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Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras

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.


Introduction
In [17] Getzler associates an ∞-groupoid γ • (g) to a nilpotent L ∞ -algebra g which generalizes the Deligne groupoid of a nilpotent differential graded Lie algebra [16,17,19,23]. The ∞-groupoid γ • (g) is equivalent to the 'integral' g [22] and to the 'nerve' Σ(g) [23]. In this paper, we consider an extension of Getzler's construction to complete L ∞ -algebras, i.e., inverse limits of towers of central extensions of nilpotent L ∞ -algebras. Our first main result is a calculation of the higher homotopy groups of γ • (g), which is new even for nilpotent g. Theorem 1.1. For a complete L ∞ -algebra g there is a natural group isomorphism B : H n (g τ ) → π n+1 (γ • (g), τ ), n ≥ 0, where g τ denotes the L ∞ -algebra g twisted by the Maurer-Cartan element τ . The group structure on H 0 (g τ ) is given by the Campbell-Hausdorff formula.
We stress that we put no boundedness or finiteness constraints on g. This will be essential for our applications. Another reason for pointing this out is that the nonnegatively graded case is well-understood by classical rational homotopy theory: Non-negatively graded complete g of finite type correspond to Sullivan models of nilpotent spaces X of finite type, via the Chevalley-Eilenberg construction C * (g). In this situation, it is known that γ • (g) is a Q-localization of X, and Theorem 1.1 recovers the result that the rational homotopy groups of a nilpotent space are dual to the indecomposables of its minimal Sullivan model [34]; see also [22, §6].
As a corollary of Theorem 1.1 we obtain models for the components of γ • (g).
Corollary 1.2. Let g be a complete L ∞ -algebra. The component of γ • (g) that contains a given Maurer-Cartan element τ is homotopy equivalent to γ • (g τ ≥0 ), where g τ ≥0 denotes the truncation of g τ . Hence, provided g is of finite type, the Chevalley-Eilenberg construction C * (g τ ≥0 ) is a Sullivan model for the component of γ • (g) that contains τ .
Our main application is within the rational homotopy theory of mapping spaces.
The key to this application is the following result.
Supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. Theorem 1.3. Let X be a connected space and let Y be a nilpotent space of finite Q-type. If A is a differential graded algebra model for X and L is an L ∞ -algebra model for Y , then there is a homotopy equivalence of Kan complexes This appears as Theorem 5.2 in the text. The following result is a direct consequence of Theorem 1.1, Corollary 1.2 and Theorem 1.3. Theorem 1.4. Let X be a connected space and let Y be a nilpotent space of finite Q-type. Let A be a commutative differential graded algebra model for X and let L be an L ∞ -algebra model for Y . There is bijection Moreover, given a map f : X → Y Q whose homotopy class corresponds to the equivalence class of a Maurer-Cartan element τ ∈ A⊗L, the Chevalley-Eilenberg construction C * ((A⊗L) τ ≥0 ) is a Sullivan model for the component Map(X, Y Q ) f of the mapping space that contains f . In particular, there are bijections For n ≥ 1 this is an isomorphism of rational vector spaces, and for n = 0 it is an isomorphism of groups where the left hand side is given the Campbell-Hausdorff group structure. The Lie bracket on the left hand side corresponds to the Whitehead product on the right hand side.
Rational models for mapping spaces have been studied before by many authors, see for instance [3,6,8,11,15,20,27] and the recent survey [33, §3.1]. Buijs-Félix-Murillo [11] prove the existence of an L ∞ -algebra structure on the space of derivations from a Quillen model for the source to a Quillen model for the target that models a given component of the (based) mapping space, but they do not give a direct description of the L ∞ -algebra structure. What is new with our approach is that we get explicit and computable L ∞ -algebra models for the components that are expressed in terms of natural constructions on models for the source and target -tensoring, twisting by a Maurer-Cartan element and truncating. Moreover, there are no constraints such as cofibrancy or minimality on the models for the source and target. In the last section we have included a number of calculational examples.
