of simplicial sets. For any n ≥0, we denote by

Δn={Δn

p}p≥0the simplicial set in which Δn

p={(j0, ..., jp) |0 ≤j0≤ ··· ≤ jp≤n}

with the usual faces and degeneracies. Given Xand Ysimplicial sets we denote by

map(X, Y)the simplicial mapping space, that is,

map(X, Y )•=Hom

sset(X×Δ•,Y).

Analogously, we denote by map∗(X, Y)the pointed mapping space Homsset∗(X×Δ•, Y)

in the category sset∗of pointed simplicial sets.

Let dgl denote the category of diﬀerential graded Lie algebras (dgl’s henceforth).

A dgl L, or (L, d)if we want to specify the diﬀerential, is n-connected, for n ≥0, if

L =L≥n. As usual, connected means 0-connected. The n-connected cover of a dgl Lis

the n-connected sub dgl L(n)given by

L(n)

p=ker dif p=n,

Lpif p>n.

We denote by 

L=L(1) the 1-connected cover.

Given a dgl Lwe denote by DerLthe dgl of its derivations in which, for each n ∈Z,

DernLare linear maps θ:L →Lof degree nsuch that

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 7

θ[a, b]=[θ(a),b]+(−1)|a|n[a, θ(b)],for a, b ∈L.

The Lie bracket and diﬀerential are given by,

[θ, η]=θ◦η−(−1)|θ||η|η◦θ, Dθ =d◦θ−(−1)|θ|θ◦d.

On the other hand, given a dgl morphism f:L →Mwe denote by Derf(L, M)the chain

complex of f-derivations in which Derf(L, M)nare linear maps θ:L →Mof degree n

such that

θ[a, b]=[θ(a),f(b)] + (−1)n|a|[f(a),θ(b)],a,b∈L.

The diﬀerential is given as before.

Following [35, §7.2], or more generally [2, §3.5], a twisted product L

×Mof the dgl’s

Land Mis a dgl structure on the underlying vector space L ×Mso that

0→L−→ L

×M−→ M→0

is an exact dgl sequence. In particular Lis a sub dgl of the twisted product.

A ﬁltration of a dgl Lis a decreasing sequence of diﬀerential Lie ideals,

L=F1⊃··· ⊃ Fn⊃Fn+1 ⊃...

such that [Fp, Fq] ⊂Fp+qfor p, q≥1. In particular, if

L1⊃··· ⊃ Ln⊃Ln+1 ⊃...

denote the lower central series of L, that is, L1=Land Ln=[L, Ln−1]for n >1, then

Ln⊂Fnfor all n.

A complete diﬀerential graded Lie algebra, cdgl henceforth, is a dgl Lequipped with

a ﬁltration {Fn}n≥1for which the natural map

L∼

−→ lim

←−−

L/F n

is a dgl isomorphism. A morphism f:L →Lbetween cdgl’s, associated to ﬁltrations

{Fn}n≥1and {Fn}n≥1respectively, is a dgl morphism which preserves the ﬁltrations,

that is, f(Fn) ⊂Fnfor each n ≥1. We denote by cdgl the corresponding category

which is complete and cocomplete [11, Proposition 3.5].

Given a dgl Lﬁltered by {Fn}n≥1, its completion is the dgl



L= lim

←−−

L/F n,

which is always complete with respect to the ﬁltration

8Y. Félix et al. / Advances in Mathematics 402 (2022) 108359



Fn=ker(



L→L/F n)

as 

L/ 

Fn=L/F n. Unless a particular ﬁltration is explicitly mentioned, the completion

of a given dgl is taken over the lower central series. This is the case, in particular, for

any cdgl L =(



L(V), d) where



L(V) = lim

←−−

L(V)/L(V)n

is the completion of the free Lie algebra L(V) generated by the graded vector space V,

see [11, §3.2] for an exhaustive treatment of this cdgl. In this instance,



Fn=

L≥n(V)=

q≥n

Lq(V),for n≥1.

By an abuse of notation, we write 

Fn=Lnso that L = lim

←−−nL/Ln.

If Ais a commutative diﬀerential graded algebra (cdga henceforth) and Lis a cdgl

with respect to the ﬁltration {Fn}n≥1, we deﬁne their complete tensor product as the

cdgl

A

⊗L= lim

←−−

A⊗(L/F n)

where the diﬀerential and the bracket in A ⊗(L/F n)are deﬁned as usual by d(a ⊗x) =

da ⊗x +(−1)|a|a ⊗dx and [a ⊗x, a⊗x] =(−1)|a||x|aa⊗[x, x].

A Maurer-Cartan element, or simply MC element, of a given dgl Lis an element

a ∈L−1satisfying the Maurer-Cartan equation da =−1

2[a, a]. We denote by MC(L)the

set of MC elements in L. Given a ∈MC(L), we denote by da=d +ad

athe perturbed

diﬀerential. This is a new diﬀerential on Lwhere dis the original one and ad is the usual

adjoint operator. The component of Lat ais the connected sub dgl Laof (L, da)given

p=ker daif p=0,

Lpif p>0.

In other terms, La=(L, da)0. Whenever Lis complete the group L0, endowed with the

Baker-Campbell-Hausdorﬀ product, acts on the set MC(L)by

xGa=eadx(a)−eadx−1

adx

(dx)=

i≥0

adi

x(a)

i!−

i≥0

adi

x(dx)

(i+1)!,x∈L0,a∈MC(L).

This is the gauge action and we denote by 

MC(L) =MC(L)/Gthe corresponding orbit

set. This is thoroughly studied in [11, §4.3] and a particular homotopically friendly

description of the gauge action can be found in [11, §5.3].

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 9

There is a pair of adjoint functors, (global) model and realization,

sset cdgl

·

,(2)

which are deeply studied in [11, Chapter 7] and are based in the cosimplicial cdgl

L•={Ln}n≥0.

For each n ≥0, see [11, Chapter 6],

Ln=

L(s−1Δn),d)

where s−1Δndenotes the desuspension s−1N∗(Δn)of the non-degenerate simplicial

chains on Δn. That is, for any p ≥0, a generator of degree p −1of s−1Δncan be

written as ai0...ipwith 0 ≤i0<··· <i

p≤n. The cofaces and codegeneracies in L•are

induced by those on the cosimplicial chain complex s−1N∗Δ•, and the diﬀerential don

each Lnis the only one (up to cdgl isomorphism) satisfying:

(1) For each i =0, ..., n, the generators of s−1Δn, corresponding to vertices, are MC

elements.

(2) The linear part of dis induced by the boundary operator of s−1Δn.

(3) The cofaces and codegeneracies are cdgl morphisms.

In particular, L0is the free Lie algebra L(a) generated by a MC element and L1is the

Lawrence-Sullivan interval [22](see also [12]).

The realization of a given cdgl Lis the simplicial set

L=Hom

cdgl(L•,L).

In particular, the set L0of 0-simplices coincides with MC(L). Moreover, see [11, §7.2],

if Ladenotes the path component of Lcontaining a, we have:

LaLa,L

a∈

MC(L)La.(3)

However, it is important to observe that the realization of a cdgl is invariant under

perturbations. That is, for any cdgl Land any a ∈MC(L),

L∼

=(L, da).(4)

Indeed, by [11, Proposition 4.28] there is a bijection 

MC(L)

∼

=

MC(L, da)which sends

the gauge class of an MC element zof Lto the gauge class of the MC element z−aof

(L, da). On the other hand, observe that Lz=(L, da)a−z. Finally, apply (3).

10 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Also, if Lis connected, and for any n ≥1, the map

ρn:πnL∼

−→ Hn−1(L),ρ[ϕ]=[ϕ(a0...n)],

is a group isomorphism [11, Theorem 7.18]. Here, the group structure in H0(L)is con-

sidered with the Baker-Campbell-Hausdorﬀ product (BCH product henceforth). We also

remark that, for each n ≥1, the nth connected cover L(n)of the realization of a con-

nected cdgl is homotopically equivalent to the realization L(n)of its nth connected

cover [11, §12.5.3].

A ﬁnal and fundamental property of the realization functor is that it coincides with any

known geometrical realization of dgl’s: if Lis a 1-connected dgl of ﬁnite type then L 

LQ, see [11, Corollary 11.17], where  · Qstands for the classical Quillen realization

functor [28]. Furthermore, see [7, Theorem 0.1], [29, Theorem 3.2] and [11, Theorem

11.13], for any cdgl L, Lis a strong deformation retract of MC•(L), the Deligne-

Getzler-Hinich simplicial realization of L[18,21].

On the other hand, the global model of a simplicial set Xis the cdgl

LX= lim

−−→

σ∈X

L|σ|.

It can be checked, see [11, Proposition 7.8], that as complete Lie algebra, LX=

L(s−1X)

where s−1Xdenotes the desuspension s−1N∗(X)of the chain complex of non-degenerate

simplicial chains on X. Moreover, the diﬀerential don LXis completely determined by

the following:

(1) The 0-simplices of Xare Maurer-Cartan elements.

(2) The linear part of dis the desuspension of the diﬀerential in N∗(X).

(3) If j:Y⊂Xis a subsimplicial set, then Lj=

Ls−1N∗(j).

If Xis a simply connected simplicial set of ﬁnite type and ais any of its vertices,

then [11, Theorem 10.2], La

Xis quasi-isomorphic to λ(X) where λis the classical Quillen

dgl model functor [28]. Moreover, see [11, Theorem 11.14], for any connected simplicial

set Xof ﬁnite type, La

Xis weakly homotopy equivalent to Q∞Xthe Bousﬁeld-Kan

Q-completion of X[4]. Recall that, whenever Xis nilpotent, its Q-completion coincides,

up to homotopy, with its classical rationalization, which we denote by XQ.

The category cdgl has a coﬁbrantly generated model structure, see [11, Chapter 8], for

which the functors in (2) become a Quillen pair with the classical model structure on

sset: a cdgl morphism f:L →Mis a ﬁbration if it is surjective in non-negative degrees

and is a weak equivalence if 

MC(f): 

MC(L)

∼

→

MC(M)is a bijection, and fa:La

→Ma

is a quasi-isomorphism for every a ∈MC(L).

Although quasi-isomorphisms are weak equivalences only in the connected case, the

generalized Goldman–Millson Theorem, see [11, Theorem 4.33], provides a criterion for

a quasi-isomorphism to be a weak equivalence in the general case: let f:L →Lbe

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 11

a cdgl morphism such that map induced by the corresponding ﬁltrations Fn/F n+1 

−→

Gn/Gn+1 is a quasi-isomorphism for any n ≥1. Then, fis a weak equivalence.

A quasi-isomorphism of connected cdgl’s of the form

(

L(V),d)

−→ L

makes of (

L(V), d)a coﬁbrant replacement of Land we say that it is a model of L. If dhas

no linear term we say that (

L(V), d)is minimal and is unique up to cdgl isomorphism.

A fundamental object in this theory is:

Deﬁnition 1.1. Let Xbe a connected simplicial set and aany of its vertices. The minimal

model (

L(V), d)of La

Xis called the minimal model of X[11, Deﬁnition 8.32]. In the same

way, a Lie model of Xis a model (non necessarily minimal) of La

If (

L(V), d)is the minimal model of Xthen, see [11, Proposition 8.35], sV ∼

=

H∗(X; Q)

and, provided Xof ﬁnite type, sH∗(

L(V), d)

∼

=π∗(Q∞X). Here, the group H0(

L(V), d)

is again considered with the BCH product. Furthermore, from all of the above, if Xis

simply connected, the minimal model of Xis isomorphic to its classical Quillen minimal

model.

Let cdgc denote the category of cocommutative diﬀerential graded coalgebras (cdgc’s

from now on). Every cdgc Cis always assumed to have a counit ε:C→Qand a

coaugmentation η:Q →C. We write C=kerε, so that C∼

=C⊕Q. The map Δ: C→

C⊗Cis deﬁned by Δx =Δx −(u ⊗x +x ⊗u), with u =η(1). A cdgc Cis n-connected,

for n ≥0, if C=C>n.

Recall the classical pair of adjoint functors

cdgc

dgl

(5)

deﬁned as follows:

Given Ca cdgc, L(C)is (L(s−1C), d), the free dgl generated by the desuspension of

Cand d =d1+d2where

d1(s−1c)=−s−1dc,

d2(s−1c)=1

2

(−1)|ai|[s−1ai,s

−1bi]withΔc=

ai⊗bi.

On the other hand, C(L)is (∧(sL), d), the free cocommutative coalgebra cogenerated

by the suspension of Land in which d =d1+d2where

12 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

d1(sv1∧···∧svn)=−



i=1

(−1)nisv1∧···∧s(dvi)∧···∧svn,

d2(sv1∧···∧svn)= 

1≤i<j≤n

(−1)|svi|ρij s[vi,v

j]∧sv1∧···∧ svi∧···∧ svj∧···∧svn,

ni=j<i |svj|and ρij is the Koszul sign of the permutation

sv1∧···∧svn→ svi∧svj∧sv1∧···∧ svi∧···∧ svj∧···∧svn.

In particular, d1(sv) =−sdv and d2(sv ∧sw) =(−1)|sv|s[v, w].

In general, Cpreserves quasi-isomorphisms [27, Proposition 4.4]. However, Lpre-

serves quasi-isomorphisms between 1-connected cdgc’s [27, Proposition 6.4] and only

between connected ﬁbrant cdgc’s of ﬁnite type [11, Proposition 2.4(2)].

The cochain algebra C∗(L)on the dgl Lis the cdga dual of C(L). Whenever Lis

connected and of ﬁnite type, this cdga has a well known description:

C∗(L)∼

=(∧(sL)#,d)

in which d =d1+d2where

d1v, sx=(−1)|v|v, sdx,

d2v, sx ∧sy=(−1)|y|+1 v, s[x, y].

Here,  , :∧(sL)×∧sL →Qdenotes the usual pairing, see for instance [14, §23].

The adjunction map α:LC(L) →Lis the unique dgl morphism (L(s−1∧+sL), d) →

Lextending the projection s−1∧+sL →s−1∧+sL/

s−1∧≥2sL∼

=L. The other adjunc-

tion β:C→CL(C)is the unique cdgc morphism C→(∧sL(C), d) lifting the inclusion

C=ss−1C⊂sL(s−1C). By [11, Proposition 2.3] αis always a quasi-isomorphism while

βis a quasi-isomorphism if Cis connected.

2. Free and pointed homotopy classes in cdgl

Throughout the next sections it will be essential to make a clear distinction between

the sets of free and pointed homotopy classes of maps, from the cdgl point of view. This

is a subtle but crucial point that we settle here. Note that this discussion is meaningless

in the classical Quillen theory as, in the simply connected category, both sets coincide.

Recall from [11, §8.3] that two cdgl morphisms ϕ, ψ:M→Lare homotopic if there is

a cdgl morphism Φ: M→L

⊗∧(t, dt)such that ε0◦Φ =ϕand ε1◦Φ =ψ. Here, ∧(t, dt)

is the free commutative graded algebra generated by the element tof degree 0and its

diﬀerential dt. Also, for i =0, 1, εi:L

⊗∧(t, dt) →Lis the cdgl morphism obtained by

sending tto i. We denote by {M, L}the set of homotopy classes of cdgl morphisms from

Mto L.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 13

For the rest of this section we ﬁx a connected cdgl Land a connected (non necessarily

reduced) simplicial set X, pointed by the 0-simplex b ∈X0. Analogously, we denote

respectively by {X, L} and {X, L}∗the set of free and pointed homotopy classes of

simplicial maps. Note that, since Lis reduced, being 0 ∈MC(L)the only 0-simplex,

every map X→Lis pointed. Then:

Proposition 2.1. {X, L}

∼

={LX, L}and {X, L}∗∼

={LX/(b), L}.

