aa r X i v : . [ m a t h . A T ] D ec MAURER-CARTAN MODULI AND MODELS FOR FUNCTION SPACES
A. LAZAREV
Abstract.
We set up a formalism of Maurer-Cartan moduli sets for L ∞ algebras and as-sociated twistings based on the closed model category structure on formal differential gradedalgebras (a.k.a. differential graded coalgebras). Among other things this formalism allows us togive a compact and manifestly homotopy invariant treatment of Chevalley-Eilenberg and Har-rison cohomology. We apply the developed technology to construct rational homotopy modelsfor function spaces. Contents
1. Introduction 11.1. Notation and conventions 22. Hinich’s closed model category structure 33. L ∞ algebras 53.1. Free homotopy 84. The Quillen-Neisendorfer closed model category structure 95. The MC moduli set and its homotopy invariance 105.1. Connected covers of dglas and MC moduli 126. Maurer-Cartan twisting 137. Examples of twisting: Chevalley-Eilenberg and Harrison cohomology 158. Rational homotopy of function spaces 18References 211. Introduction
The homotopy theory of function spaces has been much studied from various standpoints,cf. [23] for a comprehensive and up to date survey. The basic problem is as follows: giventwo spaces X and Y which are sufficiently nice (e.g. CW-complexes of finite type) describeeffectively the homotopy type of the mapping space F ( X, Y ) in terms of the homotopy typesof X and Y . From the point of view of rational homotopy theory several answers are known,[14, 5, 4], but these answers are complicated and not readily amenable to calculations. Wealso mention the recent papers [6, 7] and the preprint [1] where explicit Lie models for functionspaces were given but again, these models were not formulated in the standard framework ofderived functors of homological algebra. In [2] the homotopy groups of function spaces werecomputed in terms of Harrison-Andr´e-Quillen cohomology, however the methods of that paperdo not extend to a construction of full-fledged rational homotopy models of function spaces.In the present paper we fill this gap. Given rational models (Sullivan or Lie-Quillen) of X and Y we construct a rational model for F ( X, Y ) (as well as for the based function space F ∗ ( X, Y ))in terms of traditional derived functors (Harrison and Chevalley-Eilenberg complexes). The
Mathematics Subject Classification.
Key words and phrases.
Closed model category, Differential graded algebra, Chevalley-Eilenberg cohomology,Maurer-Cartan element, Sullivan model.The author is grateful to C. Braun, J. Chuang, V. Hinich, B. Keller and M. Markl for many useful discussionsconcerning the subject of this paper. onstruction is based on the formalism of Maurer-Cartan moduli sets and Maurer-Cartan twist-ings; some parts of this formalism are certainly known to experts but are difficult to locate inthe literature; we hope that a unified treatment presented here will be of independent interest,especially from the standpoint of algebraic deformation theory.The paper is organized as follows. In Section 2 we recall the construction of a closed modelcategory structure on the category of formal commutative differential graded algebras due toHinich; note that Hinich (as well as some other authors) prefers to work with coalgebras; wefeel that the equivalent language of formal algebras is more natural, particularly in connectionwith Maurer-Cartan sets. Section 3 contains a description of cofibrant objects in Hinich’s closedmodel category; these turn out to be formal cdgas representing L ∞ algebras. Section 4 discussesthe Neisendorfer closed model category structure which could be viewed as a localization ofHinich’s (the latter point of view is not pursued here). This is needed for the applicationsto rational homotopy theory that we have in mind. In Section 5 we introduce the notionof a Maurer-Cartan set and prove a general statement about its homotopy invariance. Thisresult underlies the modern approach to deformation theory suggested by Deligne, Feigin andDrinfeld at the end of the 1980’s; it is also related to the still older manuscript of Schlessingerand Stasheff, which has recently been published on the arXiv, [24]. Various versions of it havesince been proved by many authors. Our version is probably both the most general and simplestto prove (but it relies on the deep results of Hinich on the existence of a closed model categorystructure on formal cdgas).Section 6 is devoted to the notion of a Maurer-Cartan twisting in the setting of L ∞ algebrasand Section 7 describes how Harrison and Chevalley-Eilenberg cohomology can be describedcompactly in terms of twistings. Finally, Section 6 contains the main application of the de-veloped apparatus: the construction of a Lie-Quillen model of a function space between tworational nilpotent spaces.1.1. Notation and conventions.