During the preparation of this paper, we learned that variants of Theorem 1.4 have been obtained independently by Buijs-Félix-Murillo [12] and Lazarev [25] using different methods. degree 0. For an integer n, the n-fold suspension V [n] of a graded vector space V is defined by • (Generalized Jacobi identities) For every n ≥ 1 and all x 1 , . . . , x n ∈ g, n p=1 σ where the inner sum is over all permutations σ of {1, . . . , n} such that σ 1 < . . . < σ p and σ p+1 < . . . < σ n . The sign is given by We will write δ(x) = [x]. For n = 1, 2, 3, the generalized Jacobi identities are equivalent to In particular, g has an underlying chain complex, which we take to be homologically graded; · · · / / g 1 δ1 / / g 0 δ0 / / g −1 / / · · · The binary bracket ℓ 2 satisfies the Jacobi identity up to homotopy, the ternary bracket ℓ 3 being a contracting homotopy. For this reason an L ∞ -algebra may be thought of as a Lie algebra 'up to homotopy'.
An L ∞ -algebra concentrated in degree 0 is the same thing as an ordinary Lie algebra, because in this case ℓ r = 0 for r = 2 for degree reasons, and the generalized Jacobi identities reduce to the classical Jacobi identity. An L ∞ -algebra g is called abelian if ℓ r = 0 for r ≥ 2. An abelian L ∞ -algebra is the same thing as a chain complex (g, δ). An L ∞ -algebra is called minimal if its differential δ = ℓ 1 is zero. An L ∞ -algebra with ℓ r = 0 for r ≥ 3 is the same thing as a differential graded Lie algebra (dgl) (g, δ, [, ]).
The bar construction of an L ∞ -algebra. There is an alternative and more compact definition of L ∞ -algebras. An L ∞ -algebra structure on a graded vector space g is the same thing as a coderivation differential d on the symmetric coalgebra Λ c g [1]. The relation between d and the brackets is given by the following: write d = d 1 + d 2 + . . ., where d r is the component of d that decreases word-length by r − 1. Each d r is a coderivation and is therefore determined by its restriction d r : Λ r c g [1] → g [1]. The formula defines d in terms of the higher brackets and vice versa. The condition d 2 = 0 is equivalent to the generalized Jacobi identities for g.
The bar construction of an L ∞ -algebra g is defined to be the cocommutative differential graded coalgebra If g is a dg Lie algebra, then C * (g) coincides with the classical construction due to Quillen [31, §B.6]. The Chevalley-Eilenberg construction C * (g) is by definition the dual commutative differential graded algebra C * (g) = C * (g) ∨ . If g is of finite type, then C * (g) may be identified with (ΛV, d) where ΛV denotes the free graded commutative algebra on V = g [1] ∨ .
(2) For every r there is an N such that [g ∧ℓ ] ⊆ F r g for all ℓ > N .
(3) g is complete with respect to the topology defined by the filtration, i.e., the canonical map g → lim ← − g/F r g is an isomorphism.
We say that g is profinite if, additionally, each quotient g/F r g is finite.
Note that g is nilpotent in the sense of [17,Definition 4.2] if and only if g admits a filtration as in Definition 2.1 with F N g = 0 for some N . In particular, if g is complete then each quotient g/F r g is nilpotent, and there is a sequence of central extensions of nilpotent L ∞ -algebras A comment about (2) is in order. Nilpotence for L ∞ -algebras involves two conditions -not only should bracket expressions of large enough depth vanish, but also brackets of high enough arity. Unlike in the case of groups or Lie algebras, it is not automatic that nilpotence of the quotient in a central extension implies nilpotence of the middle term. For this reason we need to impose (2). The stronger condition [F i1 g, . . . , F i k g] ⊆ F i1+...+i k g implies both (1) and (2).