Here (b) denotes the Lie ideal generated by the Maurer-Cartan element b.

Proof. As the model and realization functors constitute a Quillen pair, the ﬁrst identity

is obvious [11, Corollary 8.2(iv)]. Nevertheless, an explicit description of this identity

will lead to the second one:

The adjunction (2) provides a bijection

Homsset(X, L)∼

=Homcdgl(LX,L)

which sends any simplicial map f:X→Lto the cdgl morphism ϕf:LX→Ldeﬁned

as follows: recall that, for p ≥0, a generator of LXof degree p −1is identiﬁed with a

non-degenerate p-simplex σ∈Xp. Then, ϕf(σ) =f(σ)(a0...p), being a0...p ∈s−1Δpthe

top generator. Note that ϕf(b) =0so it induces a map ϕf:LX/(b) →L.

Now, let H:X×Δ1→Ybe a homotopy between two simplicial maps f, g:X→L.

That is, His a simplicial map for which H|X×(0) =fand H|X×(1) =g, being (0) and

(1) the subsimplicial sets of Δ1generated by the 0-simplices 0and 1 respectively. Tak ing

into account the bijection

Homsset(X×Δ1,L)∼

=HomssetX, map(Δ1,L),

Hcorresponds to a simplicial map X→map(Δ1, L). But, see [11, Theorem 12.18]

which is precisely [1, Theorem 6.6] in our context, there is a homotopy equivalence

map(Δ1,L)L

⊗∧(t, dt).

Hence, we identify Hwith a map X→L

⊗∧(t, dt)which, again by adjunction,

corresponds to a cdgl morphism Φ: LX→L

⊗∧(t, dt). The fact that His a homotopy

between fand gtranslates to ε0◦Φ =ϕfand ε1◦Φ =ϕgso that Φis a homotopy

between ϕfand ϕg. This shows again that {X, L}

∼

={LX, L}.

Now, assume that His a pointed homotopy. That is, His constant (to the null and

only 0-simplex of L) on the subsimplicial set (b) ×Δ1of X×Δ1. Then, a careful

inspection along this identiﬁcation shows that the homotopy Φ sends bto 0(which is not

the case in general) and thus it induces a homotopy Φ: LX/(b) →L

⊗∧(t, dt) between

ϕfand ϕg. This proves that {X, L}∗∼

={LX/(b), L}.

14 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Corollary 2.2. If Lis the minimal Lie model of X, {X, L}∗∼

={L, L}. In particular,

if Lis a Lie model of the ﬁnite type simplicial set Y, {X, Q∞Y}∗∼

={L, L}.

Proof. It is known, see [11, Proposition 8.7], that the composition



−→ (LX,d

b)

−→ LX/(b)

is not only an injective quasi-isomorphism but a weak equivalence in cdgl. Hence, Lis

also weakly equivalent to LX/(b)and thus {L, L}

∼

={LX/(b), L}

∼

={X, L}∗. To ﬁnish,

recall that being La Lie model of Y, L Q∞Y.

We now translate to cdgl the action of π1Lon {X, L}∗. For it, and for any given

two connected simplicial sets Xand Y, the action •of π1(Y)on {X, Y}∗can be described

as follows: Consider the wedge [a, b] ∨Xof an interval with X, and let r:[a, b] ∨X→X

be the retraction obtained by contracting [a, b]into b. As b →Xis a coﬁbration, there

exists a homotopy equivalence g:X→[a, b] ∨Xsuch that g(b) =aand r◦gidX. Let

g=c ◦g:X→S1∨Xwhere cglues awith b. Then, given α:S1→Yand f:X→Y,

[α]∗•[f]∗is represented by (α∨f) ◦g.

We deﬁne now an analogous action of H0(L)on {LX/(b), L}. Fo r it, recall from [11,

Proposition 4.10] that, for any given element y∈L0, the exponential

eady=

n≥0

adn

is a well deﬁned automorphism of L. In fact, this is a particular instance of a more

general setting covered by Remark 3.4 below. Furthermore, if y∗zdenotes the BCH

product of elements y, z∈L0, then eady∗z=eady◦eadz, see [11, Corollary 4.12].

Deﬁnition 2.3. Given any cdgl L, the group (L0, ∗)acts on Homcdgl(L, L)by y•ϕ =

eady◦ϕ. We denote in the same way the induced action,

H0(L)×{L,L}•

−→ { L,L},[y]•[ϕ]=[eady◦ϕ].

Then, if we denote by γ:{X, L}∗∼

={LX/(b), L}the explicit bijection described in

Proposition 2.1, we prove the following which extends [11, §12.5.1]:

Theorem 2.4. The following diagram commutes,

π1L×{X, L}∗•

ρ1×γ∼

{X, L}∗

∼

=γ

H0(L)×{LX/(b),L}•{LX/(b),L}.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 15

Proof. Note that, by deﬁnition, the following commutes:

π1L×{X, L}∗

•

∼

{S1∨X, L}∗

g∗{X, L}∗.

(6)

By the naturality of the bijections in Proposition 2.1 the following also commutes:

{S1∨X, L}∗g∗

∼

{X, L}∗

∼

{LS1∨X/(b),L}Lg

∗{LX/(b),L}.

(7)

Next, as any connected simplicial set is weakly equivalent to a reduced one, we lose no

generality by assuming Xreduced with bthe only 0-simplex. Hence we may write,

LX=(



L(V⊕b),d)withV=V≥0.

Then, [11, Proposition 7.21] guarantees that dmay be chosen so that, for the perturbed

diﬀerential,

(LX,d

b)=(



L(V),d

b)

(L(b),d

b).

Here, 

denotes the coproduct in cdgl. On the other hand, L[a,b]=L1is the cdgl

(

L(a, b, x), d)in which aand bare MC elements, and

dx =ad

xb+

∞



n=0

n!adn

x(b−a)

where Bndenotes the nth Bernoulli number. Thus, as the realization functor preserves

colimits,

L[a,b]∨X=(



L(V⊕a, b, x),d).

We deﬁne,

h:(LX,d

b)−→ (L[a,b]∨X,d

a),h(b)=a, h(v)=eadx(v),v∈V,

and check that it is a cdgl morphism: obviously h(dbb) =dah(b) while, applying [11,

Proposition 4.24] and the fact that db(v) ∈

L(V)for any v∈V, we get

16 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

h(dbv)=eadx(dbv)=daeadx(v)=dah(v).

On the other hand,

Lr:(



L(V⊕a, b, x),d)→(

L(V⊕b),d)

is the identity on V⊕b, and sends ato band xto 0. Therefore, as Lr◦h =id

LX, Lg

is necessarily homotopic to hand thus, also up to homotopy,

Lg:(



L(V⊕b),d)−→ (

L(V⊕b, x),d),Lg(b)=b, Lg(v)=eadx(v).

Hence, the induced map on the quotients Lg:LX/(b) →LS1∨X/(b)is the morphism

(

L(V),d)−→ (

L(V),d)

(L(x),0),v→ eadx(v),

and therefore, the bottom row of diagram (7) becomes

{(L(x),0) 

(

L(V),d),L}Lg

∗

−→ { (

L(V),d),L}.

But {(L(x), 0)



(

L(V), d), L}

∼

=H0(L) ×{LX/(b)}and thus, the following commutes,

{LS1∨X/(b),L}

Lg∗

∼

{LX/(b),L}

H0(L)×{LX/(b)}

•

(8)

Joining diagrams (6), (7)and (8) produces the diagram of the statement. 

As {X, L}

∼

={X, L}∗/π1L, we deduce:

Corollary 2.5. If Lis a Lie model of X, {X, L}

∼

={L, L}/H0(L).

Example 2.6. Let L =(



L(u, v), 0) with |u| =|v| =0, which is the minimal model of

S1∨S1, and recall that the model of the circle is LS1=(



L(b, x), d)with |b| =−1,

db =−1

2[b, b]and dx =[x, b]. Denote by f, g:LS1→Lthe morphisms deﬁned by

f(x) =vand g(x) =eadu(v). The induced morphisms f, g:LS1/(b) =(L(x), 0) →Lare

clearly not homotopic but u•f=gwhich means that fand gare homotopic. An explicit

homotopy is given by Ψ: LS1→L

⊗∧(t, dt), ψ(x) =etadu(v)and Ψ(b) =−udt. The

corresponding maps S1→Q∞(S1∨S1)are thus homotopic but not pointed homotopic.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 17

3. Complete subgroups of complete Lie algebras

In this section all complete Lie algebras will be ungraded, or equivalently, concentrated

in degree 0, and with no diﬀerential. We denote by cl the corresponding category and,

to stress this restriction, we will denote by M(instead of L) such a general Lie algebra.

From the original work of Malcev [23], several reformulations and generalization of

the so called Malcev equivalence, a category isomorphism between complete Lie algebras

and Malcev complete groups, can be found in the literature. We brieﬂy recall it here in

the most general framework for which we refer to [16, §8.2 and §8.3] for speciﬁc details:

Given a group Gwe denote its commutator by curved brackets,

(x, y)=xyx−1y−1,

and consider its lower central series

G=G1⊃G2⊃··· ⊃ Gi⊃Gi+1 ⊃...

where Gi=(Gi−1, G)for i ≥2.

By a ﬁltration of a group Gwe mean a sequence of subgroups

G=F1⊃··· ⊃ Fn⊃Fn+1 ⊃...

such that (Fn, Fm) ⊂Fn+mfor all m, n ≥1. In particular, Gn⊂Fnfor all n ≥1. A

ﬁltered group Gis pronilpotent if the natural map G

∼

−→ lim

←−−nG/F nis an isomorphism.

On the other hand recall that Gis said to be 0-local, uniquely divisible or rational if

for any natural n ≥1the map G →G, g→ gn, is a bijection.

Deﬁnition 3.1. A group G, ﬁltered by {Fn}n≥1, is Malcev Q-complete (or simply com-

plete) if it is pronilpotent and, for each n ≥1, the abelian group Fn/F n+1 is a Q-vector

space. With the obvious notion of morphism we denote by cgrp the corresponding cat-

egory of complete groups.

Whenever no speciﬁc ﬁltration is given, the Malcev completion of a group is considered

over the lower central series.

With this notation, there is an equivalence of categories

cl ∼

=cgrp

from which we extract just the properties we will need in the most suitable form:

•The underlying set is not altered by any of the functors. Also, the underlying sets

of the ﬁltrations associated to a complete graded Lie algebra and its corresponding

Malcev complete group are the same [16, §8.2.2].

18 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

•Given Ma complete Lie algebra M, the product structure in the associated Malcev

complete group is given by the BCH product [16, §8.2.8].

•When restricted to nilpotent Lie algebras and 0-local nilpotent groups, this is the

original Malcev equivalence [16, Remark 8.3.5].

Remark 3.2. Note that the Quillen version of Malcev completion in [28, A3, Deﬁnition

3.1] is slightly more restrictive that the one above. In this deﬁnition, for a ﬁltered group G

to be complete it is also required that the associated Lie algebra be generated by G/F 2.

Hence this is a subcategory of cgrp which, in view of [28, Theorem 3.3] is equivalent to the

subcateogry of cl of complete Lie algebras Lwhich are generated by L/F 2. Nevertheless,

for nilpotent groups both deﬁnitions coincide [16, Remark 8.3.5].

In this context, we will need the following characterization of Malcev complete groups.

Theorem 3.3. A group Gis Malcev complete if and only if it is pronilpotent and 0-local.

Proof. Let Gbe a Malcev complete group and let Mbe its associated complete Lie

algebra via the Malcev equivalence. Then, after identifying the underlying sets of Gand

M, the map G →G, x → xkcorresponds to the map M→M, x → kx. Hence, since M

is a Q-vector space, this map is bijective for all k≥1and Gis 0-local.

For the other implication note ﬁrst that, given a complete Lie algebra M, the Lie

algebras Mnand M/Mnare also complete for any n ≥1. Via the Malcev isomorphism

we deduce that Gnand G/Gnare complete groups whenever Gis a complete group.

We only have to prove that Gn/Gn+1 is 0-local for all n ≥1. For it, we ﬁrst show

that the group G/Gnis torsion-free for n ≥1. Suppose that there is x ∈Gsuch that

xk≡0(modGn),

that is, xk∈Gn. We then may write

xk=



i=1

(ai,b

where ai∈Gand bi∈Gn−1. Since Gis 0-local, for each i =1, ...m we can ﬁnd ci∈G

such that ck

i=ai. Note that in G/Gn+1 the class of the elements of Gn=(G, Gn−1)are

in the center and thus one easily checks that

(ai,b

i)=(ck

i,b

i)≡(ci,b

i)k(mod Gn+1).

Then, write



i=1

(ci,b

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 19

which is a product of elements of Gnand thus, modulo Gn+1, we have:

yk=m



i=1

(ci,b

i)k

≡



i=1

(ci,b

i)k≡



i=1

(ai,b

i)=xk(mod Gn+1).

In particular, since y∈Gn,

(xy−1)k≡0(modGn+1).

Write xn=x, xn+1 =xy−1and repeat this process to obtain a sequence of elements

(xj)j≥nin Gsuch that

xj+1 ≡xj(mod Gj)

and xk

j∈Gjfor all j≥n. However, since Gis a pronilpotent group, G

∼

=lim

←−− G/Gnand,

denoting xjthe class of xjin G/Gj, we may then identify the sequence x0=(xj)j≥n

with an element of Gfor which

x0≡xi(mod Gj) for all i≥j.

In particular, xk

0∈Gjfor all j≥n, which implies that xk

0=0. Since Gis 0-local, we

deduce that x0=0. Therefore, the sequence (xj)j≥nis identically zero. In particular,

x ∈Gn, and therefore G/Gnis torsion-free.

Next, since G/Gnis nilpotent and torsion-free, by [20, Theorem 2.2], the map G/Gn→

G/Gn, x→ xk, is injective for any natural k≥1. On the other hand, it is clearly

surjective as Gis 0-local and we conclude that G/Gnis 0-local, for any n ≥1.

Finally, consider the short exact sequence

Gn/Gn+1 →G/Gn+1 →G/Gn

where the two right groups are 0-local and all of them are nilpotent. Then, [20, Corollary

2.5] implies that Gn/Gn+1 is also 0-local. 

We now brieﬂy describe the exponential and logarithm maps for a given complete Lie

algebra Mand refer to [15, §2.4] for details. Given UM the universal enveloping algebra

of Mconsider the ideal Iof UM generated by M. By completing with respect to the

ﬁltration given by I0=UM, I1=Iand In=In−1I, n ≥2, we get



UM = lim

←−−

n≥0

UM/Inand 

I= lim

←−−

n≥1

I/In,

together with the well known bijections (see for instance [32, Chapter 4]), inverses of

each other,

20 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359



I∼

exp

1+

log

given by

exp(x)=ex=

n≥0

n!and log(1 + x)=

n≥1

(−1)n+1 xn

Now, the diagonal Δ: UM →UM clearly sends Into Jn=i+j=nIi⊗Ijand thus it

induces an algebra morphism denoted in the same way

Δ: 

UM →

UM

⊗

UM = lim

←−−

(UM ⊗UM)/Jn.

The primitive and grouplike elements of UM are respectively deﬁned by

P={x∈

I, Δx=x⊗1+1⊗x}and G={1+y∈1+

I, Δ(1 + y)=(1+y)⊗(1 + y)}.