We work in the category of Z -graded vector spaces over afield k of characteristic zero, and we avoid mentioning k explicitly. When considering models fortopological spaces the field k is understood to be Q . Differential graded algebras will have coho-mological grading with upper indices and differential graded Lie algebras will have homologicalgrading with lower indices, unless indicated otherwise. The degree of a homogeneous element x in a graded vector space is denoted by | x | . The suspension Σ V of a homologically gradedvector space V is defined by the convention Σ V i = V i − ; for a cohomologically graded space theconvention is as follows: Σ V i = V i +1 . The functor of taking the linear dual takes homologicallygraded vector spaces into cohomologically graded ones so that ( V ∗ ) i = ( V i ) ∗ ; further we willwrite Σ V ∗ for Σ( V ∗ ); with this convention there is an isomorphism (Σ V ) ∗ ∼ = Σ − V ∗ .The adjective ‘differential graded will be abbreviated as ‘dg’. A (commutative) differentialgraded (Lie) algebra will be abbreviated as (c)dg(l)a. We will often invoke the notion of a formal (dg) vector space; this is just an inverse limit of finite-dimensional vector spaces. Herethe notion of formality is understood in the sense of a ‘formal neighborhood’ rather than ‘beingquasi-isomorphic to the cohomology’; in a couple of places where this clashes with the standardterminology of rational homotopy theory the distinction is specifically spelled out. An exampleof a formal space is V ∗ , the k -linear dual to a discrete vector space V . A formal vector spacecomes equipped with a topology and whenever we deal with a formal vector space all linearmaps from or into it will be assumed to be continuous; thus we will always have V ∗∗ ∼ = V . Allof our unmarked tensors are understood to be taken over k . The tensor product V ⊗ W oftwo formal spaces is understood to be the completed tensor product (and so, it will again beformal). If V is a discrete space and W = lim ← W i is a formal space we will write V ⊗ W forlim ← V ⊗ W i ; thus for two discrete spaces V and U we have Hom( V, U ) ∼ = U ⊗ V ∗ .For two topological spaces X and Y we will write [ X, Y ] for the set of homotopy classesof maps X → Y ; if X and Y are pointed spaces then [ X, Y ] ∗ will denote the set of pointedhomotopy classes of such maps. . Hinich’s closed model category structure
Consider the category of dg cocommutative coassociative coalgebras which are cocomplete inthe sense that the filtration given by the kernels of the iterated comultiplication is exhaustive.For a homologically graded coalgebra X there is defined a dgla L ( X ) which is the free Liealgebra on Σ X and the differential is induced in the standard way by the comultiplication on X and by the internal differential on X . Then Hinich proved in [15] that this category couldbe turned into a closed model category where the weak equivalences are those maps X → Y ofdg coalgebras for which L ( X ) → L ( Y ) are quasi-isomorphisms of dglas.Note that the linear dual to a cocomplete coalgebra X is a (non-unital) algebra X ∗ which is formal in the sense that X ∗ = lim ← X ∗ n ;here X ∗ n is the cokernel of the n-fold multiplication map X ∗⊗ n → X ∗ . The filtration by thekernels of the maps X ∗ → X ∗ n will be called the canonical filtration on X ∗ ; it is complete andHausdorff. Some of the elementary properties of formal cdgas are discussed in the appendix to[16].It follows that there is a closed model category structure on formal cdgas which we willexplicitly describe; this category will be denoted by F Alg . Thus, an object in F Alg is a formalnon-unital cdga and morphisms are required to be continuous with respect to the profinitetopology. Note that sometimes it is more convenient to consider the category of unital and augmented formal cdgas; the morphisms will then be required to respect the unit and theaugmentation. The latter category is clearly equivalent to the category of F Alg . Indeed anon-unital formal cdga A determines an augmented unital one: ˜ A = A ⊕ k , obtained from A byadjoining a unit and conversely, the augmentation ideal B + of an augmented formal cdga B isa non-unital formal cdga. We will use the term ‘formal cdga’ to mean ‘non-unital formal cdga’unless stated otherwise. Note that the canonical filtration on A corresponds to the filtration on˜ A by the powers of the maximal ideal in ˜ A .Any formal cdga determines a dgla as follows. Definition 2.1.
Let A be a formal cdga and set L ( A ) to be a dgla whose underlying space isthe free Lie algebra on Σ A ∗ and the differential d is defined as d = d I + d II ; here d I is inducedby the internal differential on A and d II is determined by its restriction onto Σ A ∗ which is inturn induced by the product map A ⊗ A → A . Remark 2.2.
Note that since A is formal its dual A ∗ is discrete and thus, the dgla L ( A ) is aconventional dgla (with no topology). The construction L ( A ) is the continuous version of theHarrison complex associated with a cdga. Definition 2.3.