Complete L ∞ -algebras and Sullivan algebras. Recall [14, §12] that a Sullivan algebra is a commutative differential graded algebra of the form (ΛV, d) where V is concentrated in positive cohomological degrees and admits a filtration Proposition 2.2. The Chevalley-Eilenberg construction C * (L) is a finite type Sullivan algebra if and only if L is a non-negatively graded, finite type, profinite L ∞algebra. Conversely, any finite type Sullivan algebra is isomorphic to C * (L) for some L.
Proof. We may identify C * (L) = (ΛV, d) where V = L[1] ∨ . Then V is concentrated in positive cohomological degrees if and only if L is non-negatively graded. Suppose given a filtration V (r) of V that exhibits (ΛV, d) as a Sullivan algebra. Since V is of finite type, we may assume that each V (k) is finite. If necessary, this may be achieved by setting W (k) = V (0) ≤k + V (1) ≤k−2 + V (2) ≤k−4 + . . .. If we let F r L ⊆ L = V [1] ∨ be the annihilator of V (r) we get a decreasing filtration Each quotient L/F r L is finite since V (r) is finite. The condition dV (r + 1) ⊆ ΛV (r) translates into the condition [F r L, L, . . . , L] ⊆ F r+1 L. Since L/F r L is finite we have (L/F r L) >p = 0 for some p. Since it is non-negatively graded this implies that brackets of arity > p + 2 vanish in L/F r L.
Maurer-Cartan elements. If g is a complete L ∞ -algebra the following series converges for every τ ∈ g −1 Here we write [τ ∧k ] for [τ, . . . , τ ] (k copies of τ ). If F (τ ) = 0 then τ is called a Maurer-Cartan element, and we let MC(g) denote the set of all such elements. Given τ ∈ MC(g) the series converge and define a new L ∞ -algebra structure on the underlying graded vector space of g [17,Proposition 4.4]. Denote this L ∞ -algebra by g τ .
Extension of scalars. Given a commutative differential graded algebra (cdga) A and an L ∞ -algebra g, we can extend scalars and form a new L ∞ -algebra A ⊗ g with differential and brackets defined by If g is a complete L ∞ -algebra and A is a cdga, we define a new complete L ∞algebra This is the completion of the tensor product A ⊗ g with respect to the topology defined by the filtration F r (A ⊗ g) = A ⊗ F r g. Note that we have the relation A⊗(B⊗g) ∼ = (A ⊗ B)⊗g. Convolution L ∞ -algebras. Let C be cocommutative differential graded coalgebra (cdgc) and let L be an L ∞ -algebra. The convolution L ∞ -algebra Hom(C, L) has differential and brackets defined by where ℓ r : L ⊗r → L denotes the r th higher bracket in L and ∆ (r) the iterated coproduct in C. The latter is defined inductively by ∆ (2) = ∆ : If L is complete with respect to a filtration F r L, then Hom(C, L) is complete with respect to the induced filtration F r Hom(C, L) = Hom(C, F r L). Similarly, if C is conilpotent, meaning C = ∪C(r) where C(r) = ker(∆ (r) ), and ∆ denotes the then Hom(C, L) is complete with respect to the filtration F r Hom(C, L) = Hom(C(r), L). If we let A = C ∨ be the dual cdga, there is a natural morphism of complete L ∞ -algebras which is an isomorphism if C is of finite type and L is profinite.
In this section we will review the definition and some fundamental properties of the ∞-groupoid γ • (g) associated to a complete L ∞ -algebra g.