Then, exp and log restrict to isomorphisms between Pand G. However, see [15, Propo-

sition 2.3], the inclusion M→Pis in fact an isomorphism and thus we have:

M∼

exp

log

Now, the multiplication on 

UM restricts to a product ·on Gfor which it becomes a

group. This in turn induces a group structure on Mvia the classical Baker-Campbell-

Hausdorﬀ product,

x∗y=log(ex·ey),x,y∈M.

To ﬁnish the section we make a crucial observation which, via the exponential, let us

identify certain complete Lie algebras of 0-derivations of a given cdgl Lwith a group of

automorphisms of L.

Remark 3.4. Let Lbe a cdgl with respect to the ﬁltration {Fn}n≥1and let M⊂Der0L

be a sub Lie algebra of 0-derivations of Lwhose elements increase the ﬁltration degree.

That is, θ(Fn) ⊂Fn+1 for any θ∈Mand any n ≥1. Note that, in this case, M

is a complete Lie algebra with respect to the ﬁltration {Fn}n≥1where Fn={θ∈

M, θ(Fm) ⊂Fm+nfor all m}. Then, the injection

M→Hom(L, L)

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 21

extends to an algebra morphism, I→Hom(L, L) where the product on the endomor-

phisms of Lis given by composition. More generally, by our assumption on M, the above

injection induces, also by composition on the right hand side, injections of the form

In→Hom(L, F n),n≥1,

which deﬁne a map



I= lim

←−−

n≥1

I/In−→ lim

←−−

n≥1

Hom(L, L)/Hom(L, F n).(9)

However, note that

lim

←−−

n≥1

Hom(L, L)/Hom(L, F n)∼

=lim

←−−

n≥1

Hom(L, L/F n)∼

=Hom(L, lim

←−−

n≥1

L/F n)∼

=Hom(L, L)

and thus, (9) becomes a map 

I−→ Hom(L, L)which we extend to

ξ:1+

I−→ Hom(L, L)

by sending 1to idL. Observe that, by restricting ξto the grouplike elements we obtain

an injection of Ginto the linear (non necessarily compatible with the diﬀerential) auto-

morphisms of L. Then, given θ∈M, we abuse of notation and write eθ=ξ◦exp(θ),

that is,

eθ=

n≥0

θn

where the product is now composition. This is then a well deﬁned linear automorphism

of L. It is easy to check that it commutes with the Lie bracket in general, and with the

diﬀerential whenever θis a cycle, see for instance [11, Proposition 4.10].

Thus, if M⊂Der0L ∩ker D, the group (M, ∗), with the BCH product, is identiﬁed,

via the exponential, with a subgroup of aut(L).

4. Evaluation ﬁbrations

Given Xand Ysimplicial sets, the evaluation ﬁbration is given by

map∗(X, Y )−→ map(X, Y )ev

−→ Y

where ev denotes the evaluation at the base point. We will need a particular version of

certain results, some of them already known under a diﬀerent perspective, describing

cdgl models of evaluation ﬁbrations.

22 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Given any cdgc Cand any dgl L, consider the dgl structure on Hom(C, L)with the

usual diﬀerential Df =d ◦f−(−1)|f|f◦d, and the convolution Lie bracket given by

[f,g]=[,]◦(f⊗g)◦Δ,

that is,

[f,g](c)=

(−1)|g||ci|[f(ci),g(c

i)],with Δc=

ci⊗c

Remark 4.1. If Lis a cdgl then Hom(C, L)is also complete. Indeed, let {Fn}n≥1be a

ﬁltration for which L

∼

=lim

←−−nL/F nand consider the ﬁltration {Gn}n≥1of Hom(C, L)

given by Gn=Hom(C, Fn). Then,

Hom(C, L)∼

=lim

←−−

Hom(C, L/F n)∼

=lim

←−−

Hom(C, L)/Gn.

The same applies to note that Hom(C, L)is also a sub cdgl of Hom(C, L)and there is a

cdgl isomorphism,

Hom(C, L)∼

=Hom(C,L)

×L, (10)

where both Land Hom(C, L)are sub cdgl’s and [x, f] =ad

x◦ffor x ∈Land f∈

Hom(C, L). In particular, this twisted product is naturally a complete dgl.

Finally observe that the homotopy type of the cdgl Hom(C, L)is an invariant of

the homotopy type of C. Indeed, if ϕ:C

−→ Cis a cdgc quasi-isomorphism, the cdgl

quasi-isomorphism ϕ∗:Hom(C, L)



−→ Hom(C, L) trivially induces quasi-isomorphisms

Hom(C, L)/Gn

−→ Hom(C, L)/Gn. Thus, by the generalized Goldman-Millson Theo-

rem, see §1, ϕ∗is a weak equivalence of cdgl’s.

For the remaining of this section we ﬁx a connected cdgl Land a connected simplicial

set Xof ﬁnite type whose minimal model is denoted by L. Recall from §1the notation

{Ln}≥1for the ﬁltration associated to Land that Cdenotes the chain coalgebra functor

deﬁned in (5). We begin with a reformulation of [11, Proposition 12.25]:

Theorem 4.2. The realization of the short exact cdgl sequence

0→lim

−−→

Hom(C(L/Ln),L)−→ lim

−−→

Hom(C(L/Ln),L)ev

−→ L→0

has the homotopy type of the ﬁbration sequence

map∗(X, L)−→ map(X, L)ev

−→  L.

In other words, there is a commutative diagram

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 23

map∗(X, L)map(X, L)ev L

lim

−−→nHom(C(L/Ln),L)



lim

−−→nHom(C(L/Ln),L)



L,



where the vertical maps are homotopy equivalences.

Here, the direct limit is taken in cdgl and lim

−−→nHom(C(L/Ln), L)

→Lis induced

by evaluating at 1 ∈Qevery morphism of each Hom(C(L/Ln), L).

Proof. Recall from [11, Theorem 12.18] or [1, Theorem 6.6] that for any cdga model A

of any simplicial set Xand any cdgl Lthere is a weak homotopy equivalence

A

⊗Lmap(X, L)

natural on any possible choice. Moreover, see [11, Proposition 12.25], there is a commu-

tative diagram of simplicial sets

map∗(X, L)map(X, L)ev L

A+

⊗L



A

⊗L



L,



where the vertical maps are homotopy equivalences and the bottom row is just the

realization of the cdgl short exact sequence A+

⊗L →A

⊗L →L.

On the other hand, if Ais a connected cdga of ﬁnite type, the cdgl A

⊗L(respec.

A+

⊗L) is naturally isomorphic to Hom(A, L) (respec. Hom(A, L)). Indeed, the follow-

ing chain of linear isomorphisms,

A

⊗L= lim

←−−

(A⊗L/F n)∼

=lim

←−−

Hom(A,L/Fn)∼

=Hom(A,lim

←−−

L/F n)∼

=Hom(A,L),

preserves diﬀerentials and Lie brackets.

Now, we choose

A= lim

−−→

C∗(L/Ln)

which, by [11, Theorem 10.8], is a Sullivan model of X. Observe that since each L/Ln

is of ﬁnite type, so is C∗(L/Ln). Then, as complete tensor products commute with

inductive limits, the theorem follows from the isomorphism

A

⊗L∼

=lim

−−→

C∗(L/Ln)

⊗L∼

=lim

−−→

Hom(C(L/Ln),L)

24 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

and its restriction to

A+

⊗L∼

=lim

−−→

C+(L/Ln)

⊗L∼

=lim

−−→

Hom(C(L/Ln),L)

where C+denotes the augmentation ideal of C∗.

If Xis a nilpotent simplicial set of ﬁnite type the above result is drastically simpliﬁed.

For it, the following observation is important.

Remark 4.3. Although outside of the classical Quillen theory for simply connected spaces,

Neisendorfer deﬁned in [27, §7] a “Lie model” of a nilpotent, ﬁnite type complex Xas

any dgl (not complete) quasi-isomorphic to L(A). Here, Lis the functor deﬁned in

(5) and, as in the proof of Theorem 4.2, Astands for a ﬁnite type Sullivan model of

X. On the other hand, if Lis the minimal (cdgl) model of X, then Land L(A)are

quasi-isomorphic, see [11, Theorem 10.2]. Moreover, as stated in §1, L XQwhich

is also of the homotopy type of the cdga realization of C∗L(A). For a compendium of

the geometrical realization in the commutative side see for instance [15, §1.6].

In what follows Hom(C(L), L)

×Lwill always denote the twisted product arising from

the isomorphism (10)in Remark 4.1. The following extends [5, Corollary 15] the following

result we consider, as in (10), the twisted product Hom(C(L), L)

×L.

Corollary 4.4. Let Lbe a Lie model of the nilpotent simplicial set Xof ﬁnite type. Then,

the realization of the short exact cdgl sequence

0→Hom(C(L),L)−→ Hom(C(L),L)

×L−→ L→0

has the homotopy type of the ﬁbration sequence

map∗(X, L)−→ map(X, L)ev

−→  L.

Proof. As before, this ﬁbration sequence has the homotopy type of the realization of the

cdgl sequence A+

⊗L →A

⊗L →Lwhere Ais the Sullivan minimal model of X. As A

is of ﬁnite type this sequence becomes

0→Hom(A

+,L)−→ Hom(A

+,L)

×L−→ L→0.

Now, see the properties of the Quillen pair (5)in §1, since Ais a connected cdgc, it

is quasi-isomorphic to CL(A). Observe, that L(A)is the Neisendorfer model of X

which by Remark 4.3, is quasi-isomorphic to any given cdgl model of X. Hence, since C

preserves quasi-isomorphisms,

ACL(A)C(L).

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 25

This implies, in view of the last observation in Remark 4.1, that the above cdgl sequence

is homotopy equivalent to

0→Hom(C(L),L)−→ Hom(C(L),L)

×L−→ L→0

and the corollary follows. 

We now consider the restriction of the evaluation ﬁbration to a given path component

of map(X, L),

map∗

f(X, L)−→ mapf(X, L)ev

−→  L,

determined by a map f:X→L. Note that the ﬁber is the non-connected complex of

pointed homotopy classes of pointed maps freely homotopic to f.

Remark 4.5. Given a Lie model Lof a simplicial set X, the set π0map(X, L)of

path components of map(X, L)is the set of free homotopy classes {X, L} which, by

Corollary 2.5, is in bijective correspondence with the set {L, L}/H0(L). In the same

way, by Corollary 2.2, the set π0map∗(X, L)of pointed homotopy classes {X, L}∗is

identiﬁed with {L, L}.

Now, if Xis nilpotent, by Corollary 4.4,

Hom(C(L),L)map(X, L)andHom(C(L),L)map∗(X, L).

As a result,

π0map(X, L)∼

={L,L}/H0(L)∼

=

MCHom(C(L),L)

while, on the other hand,

π0map∗(X, L)∼

={L,L}∼

=

MCHom(C(L),L).

These correspondences can be explicitly described:

Let q:C(L) →Lbe the degree −1 linear map deﬁned by q(sx) =xif x ∈Land

q(∧≥2sL) =0. Denote in the same way the only possible extension of qto C(L)by

q(1) =0. Now choose a map f:X→Lwhich can be assumed to be pointed since L

is connected. Then, the pointed homotopy class [f]∗is identiﬁed to the homotopy class

of a cdgl morphism ϕ:L→Lwhich corresponds to the gauge class of the MC element

ϕ=ϕ◦q:C(L)−→ L. (11)

On the other hand, the free homotopy class [f] corresponds to the class of ϕin

{L, L}/H0(L)which is identiﬁed to the gauge class of the MC element

ϕ=ϕ◦q:C(L)−→ L. (12)

26 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

With this notation, consider the component Hom(C(L), L)ϕ→Lof Hom(C(L), L) →

Lat the MC element ϕ. Recall from §1that Hom(C(L), L)ϕis, by deﬁnition, the con-

nected cover of (Hom(C(L), L), Dϕ) where Dϕis the diﬀerential Din Hom(C(L), L)

perturbed by ϕ. Then we have:

Proposition 4.6. The realization of the cdgl morphism

Hom(C(L),L)ϕ−→ L

has the homotopy type of the ﬁbration

mapf(X, L)ev

−→  L.

Proof. Recall that Hom(C(L), L)ϕ Hom(C(L), L)ϕ. To ﬁnish, note that in view

of Corollary 4.4 and Remark 4.5, Hom(C(L), L)ϕ→Lis of the homotopy type of

mapf(X, L)

−→  L.

Remark 4.7. (1) Observe that Hom(C(L), L)ϕ→Lis not in general a surjective mor-

phism, i.e., it is not a cdgl ﬁbration. Indeed, see (10), in

(Hom(C(L),L),D

ϕ)∼

=(Hom(C(L),L)

×L, Dϕ),

Dϕ(x) =dx +[ϕ, x] =dx −(−1)|x|adx◦ϕfor any x ∈L. Therefore, elements of L0are

not, in general, Dϕ-cycles and thus, they do not belong to Hom(C(L), L)ϕ.

(2) Note also that, in view of Corollary 4.4 and Remark 4.5, the ﬁber of the map

Hom(C(L), L)ϕ→Lis the non-connected complex

[ψ]Hom(C(L),L)ψ

where: [ψ] runs through homotopy classes of morphisms ψ:L→Lsuch that, when

considered ψin Hom(C(L), L), this is a MC element gauge related to ϕ. A short com-

putation shows that this amounts to the existence of x ∈L0such that eadx◦ϕ =ψ.

In other terms, and with the notation in Deﬁnition 2.3, [ψ] =[x]•[ϕ]with [x] ∈H0(L).

Each of these components is of the homotopy type of the corresponding component of

map∗

f(X, L).

Remark 4.8. In all of the above, if Lis a Lie model of a connected simplicial set Yof ﬁnite

type, then the ﬁbrations realized in Theorems 4.2, Corollary 4.4 and Proposition 4.6 are,

respectively, the ﬁbration

map∗(X, Q∞Y)−→ map(X, Q∞Y)ev

−→ Q∞Y

and each of its components.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 27

5. Derivations models of evaluation ﬁbrations

The results in the past section can be expressed in terms of derivations. To avoid

excessive technicalities we assume Xnilpotent in what follows. Again Ldenotes a Lie

model of Xand Lis a connected cdgl. Given a cdgl morphism ϕ:L→L, consider the

dgl morphism,

ϕ=ϕα:LC(L)−→ L,

with αthe adjunction map, see §1. Next, endow

s−1Der ϕ(LC(L),L)

with the desuspended diﬀerential,

D(s−1θ)=−s−1[d, θ]=−s−1(d◦θ−(−1)|θ|θ◦d),

and a Lie bracket,

[s−1γ,s−1η]=s−1θ,

where θis ﬁrst deﬁned on s−1C(L)by

−[,]◦(γs−1⊗ηs−1)◦Δ◦s,

and then extended to LC(L)as a ϕ-derivation.

Consider the twisted product

(s−1Der ϕ(LC(L),L)

×L, D)

where s−1Der ϕ(LC(L), L)is a sub dgl and

[x, s−1θ]=(−1)|x|s−1(adx◦θ),Dx=dx −s−1(adx◦ϕ◦s),x∈L.

Here, adx◦ θand adx◦ ϕ◦sdenote the ϕ-derivations which are these compositions on

s−1C(L), and dis the diﬀerential on L. Then, Corollary 4.4 translates to the following

which extends [6, Theorem 3]:

Theorem 5.1. For any cdgl morphism ϕ:L→L,

0→s−1Der ϕ(LC(L),L)−→ s−1Der ϕ(LC(L),L)

×L−→ L→0 (13)

is a cdgl short exact sequence whose realization has the homotopy type of the ﬁbration

sequence

28 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

map∗(X, L)−→ map(X, L)ev

−→  L.(14)

Proof. As perturbing a cdgl does not interfere with its realization, see (4), Corollary 4.4

amounts to say that (14)is homotopy equivalent to the realization of the cdgl sequence,

0→(Hom(C(L),L),D

ϕ)−→ (Hom(C(L),L)

×L, Dϕ)ev

−→ L→0.

