ON EXPONENTIAL GROUPS AND MAURER–CARTAN SPACES

. The purpose of this note is to give a concise account of some fundamental properties of the exponential group and the Maurer–Cartan space associated to a complete dg Lie algebra. In particular, we give a direct elementary proof that the Maurer–Cartan space is a delooping of the exponential group. This leads to a short proof that the Maurer–Cartan space functor is homotopy inverse to Quillen’s functor from simply connected pointed spaces to positively graded dg Lie algebras.


Introduction
The Maurer-Cartan space, or nerve, MC • (L) of a dg Lie algebra L, introduced in the context of deformation theory by Hinich [12] and studied extensively by Getzler [9], has come to play a significant role in rational homotopy theory (see e.g.[2,14,5,21]).
The main purpose of this note is to record an elementary proof that the Maurer-Cartan space MC • (L) is a delooping of the exponential group exp • (L) for arbitrary complete dg Lie algebras L. As a corollary, we observe that this leads to a quick proof that the Maurer-Cartan space functor is inverse to Quillen's equivalence λ from the rational homotopy category of simply connected pointed spaces to the homotopy category of positively graded dg Lie algebras [19].This recovers the main result of [7] and leads to a new, more direct, proof of the Baues-Lemaire conjecture [1,Conjecture 3.5], first solved by Majewski [16].
Along the way, we give streamlined proofs of other fundamental properties, e.g., that MC • (−) takes surjections to Kan fibrations, that gauge equivalence agrees with homotopy equivalence, that MC • (L) is equivalent to the nerve of the Deligne groupoid for non-positively graded L, and we give an efficient computation of the homotopy groups together with Whitehead products and the action of the fundamental group.

Complete dg Lie algebras
Let k be a field of characteristic zero.By a cochain complex we mean an unbounded cochain complex V of k-vector spaces, We use the standard convention V k = V −k to view a cochain complex as a chain complex when convenient.A complete filtered cochain complex is a cochain complex V equipped with a filtration of subcomplexes A complete dg Lie algebra is a complete filtered cochain complex L together with a Lie bracket L ⊗ L → L that is compatible with the filtration in the sense that [F p L, F q L] ⊆ F p+q L for all p, q.Nilpotent dg Lie algebras equipped with the lower central series filtration are basic examples of complete dg Lie algebras.
Let G(L) denote the group exp(L 0 ), i.e., the group with underlying set L 0 and multiplication given by the Baker-Campbell-Hausdorff formula.The gauge action of x ∈ G(L) on τ ∈ MC(L) is given by the well-known formula A direct verification that this defines a group action can be found in e.g.[5, §4.3].
Lemma 2.1.The stabilizer G(L) τ of a Maurer-Cartan element τ in L under the gauge action may be identified with the subgroup for all k by induction, so d τ x = 0 by completeness.
We denote the set of gauge equivalence classes of Maurer-Cartan elements by The only non-trivial fact we will need about MC(L) is the following light version of the Goldman-Millson theorem [11].We give a short proof for completeness.Lemma 2.2.If f : L → L ′ is a surjective morphism of complete dg Lie algebras such that H 1 (F n I/F n+1 I) = 0, where I is the kernel of f and = y and assume by induction that we have found x 1 , . . ., x n ∈ L 0 satisfying (1) where the last congruence follows since c ∈ F n I. Setting x n+1 = x n + c finishes the induction.By completeness, this defines an element x ∈ L 0 such that τ = x • ρ.
Lemma 3.1.The unit map k → Ω n is a quasi-isomorphism for every n.
Proof.The unit map of Ω 1 ∼ = k[t, dt] is easily seen to be a quasi-isomorphism if and only if k has characteristic zero.Since Ω n is isomorphic to Ω ⊗n 1 , the claim for n = 1 follows from the Künneth theorem.