The nerve of an L ∞ -algebra. Let Ω • denote the simplicial de Rham algebra; it is the simplicial cdga with n-simplices Note that although Ω n is not of finite type, it is bounded; Since MC commutes with inverse limits we have that In particular, for nilpotent g this definition agrees with [17,23]. Getzler's ∞-groupoid γ • (g). For each n, the elementary differential forms span a subcomplex C n ⊆ Ω n . The cochain complex C n is isomorphic to the normalized cochains N * (∆[n]) on the standard n-simplex. The subcomplexes C n assemble into an inclusion of simplicial cochain complexes C • ⊆ Ω • . Moreover, there is a projection P • : Ω • → C • , and a contraction s • : see [17]. Getzler introduces the simplicial subset The definition of γ • (g) extends to complete g by setting It follows from [17,Corollary 5.11] that the inclusion γ • (g) ⊆ MC • (g) is a homotopy equivalence of Kan complexes. We now will list some properties of γ • (g) that will be useful to us. [1]) denotes the Dold-Kan construction on the chain complex g [1].
induces a Kan fibration of simplicial sets 4. The homotopy groups of γ • (g) This section contains the proof of Theorem 1.1 and is the core of the paper. For g a complete L ∞ -algebra, we will define the natural map and prove that it is an isomorphism. We will first deal with the case τ = 0. Then in Proposition 4.9 we will show how to reduce everything to this case.
We note that the simplicial faces of ω r are given by ∂ i (ω r ) = ω if i = r and zero otherwise. Furthermore, with respect to the de Rham differential, we have that dω r = (−1) r ω. As another notational device, let us write ∂γ = (∂ 0 γ, . . . , ∂ n γ) for the simplicial boundary of an n-simplex γ.
The element λ is obtained as a limit λ = lim k→∞ λ k where the elements λ k are defined iteratively by . Then it follows that the same is true for λ k+1 . Indeed, for degree reasons [λ ∧ℓ k ] = 0 for ℓ ≥ 3, so the iterative formula (4) reduces to This finishes the inductive step.
Proof. Let α, β ∈ g n be two cycles. Then we would like to show that This means that we have to find an element λ ∈ γ n+2 (g) with simplicial boundary We claim that satisfies the requirements. Indeed, λ has the correct simplicial boundary. One calculates that (d + δ)λ = 0, and since n ≥ 1 we have that [λ ∧ℓ ] = 0 for ℓ ≥ 2 for degree reasons. Therefore λ ∈ γ n+2 (g).
For a complete L ∞ -algebra g, the zeroeth homology H 0 (g) is an ordinary (ungraded) Lie algebra, and it can be given a group structure via the Campbell-Hausdorff formula α · β = log(e α e β ). The first few terms of this formula are given by  [32]. The series converges as g is complete.
Proposition 4.4. The map B 0 : H 0 (g) → π 1 (g) is a group homomorphism when H 0 (g) is given the Campbell-Hausdorff group structure.
Fibration sequences. Let 0 → g ′ µ → g ǫ → g ′′ → 0 be a short exact sequence of complete L ∞ -algebras. Then there is an associated fibration sequence whence a long exact sequence of homotopy groups On the other hand, the short exact sequence also induces a long exact sequence in homology Proposition 4.5. Let 0 → g ′ µ → g ǫ → g ′′ → 0 be a short exact sequence of complete L ∞ -algebras. The diagram commutes for all n ≥ 1. If g ′ is abelian, then it commutes for n = 0 also.
Proposition 4.6. Let {g (r) } r be a tower of surjections of complete L ∞ -algebras and let g = lim ← − g (r) . The diagram Proof. The right square commutes by naturality of the map B n . To show that the left square commutes, we must first recall how the kernels of the maps f H and f π are identified with the respective lim ← − 1 -groups.
Let [(α r ) r ] be an element in the kernel of f H : ). This means that each α r ∈ g (r) n is a boundary; say δβ r = α r . Since pα r+1 = α r for all r, the element pβ r+1 − β r is a cycle. The identification is effected by sending the class [(α r ) r ] to the equivalence class represented by the sequence ([pβ r+1 − β r ]) r ∈ r H n+1 (g (r) ).
Let [(α r ) r ] be an element in the kernel of f H . The map B n : H n (g) → π n+1 (g) sends [(α r ) r ] to the class [(a r ) r ] where a r := α r ⊗ ω n+1 . By commutativity of the right square this class belongs to the kernel of f π . To prove commutativity of the left square, it suffices to prove that a filler c r may be chosen such that (7) [∂ 2 c r ] = (−1) n [(pβ r+1 − β r ) ⊗ ω n+2 ] ∈ π n+2 (g (r) ), for all r ≥ 0.