Recall that Dϕstands for the perturbed diﬀerential by the MC elements ϕas in (11)

and (12) respectively. Next, as in [6, Theorem 3] we show that

Γ: s−1Der ϕ(LC(L),L)∼

−→ (Hom(C(L),L),D

ϕ),Γ(s−1θ)(c)=(−1)|θ|θ(s−1c),

(15)

is a dgl isomorphism, which then exhibit s−1Der ϕ(LC(L), L)as a complete dgl.

It is trivially a linear isomorphism and a straightforward computation shows it com-

mutes with the Lie brackets. It remains to prove that Γcommutes with diﬀerentials. For

it, let θ∈Der ϕ(LC(L), L)and c ∈C(L)with Δ(c) =ici⊗c

On the one hand,

DϕΓ(s−1θ)(c)=dΓ(s−1θ)(c)−(−1)|θ|+1Γ(s−1θ)(dc)+[ϕ, Γ(s−1θ)](c)

=(−1)|θ|dθ(s−1c)+θ(s−1dc)+



(−1)|ci|(|θ|+1)+|θ|

[ϕ(ci),θ(s−1c

i)].

On the other hand,

ΓD(s−1θ)(c)=(−1)|θ|(d◦θ−(−1)|θ|θ◦d)(s−1c)

=(−1)|θ|dθ(s−1c)−θ(d1s−1c)−θ(d2s−1c)

=(−1)|θ|dθ(s−1c)+θ(s−1dc)−θ1

2

(−1)|ci|[s−1ci,s

−1c

i]

=(−1)|θ|dθ(s−1c)+θ(s−1dc)+

(−1)|ci|+|θ|(|ci|+1)[ϕ(ci),θ(s−1c

i)],

and both expressions coincide. In the last equality we use that C(L)is cocommutative,

θis a ϕ-derivation and ϕ(s−1c) =−ϕ(c)for any c ∈C(L).

To ﬁnish, note that the isomorphism Γ trivially extends to

(Hom(C(L),L)

×L, Dϕ)∼

=(s−1Der ϕ(LC(L),L)

×L, D).(16)

Note that, in view of the isomorphisms (15)and (16), together with Remark 4.1, the

objects in (13)are cdgl’s and the maps are cdgl morphisms. 

As a consequence we can easily express each component of (14)in terms of deriva-

tions. Again, let mapf(X, L)denote the component of map(X, L) containing the

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 29

map f:X→L, which corresponds, via Remark 4.5, to an element in {L, L}/H0(L)

represented by the homotopy class of a cdgl morphism ϕ:L→L.

Corollary 5.2. The realization of the cdgl morphism

(s−1Der ϕ(LC(L),L)

×L, D)0−→ L

has the homotopy type of the ﬁbration

mapf(X, L)ev

−→  L.

Recall from §1that given Many cdgl, M0stands for its connected cover, or equiva-

lently, its component at the MC element 0.

Proof. This is an immediate application of Proposition 4.6 and the isomorphism (16). 

We ﬁnish by describing, also in terms of derivations, the homotopy long exact sequence

of the evaluation ﬁbration. For it, ﬁx a pointed map f:X→Las the base point of

both, map∗

f(X, L)and mapf(X, L).

On the other hand, let Lbe a Lie model of Xand let ϕ:L→Lbe a cdgl morphism

whose homotopy class in {L, L}identiﬁes the pointed homotopy class [f]∗. Recall again

that, by Remark 4.5, this morphism corresponds to the free homotopy class [f]in the

orbit set {L, L}/H0(L). Consider the twisted chain complex

(Derϕ(L,L)

×sL, D)

which has Derϕ(L, L)as subcomplex and

Dsx =−sdx +ad

x◦ϕ, x ∈L.

Then, we prove the following which, in particular, recovers [11, Theorem 12.35]:

Theorem 5.3. For the chosen basepoints, the homotopy long exact sequence of the ﬁbration

map∗

f(X, L)−→ mapf(X, L)ev

−→  L(17)

is isomorphic to the homology long exact sequence of

0→Derϕ(L,L)0−→ (Derϕ(L,L)

×sL)0−→ sL →0.

In particular,

π∗(map∗

f(X, L),[f]∗)∼

=H∗Derϕ(L,L)and π∗mapf(X, L)∼

=H∗(Derϕ(L,L)

×sL).

30 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Proof. On the one hand, since the adjunction map α:LC(L)



→Lis a quasi-

isomorphism and Lis a Lie model (and thus coﬁbrant), the map of chain complexes,

α∗:Der

ϕ(L,L)

−→ Der ϕ(LC(L),L),(18)

is also a quasi-isomorphism, see for instance [6, Lemma 6]. Extend it to another quasi-

isomorphism

α∗×idsL :(Der

ϕ(L,L)

×sL, D)

−→ (Der ϕ(LC(L),L)

×sL, D) (19)

by setting

Dsx =−sdx +θ, x ∈L,

where θis the ϕ-derivation which extends the map adx◦ϕ◦s:s−1C(L) →L. In partic-

ular, θcoincides with adx◦ϕon L∼

=s−1sL. This shows that (19)is indeed a chain map

and thus, a quasi-isomorphism. Fro m (18)and (19)we obtain a commutative diagram

of chain complexes where the top sequence is the suspension of (13):

0Der ϕ(LC(L),L)(Der ϕ(LC(L),L)

×sL, D)sL 0

0Derϕ(L,L)



(Derϕ(L,L)

×sL, D)



sL 0

Thus, by Corollary 5.2, the homology long exact sequence of the component at 0of the

top row is precisely the homotopy long exact sequence of (17). For it note that, by Re-

mark 4.7(2), the component of map∗

f(X, L) containing the pointed map fhas the homo-

topy type of Hom(C(L), L)ϕand (Hom(C(L), L), Dϕ)

∼

=s−1Der ϕ(LC(L), L). 

6. Complete Lie algebras of derivations and twisted products

Given La dgl (non necessarily complete in principle), the following twisted products

are key tools for our main results:

Given a cdgc Cconsider the twisted product

Hom(C, L)−→ Hom(C, L)

×DerL−→ DerL(20)

in which both terms are sub dgl’s and

[θ, f ]=θ◦f, θ ∈DerL, f ∈Hom(C, L).

This dgl sequence, and its “dual” Coder(C)

×Hom(C, L), have already proven to be

useful in the modeling of certain simply connected spaces of homotopy automorphisms,

see [2, §3] and [3, §4].

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 31

Also, consider the twisted product

DerL−→ DerL

×sL −→ sL (21)

where sL is an abelian Lie algebra and

Dsx =−sdx +ad

x,[θ, sx]=(−1)|θ|sθ(x),x∈L, θ ∈DerL.

Finally, let

L−→ L

×DerL−→ DerL(22)

be the twisted product in which both terms are sub dgl’s and [θ, x] =θ(x)for any

θ∈DerLand x ∈L.

If Lis a cdgl we observed in Remark 4.1 that Hom(C, L)and any of its isomorphic

forms, see for instance (10)or (16), are complete dgl’s. However, DerLis not complete

in general and, even if it is, Hom(C, L)

×DerLmay fail to be so. The same applies to the

twisted products DerL

×sL and L

×DerLjust deﬁned.

The following illustrates this situation.

Example 6.1. (1) Consider the cdgl L =(



L(x, y), 0) where |x| =|y| =0. Observe that

DerL =Der

0Land deﬁne θ, γ∈Der(L)by

θ(x)=y, θ(y)=0,γ(x)=−1

2x, γ(y)= 1

2y.

A short computation shows that [γ, θ] =θand thus adn

γ(θ) =θfor any n ≥1. In

particular θlives in the kernel of the canonical map DerL →lim

←−−n≥1DerL/(DerL)n

which, therefore, prevents DerLto be complete.

(2) On the other hand, consider an odd dimensional sphere Snwhose model is the

abelian Lie algebra L =(L(x), 0) with xof degree n −1. In this case C(L) =(∧sx, 0).

Then, map(Sn

Q, Sn

Q)is modeled by Hom(C(L), L)which is an abelian Lie algebra gen-

erated by x, corresponding to 1 → x, and an element zof degree −1, corresponding to

sx → x. On the other hand DerLis an abelian lie algebra generated by idL, a degree 0

derivation which we denote by θ. Hence, in this case, (20)takes the form

Span{x, z}−→Span{x, z, θ}−→Span{θ}

where, in the middle, [θ, x] =xand [θ, z] =z. Observe then that the bracket

does not respect the usual ﬁltration in the non twisted product. Note also that,

even though Hom(C(L), L)and DerLare obviously complete as they are abelian,

Hom(C(L), L) 

×DerLis not (with respect to the usual bracket ﬁltration) as its comple-

tion is easily seen to be just Span{θ}. That is, the completion of (20) becomes

32 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Hom(C(L),L)0

−→ DerLid

−→ DerL.

This also illustrates that the completion functor is not left exact in general.

(3) Finally, with Las in (2) above, observe that the twisted dgl DerL

×sL becomes

the dgl Span{θ, sx}, with zero diﬀerential and [θ, sx] =sx, whose completion is DerL =

Span{θ}. An analogous argument shows that L

×DerLfails also to be complete with

respect to the usual ﬁltration.

The purpose of this section is to overcome these obstacles by imposing some restric-

tions which are suﬃciently mild for our goals.

From now on we ﬁx L =(



L(V), d)a connected, minimal, free cdgl and a ﬁnite ﬁltration

of Vby graded subspaces:

V=V0⊃··· ⊃ Vi⊃Vi+1 ⊃··· ⊃ Vq=0.(23)

We ﬁrst reﬁne the usual ﬁltration in Lby considering, for n ≥1and p ≥0,



Ln,p(V) = Span{a1,[a2,[...,[an−1,a

n]...∈

Ln(V),a

i∈Vαiand n

i=1αi=p}.

Observe that 

Ln,p(V) =0for p ≥nq. Then, for n ≥1and 0 ≤p ≤nq −1, deﬁne

Fn,p =

Ln,p(V)⊕

L≥n+1(V)

and note that,



L(V)=F1,0⊃F1,1⊃··· ⊃ F1,q−1

⊃F2,0⊃F2,1⊃··· ⊃ F2,2q−1

...............................................

⊃Fn,0⊃Fn,1⊃··· ⊃ Fn,nq−1

...............................................

Remark that Fn,p ranks

m=q+···+(n−1)q+p+1= (n−1)nq

2+p+ 1 (24)

in the order given by this chain of inclusions.

Deﬁnition 6.2. For n, pand mas above deﬁne

Fm=Fn,p.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 33

Proposition 6.3. The sequence {Fm}m≥1is a ﬁltration of L. Furthermore, Lis complete

with respect to this ﬁltration.

Proof. Observe that [Fn,p, Fr,s] ⊂Fn+r,p+s. Now, Fn,p and Fr,s rank respectively

(n−1)nq

2+p +1and

(r−1)rq

2+s +1in the ﬁltration order. The sum of these integers is

smaller than

(n+r−1)(n+r)q

2+p +s +1 which is the position of Fn+r,p+sin that order.

This shows that the Lie bracket is ﬁltration preserving.

On the other hand, since the diﬀerential of Lis decomposable each Fmis a diﬀerential

ideal. Moreover, we have

dF m⊂Fm+1.

We now check that Lis complete respect to this ﬁltration. Since ∩n,pFn,p =0the

natural map



L(V)−→ lim

←−−

m

L(V)/F m

is injective. Finally, write a given element in lim

←−−n,p 

L(V)/F n,p as a series,



n,p

xn,p with xn,p ∈Fn,p,

and note that this is a well deﬁned element in 

L(V) since, for each m ≥1, n,p xn,p

contains only a ﬁnite sum in 

Lm(V). This implies the surjectivity of the map above. 

Note that considering in Lthe usual ﬁltration {

L≥m(V)}m≥1for which it is complete,

the identity (L, {

L≥m(V)}m≥1) →(L, {Fm}m≥1)is a cdgl morphism.

Deﬁnition 6.4. Denote by DerL ⊂DerLthe connected sub dgl in which,

Der≥1L=Der

≥1L,

Der0L={θ∈Der0L∩ker D, θ(Vi)⊂Vi+1 ⊕

L≥2(V) for all i}.

Note that this is a well deﬁned Lie algebra and, since the diﬀerential in Lis decomposable,

it is a diﬀerential sub Lie algebra.

Using the new ﬁltration on Lwe ﬁrst construct a ﬁltration of Der0Lfor which it

becomes a cdgl. For it, note that

Der0L={θ∈Der0L∩ker D, θ(Fm)⊂Fm+1 for all m}.

Hence, we ﬁlter Der0Lby,

34 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Fn={θ∈Der0L, θ(Fm)⊂Fm+n,for all m},n≥1,(25)

and a straightforward computation proves:

Lemma 6.5. Der0Lis complete with respect to {Fn}n≥1.

To ﬁnish, we “extend” this ﬁltration on Der0Lto DerL by means of the following

procedure:

Consider ﬁrst the subspaces of Der≥1L,

Mn={θ∈Der≥1L, θ(L)⊂L≥n+1},n≥1,

and choose a reﬁnement of this sequence by constant subspaces

M1=J1,0=J1,1=···=J1,q−1⊃

M2=J2,0=J2,1=···=J2,2q−1⊃

..................................................

Mn=Jn,0=Jn,1=···=Jn,nq−1⊃

..................................................

As in (24), Jn,p ranks m =(n−1)nq

2+p +1in the order given by this chain, and deﬁne

Jm=Jn,p.

The only purpose of extracting this reﬁnement is simply to assure that,

[Fn,Jm]⊂Jm+n,n,m≥1.

Finally, for any n ≥1 deﬁne the graded vector space Enas follows:

p=⎧

⎪

⎨

⎪

⎩

Fnif p=0,

Jn−p

pif 1 ≤p≤n−1,

DerpLif p≥n.

(26)

Proposition 6.6. DerL is a cdgl with respect to the ﬁltration {En}n≥1.

Proof. Note that, E1=F1⊕Der≥1L =DerL. Also, since the diﬀerential in Lis de-

composable, DJm

p⊂Jm+1

p−1for p ≥2, and DJm

1⊂Fm+1, for all m ≥1. Finally

[Jn, Jm] ⊂Jn+mwhile, as noted above, [Fn, Jm] ⊂Jm+nfor n, m ≥1. With this data

one easily checks that {En}≥1is a ﬁltration of DerL.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 35

Since, for p ≥1and n ≥1, we have

p=DerpLif n≤p,

Jn−p

pif n>p,

it follows that, for p ≥1,

lim

←−−

(DerL/En)p= lim

←−−

DerpL/Jn

p=Der

pL.

On the other hand, for p =0, and in view of Lemma 6.5,

lim

←−−

(DerL/En)0= lim

←−−

Der0L/ Fn=Der0L, 

Next, given f:(



L(V), d)

∼

→(

L(V), d)an element of aut L, denote by

f∗:V∼

−→ V

the induced automorphism on the indecomposables. Then, we prove:

Proposition 6.7. expDer0L={f∈aut(L), (f∗−idV)(Vi) ⊂Vi+1 for al l i}.

Proof. Let θ∈Der0Land call f=eθ. Then f−idL=n≥1θn

n!which clearly satisﬁes

(f−idL)(Vi) ⊂Vi+1 ⊕

L≥2(V)for all i. Hence (f∗−idV)(Vi) ⊂Vi+1 also for all i.