Some simplicial homotopy theory
In this section we review some basic simplicial homotopy theory, organized in a way suitable for our applications.First let us recall the following definitions.(1) A surjective map of simplicial sets p : E → B is called a fibre bundle with fibre F if for every simplex σ : ∆ n → B the left vertical map in the pullback σ * (E) (2) Let G be a simplicial group.A principal G-fibration, or principal G-bundle, is a simplicial map p : E → B in which E has a free G-action such that p induces an isomorphism E/G ∼ = B.
(1) Every simplicial group G is a Kan complex.(2) Every principal G-bundle is a fibre bundle with fibre G.
(3) Every fibre bundle with fibre a Kan complex is a Kan fibration.
Let G be a simplicial group acting on a simplicial set X.For a vertex v ∈ X 0 , the orbit Gv is the subspace of X which in level n is the orbit of the degenerate n-simplex s n 0 (v) under the action of G n .Similarly, the stabilizer G v is the simplicial subgroup of G which in level n is the stabilizer of s n 0 (v).Proposition 4.3.Let G be a simplicial group acting on a simplicial set X.
(1) For every vertex v ∈ X 0 , the orbit Gv is a Kan complex and the map is an isomorphism if and only if the orbit space X/G is discrete and the stabilizer G v is discrete for every v ∈ X 0 .If this holds and moreover G is contractible, then X is weakly equivalent to the nerve of the groupoid associated to the action of G 0 on X 0 . Proof.
. This proves the first part of the second statement.The statement about X being Kan follows since disjoint unions of Kan complexes are Kan.If G is connected, then so is Gv, and this implies the second part.
For the third statement, note that X/G being discrete is an obvious necessary condition.Assuming X/G discrete, the decompositions of X 0 and X as coproducts over all vertices v of the orbits G 0 v and Gv, respectively, reduce the statement to a model for the homotopy orbit space X 0 / / G 0 .Another model for the homotopy orbit space is the bar construction B * , G 0 , X 0 , which is isomorphic to the nerve of the groupoid associated to the action of G 0 on X 0 .Proposition 4.4.Let G and G ′ be simplicial groups acting on simplicial sets X and X ′ , respectively, and assume that the orbit spaces X/G and X ′ /G ′ are discrete.
Let ϕ : G → G ′ be a map of simplicial groups and f : X → X ′ a map of G-spaces.If ϕ is a Kan fibration, then so is f .Proof.It follows from the decompositions of X and X ′ as coproducts of orbits (Proposition 4.3 (2)) that it is enough to verify that Gv → G ′ f (v) is a Kan fibration for every v ∈ X 0 .By Proposition 4.3(1), the vertical maps in the diagram are surjective Kan fibrations.The lemma below then implies that the bottom horizontal map is a Kan fibration if ϕ is.
− → Z are simplicial maps such that p and qp are Kan fibrations and p is surjective, then q is a Kan fibration.

Exponential groups and Maurer-Cartan spaces
If L is a complete dg Lie algebra, then we can form the simplicial complete dg Lie algebra Ω obtained by applying exp(−) levelwise to the simplicial complete Lie algebra Z 0 Ω • (L).The nerve or Maurer-Cartan space is the simplicial set The levelwise gauge action defines an action of the simplicial group • (L) on MC • (L).We will now apply the results of the previous section to this action in order to deduce a number of basic properties of the Maurer-Cartan space.
Lemma 5.1.Let L be a complete dg Lie algebra.
(1) The simplicial group , the contractibility of which follows from Lemma 3.2.Towards the second claim, note that the orbit space is the simplicial set MC(Ω • (L)) obtained by applying MC(−) levelwise.The map η Theorem 5.2.Let L be a complete dg Lie algebra.
(1) The canonical map is an isomorphism.Hence, MC • (L) is a Kan complex and the orbit G • (L)τ agrees with the path component MC • (L) τ containing τ .In particular, there is a natural bijection i.e., two Maurer-Cartan elements are gauge equivalent if and only if they belong to the same path component of the Maurer-Cartan space.
(2) For every τ ∈ MC(L), the map given by acting on τ , is an isomorphism if and only if L = L ≥0 .If this holds, then MC • (L) is weakly equivalent to the nerve of the Deligne groupoid, i.e., the groupoid associated to the action of G(L) on MC(L).
Proof.This is immediate from Lemma 5.1 and Proposition 4.   [5,Lemma 11.11]).These decompositions should be compared to the decomposition in Theorem 5.2(1), which is less refined but sufficient for proving the Kan property.Unlike Getzler's proof [9], the proof given here does not apply to L ∞ -algebras L, because the groups G • (L) and exp • (L) are not available.The suggestion [13, Note 8.2.7] is close in spirit to the proof given here.