The element is obtained as a limit λ = lim k→∞ λ k , where λ k is defined by the iterative formula for a certain operator 1 F : Ω * 3 → Ω * −1

3
. In the first iteration, a calculation yields Observe that since α r = δβ r and pβ r+1 − β r is a cycle, we have that 1 ⊗ Ω 2 3 . By an induction that we leave to the reader, the same is true for all k: Since also ∂ 2 (λ) ∈ γ 2 (g (r) ) this implies that ∂ 2 (λ) = ξ ⊗ ω 2 for some cycle ξ ∈ g (r) 1 . Moreover, ξ and pβ r+1 − β r differ by a boundary, whence by well-definedness of the map B 1 : H 1 (g (r) ) → π 2 (g (r) ). Thus (7) is satisfied for c r = λ, and this finishes the proof. is an isomorphism of groups for all n ≥ 0, where H 0 (g) is given the Campbell-Hausdorff group structure. 1 F = P 3 h 2 3 + s 3 in the notation of [17].
Proof. For abelian g this follows from the fact that γ • (g) is isomorphic to the Dold-Kan construction on the suspended chain complex g [1] [17, Proposition 5.1]. For nilpotent g the claim follows by repeated application of Proposition 4.5 and the five lemma. Finally, for complete g it follows from Proposition 4.6 and the five lemma.
The components of γ • (g). So far, we have only been concerned with the homotopy groups of γ • (g) at the base-point 0. The arguments given above can be adapted to work for an arbitrary base-point τ ∈ MC(g). Alternatively, the following will reduce everything to the base-point 0. Recall the definition of the twisted L ∞ -algebra g τ from §2.
Lemma 4.8. Let g be a complete L ∞ -algebra and let τ ∈ MC(g). Then Proof. After writing them out, the conditions σ ∈ MC(g τ ) and σ + τ ∈ MC(g) both turn out to be equivalent to n,k≥0 (where an empty bracket is interpreted as zero).
If X is a Kan complex and v is a vertex of X, then let X v denote the simplicial subset of X consisting of the simplices all of whose vertices are v. In other words, an n-simplex x belongs to X v if and only if ∂ n−i The simplicial set X v is a Kan complex and it is reduced in the sense that it has only one vertex. The inclusion X v ⊆ X induces a homotopy equivalence between X v and the connected component of X containing v. Proposition 4.9. Let g be a complete L ∞ -algebra. For every Maurer-Cartan element τ in g there is an isomorphism of of reduced Kan complexes Proof. Since, evidently, g τ⊗ Ω n = (g⊗Ω n ) τ it follows from Lemma 4.8 that nsimplices x of γ • (g τ ) 0 correspond bijectively to n-simplices τ + x of γ • (g) τ , and it is obvious that this bijection respects face and degeneracy maps. Theorem 4.10. Let g be a complete L ∞ -algebra. For any Maurer-Cartan element τ ∈ MC(g) and every n ≥ 0 the map , is an isomorphism of groups, where H 0 (g τ ) is given the Campbell-Hausdorff group structure.
Proof. This follows by combining Theorem 4.7 and Proposition 4.9.
For each integer m the truncation g ≥m of g is the sub L ∞ -algebra whose underlying chain complex is The inclusion g ≥m → g induces an isomorphism in homology H n (g ≥m ) ∼ = → H n (g) for n ≥ m.
Corollary 4.11. Let g be a complete L ∞ -algebra. For any Maurer-Cartan element τ ∈ MC(g), the inclusion g τ ≥0 ⊆ g τ induces a homotopy equivalence of reduced Kan complexes In particular, if g is of finite type, the Chevalley-Eilenberg construction C * (g τ ≥0 ) is a Sullivan model for the connected component of γ • (g) containing τ .