Conversely, let f∈aut(L)be such that f(v) −v∈Vi+1 ⊕

L≥2(V)for all v∈Viand

all i. Deﬁne a linear morphism,

θ:V−→ L, θ(v)=

n≥1

(−1)n+1 (f−idL)n(v)

Recursively, one sees that the component of (f−idL)n(v)in 

Lm,r(V)is zero for nlarge

enough. Thus, θis well deﬁned and can be extended to a derivation θ∈Der0(L). Finally,

since the formula that deﬁnes θis that of log f, computing eθrecovers f.

Next, recall from Remark 4.1 that, for any cdgc Cand any cdgl L, Hom(C, L)is

complete with respect to the natural ﬁltration given by that of L. However, we will need

a new ﬁltration which is “compatible” with the one we just constructed for DerL.

Deﬁnition 6.8. Let Cbe a cdgc and L =(



L(V), d)be a (connected) minimal cdgl ﬁltered

by {Fm}m≥1of Deﬁnition 6.2. For m ≥1, deﬁne the non negatively graded vector space

Imby,

Im=⊕k≥0Hom(C, F m−k

k),

36 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

where Fn=Lfor n ≤0. Note that each of the terms in this direct sum contains, in

general, elements of any degree.

Proposition 6.9. The sequence {Im}m≥1is a ﬁltration of Hom(C, L)for which it becomes

a cdgl.

Proof. Clearly I1=Hom(C, L), Im+1 ⊂Imand DIm⊂Imfor all m. On the other

hand, given ϕ ∈Hom(C, Fn−i

i) ⊂In, ψ∈Hom(C, Fm−j

j) ⊂Imand c ∈Cwith

Δ(c) =ici⊗c

[ϕ, ψ](c)=±

[ϕ(ci),ψ(c

i)] ∈Fn+m−i−j

i+j.

This implies that [In, Im] ⊂In+mand therefore {Im}m≥1is a ﬁltration. A degree-

wise analogous argument to that in Remark 4.1 shows that Hom(C, L)is complete with

respect to this new ﬁltration. 

Next, in the restriction of the twisted product in (20)to

Hom(C, L)

×DerL,

consider {Jn}n≥1={In×En}n≥1with {In}n≥1of Deﬁnition 6.8 and {En}n≥1as in

(26). Then, a direct inspection proves:

Proposition 6.10. The sequence {Jn}n≥1is a ﬁltration of Hom(C, L)



×DerL for which

it becomes a cdgl. 

Furthermore, from Propositions 6.6, 6.9 and 6.10, and with respect to the correspond-

ing ﬁltrations, we deduce:

Corollary 6.11. The maps

Hom(C, L)−→ Hom(C, L)

×DerL −→ DerL

form a short exact sequence of cdgl morphisms and thus, it is a cdgl ﬁbration. 

Finally, in the twisted products

DerL 

×sL and L

×DerL,

obtained as restrictions of those in (21)and (22), we consider the respective sequences

{En×sF n}n≥1and {Fn×En}n≥1.

By procedures analogous to those used in this section one obtains:

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 37

Proposition 6.12. These sequences are ﬁltrations for which DerL

×sL and L

×DerL, are

cdgl’s. 

7. Nilpotent monoids of homotopy automorphisms and classifying ﬁbrations

Whenever Xis a simply connected ﬁnite complex with minimal model L, it is well

known (see [31,35], [2]in the ﬁberwise context or [3]for the relative case) that the

ﬁbration,

XQ−→ Baut∗

1(XQ)−→ Baut1(XQ)

is modeled by a dgl ﬁbration sequence of the form

Lad

−→ 

DerL−→ 

DerL

×sL,

where 

DerL

×sL is the restriction of the twisted product (21). Again for Xsimply con-

nected, the realization of the above ﬁbration is the simply connected cover of the universal

classifying ﬁbration

XQ−→ Baut∗(XQ)−→ Baut(XQ).

However, in general and even if Xis simply connected, this classifying ﬁbration cannot

be modeled. Indeed, as the following result shows, Baut∗(XQ)and Baut(XQ)do not lie,

in general, in the image of the realization functor. In particular, they are not nilpotent

spaces.

Proposition 7.1. For any n ≥1, neither Baut∗(Sn

Q)nor Baut(Sn

Q)have the homotopy

type of the realization of any connected cdgl.

Proof. Suppose Baut∗(Sn

Q) Lfor a given connected cdgl L. Then there is a group

isomorphism

π0aut∗(Sn

Q)∼

=π1Baut∗(Sn

Q)∼

=H0(L)

where in the latter, the group structure is given by the BCH product. However, by

Remark 4.5, π0map∗(Sn

Q, Sn

Q) ={L(x), L(x)}. Hence, π0aut∗(Sn

Q)is just the automor-

phisms of a one dimensional vector space which is identiﬁed with the multiplicative group

Q∗=Q −{0}.

Now, assume there is an isomorphism ψ:Q∗∼

→H0(L)and let a =ψ(2). Recall that,

for the BCH product, μa ∗νa =(λ +μ)afor μ, ν∈Q. Thus, given λ ∈Q∗with ψ(λ) =1

we have ψ(λ2) =1

2a ∗1

2a =a =ψ(2). Then λ2=2which is a contradiction. The same

argument also works for the non-pointed case. 

38 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Despite this result we will be able to realize distinguished classifying ﬁbrations, also

by means of cdgl’s of derivations.

Let Xbe a connected ﬁnite complex and let π0aut(X)be the group of homotopy

classes of self homotopy equivalences of X, often denoted by E(X). In the same way,

π0aut∗(X)is the group E∗(X)of based homotopy classes of pointed self homotopy

equivalences of X. Denote by

ζ:E∗(X)−→ E(X) (27)

the natural surjection which induces the bijection E∗(X)/π1(X)

∼

=E(X).

Deﬁnition 7.2. For a given subgroup G ⊂E(X)we consider the sub monoids autG(X) ⊂

aut(X)and aut∗

G(X) ⊂aut∗(X) deﬁned by

autG(X)={f∈aut(X),[f]∈G},aut∗

G(X)={f∈aut∗(X),[f]∈G}.

Note that π0autG(X) =Gwhile π0aut∗

G(X)is the subgroup G∗⊂E∗(X) consisting

of pointed homotopy classes of pointed homotopy automorphisms in aut∗

G(X). In other

terms,

G∗=ζ−1(G)={[f]∗∈E∗(X),with [f]∈G}.

Thus, note that G∗is preserved by the action of π1(X)and

G∗/π1(X)∼

=G.

Then, the evaluation ﬁbration map∗(X, X) −→ map(X, X) −→ Xrestricts to a ﬁbration

aut∗

G(X) →autG(X) →Xwhich is extended on the right to provide a ﬁbration sequence

X−→ Baut∗

G(X)−→ BautG(X).

This ﬁbration sequence, see [17], is universal with respect to ﬁbrations with ﬁber Xand

whose image of the holonomy action is contained in G. In other terms, the set

FibG(X, B) (28)

of equivalence classes of ﬁbration sequences X→E→Bover Bfor which the image of

the natural map π1(B) →E(X) lies in G, is in bijective correspondence with the set of

(free) homotopy classes {B, BautG(X)}.

The counterpart of the above in the pointed setting is the following:

Deﬁnition 7.3. Given π⊂E∗(X)a subgroup of pointed homotopy classes of pointed

equivalences, deﬁne

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 39

aut∗

π(X)={f∈aut∗(X),[f]∗∈π}.

Remark 7.4. Note that in general, given G ⊂E(X)and π⊂E∗(X), aut∗

G(X)and

aut∗

π(X)are diﬀerent submonoids of aut∗(X). In particular, for the trivial subgroup

G ={1} ⊂E(X), aut∗

1(X)is the non connected monoid of pointed maps freely homotopic

to idX. However, for π={1} ⊂E∗(X), aut∗

1(X)is the connected monoid of pointed maps

based homotopic to idX. In what follows, and to avoid confusion, we always make clear

whether the considered subgroup is taken from E(X)or E∗(X).

Recall that a ﬁbration of pointed spaces is pointed if it has a section. Then, see for

instance [17, Theorem 2.2], there exists a “universal” pointed ﬁbration,

X−→ Z−→ Baut∗

π(X) (29)

such that: the set

Fib∗

π(X, B)

of equivalence classes of pointed ﬁbrations F→E→Bover B, with ﬁber homotopically

equivalent to X, and for which the image of the natural map π1(B) →E∗(F) lies in π,

is in bijective correspondence with the set of free homotopy classes {B, Baut∗

π(X)}.

From this point on, we ﬁx a ﬁnite nilpotent complex X. We also recall that an action

of a group Gon an abelian group His nilpotent if the lower central series of the action,

Γ0⊃··· ⊃ Γi−1⊃Γi⊃...

where Γ0=Hand Γiis generated by {gh −h, g∈G, h ∈Γi−1}, is ﬁnite. That is, Γq=0

for some q.

Let ﬁrst Gbe a subgroup of E(X)which acts nilpotently on H∗(X). Then, see

[13, Theorem D], both BautG(X)and Baut∗

G(X)are nilpotent spaces and thus G

∼

π1BautG(X)and G∗∼

=π1Baut∗

G(X)are nilpotent groups. Let

Γ0=H∗(X)⊃··· ⊃ Γi−1⊃Γi⊃··· ⊃ Γq=0

be the central series of the G-action. That is, Γiis generated by f∗(α) −α, with [f] ∈G

and α∈Γi−1.

Deﬁnition 7.5. Deﬁne K⊂E(X)as the subgroup of self homotopy equivalences which

“stabilize” the above series. That is,

K={[f]∈E(X),f

∗induce the identity on Γi/Γi+1, for all i}.

Accordingly, deﬁne

40 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

K∗=ζ−1(K)={[f]∗∈E∗(X),with [f]∈K}

and note that G ⊂K⊂E(X)and G∗⊂K∗⊂E∗(X). Observe that, by [13, Theorem

D], both Kand K∗are nilpotent groups.

On the one hand the following straightforward homological reformulation of [24, The-

orem 3.3] characterizes the rationalization of K:

Theorem 7.6. The rationalization KQof Kis the subgroup of E(XQ)which stabilize the

following series,

Γ0

Q=H∗(XQ)=H∗(X)Q⊃··· ⊃ Γi−1

Q⊃Γi

Q⊃··· ⊃ Γq

Q=0.(30)

In other terms, a class [f]of E(XQ)is in KQif it acts trivially on each Γi−1

Q/Γi

Qfor all

i ≥0. Moreover, the map K→KQ, [f] → [fQ], is the rationalization. 

As Gis a nilpotent subgroup of the nilpotent group K, and rationalization is an exact

functor on nilpotent groups [20, Theorem 2.4], the rationalization GQof Gis a subgroup

of KQand the map G →GQ, [f] → [fQ], is also the rationalization.

On the other hand, a completely analogous procedure exhibits K∗

Qcontaining G∗

Qand

establishes that the map G∗→G∗

Q, [f]∗→ [fQ]∗is the rationalization.

Remark 7.7. This deﬁnes maps of monoids,

autG(X)−→ autGQ(XQ),aut∗

G(X)−→ aut∗

GQ(XQ),

both sending a map to its rationalization. Moreover, and this is crucial in the next result,

remark that, componentwise, these maps are also the rationalizations as, for each map

f:X→X, pointed in the second case,

mapfQ(XQ,X

Q)mapf(X, X)Q,map∗

fQ(XQ,X

Q)map∗

f(X, X)Q.

From all of the above, we may easily deduce:

Proposition 7.8. Let Xbe a ﬁnite nilpotent complex and G ⊂E(X)be a subgroup which

acts nilpotently on H∗(X). Then, the rationalization of the classifying ﬁbration sequence

X−→ Baut∗

G(X)−→ BautG(X) (31)

has the homotopy type of the classifying ﬁbration sequence

XQ−→ Baut∗

GQ(XQ)−→ BautGQ(XQ).(32)

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 41

Proof. Observe that the rationalization of (31)

XQ−→ Baut∗

G(X)Q−→ BautG(X)Q

is a ﬁbration sequence which lies in FibGQXQ,

BautG(X)Qand thus, by the classi-

fying property of (32), it ﬁts in a commutative diagram as follows:

XQBaut∗

G(X)QBautG(X)Q

XQBaut∗

GQ(XQ)BautGQ(XQ).

(33)

This can be extended to a homotopy commutative diagram of the form

aut∗

G(X)Q



autG(X)Q



XQBaut∗

G(X)QBautG(X)Q

aut∗

GQ(XQ)autGQ(XQ)XQBaut∗

GQ(XQ)BautGQ(XQ),

where both rows are again ﬁbration sequences and, in view of Remark 7.7, the ﬁrst two

vertical arrows are homotopy equivalences. Hence, all the vertical arrows in (33)are also

homotopy equivalences and the proposition follows. 

Now, let L =(



L(V), d)be the minimal Lie model of X. In view of Corollary 2.2,

{XQ, XQ}∗∼

={L, L}. In particular, every pointed homotopy equivalence XQ



→XQ

in E∗(XQ)is identiﬁed with a homotopy class of a quasi-isomorphism L

∼

→Lwhich is

necessarily an automorphism as Lis connected [11, Theorem 3.19].

Deﬁnition 7.9. Denote by

G⊂aut(L)/∼,

the subgroup of homotopy classes of automorphisms of Lthat corresponds to G∗

Q⊂

E∗(XQ) under the isomorphism {L, L}

∼

={XQ, XQ}∗.

As G∗

Q/π1(XQ)

∼

=GQ, by Corollary 2.5 we deduce that Gis invariant by the H0(L)

action and

G/H0(L)∼

=GQ.

Denote also,

autG(L)={ϕ∈aut(L),[ϕ]∈G},

42 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

which is clearly a subgroup of aut(L). Accordingly, for K⊂E(X)as in Deﬁnition 7.5,

deﬁne

K⊂aut(L)/∼

isomorphic to K∗

Q⊂E∗(XQ)and

autK(L)={ϕ∈aut(L),[ϕ]∈K}.

The counterpart of these groups are the following fundamental Lie algebras of deriva-

tions:

Deﬁnition 7.10. Deﬁne DerKL ⊂DerLas the connected cdgl where

DerK

≥1L=Der

≥1Land DerK

0L={θ∈Der0Lsuch that Dθ =0andeθ∈autK(L)}.

Remark 7.11. Observe that DerKLis in fact a well deﬁned cdgl: recall that L =(



L(V), d)

is the minimal model of X. By the isomorphism s−1

H∗(XQ)

∼

=V, the series (30)of

Theorem 7.6 is then of the form

V=V0⊃··· ⊃ Vi⊃Vi+1 ⊃··· ⊃ Vq=0

and therefore,

autK(L)={f∈aut(L),(f∗−idV)(Vi)⊂Vi+1 for all i}.

We then apply Proposition 6.7 to conclude that DerKLis precisely DerL of Deﬁnition 6.4

for the above ﬁltration of V. Hence, by Proposition 6.6, DerKLis a well deﬁned cdgl.

Moreover, in view of Remark 3.4, the group (DerK

0L, ∗), endowed with the BCH product,

is isomorphic to autK(L).

Deﬁnition 7.12. Deﬁne DerGL ⊂DerKLby

DerG

≥1L=Der

≥1Land DerG

0L={θ∈Der0Lsuch that Dθ =0andeθ∈autG(L)}.

Note again that the group (DerG

0L, ∗)is identiﬁed, via Remark 3.4, with autG(L)which

is a subgroup of autK(L). Thus, DerG

0Lis a subgroup of DerK

0Lfor the BCH product.