A morphism f : A → B in F Alg is called (1) a weak equivalence if L ( f ) : L ( B ) → L ( A ) is a quasi-isomorphism of dglas; (2) a fibration if f is surjective; if, in addition, f is a weak equivalence then f is called an acyclic fibration ; (3) a cofibration if f has the left lifting property with respect to all acyclic fibrations. Thatmeans that in any commutative square A f (cid:15) (cid:15) / / C g (cid:15) (cid:15) A > > ⑦⑦⑦⑦ / / D where g is an acyclic fibration there exists a dotted arrow making the whole diagramcommutative. Theorem 2.4.
The category F Alg is a closed model category with fibrations, cofibrations andweak equivalences defined as above. roof. One has to observe only that the category F Alg is anti-equivalent to the category ofcocomplete coalgebras of Hinich: given any cocomplete dg coalgebra X , its linear dual X ∗ is aformal cdga, and the continuous dual to a formal cdga is a cocomplete dg coalgebra. (cid:3) The functor L from F Alg to the category L ie of dglas admits an adjoint functor C : L ie → F Alg defined as follows. For a dgla g set C ( g ) = ˆ S Σ − g ∗ , the completed symmetric algebraon g ∗ . The differential d on C ( g ) is defined as d = d I + d II ; here d I is induced by the internaldifferential on g and d II is determined by its restriction onto Σ − g ∗ which is in turn inducedby the bracket map g ⊗ g → g . Remark 2.5.
The formal cdga C ( g ) is otherwise known as the Chevalley-Eilenberg cohomologycomplex of the dgla g . Remark 2.6.
We will occasionally need the following mild generalization of the functors C and L . Let A be a cdga. For a dgla g we will call A ⊗ g an A -linear dgla ; similarly if B is a formalcdga then A ⊗ B is an A -linear formal cdga . The morphisms of A -linear dgla and formal cdgasare required to be morphisms of dg A -modules. We define: C A ( A ⊗ g ) := ˆ S A Σ − ( A ⊗ g ∗ ) ∼ = A ⊗ ˆ S Σ − g ∗ ∼ = A ⊗ C ( g ); the differential in C A ( A ⊗ g ) is the tensor product of the differential in A and in C ( g ) . Thus, C A ( A ⊗ g ) is an A -linear formal cdga. Likewise L A ( A ⊗ B ) is the A -linear dgla A ⊗ L ( B ) .Clearly the functors C A and L A are adjoint functors between the categories of A -linear dglasand A -linear dglas. The category L ie is itself a closed model category where weak equivalences are quasi-isomorphisms of dglas and fibrations are surjective maps.The following result is proved in [15]. Theorem 2.7. • For any formal cdga A and a dgla g here is a natural isomorphism of sets Hom F Alg ( L ( A ) , g ) ∼ = Hom Lie ( C ( g ) , A ) . • The functor L converts fibrations and acyclic fibrations in F Alg into cofibrations andacyclic cofibrations in L ie respectively. • The functor C converts fibrations and acyclic fibrations in L ie into cofibrations andacyclic cofibrations in F Alg respectively. • The adjunction maps p = p ( g ) : L C ( g ) → g and q = q ( A ) : C L ( A ) → A are weakequivalences of dglas and formal cdgas respectively; thus the functors C and L induceinverse equivalences of the homotopy categories of F Alg and L ie . An important special case of a weak equivalence in F Alg is a filtered quasi-isomorphism . Definition 2.8. • An admissible filtration on a formal cdga A is a filtration of the form A = F ( A ) ⊃ F ( A ) ⊃ . . . which is Hausdorff, i.e. T p> F p ( A ) = 0 . In addition, an admissiblefiltration is required to be multiplicative , i.e. F p ( A ) · F q ( A ) ⊂ F p + q ( A ) . The associatedgraded cdga is defined as Gr( A ) = L p> F p ( A ) /F p +1 ( A ) . • A map f : A → B in F Alg is a filtered quasi-isomorphism if f induces a quasi-isomorphism on the associated graded cdgas for some admissible filtrations on A and B . We will refer to a formal cdga endowed with an admissible filtration as simply ‘filtered formalcdga’. The canonical filtration on a formal cdga is an example of an admissible filtration. Notethat an admissible filtration F p on a formal cdga A is always complete , i.e. lim −→ A/F p ( A ) = A owing to the vanishing of lim −→ on the category of formal vector spaces.We have the following result proved in the language of coalgebras in [15], Proposition 4.4.4. roposition 2.9. A filtered