Proof. By Proposition 4.9 we may assume that τ = 0 without loss of generality. It is clear that the image of γ • (g ≥0 ) → γ • (g) is contained in the simplicial subset γ • (g) 0 . We need to show that the resulting map is a homotopy equivalence. Since both the source and target are reduced Kan complexes, this happens if and only if the map induces an isomorphism on π n (at the unique base-point) for all n ≥ 1. Since g ≥0 → g induces an isomorphism in homology in non-negative degrees, Theorem 4.7 together with naturality of the map B n finishes the proof; for all n ≥ 0 H n (g ≥0 ) The last statement follows from Proposition 3.2.
Naturality. Let us be more explicit about the naturality of the map B τ n in Theorem 4.10. Given an L ∞ -morphism f : g → h between complete L ∞ -algebras there is an induced map f * : MC(g) → MC(h) given by Moreover, for each τ ∈ MC(g) there is an induced L ∞ -morphism f τ : g τ → h f * (τ ) given by Naturality means that the following diagram is commutative: We leave the verification to the reader.

Application: rational models for mapping spaces
In this section, we will show that the mapping space Map(X, Y Q ) is homotopy equivalent to γ • (A⊗L) where A is a cdga model for X and L is a profinite L ∞algebra model for Y . Once this is established, Theorem 1.4 will follow by applying Theorem 1.1 and Corollary 1.2 to the L ∞ -algebra A⊗L.
The simplicial deRham algebra Ω • gives rise to a (contravariant) adjunction between simplicial sets and commutative differential graded algebras Ω(X) = Hom sSet (X, Ω • ), B = Hom cdga (B, Ω • ). The cdga Ω(X) is the Sullivan-deRham algebra of polynomial differential forms on X, and B is the spatial realization of B. It is a fundamental result in rational homotopy theory that the adjunction induces an equivalence between the homotopy categories of nilpotent rational spaces of finite Q-type and minimal algebras of finite type, see [4,34]. The category of cdgas is enriched in simplicial sets via see [4, §5], and the category of simplicial sets is enriched in itself via It is natural to ask to what extent the adjunction above is compatible with the simplicial enrichments. This has been answered by Brown and Szczarba. Furthermore, the functor Map cdga (B, −) takes quasi-isomorphisms between commutative differential graded algebras to homotopy equivalences between Kan complexes.
We give a short proof using nerves of L ∞ -algebras below. The following is a consequence.
Theorem 5.2. Let X be a connected simplicial set and let Y be a nilpotent space of finite Q-type. If L is a profinite L ∞ -model for Y and A is a cdga model for X, then there is a homotopy equivalence of Kan complexes It is straightforward to check that f is a simplicial map, and we set µ(τ ) = f . The map µ is evidently an isomorphism for X = ∆[n]. Since − ⊗ L commutes with finite limits, the functor MC(Ω(−) ⊗ L) takes finite colimits to limits. The functor Hom sSet (−, MC • (L)) preserves all limits, so it follows that µ is an isomorphism for finite X. On the other hand, if L is finite then − ⊗ L commutes with all limits and in this case we get an isomorphism for arbitrary X. Finally, for complete L the map µ is defined as the map induced on inverse limits and by the above this is an isomorphism if X is finite or if each L/F r L is finite. Proof. We define the map ϕ for nilpotent L first. On n-simplices, the map ϕ n is defined as the composite The proof that ϕ is a weak equivalence is by reduction to the abelian case. For abelian L, the map ϕ is a weak equivalence because it follows from Proposition 3.3 that ϕ is equivalent to the natural map ) is a product of rational Eilenberg-MacLane spaces, and by standard arguments the homotopy groups of the right hand side are given by H * (X; Q) ⊗ H * (L) [1]. Next, we note that both source and target of ϕ take central extensions of L ∞ -algebras to principal fibrations, so we can induct on a composition series for L to conclude that ϕ is a weak equivalence for all nilpotent L ∞ -algebras L. For complete L, we get a levelwise weak equivalence of towers of fibrations Hence, we get a weak equivalence when passing to inverse limits. This finishes the proof of the first statement. The proof that MC • (−⊗L) takes quasi-isomorphisms to weak equivalences is proved by a similar reduction to the abelian case.