Then, the precise statement of Theorem 0.1 is:

Theorem 7.13. The maps

Lad

−→ DerGL−→ DerGL

×sL

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 43

constitute a cdgl ﬁbration sequence whose realization is homotopy equivalent to the clas-

sifying ﬁbration

XQ−→ Baut∗

GQ(XQ)−→ BautGQ(XQ).

Here, ad denotes the adjoint operator and (DerGL

×sL, D)is the twisted product (21),

i.e., sL is an abelian Lie algebra and

Dsx =−sdx +ad

x,[θ, sx]=(−1)|θ|sθ(x),x∈L, θ ∈DerGL.

Observe that combining this with Corollary 4.4 we obtain:

Corollary 7.14. The realization of the cdgl sequence

Hom(C(L),L)→Hom(C(L),L)→Lad

→DerGL→DerGL

×sL (34)

is of the homotopy type of

map∗(XQ,X

Q)→map(XQ,X

Q)ev

→XQ→Baut∗

GQ(XQ)→BautGQ(XQ).

In particular, restricting to the corresponding components in the above cdgl’s produces a

Lie model of the ﬁbration sequence,

aut∗

GQ(XQ)→autGQ(XQ)ev

→XQ→Baut∗

GQ(XQ)→BautGQ(XQ).

We now focus on a subgroup π⊂E∗(X)of pointed homotopy classes of pointed

equivalences which acts nilpotently on π∗(X). Then, by [13, Theorem C], Baut∗

π(X)is

a nilpotent complex and thus πis a nilpotent group. As in the homological case, the

rationalization πQof πis similarly described as a subgroup of E∗(XQ), this time with

the aid of [24, Theorem 3.3]. The analogs of Deﬁnitions 7.9 and 7.12 are:

Deﬁnition 7.15. Deﬁne

Π⊂aut(L)/∼,

as the subgroup of homotopy classes of automorphisms of Lwhich corresponds to πQ⊂

E∗(XQ) under the isomorphism {L, L}

∼

={XQ, XQ}∗of Corollary 2.2.

Consider also the subgroup of aut(L)given by

autΠ(L)={ϕ∈aut(L),[ϕ]∈Π},

and deﬁne DerΠL ⊂DerLby,

DerΠ

≥1L=Der

≥1Land DerΠ

0L={θ∈Der0Lsuch that Dθ =0andeθ∈autΠ(L)}.

44 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

As πQacts nilpotently on π∗(XQ)it does so on H∗(XQ)(see [19, Theorem 2.1]). Hence,

autΠ(L) ⊂autK(L)and thus, again by Remark 3.4, DerΠ

0Lis a well deﬁned subgroup

of DerK

0Lfor the BCH product.

Finally, we will also assume that πQis preserved by the action of π1(XQ), or equiva-

lently, Πis preserved by the action of H0(L). Then, Theorem 0.2 reads:

Theorem 7.16. The twisted product

L−→ L

×DerΠL−→ DerΠL

constitutes a cdgl ﬁbration sequence whose realization is homotopy equivalent to the clas-

sifying ﬁbration

XQ−→ ZQ−→ Baut∗

πQ(XQ).

Here L

×DerΠLis the restriction of the twisted product in (22). That is, both terms

are sub dgl’s and [θ, x] =θ(x)for any θ∈DerΠLand x ∈L.

Remark 7.17. Note that, with the notation and nomenclature of the classical refer-

ence [26, §7], this theorem proves in particular that the geometric bar construction

B(∗, aut∗

π(X), X)has the rational homotopy type of the realization of L

×DerΠL. Indeed

in the universal classifying ﬁbration (29), Zis precisely B(∗, aut∗

π(X), X).

We point out that the hypothesis of Theorems 7.13 and 7.16 cannot be weakened.

Let Kbe a subgroup of E∗(XQ)or E(XQ)acting nilpotently on π∗(XQ)or H∗(XQ),

and which does not necessarily arise as the rationalization of a nilpotent action on the

homotopy or homology groups of X. Then, as shown in the following example, and even

for Kabelian and Xsimply connected, Baut∗

K(XQ)and BautK(XQ)do not lie in

general in the image of the realization functor.

Example 7.18. Let X=Sn∨Sn, with n ≥2, whose minimal model is M=(L(x, y), 0)

with |x| =|y| =n −1. Since XQis simply connected, free and pointed classes coincide.

Consider the subgroup K⊂E∗(XQ) =E(XQ) generated by the homotopy class of the

automorphism ϕ:M∼

→Mgiven by ϕ(x) =x +yand ϕ(y) =y. Obviously, Kacts

nilpotently on both π∗(XQ)and H∗(XQ)as its central series van ish es at the second

stage. Now, if Lis a cdgl for which L Baut∗

K(XQ)it follows that H0(L), endowed

with the BCH group structure, must coincide with Kwhich is in turn isomorphic to Z.

However, a similar argument to the one in Proposition 7.1 shows that this is not possible.

The same applies to BautK(XQ).

Remark 7.19. Although all spaces involved in Theorems 7.13 and 7.16 are connected and

nilpotent, it is important to stress that it is absolutely essential to work in cdgl and

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 45

not in the model category deﬁned in [27]whose objects are connected (not complete)

dgl’s. Among others, this is the main obstruction: ﬁbrations in this model structure are

surjective morphisms but there are ﬁbrations of nilpotent complexes which cannot be

modeled by surjective morphisms of connected dgl’s. A simple example of this kind of

ﬁbrations is given by the universal cover of the circle R →S1. However, it can be modeled

by the following cdgl ﬁbration in which we must allow elements of negative degrees in

the source:

ϕ:(



L(x, y),d)−→ (L(x),0),ϕ(x)=x, ϕ(y) = 0 where |x|=0,dx=y.

Note that the ﬁber of this morphism is given by,

ker ϕ=(



L(an)n≥0,d),a

n=ad

x(y).

Since each anis a Maurer-Cartan element the realization of this cdgl is, as it should,

a discrete countable space. Other examples of this kind are given by the maps XQ→

Baut∗

GQ(XQ)and XQ→Zof Theorems 7.13 and 7.16 respectively, whose homotopy

ﬁbers autGQ(XQ)and aut∗

πQ(XQ)are non-connected.

8. Modeling free and pointed classifying ﬁbrations

In this section we complete the proofs of Theorems 7.13 and 7.16. We therefore adopt

(and ﬁx) the notation in §7and start with the following:

Lemma 8.1. The maps

Lad

−→ DerGL−→ DerGL

×sL,

constitute a well deﬁned cdgl sequence.

Proof. We begin by proving that DerGLis in fact a well deﬁned cdgl. First, we see that

DerG

0Lis a complete Lie algebra. Up to this point, it is simply a subgroup of DerK

0Lfor

the BCH product. However, recall from Remark 7.11 that DerK

0Lcoincides with Der0L

of Deﬁnition 6.4 which, by Lemma 6.5, is complete with respect to the ﬁltration {Fn}n≥1

in (25). For each n ≥1denote Fn

G=Der

0L ∩Fnand note that this a subgroup of DerG

for the BCH product. Moreover, initially just as groups,

DerG

0L= lim

←−−

n≥1

DerG

0L/ Fn

and thus DerG

0Lis a pronilpotent group.

We check now that DerG

0Lis a 0-local group. Equivalently, we see that autG(L)is

0-local. First, for each k≥1, the map ϕ → ϕkof autG(L)is injective since this is a

46 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

subgroup of autK(L)which is 0-local. For the same reason, given ϕ ∈autGLand k≥1,

there exists ξ∈autK(L)such that ξk=ϕ. We ﬁnish by checking that ξlives in autG(L).

Since Gis 0-local there exists ψ∈autG(L)such that ψk∼ϕ. Thus, ξk∼ψkand, since

Kis 0-local, it follows that ξ∼ψso that ξ∈autG(L).

The pronilpotent and 0-local character of DerG

0Llet us apply Theorem 3.3 to conclude

that this is a (Malcev) complete subgroup of DerK

0Land thus, by the Malcev category

isomorphism (see §3), DerG

0Lis a complete Lie algebra with respect to the ﬁltration Fn

We check now that D(Der1L) ⊂DerG

0Lby observing that,

eDη ∼id L, for any η∈Der1L.

On the one hand, since the diﬀerential in L =(



L(V), d)is decomposable (Dθ)(V) ⊂



L≥2(V)and thus eDη is always well deﬁned. On the other hand, deﬁne a derivation ˜ηin

∧(t, dt)

⊗Lby

˜η(tn⊗x)=tn+1 ⊗η(x),˜η(tndt ⊗x)=−tn+1dt ⊗η(x),n≥0,x∈L,

and consider the cdgl morphism

Φ=eD˜η◦ι:L→∧(t, dt)

⊗L

where ι(x) =1 ⊗x. A straightforward computation shows that ε0◦Φ =id

Land ε1◦Φ =

eDη so that eDη ∼idL.

All of the above shows that DerGL, and thus DerGL

×sL, are sub dgl’s of DerKLand

DerKL

×sL respectively, which are complete by Propositions 6.6 and 6.12. Hence, DerGL

and DerGL

×sL are also complete for the induced ﬁltrations and all the objects in the

statement are cdgl’s.

We ﬁnish by checking that the image of the adjoint operator lies in DerGL. By deﬁ-

nition, this amounts to say that, for each x ∈L0, the homotopy class of eadxlives in G.

But this is trivial as, with the notation in Deﬁnition 2.3, [eadx] =[x]•[idL] ∈Gsince G

is closed by the action of H0(L). 

Next, consider the restriction of (20)to

0→Hom(C(L),L)−→ Hom(C(L),L)

×DerGL−→ DerGL→0.(35)

Recall that both terms in the middle are sub dgl’s and [θ, f] =θ◦fwith θ∈DerGLand

f∈Hom(C(L), L).

Fix the MC element q:C(L) →Lof Hom(C(L), L) deﬁned in Remark 4.5. With

the notation in (12), q=idLas it represents the homotopy class of the identity in

{L, L}/H0(L)which, in turn, is identiﬁed with idXQ.

Note that (35)is a cdgl ﬁbration sequence, so is its realization,

Hom(C(L),L)−→Hom(C(L),L)

×DerGL−→DerGL.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 47

We restrict this ﬁbration, in which the base DerGLis connected, to the path component

of the total space Hom(C(L), L)



×DerGLcontaining the 0-simplex q. We then obtain

another ﬁbration

F−→  Hom(C(L),L)

×DerGLq−→  DerGL,(36)

for which we identify its ﬁber and total space in the next results.

Lemma 8.2. F=[ϕ]∈G/H0(L)Hom(C(L), L)ϕ.

Proof. Observe that Fdoes not have to be connected as it is formed by all the path

components of Hom(C(L), L)lying in Hom(C(L), L)



×DerGLq. That is, and with

notation in Remark 4.5,

F=ϕHom(C(L),L)ϕ,

where ϕ:L →Lruns through the homotopy classes of cdgl morphisms such that the

MC element ϕ=ϕq of Hom(C(L), L)is gauge related to qwhen considered as MC

elements of Hom(C(L), L)



×DerGL. That is, ϕand qrepresent the same element in



MC(Hom(C(L), L) 

×DerGL).

Now, again in view of Remark 4.5, ϕand qare already gauge related in Hom(C(L), L)

if and only if ϕ ∼idL. If this is not the case, but still ϕis gauge related to qin

Hom(C(L), L) 

×DerGL, there must be a derivation,

θ∈DerG

0Lsuch that θGq=ϕ.

That is,

eadθ(ϕ)−eadθ−1

adθ

(Dθ)=q.

Since θis a cycle and ϕ=ϕq, this becomes,

eθ=ϕ, that is, ϕ∈autGL.

Conversely, given an automorphism ϕ ∈autGLconsider θ=logϕ ∈DerG

0Land one

easily checks, as above, that θGq=ϕ.

Lemma 8.3. (Hom(C(L), L) 

×DerGL)qL.

Proof. Consider the map,

γ:L−→ (Hom(C(L),L)

×DerGL, Dq),γ(x)=−x+ad

x,x∈L.

48 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Note that in view of (10), Hom(C(L), L)

∼

=Hom(C(L), L)

×L. Also, as shown in

Lemma 8.1, adx∈DerGLfor any x ∈L. Thus, γis well deﬁned.

On the other hand, recall from (10)and (35)that, in

Hom(C(L),L)

×DerGL∼

=Hom(C(L),L)

×L

×DerGL,

the diﬀerential Dqis deﬁned as:

Dqf=Df +[q, f ],f∈Hom(C(L),L),

Dqx=dx +[q, x]=dx −(−1)|x|adx◦q, x ∈L,

Dqθ=Dθ +[q, θ]=Dθ −(−1)|θ|θ◦q, θ ∈DerGL.

Thus, a simple computation shows that γcommutes with diﬀerentials. We then show

that the component of γat 0,

γ:L

−→ (Hom(C(L),L)

×DerGL, Dq)0=(Hom(C(L),L)

×DerGL)q

is a quasi-isomorphism.

We recall again that q=idLwith the notation in (12). In this particular case, the

cdgl isomorphism in (15)and the quasi-isomorphism of chain complexes in (18) become

respectively,

Γ: s−1Derα(LC(L),L)∼

−→ (Hom(C(L),L),D

q)andα∗:DerL

−→ Derα(LC(L),L).

The composition Γ ◦s−1α∗is then a quasi-isomorphism of chain complexes

DerL

−→ (Hom(C(L),L),D

which trivially extends, by the identity on L

×DerG, to a quasi-isomorphism

(s−1DerL

×L

×DerGL, 

D)

−→ (Hom(C(L),L)

×L

×DerGL, Dq)

where, on the left,



Ds−1η=−s−1Dη, η ∈DerL,



Dx =dx −s−1adx,x∈L,



Dθ =Dθ −s−1θ, θ ∈DerGL.

Obviously γfactors through this map,

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 49

Lγ

(s−1DerL

×L

×DerGL, 



(Hom(C(L),L)

×L

×DerGL, Dq),

where, again, γ(x) =−x +ad

x. It is enough then to check that the component of γat 0,

γ:L

−→ (s−1DerL

×L

×DerGL, 

D)0

is a quasi-isomorphism. Let s−1η+x +θbe a 

D-cycle in s−1DerL

×L

×DerGLof non-

negative degree. That is, dx =Dθ =0and Dη +adx+θ=0. Then,

s−1η+x+θ+γ(x)=s−1η+θ+ad

x=−

Dη.

This shows that H(γ)is surjective. Finally, if x ∈Lis a cycle for which γ(x) =

D(s−1η+

y+θ), then −dy =xand H(γ)is also injective.

Furthermore, observe that the projection,

ρ:(Hom(C(L),L)

×DerGL)q

−→ L(37)

is a retraction of γand therefore, it is also a quasi-isomorphism. 

As a crucial consequence of the past results we get:

Proposition 8.4. There exists a ﬁbration sequence

Fξ

−→  Lad

−→  DerGL

such that ξis homotopy equivalent to the evaluation ﬁbration ev: autGQ(XQ) →XQ.

Proof. With γas in Lemma 8.3, consider the commutative diagram

(Hom(C(L),L)

×DerGL)qDerGL.



whose realization

FHom(C(L),L)

×DerGLqDerGL.

L



γ

ad

50 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

exhibits a factorization of adas a weak equivalence followed by the ﬁbration in (36).

Next, deﬁne ξas the composition

Hom(C(L),L)

×DerGLq



ρ

L

with ρas in (37). But, by Lemma 8.2, the restriction of ξto each path component of F,

ξ:Hom(C(L),L)ϕ−→  L,ϕ∈autG(L),

is precisely the realization of the projection,

Hom(C(L),L)ϕ−→ L.