Homotopy groups
In this section, we compute the homotopy groups of the exponential group and the Maurer-Cartan space.Since the homotopy groups of exp • (L) are the same as those of the underlying simplicial set Z 0 Ω • (L), the computation for exp • (L) reduces to the case of abelian L. The homotopy groups of MC • (L) can then be computed using the fact that ΩMC • (L) τ is weakly equivalent to exp • (L τ ) by Theorem 5.2 (2).
Consider the differential form ω n ∈ Ω n n defined by It satisfies dω n = ∂ω n = 0 and ∆ n ω n = 1.Here and below, ∂ denotes the alternating sum i (−1) i ∂ i of the face maps.Proposition 6.1.Let V be a complete filtered cochain complex, consider the simplicial cochain complex Ω • (V ) = Ω • ⊗V , and let k be an integer.
(1) The simplicial vector space Proof.The first two claims are direct consequences of Lemma 3.1 and Lemma 3.2.Towards the third claim, let Ω k , Z k , B k , and H k , denote the simplicial vector spaces of k-dimensional cochains, cocycles, coboundaries, and cohomology, respectively, of Ω • (V ).The first statement together with the long exact sequence of homotopy groups associated to the short exact sequence of simplicial vector spaces ) for all integers k, ℓ.In particular, π 0 (B k ) = 0. Similarly, using the second statement and the sequence Combining the above facts yields natural isomorphisms Explicitly, the isomorphism π ℓ (Z k ) → π ℓ−1 (Z k−1 ) for ℓ > 0 sends an element of the form [dz] to [∂z].Using this and the equations , whence B 0 is zero and in particular discrete.Theorem 6.2.There is a natural isomorphism of groups and for every k ≥ 1 a natural isomorphism of abelian groups compatible with Samelson products and the actions of the groups in (1).In both cases, the map is given by Proof.The underlying simplicial set of exp • (L) is Z 0 Ω • (L) so Proposition 6.1 yields a natural bijection H k (L) → π k exp • (L) for all k ≥ 0, which is an isomorphism of abelian groups for k > 0. It remains to identify the group structure on π 0 (exp • (L)) and its action on the higher homotopy groups that come from the simplicial group structure on exp • (L).To do this, one can argue using the zig-zag of H 0 -isomorphisms Here L k ⊆ L is defined by The induced maps on π 0 (exp • (−)) are group homomorphisms and, by what we have just seen, bijections.Since exp • (H 0 (L)) is isomorphic to the discrete simplicial group exp(H 0 (L)), the claim about the group structure follows.The claim about the action on higher homotopy groups can be verified similarly by considering the zig-zag of H k -isomorphisms with the actions of the 0-cocycles of the dg Lie algebras in (2).The statement about Samelson products is proved in Corollary 7.2 below.
Corollary 6.3.For every τ ∈ MC(L) there is a natural isomorphism of groups and for every k ≥ 1 a natural isomorphism of abelian groups compatible with the actions of the groups in (3) and with Whitehead products up to a sign.In both cases, the map is given by Proof.This follows from Theorem 6.2 and Theorem 5.2(2) together with the fact that the canonical isomorphism π * +1 (X) → π * (ΩX) takes Whitehead products to Samelson products up to a sign (see [6, p.197]) and is compatible with the relevant group actions.To get the explicit formula, one can compute the connecting homomorphism ∂ : The key observation is that the gauge action of Remark 6.4.This simplifies the computation of the homotopy groups of MC • (L) given in [2] in the case when L is a dg Lie algebra.It also enhances the computation by showing that the isomorphism is compatible with Whitehead products (up to a sign) and the action of the fundamental group.
Theorem 7.1.Let L be a complete dg Lie algebra.The map induces a natural quasi-isomorphism of dg Lie algebras where L • denotes the simplicial Lie algebra Z 0 Ω • (L).
Proof.Elements of Ω 0 n (L) are of the form ξ = n i=0 ω i ⊗ x i , where ω i ∈ Ω i n and Since ∆ n dω n−1 = ∆ n−1 ∂ω n−1 by Stokes' theorem, this shows that I(∂ξ) = dI(ξ), so the restriction of I to N Z 0 Ω • (L) is a chain map.The verification that I respects Lie brackets uses the formula which holds as both sides compute the integral of the differential (p + q)-form α × β over ∆ p × ∆ q , on one hand using the Fubini theorem and on the other hand using the standard decomposition of ∆ p × ∆ q into a union of (p+ q)-simplices (cf.e.g.[15, pp. 243-244]).Using ∆ n ω n = 1, one sees that the map induced by I in homology is left inverse to the isomorphism f in Proposition 6. 1(3).This implies that I is a quasi-isomorphism and, as a by-product, that f respects Lie brackets.Corollary 7.2.Let L be a complete dg Lie algebra.The isomorphism takes Lie brackets to Samelson products.
Proof.By Curtis' formula for Samelson products [6, p.197], we have that ( 5) where the product over the (p, q)-shuffles (µ, ν) is taken in a certain order that will turn out not to matter for us.The commutator with respect to the BCH product satisfies [a, b] BCH = [a, b] + (higher terms).All products of length > 2 formed out of s ν ω p and s µ ω q are zero since Ω k p+q = 0 for k > p + q, so it follows that the higher terms vanish when a = s ν ω p ⊗ x and b = s µ ω q ⊗ y.Similarly, since the BCH product * satisfies a * b = a + b + (higher terms), the product in (5) reduces to a sum.This shows that the Samelson product on the homotopy groups of exp • (L) agrees with the Lie bracket (4) under the identification π k (exp • (L)) = H k (N L • ).Since f is inverse to H * (I) and the latter preserves Lie brackets by the previous proposition, the result follows.