Examples
Finite rational cohomology or homotopy. If H * (X; Q) or π * (Y ) ⊗ Q is finite, then for a fixed map g : X → Y , composition with the Q-localization map r : Y → Y Q induces a rational homotopy equivalence If X has finite dimensional rational cohomology, we can find a finite dimensional cdga model A (see e.g. [14, Example 6, p.146]). Likewise, if π * (Y ) ⊗ Q is finite, then we can find a finite dimensional nilpotent L ∞ -algebra model L for Y . In either of these situations, A⊗L = A ⊗ L, and given a Maurer-Cartan element τ ∈ MC(A ⊗ L) that represents the map rg : X → Y Q , the truncated and twisted L ∞ -algebra A ⊗ L τ ≥0 is an L ∞ -algebra model for the component Map(X, Y ; g). Equivalently, the Chevalley-Eilenberg construction C * (A⊗L τ ≥0 ) is a (not necessarily minimal) Sullivan model for Map(X, Y ; g). In particular, we have an isomorphism of graded Lie algebras Here, π 1 ⊗ Q is interpreted as the Malcev completion of the nilpotent group π 1 [31, §A3], and H 0 (A ⊗ L τ ) is given the Campbell-Hausdorff group structure.
Maps into Eilenberg-MacLane spaces. A simply connected space Y is rationally homotopy equivalent to a product of Eilenberg-MacLane spaces if and only if L = π * (ΩY )⊗Q, viewed as an L ∞ -algebra with zero differential and zero brackets, is an L ∞ -model for Y . In this case A⊗L is abelian for any cdga A, and A⊗L τ = A⊗L for any τ . By choosing a quasi-isomorphism of chain complexes A → H * (A) (not necessarily a cdga morphism) we get a quasi-isomorphism of abelian L ∞algebras A ⊗ L → H * (A) ⊗ L. Therefore, all components of Map(X, Y Q ) are rationally homotopy equivalent and they have a minimal abelian L ∞ -algebra model (H * (X; Q) ⊗ π * (ΩY )) ≥0 . So each component is a product of rational Eilenberg-MacLane spaces whose homotopy groups are given by Inclusions of complex projective spaces. Consider the standard inclusion i : CP n → CP m where m ≥ n ≥ 1. As a cdga model for CP n we may choose the cohomology A minimal L ∞ -algebra model for CP m is given by where the only non-vanishing bracket is The twisting cochain τ = x ⊗ α ∈ A ⊗ L represents the inclusion i : CP n → CP m . Thus, an L ∞ -model for the component Map(CP n , CP m ; i) is given by the finite dimensional L ∞ -algebra g = A ⊗ L τ ≥0 , and the Chevalley-Eilenberg construction C * (g) is a Sullivan model. A basis for g is given by The L ∞ -algebra structure is described by Since β is central in L, it follows that the elements x r ⊗ β are central in g, so the only possible non-zero brackets are described by the above. Note that we get zero if r ≤ m − n since x n+1 = 0. In particular, the differential is zero if m > n, and in this case the Chevalley-Eilenberg construction C * (g) is a minimal Sullivan model for Map(CP n , CP m ; i). It has the following description: Λ(z, w m−n , w m−n+1 , . . . , w m ), dz = 0, dw r = z r+1 , |z| = 2, |w r | = 2r + 1.