However, in view of Proposition 4.6, the realization of this cdgl morphism is homotopy

equivalent to the evaluation map

ev: mapf(XQ,X

Q)−→ XQ

with [f] ∈GQcorresponding to the homotopy class of ϕ. That is, we have a homotopy

commutative diagram,

autGQ(XQ)ev XQ



ξL.



Remark 8.5. In the proof of Theorem 7.13 we will also need the following observation of

general nature: Let

0→ker p−→ Mp

−→ L→0

be a ﬁbration of connected cdgl’s and

ker p−→Mp

−→  L

the realization ﬁbration. Let x ∈L0, y∈M0with p(y) =x, and consider the automor-

phism eady:kerp

∼

−→ ker p. Its realization

eady:ker p

−→  ker p

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 51

is precisely, up to homotopy, the homotopy equivalence of the ﬁber induced by the ele-

ment [x] ∈π1L. In view of the deﬁnition of the holonomy action for Kan ﬁbrations (see

for instance [25, Corollary 7.11]), this is a simple exercise using just the deﬁnition of the

realization functor and the identiﬁcation of the elements xand ywith the corresponding

1-simplices of Land M.

In other words, the natural map π1L →Eker psends [x] ∈H0(L)to the homotopy

class of eady.

We are now able to complete the:

Proof of Theorem 7.13.We ﬁrst check that

Lad

−→ DerGL−→ DerGL

×sL, (38)

is a ﬁbration sequence. For it, and as in the classical, simply connected case [35,

§VII.4(1)], consider the twisted product of Land DerGL

×sL,

L−→ (L

×DerGL

×sL, 

D)−→ DerGL

×sL (39)

where, in the middle,

[θ, x]=θ(x),[sx, y]=0,

Dsx =−sdx−x+adx,

Dθ =Dθ, x, y ∈L, θ ∈DerGL.

In view of §6, this is a cdgl ﬁbration.

On the one hand, since the inclusion i:Der



→(L

×DerGL

×sL, 

D)is a quasi-

isomorphism, the diagram

(L

×DerGL

×sL, 

DerGL



DerGL

×sL

(40)

is a factorization of DerGL →DerGL

×sL as a weak equivalence followed by a ﬁbration.

On the other hand, the morphism

:(L

×DerGL

×sL, 

D)

−→ DerGL,

(x)=ad

x,(θ)=θ, (sx)=0,x∈L, θ ∈DerGL,

is a left inverse of iand thus, it is also a quasi-isomorphism which makes commutative

this diagram,

52 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

(L

×DerGL

×sL, 





Lad DerGL.

These two facts exhibit (38)and (39)as homotopy equivalent ﬁbration sequences.

Next we check that the realization of (38),

Lad

−→  DerGL−→DerGL

×sL,(41)

is a ﬁbration in FibGQ(XQ, DerGL

×sL), see (28). For it, note that,

π1DerGL

×sL=H0(DerGL

×sL)=H0(DerGL)/Im H0(ad).(42)

Hence, by Remark 8.5 and since exp(DerGL) =aut

G(L), the image of the holonomy ac-

tion π1DerGL

×sL →EL

∼

=E(XQ)is precisely G/H0(L)

∼

=GQas required. Therefore,

(41)is obtained as the pullback of the universal ﬁbration over a certain map,

XQBaut∗

GQ(XQ)BautGQ(XQ)

L



adDerGL DerGL

×sL.

Finally, by Proposition 8.4, this diagram ﬁts in a larger one

autGQ(XQ)ev XQBaut∗

GQ(XQ)BautGQ(XQ)



ξL



adDerGL DerGL

×sL,

where, again, both arrows are ﬁbration sequences. Thus, the last two vertical maps are

homotopy equivalences, and the proof is complete. 

We now turn to the proof of Theorem 7.16 for which the analogue of Lemma 8.1 reads:

Lemma 8.6. The maps

L−→ L

×DerΠL−→ DerΠL,

constitute a well deﬁned cdgl sequence.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 53

Proof. The same argument in the proof of Lemma 8.1 shows that DerΠLis a subdgl of

DerKL. Therefore, by Propositions 6.6 and 6.12 respectively, DerΠLand L

×DerΠLare

well deﬁned cdgl’s. 

Proof of Theorem 7.16.To avoid excessive notation we denote by autπQ(XQ)the

monoid autζ(πQ)(XQ)with ζas in (27). That is,

autπQ(X)={f∈aut(XQ),[f]∈ζ(πQ)}.

In view of Deﬁnitions 7.2 and 7.3 note that, since πQis preserved by the action of

π1(XQ),

aut∗

ζ(πQ)(XQ)=aut

∗

πQ(XQ).

On the other hand, invoking again [19, Theorem 2.1], πQand therefore ζ(πQ)act

nilpotently on H∗(XQ). Hence, we may apply Theorem 7.13 to obtain a homotopy com-

mutative diagram,

Baut∗

πQ(XQ)pBautπQ(XQ)

DerΠL



DerΠL

×sL



where pis the classifying ﬁbration. On the other hand, combining [26, Proposition 7.8]

and [17, Remark 1.2], we see that the classifying ﬁbration ZQ→Baut∗

πQ(XQ)in Theo-

rem 7.16 sits in a homotopy pullback,

Baut∗

πQ(XQ)BautπQ(XQ)

ZQBaut∗

πQ(XQ).

Thus, since the realization functor preserves homotopy limits, the above square is ho-

motopy equivalent to the realization of the cdgl homotopy pullback of the morphisms

DerΠL−→ DerΠL

×sL ←− DerΠL

To compute this homotopy pullback factor any of the above morphisms as in (40)and

then take the pullback of the resulting ﬁbration (L

×DerΠL

×sL, 

D) →DerΠL

×sL and

DerΠL −→ DerΠL

×sL,

54 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

(L

×DerΠL

×sL, 

D)DerΠL

×sL

L

×DerΠLDerΠL.

To conclude note that the realization of L, the ﬁber of the bottom morphism, must

be homotopy equivalent to XQ, the homotopy ﬁber of the classifying ﬁbration Z→

Baut∗

πQ(XQ). 

9. Some consequences and examples

In this section, and unless explicitly stated otherwise, we adopt the notation in §7

and the assumptions in Theorems 7.13 and 7.16.

The ﬁrst immediate consequence, which is Corollary 0.3, is a description of the ratio-

nalization of πand G. In what follows and as usual, H0is considered as a group with

the BCH product.

Theorem 9.1. πQ∼

=H0(DerΠL)and GQ∼

=H0(DerGL)/Im H0(ad).

Proof. Trivially, πQ∼

=π1BautπQ(XQ)

∼

=π1DerΠL

∼

=H0(DerΠL). The second asser-

tion is also immediate in view of (42). 

Remark 9.2. Observe that [30, Proposition 12], is the particular instance of Theorem 9.1

for the distinguished subgroups EH(XQ) ⊂E(XQ)and Eπ(XQ) ⊂E∗(XQ)of those

classes that induce the identity on H∗(XQ)and π∗(XQ) respectively. In this case DerG

consists of derivations θcommuting with the diﬀerential and such that eθis an automor-

phism of L =(



L(V), d) inducing the identity on V. That is, θ(V) ∈

L≥2(V). Analogously,

DerΠ

0Lare those derivations θcommuting with the diﬀerential and such that eθinduces

the identity on H∗(L).

Another immediate application concerns the homotopy nilpotency of aut∗

πQ(XQ)and

autGQ(XQ). Recall that given an H-group Y, its nilpotency nil Y, is the least integer

n ≤∞for which the (n + 1)th homotopy commutator of Yis homotopically trivial. On

the other hand, for any dgl Lwe denote by nil Lthe usual nilpotency index. Then:

Proposition 9.3. nil aut∗

πQ(XQ) =nilH(DerΠL)and nil autGQ(XQ) =nilH(DerGL

×sL).

Proof. By [30, Theorem 3], nil aut∗

πQ(XQ)and nil autGQ(XQ)coincide, respectively, with

the iterated Whitehead product length of π∗Baut∗

πQ(XQ)and π∗BautGQ(XQ). Finally,

apply Theorems 7.13 and 7.16 taking into account that, as Lie algebras, H∗(M)

∼

π∗+1Mfor any connected cdgl M[11, §12.5.2]. 

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 55

We now consider, again for Xnilpotent, the monoid aut1(X)of self equivalences

homotopic to idX, which corresponds to choosing G ⊂E(X)the trivial group. Note (see

Deﬁnition 7.2 and Remark 7.4) that the submonoid aut∗

1(X) ⊂aut1(X)is not connected

in general. In this case, Theorem 7.13 asserts that the rational homotopy type of the

classifying ﬁbration

X−→ Baut∗

1(X)−→ Baut1(X) (43)

is modeled by the cdgl ﬁbration sequence

Lad

−→ DerIL−→ DerIL

×sL,

where I ⊂aut L/ ∼is the subgroup of homotopy classes of automorphisms for which the

orbit group is trivial: I/H0(L) ={1}. In this particular instance:

Proposition 9.4.

(i) DerIL =Der

≥1L ⊕R0where R0=D(Der1L) +adL0.

(ii) DerIL

×sL 



DerL

×sL where, as usual, this is the 1-connected cover of DerL

×sL.

Note that 

DerL

×sL≥2=(DerL

×sL)≥2and



DerL

×sL1={(θ, sx)∈Der1L×sL0,Dθ=−adx}.

Proof. (i) By Theorem 9.1, H0(DerIL)/Im H0(ad) =0, i.e. H0(DerIL) =ImH0(ad).

That is, for any x ∈DerI

0L, [x] =[ady]for some y∈L0. In other words x =Dφ +ady

for some φ ∈Der1L. This translates to DerI

0L =R0.

(ii) In view of (i) this is straightforward. One can also argue that, as Baut1(X)is

simply connected, the realization of DerIL

×sL is of the homotopy type of its universal

cover. To ﬁnish, recall from §1that taking 1-connected covers commutes with realization

and note that the 1-connected cover of DerIL

×sL is precisely



DerL

×sL.

Interesting consequences of this result are the following in which aut1(L)denote the

group of automorphisms of Lhomotopic to the identity.

Corollary 9.5. For any connected, minimal cdgl Lthe exponential restrict to a group

isomorphism

exp: D(Der1L)∼

−→ aut1(L).

Proof. The exponential exp : DerI

∼

→autI(L) induces an isomorphism

exp: DerI

0L/ ad L0

∼

−→ autI(L)/ead L0.

56 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

However, by (i) of Proposition 9.4, DerI

0L/ ad L0=D(Der1L). On the other hand, via the

action of H0(L)on DerI

0L(see Deﬁnition 2.3), autI(L)/ead L0is precisely aut1(L). 

Remark 9.6. The referee kindly brought up to our attention that this result is the combi-

nation of Propositions 6.3 and 6.5 of [34]in the “cdgl context”. Indeed, given a connected

Sullivan algebra Athese results assert that the automorphisms of Awhich are homotopic

to the identity are precisely the inner automorphisms exp D(Der1(A).

We easily deduce that the logarithm takes homotopic automorphisms to homologous

derivations:

Corollary 9.7. Let G ⊂aut(L)be a complete subgroup. Two automorphisms f, g∈Gare

homotopic if and only if log(f) ∗− log(g)=Dη with η∈Der1L.

Proof. f∼gif and only if fg−1∼idL. By Corollary 9.5 this amounts to say that

log(fg−1) =log(f) ∗− log(g)is in D(Der1L). 

Remark 9.8. Observe that, when Xis assumed to be simply connected, we recover from

Proposition 9.4 the classical result with which we started Section 7. Indeed, in this case,

aut∗

1(X)is also connected and thus (43)is a ﬁbration sequence of simply connected

spaces. On the other hand, Lis just the classical 1-connected Quillen minimal model of

X. Hence, applying Proposition 9.4 we deduce that,

DerIL=



DerL, DerIL

×sL =



DerL

×sL,

and thus, (43)is modeled by

Lad

−→ 

DerL−→



DerL

×sL.

Next, recall that given G ⊂E(X)and π⊂E∗(X) there are ﬁbrations

aut1(X)−→ autG(X)−→ Gand aut∗

1(X)−→ aut∗

π(X)−→ π(44)

which extend to

Baut1(X)−→ BautG(X)−→ BG and Baut∗

1(X)−→ Baut∗

π(X)−→ Bπ. (45)

Here, according to Deﬁnition 7.3 and Remark 7.4, aut∗

1(X)is the connected monoid of

pointed maps based homotopic to idX.

Then, under the conditions of Theorems 7.13 and 7.16, we obtain the following, which

in particular proves Corollary 0.4:

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 57

Theorem 9.9. The ﬁbration sequences

Baut1(XQ)→BautGQ(XQ)→BGQand Baut∗

1(XQ)→Baut∗

πQ(XQ)→BπQ

have, respectively, the homotopy type of the realization of the cdg l ﬁbrations,



DerL

×sL →DerGL

×sL →(DerGL

×sL)/(



DerL

×sL)

and



DerL→DerΠL→DerΠL/ 

DerL.

Remark 9.10. A short computation let us observe that

(DerGL

×sL)/(



DerL

×sL)=Der

0L⊕R1and DerΠL/

DerL=Der

0L⊕S1

where R1and S1denote a complement of the cycles in degree 1of DerGL

×sL and DerΠL

respectively.

Proof. Note that the ﬁbrations sequences in (45)can also be obtained by ﬁbring

BautG(X)and Baut∗

π(X)over their ﬁrst Postnikov stage. On the other hand, given

Ma connected cdgl and n ≥0, consider the cdgl ﬁbration

M>n ⊕Zn−→ M−→ M/(M>n ⊕Zn)

where Zn⊂Mnis the subspace of cycles. Then [11, Proposition 12.43], the realization

of this sequence is of the homotopy type of the ﬁbration of Mover its nth Postnikov

stage. The result follows from choosing n =1and Meither DerGL

×sL or DerΠL.

Indeed, for M=Der

GL

×sL, it follows that M>1⊕Z1=



DerL

×sL. In the same way, for

M=Der

ΠL, one checks that M>1⊕Z1=

DerL.

The ﬁbrations connecting (44)and (45),

autG(X)−→ G−→ Baut1(X)and aut

∗

π(X)−→ π−→ Baut∗

1(X),

can also be modeled. For it, consider ﬁrst the dgl twisted product,

Hom(C(L),L)−→ Hom(C(L),L)

×DerL

×sL −→ DerL

×sL,

deﬁned as follows: on the one hand, the structure in Hom(C(L), L)

×DerLis the usual,

i.e., that of (20). On the other hand, considering the isomorphism Hom(C(L), L)

∼

Hom(C(L), L)

×Lin (10), deﬁne

58 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

Dsx =x−sdx +ad

x,[sx, f ]=0,[θ, sx]=(−1)|θ|sθ(x),[sx, y]=1

2s[x, y],

for x, y∈L, f∈Hom(C(L), L), and θ∈DerL. Note that s[x, y] =[sx, y] +(−1)|x|[x, sy]

which facilitates checking that Drespects the bracket. As warned in §6this may not be

a cdgl sequence but the restriction to

Hom(C(L),L)−→ Hom(C(L),L)



DerL

×sL −→



DerL

×sL

is so, as the left hand side is 1-connected and thus, complete. Therefore, its realization

Hom(C(L),L)−→Hom(C(L),L)



DerL

×sL−→



DerL

×sL

is a ﬁbration and thus, by restricting the total space to the following particular set of

path components we get a ﬁbration,

F−→ [ϕ]∈G/H0(L)Hom(C(L),L)



DerL

×sLϕ−→ 



DerL

×sL.(46)

Then we prove (cf. [2, Theorem 1.1]):

Proposition 9.11. The ﬁbration (46) has the rational homotopy type of

autG(X)−→ G−→ Baut1(X).