Relation to Quillen's functor
Quillen [19] defined a functor λ : Top 2 → DGL 1 from the category of simply connected pointed spaces to the category of positively graded dg Lie algebras over Q and showed it induces an equivalence after localizing with respect to the rational homotopy equivalences and the quasi-isomorphisms, respectively.If L is a positively graded dg Lie algebra over Q, then the geometric realization |MC • (L)| is simply connected and it is natural to ask whether λ|MC • (L)| is quasi-isomorphic to L. For L of finite type, this statement is easily seen to be equivalent to a conjecture formulated by Baues-Lemaire in 1977 [1, Conjecture 3.5], which was proved by Majewski in 2000 [16].Félix-Fuentes-Murillo [7] recently gave a proof that does not require L to be of finite type.However, this proof is indirect as it relies on identifying MC • (L) with the realization L of [5, §7] up to homotopy.A corollary of Theorem 5.2 and Theorem 7.1 is the following more direct proof.Theorem 8.1.Let L be a positively graded dg Lie algebra over Q.There is a natural zig-zag of quasi-isomorphisms of dg Lie algebras Proof.We adopt Quillen's setup [19, p.211].The simplicial set MC • (L) is 2-reduced as L is positively graded, so the unit of the adjunction between Top 2 and the category of 2-reduced simplicial sets induces a natural quasi-isomorphism ( 6) Our goal is now to construct a natural quasi-isomorphism from the source of (6) to L. Let L • denote the simplicial nilpotent Lie algebra Z 0 Ω • (L).Theorem 5.2(2) implies that there is a natural weak equivalence of reduced simplicial groups where G denotes the Kan loop group (cf.e.g.[10, §V.5]).Since L • is levelwise nilpotent, Corollary 3.9 and (2.7) in Appendix A of [19] give natural isomorphisms where P U and G U denote primitives and group-like elements, respectively, in the completed universal enveloping algebra.Hence, we may identify the simplicial group exp(L • ) with G U L • and regard the weak equivalence (7) as a map Now, G is the right adjoint in a Quillen equivalence between reduced simplicial groups and reduced simplicial complete Hopf algebras ( [19,Theorem II.4.8]).Since all objects in the latter category are fibrant and since Kan loop groups are always cofibrant, this implies that the adjoint of (8), is a weak equivalence.This means that the induced map on primitives, is a weak equivalence (cf.[19,Theorem II.4.7]).Applying normalized chains, we obtain natural quasi-isomorphisms where the last map is the quasi-isomorphism from Theorem 7.1.

Quillen's universal dg coalgebra bundle and the gauge action
We end this note with a remark about the relation to Quillen's theory of dg coalgebra bundles [19, Appendix B §5- §6].The gauge action does not appear explicitly in Quillen's work, but it is implicit in a sense that we will now explain.On p.291, Quillen defines the dg Lie algebra sL#L, whose underlying graded vector space is sL ⊕ L, where sL is the suspension of L and where the bracket and differential are determined by the equations [sx, sy] = 0, [sx, y] = s[x, y], dsx = x − sdx, and the requirement that L is a dg Lie subalgebra.He then constructs a universal principal dg coalgebra bundle where CL is defined to be U (sL#L) ⊗ UL k (and then shown to be isomorphic to the standard complex for computing Lie algebra homology).Observing that exp • (L) and MC • (L) are isomorphic to the coalgebra realizations U L and CL , respectively (cf.[4, §3.3]), and that Z 0 (sL#L) is isomorphic to L 0 , one can rediscover the gauge action as the action of the simplicial group and Ω n = ker(∂ n 0 ) has zero cohomology by Lemma 3.1, so MC(ǫ n ) is a bijection.It follows that MC(η • ) : MC(L) → MC(Ω • (L)) is a levelwise bijection.The third claim follows from Lemma 2.1 and Proposition 6.1(4) below.

3 . 5 . 3 .
Remark The fact that MC • (L) is a delooping of exp • (L) is shown for positively graded L of finite type in[4, Corollary 3.10], but the proof relies on results of[8,  Chapter 25] that are not applicable if L is unbounded or not of finite type.A version of the statement is embedded in [18, §4.7], but a direct elementary proof has been missing from the literature as far as we know.Theorem 5.4.If f : L → L ′ is a surjective morphism of filtered complete dg Lie algebras, then the induced map MC • (L) → MC • (L ′ ) is a Kan fibration.Proof.Apply Proposition 4.4 to the induced maps G