On the other hand, if m = n, then we have a non-zero differential [1 ⊗ α] τ = (n+1)x n ⊗β, and hence π * (Map(CP n , CP n ); 1)⊗Q is concentrated in odd degrees (as predicted by Halperin's conjecture, see below) with precisely one basis element each in the degrees 3, 5, . . . , 2n + 1. The minimal Sullivan model for Map(CP n , CP n ; 1) is therefore an exterior algebra on the dual of the rational homotopy with zero differential: On the Halperin conjecture. Let X be an F 0 -space, i.e., a simply connected space with evenly graded rational cohomology such that both dim Q H * (X; Q) < ∞ and dim Q π * (X) ⊗ Q < ∞. Then X is formal and the cohomology algebra admits a presentation H * (X; where x 1 , . . . , x n are evenly graded generators and f 1 , . . . , f n is a regular sequence [21]. Halperin's conjecture says that for such spaces X, the component aut 1 (X) of the space of homotopy self-equivalences of X that contains the identity map is rationally homotopy equivalent to a product of odd dimensional spheres. Equivalently, the rational homotopy groups π * (aut X, 1) ⊗ Q are concentrated in odd degrees. Using the model in Theorem 1.4, we get a new proof of the following.
Theorem 6.1. (Meier [29]) Let X be an F 0 -space. Then the Halperin conjecture holds for X if and only if the cohomology algebra H * (X; Q) admits no derivations of negative degree.
Since L even is central in L, we have that D π (A ⊗ L even ) = 0. Thus, H odd (A ⊗ L, D π ) = ker(D π : A ⊗ L odd → A ⊗ L even ).
The differential D π is A-linear, and a calculation yields Therefore, an element i p i ⊗ α i ∈ A ⊗ L odd belongs to the kernel of D π if and only if But this is true if and only if i p i ∂ ∂xi defines a derivation of A. Thus, we obtain an isomorphism ker(D π : A ⊗ L odd → A ⊗ L even ) ∼ = DerA, This shows that H n (A ⊗ L, D π ) = 0 for all odd n > 0 if and only if A admits no negative (in cohomological grading) derivations.
Homotopy automorphisms of formal and coformal spaces. In [1], the following characterization of spaces that are both formal and coformal was established.
Theorem 6.2 ([1]). The following are equivalent for a connected nilpotent space X of finite Q-type: (1) X is both formal and coformal.
(2) X is formal and H * (X; Q) is a Koszul algebra.
That the cohomology H * (X; Q) is a Koszul algebra means that it is generated by elements x i modulo quadratic relations i c ij x i x j = 0 such that Tor A s,t (Q, Q) = 0 for s = t, where the extra grading on Tor is induced by wordlength in the generators x i . That π * (ΩX) ⊗ Q ∼ = H * (X; Q) !Lie means that, as a graded Lie algebra, π * (ΩX) ⊗ Q is generated by classes α i dual to x i modulo the orthogonal relations: a relation The component of the mapping space Map(X, X) that contains a fixed homotopy self-equivalence is equal to the same component of aut(X), since any map homotopic to a homotopy equivalence is itself a homotopy equivalence. Moreover, π 1 (aut(X), 1 X ) is an abelian group as aut(X) is a monoid. By combining Theorem 1.4 and Theorem 6.2, we obtain the following theorem. Theorem 6.3. Let X be formal and coformal nilpotent space such that either dim Q H * (X; Q) < ∞ or dim Q π * (X) ⊗ Q < ∞. Then there is a finite basis x 1 , . . . , x n for the indecomposables of the cohomology algebra H * (X; Q) and a dual basis α 1 , . . . , α n for the indecomposables of the homotopy Lie algebra π * (ΩX) ⊗ Q. Setting κ = x 1 ⊗ α 1 + . . . + x n ⊗ α n ∈ H * (X; Q) ⊗ π * (ΩX), the derivation [κ, −] is a differential, and there are isomorphisms π k+1 (aut(X), 1 X ) ⊗ Q ∼ = H k (H * (X; Q) ⊗ π * (ΩX), [κ, −]), k ≥ 0.
In [2], we use this result to calculate the rational homotopy groups of the space of self-equivalences of highly connected manifolds.