Proof. An analogous argument to that of Lemma 8.2 shows that

F=[ϕ]∈G/H0(L)Hom(C(L),L)ϕ,

which, in view of §4, has the homotopy type of autGQ(XQ).

Now, by construction, the total space of (46)has as many path components as the

order of GQ. Finally, an analogous argument to that of Lemma 8.3 proves that, for each

ϕ,

Hom(C(L),L)



DerL

×sLϕ0.

This shows that each of the components is homotopically trivial and the proposition

follows. 

On the other hand, consider the restriction of the twisted product (20)to

Hom(C(L),L)−→ Hom(C(L),L)

×

DerL−→ 

DerL

which, in view of §6, stays in cdgl. Its realization is a ﬁbration

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 59

Hom(C(L),L)−→Hom(C(L),L)

×

DerL−→



DerL

and we restrict again to certain path components of the total space to obtain another

ﬁbration,

F−→ [ϕ]∈ΠHom(C(L),L)



DerLϕ−→ 



DerL.(47)

Similar arguments to those of Proposition 9.11 prove:

Proposition 9.12. The ﬁbration (47) has the rational homotopy type of

aut∗

π(X)−→ πQ−→ Baut∗

1(X).

We ﬁnish with two examples which cover a wide spectrum. We see how any ﬁnitely

generated rational group of nilpotency index ncan be realized as a subgroup of self ho-

motopy equivalences of the rationalization of a suitable ﬁnite complex, acting nilpotently

on the homology of the complex with the same nilpotency index. The corresponding clas-

sifying group is explicitly described in terms of derivations. On the other hand, any such

group can also be realized as a subgroup of self homotopy equivalences acting nilpotently

on the homotopy groups of the complex with the same nilpotency index.

For that we ﬁrst recall how to describe in simple terms the exponential and the

logarithm in the ﬁnitely generated nilpotent case. Denote by T(n)the group of n ×n

strictly triangular matrices with rational entries, i.e., matrices (mij)with mij =0if i ≤j.

In T(n)we consider the Lie bracket given by commutators. On the other hand, denote

by U(n)the group of n ×nunitriangular matrices also over Q, that is, lower triangular

matrices where all entries in the diagonal are 1. Then, any ﬁnitely generated nilpotent 0-

local group and any ﬁnitely generated nilpotent Lie algebra can be respectively embedded

in U(n)and T(n)so that their logarithm and exponential maps are just the restriction

of the classical bijections

U(n)∼

log

T(n).

exp

Example 9.13. Let Mbe a ﬁnitely generated nilpotent Lie algebra, with nilpotency index

less than or equal to n, and concentrated in degree 0. By the above observation, Mcan

be embedded in T(n). Consider the ﬁnite nilpotent complex X=∨n

j=1Smwith m >1,

whose minimal model is L =(L(y1, ..., yn), 0) where |yj| =m −1for all j. Note that,

for degree reasons, any derivation θ∈Der0Lis necessarily of the form

θ(yj)=



k=1

λjk yk,λ

jk ∈Q.

That is, θis identiﬁed with the matrix (λjk) considering

60 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

⎛

⎝

θ(y1)

θ(yn)⎞

⎠=θ⎛

⎝

⎞

⎠.(48)

Hence, Mis a sub Lie algebra of Der0(L)which, being in T(n), it determines a decreasing

ﬁltration of length at most nof V= Span{y1, ..., yn}

∼

=s−1

H∗(XQ),

V=V0⊃··· ⊃ Vi⊃Vi+1 ⊃··· ⊃ Vq=0,(49)

where Vi=M(Vi−1), i ≥1.

On the other hand, if we denote by G =exp(M), we may also identify any matrix

ϕ =eθ∈Gwith an automorphism of Las in (48):

ϕ:L∼

−→ L, ⎛

⎝

ϕ(y1)

ϕ(yn)⎞

⎠=ϕ⎛

⎝

⎞

⎠.

Note that aut(L) =aut(L)/ ∼so that G =aut

G(L)is identiﬁed with the rational sub-

group Gof E(XQ)whose action in H∗(XQ) produces, as central series, the suspension of

the ﬁltration (49)of s−1

H∗(XQ)(see Deﬁnitions 7.5 and 7.9). That is, Gacts nilpotently

on H∗(XQ).

In other terms, M=Der

0Land, by Theorem 7.13, the sequence

Lad

−→ DerGL−→ DerGL

×sL

is a Lie model of the classifying ﬁbration

XQ−→ Baut∗

G(XQ)−→ BautG(XQ).

Proceeding exactly in the converse direction, any nilpotent rational group Gof E(XQ)

can be identiﬁed with a rational subgroup of U(n)and thus, with a subgroup G =aut

G(L)

of automorphisms of L. In turn, this determines the nilpotent Lie algebra M=log(G) ⊂

T(n)so that DerG

0L =M.

Just to illustrate how to proceed in a particular instance, choose for example

M=T(3) = 000

α00

βγ0, α,β,γ ∈Q

and let X=Sm∨Sm∨Sm, with m >1whose minimal model is L =(L(y1, y2, y3), 0).

Every θ∈Macts in s−1

H∗(X)by θ(y1) =0, θ(y2) =αy1and θ(y3) =βy1+γy2. This

determines the ﬁltration

s−1

H∗(X) = Span{y1,y

2,y

3}⊃Span{y1,y

2}⊃Span{y1}⊃0.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 61

Then, the group G =aut

G(L) =expM=U(3) is given by the automorphisms

{ϕ:L∼

→L, ϕ(y1)=λy1,ϕ(y2)=λy1+y2,ϕ(y3)=μy1+ρy2+y3λ, μ, ρ ∈Q},

and is identiﬁed with a subgroup G ⊂E(XQ)which acts nilpotently on H∗(XQ)with

nilpotency index 3. Note that M=Der

0L.

Example 9.14. We consider now the dual setting: let Πbe a ﬁnitely generated, nilpotent,

rational group, with nilpotency index less than or equal to n ≥1. By the observation

above, Πis embedded in U(n). Consider the complex Y=n

j=1 Smwith m ≥1odd

whose Sullivan minimal model is A =(∧V, 0) with V= Span{x1, ..., xn}concentrated

in degree m. Then, any map YQ→YQis necessarily modeled by a cdga morphism

ψ:A →Awhich might be identiﬁed with an n ×nmatrix by imposing

⎛

⎝

ψ(x1)

ψ(xn)⎞

⎠=ψ⎛

⎝

⎞

⎠.

Note also that aut(A) =aut(A)/ ∼. Hence, the group Πis identiﬁed with a subgroup

πof E∗(YQ)acting nilpotently on π∗(YQ)

∼

=Vand thus, acting also nilpotently on

H∗(YQ)

∼

=A. Observe that the action of π1(YQ)(which is trivial or abelian if Yis a

torus) on {YQ, YQ}∗is trivial and thus E∗(YQ) =E(YQ).

By [11, Theorem 10.2] the cdgl



L(A)

is a Lie model of Ywhich turns out to be minimal. Indeed, in view of the explicit

deﬁnition of Lin (5), denote

yi1...is=s−1(xi1...x

is)#,with 1 ≤i1<···<i

s≤n1≤s≤n,

and observe that



L(A)=(



L(yi1...is),d),

where the diﬀerential is quadratic and given as follows: ﬁxed integers 1 ≤i1<···<i

s≤

nand let Ebe the set of decompositions of {i1, ..., is}in two disjoint tuples, {j1, ..., jp}

and {k1, ..., kq}, with j1<··· <j

pand k1<···<k

q. Then,

d(yi1...is)=1

2

εE[yj1...jp,y

k1...kq],

where εEdenotes the sign of the permutation

62 Y. Félix et al. / Advances in Mathematics 402 (2022) 108359

i1,...,i

s→ j1,...,j

p,k

1,...,k

Remark that, whenever Yis not the torus, i.e., m >1, then 

L(A) =L(A)as

no generator of degree 0 appears. Call L =

L(A)and note that the map ψ→

homotopy class of 

L(ψ) deﬁnes a group isomorphism

aut(A)∼

=aut(L)/∼(50)

which exhibits Πas a subgroup of aut(L)/ ∼. We may then consider autΠ(L), which

is a complete, possibly non nilpotent group, and thus diﬀerent from Πin general. By

Theorem 7.16 the corresponding cdgl DerΠ(L)is a Lie model of Bautπ(YQ).

For instance, choose

Π=U(2) = 10

λ1,λ∈Q(51)

which is isomorphic to Q, and corresponds to a group πof self homotopy equivalences

of the rational torus TQ=S1

Q×S1

Qwhose Sullivan minimal model is A =(∧(x1, x2), 0).

As explained before, the action of Πon Span{x1, x2}, of nilpotency index 2, corresponds

to an action of πon π∗(TQ)which has the following central series:

π∗(TQ)∼

=Span{x1,x

2}⊃Span{x1}⊃0.

As noted, the minimal Lie model of TQis the cdgl

L=

L(A)=(



L(y1,y

2,y

12),d),|y1|=|y2|=0,|y12|=1,dy

12 =[y1,y

2].

Another easy computation shows that for every ψ∈Πas in (51), 

L(ψ)is the automor-

phism of Lgiven by the 3 ×3matrix

B=100

λ10

001

.That is, ⎛

⎜

⎝

L(ψ)(y1)



L(ψ)(y2)



L(ψ)(y12)

⎞

⎟

⎠=By1

y12 .(52)

Now, in view of (50), any automorphism in autΠ(L)is homotopic to 

L(ψ)for some

ψ∈Π. Hence, by Corollary 9.7, and since the diﬀerential is trivial in L,

logautΠ(L)=log(Π)+D(Der1L)=log(Π).

Note that for any matrix Bas in (52),

log(B)=000

λ00

000

.

Y. Félix et al. / Advances in Mathematics 402 (2022) 108359 63

Thus, the Lie algebra log(Π) is identiﬁed to the 3 ×3 matrices of this sort which, in turn,

deﬁne the corresponding derivations of degree 0in L. Denote M=log(Π)and observe

that the induced ﬁltration of Mon Span{y1, y2, y12}is

Span{y1,y

2,y

12}⊃Span{y1}⊃0.

Finally, by Theorem 7.16, DerMLis a Lie model of Baut∗

π(TQ).

References

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Rational homotopy type of relative universal fibrations

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Full-text available

Nov 2023

Realization of Lie algebras of derivations and moduli spaces of some rational homotopy types

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Jun 2022

We construct Lie algebras of derivations (and identify their geometrical realization) whose Maurer-Cartan sets provide moduli spaces describing the classes of homotopy types of rational spaces sharing either the same homotopy Lie algebra, homology or cohomology.

Classification of fibrations with a prescribed structural group

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Feb 2021

Mario Fuentes Rumí

We classify fibrations with a prescribed action of the fundamental group of the base on the fibre.

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Mar 2022

We construct an algebraic model for the rational homotopy type of the classifying space $B\operatorname{aut}(X)$ for fibrations with fiber a simply connected finite CW-complex $X$. The ingredients are a nilpotent dg Lie algebra $\mathfrak{g}(X)$ in the category of algebraic representations of a certain reductive algebraic group $R(X)$ together with an arithmetic subgroup $\Gamma(X)$ of $R(X)$. The algebraic model reduces the computation of the rational cohomology ring of $B\operatorname{aut}(X)$ to the computation of Chevalley-Eilenberg cohomology of dg Lie algebras and cohomology of arithmetic groups with coefficients in algebraic representations. This has strong consequences for the structure of this ring.

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Full-text available

Dec 2020

Since the birth of rational homotopy theory, the possibility of extending the Quillen approach – in terms of Lie algebras – to a more general category of spaces, including the non-simply connected case, has been a challenge for the algebraic topologist community. Despite the clear Eckmann-Hilton duality between Quillen and Sullivan treatments, the simplicity in the realization of algebraic structures in the latter contrasts with the complexity required by the Lie algebra version. In this book, the authors develop new tools to address these problems. Working with complete Lie algebras, they construct, in a combinatorial way, a cosimplicial Lie model for the standard simplices. This is a key object, which allows the definition of a new model and realization functors that turn out to be homotopically equivalent to the classical Quillen functors in the simply connected case. With this, the authors open new avenues for solving old problems and posing new questions.

Homotopy theory of complete Lie algebras and Lie models of simplicial sets

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Sep 2018

In a previous work, by extending the classical Quillen construction to the non-simply connected case, we have built a pair of adjoint functors, model and realization, (Equation presented) between the categories of simplicial sets and complete differential graded Lie algebras. This paper is a follow-up of this work. We show that when X is a finite connected simplicial set, then <£X> coincides with ℚ ∞ X ⁺ , the disjoint union of the Bousfield-Kan completion of X with an external point. We also define a model category structure on cDGL making £ and <-> a Quillen pair, and we construct an explicit cylinder. In particular, these functors preserve homotopies and weak equivalences and therefore, this gives the basis for developing a cDGL rational homotopy theory for all spaces.

Maurer–Cartan elements in the Lie models of finite simplicial complexes

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In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Lie models of simplicial sets and representability of the Quillen functor

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We associate to any simplicial set $K$ a complete differential graded Lie algebra $\mathfrak{L}(K)$ that represents its rational homotopy type. The family $\mathfrak{L}_\bullet=\{\mathfrak{L} (\Delta^n)\}_{n\ge 0}$ is endowed with a cosimplicial differential graded Lie algebra structure that is an analogue of the simplicial algebra $A_{PL}(\Delta^\bullet)$. Then, a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets is defined by $\langle L\rangle = {\bf DGL}(\mathfrak{L}_\bullet, L)$. We prove that $\mathfrak{L} (K)$ contains a model of each component of $K$. When $K$ is a 1-connected, finite simplicial complex the Quillen model of $K$ can be extracted from $\mathfrak{L} (K)$. When $K$ is connected then, for a perturbed differential $\partial_a$, $H_0(\mathfrak{L} (K),\partial_a)$ is the Malcev Lie completion of $\pi_1(K)$.

A $\operatorname{dg}$ Lie model for relative homotopy automorphisms

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Rational Models for Automorphisms of Fiber Bundles

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Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan

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Daniel Tanré

Representing the Deligne-Hinich-Getzler $\infty$-groupoid

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Daniel Robert-Nicoud

The goal of the present paper is to introduce a smaller, but equivalent version of the Deligne-Hinich-Getzler $\infty$-groupoid associated to a homotopy Lie algebra. In the case of differential graded Lie algebras, we represent it by a universal cosimplicial object.

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We calculate the higher homotopy groups of the Deligne-Getzler infinity-groupoid associated to a nilpotent L-infinity-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

A formula for topology/deformations and its significance

Article

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The formula is ∂e = (ade)b + ∞σi=0 Bi/i!(ade)i(b - a); with ∂a + 1/2 [a; a] = 0 and ∂b + 1/2 [b; b] = 0, where a, b and e in degrees -1, -1 and 0 are the free generators of a completed free graded Lie algebra L[a; b; e]. The coeffcients are defined by x/ex- 1 = ∞σn=0 Bn/n!xn: The theorem is that this formula for ∂ on generators extends to a derivation of square zero on L[a; b; e]; the formula for ∂e is unique satisfying the first property, once given the formulae for ∂a and ∂b, along with the condition that the "flow" generated by e moves a to b in unit time. The immediate significance of this formula is that it computes the infinity cocommutative coalgebra structure on the chains of the closed interval. It may be derived and proved using the geometrical idea of at connections and one-parameter groups or ows of gauge transformations. The deeper significance of such general DGLAs which want to combine deformation theory and rational homotopy theory is proposed as a research problem.

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