quasi-isomorphism between filtered formal cdgas is necessarily aweak equivalence. (cid:3) L ∞ algebras The notion of a weak equivalence in F Alg is somewhat obscure and one wants to have amore explicit characterization of them. In this section we will give such a characterization for cofibrant cdgas and also show that the latter are, effectively, the same as L ∞ algebras. Definition 3.1. • Let ( V, d ) be a (homologically) graded dg vector space; then an L ∞ algebra structure on V is a continuous derivation m of the completed symmetric algebra ˆ S Σ − V ∗ such that m has no constant and linear terms and ( m + d ) = 0 . We will call ˆ S Σ − V ∗ supplied withthe differential m + d the representing formal cdga of V . The homogeneous componentsof m will be denoted by m i , so that m i : ˆ S i Σ − V ∗ → Σ − V ∗ . If V itself has vanishingdifferential, then ( V, m ) is called a minimal L ∞ algebra. • For two L ∞ algebras ( V, m V ) and ( U, m U ) an L ∞ morphism U → V is a continuousmap of formal cdgas ˆ S Σ − V ∗ → ˆ S Σ − U ∗ . It is called an L ∞ quasi-isomorphism if itinduces a quasi-isomorphism of dg vector spaces Σ − U ∗ → Σ − V ∗ (or equivalently of dgvector spaces U → V ). One of the central results about L ∞ algebras is the decomposition theorem for L ∞ algebras(Lemma 4.9 of [19]) which was later generalized and reproved by several authors. To formulateit, let us introduce an L ∞ algebra represented by the formal cdga L ( x, y ) = ˆ S ( x, y ) with | y | = | x | + 1 and m ( x ) = y . This is an elementary linear contractible L ∞ algebra. A generallinear contractible L ∞ algebra is a direct product of elementary ones (this corresponds to thetensor product of representing formal cdgas). Then one has the following result whose proofcould be found in the mentioned paper of Kontsevich; a non-inductive proof, also valid in theoperadic generality is contained in [9]. Theorem 3.2.
Any L ∞ algebra ( V, m ) is L ∞ isomorphic to a direct product of a linear con-tractible L ∞ algebra and a minimal one. The latter is determined uniquely up to a non-canonicalisomorphism. (cid:3) Proposition 3.3. (1)
The map ˆ S Σ − V ∗ → ˆ S Σ − U ∗ representing an L ∞ quasi-isomorphism ( U, m U ) → ( V, m V ) is a weak equivalence in F Alg ; conversely any weak equivalence ˆ S Σ − V ∗ → ˆ S Σ − U ∗ represents an L ∞ quasi-isomorphism of the corresponding L ∞ algebras. (2) Any formal cdga ˆ S Σ − V ∗ determining an L ∞ structure on a space V is a cofibrant objectin F Alg . Conversely, any cofibrant formal cdga is a formal cdga ˆ S Σ − V ∗ representingan L ∞ algebra.Proof. Note first, that an L ∞ quasi-isomorphism ˆ S Σ − V ∗ → ˆ S Σ − U ∗ is a filtered quasi-isomorphism with respect to the canonical filtrations on ˆ S Σ − V ∗ and ˆ S Σ − U ∗ and it follows byProposition 2.9 that it is a weak equivalence. Conversely, let f : ˆ S Σ − V ∗ → ˆ S Σ − U ∗ be a weakequivalence between two formal cdgas; such a map is clearly an L ∞ morphism and we need toshow that f induces a quasi-isomorphism on the spaces of indecomposables: Σ − V ∗ → Σ − U ∗ .Since f is a weak equivalence the dgla map L ( f ) : L ( ˆ S Σ − V ∗ ) → L ( ˆ S Σ − U ∗ )is a quasi-isomorphism.Let us now compute the homology of L ( ˆ S Σ − V ∗ ) and L ( ˆ S Σ − U ∗ ). The filtration by thepowers of the augmentation ideal on ˆ S Σ − V ∗ determines an increasing filtration on L ( ˆ S Σ − V ∗ ). ince the internal differential of the formal cdga ˆ S Σ − V ∗ vanishes on its associated graded it fol-lows that the differential d in the spectral sequence converging to the homology of L ( ˆ S Σ − V ∗ )is the Harrison differential d II (see Definition 2.1). Since ˆ S Σ − V ∗ is a (completed) free gradedcommutative algebra the homology with respect to this differential reduces to V . Thus, thespectral sequence collapses and H ∗ [ L ( ˆ S Σ − V ∗ ] ∼ = H ∗ ( V ) . Arguing similarly, we obtain H ∗ [ L ( ˆ S Σ − U ∗ ] ∼ = H ∗ ( U )and since the spectral sequences for L ( ˆ S Σ − V ∗ ) and L ( ˆ S Σ − U ∗ ) map into one another under L ( f ), we conclude that f induces a quasi-isomorphism Σ − V ∗ → Σ − U ∗ as required. Part (1)is therefore proved.Let us prove (2). Consider an L ∞ algebra ( V, m ) and recall that by the decomposition theoremit is isomorphic to a product of linear contractible L ∞ algebras and a minimal one. Since linearcontractible L ∞ algebras are obviously represented by cofibrant formal cdgas we may assumethat ( V, m ) is minimal from the start. Consider its representing formal cdga A = ˆ S Σ − V ∗ andthe adjunction map q : C L ( A ) → A . The latter map represents an L ∞ quasi-isomorphism andso it has a quasi-inverse i : A → C L ( A ). We see that q ◦ i represents a quasi-isomorphism of A onto itself which is chain homotopic to the identity when restricted onto the indecomposablesof A ; since A represents a minimal L ∞ algebra q ◦ i must be the identity and so A is a retractof C L ( A ). Since the latter is cofibrant so is A .Conversely, for any cofibrant formal cdga A there is a surjective weak equivalence C L ( A ) → A which will necessarily admit a section (since A is cofibrant) exhibiting A as a retract of aformal cdga representing an L ∞ algebra. Finally, it is clear that a (continuous) retract of aformal power series algebra is a formal power series algebra itself which concludes the proof of(2). (cid:3) Remark 3.4. (1)
We see, therefore, that as long as we are interested in the homotopy theoretical problemsin the category F Alg , we can restrict ourselves to considering only those formal cdgaswhich represent L ∞ algebras; a weak equivalence between such objects will then be ausual L ∞ quasi-isomorphism. One could go even further and consider only those formalcdgas representing minimal L ∞ algebras; a weak equivalence will in this case reduce toa mere isomorphism. (2) It is not true, in general, that any weak equivalence in F Alg must be a filtered quasi-isomorphism with respect to canonical filtrations (although for cofibrant formal cdgas itis true by Proposition 3.3). Here is a simple example. Let A be the formal cdga spannedby a single vector x with | x | = 1 and with zero multiplication (the augmented version of A is the algebra ˜ A = k [ x ] /x ). Then it is easy to see that C L ( A ) is the formal cdga B = k [ x, y ] + with | y | = 1 and d ( y ) = x . Further, the obvious map f : B → A sending y to zero is a weak equivalence in F Alg , but it is not a filtered quasi-isomorphism withrespect to the canonical filtration on B . The ‘correct’ filtration on B is specified by F ( B ) = B and F p ( B ) = ( yx p − , x p ) , p ≥ , the ideal in k [ x, y ] generated by yx p − and x p rather than the p th power of the maximal ideal. Under this filtration the map f is afiltered quasi-isomorphism. (3) On the other hand, it is true that if f : A → B is a weak equivalence in F Alg then A and B can be connected by a zig-zag of maps which are filtered weak equivalences (butnot necessarily with respect to the canonical filtrations). Indeed, we have the following ommutative diagram: C L ( A ) C L ( f ) / / (cid:15) (cid:15) C L ( B ) (cid:15) (cid:15) A f / / B where the vertical arrows are the adjunction morphisms. These adjunction morphismsare filtered quasi-isomorphisms by [15] , Proposition 4.4.3 and the upper horizontal mapis a filtered quasi-isomorphism as a weak equivalence between cofibrant formal cdgas. Next we will discuss the notion of a homotopy in the category F Alg . To this end, consider thecdga k [ z, dz ] the free graded commutative algebra on the generators z and dz with | z | = 0 , | dz | =1 and d ( z ) := dz ; this is the familiar de Rham algebra of forms on an interval. Specializing z to1 and 0 gives cdga maps | , | : k [ z, dz ] → k . Given a formal cdga A , we will consider A [ z, dz ],the tensor product of A and k [ z, dz ]. There are two specialization maps A [ z, dz ] → A which wewill denote | and | as above. Definition 3.5. (1)
Let
A, B be formal cdgas and f, g : A → B be maps in F Alg . Then f and g are said tobe Sullivan homotopic if there exists a continuous cdga map h : A → B [ z, dz ] such that | ◦ h = f and | ◦ h = g . (2) Two formal cgdas A and B are called Sullivan homotopy equivalent if there are maps f : A → B and g : B → A such that f ◦ g and g ◦ f are Sullivan homotopic to id B and id A respectively. Recall that one also have a similar notion of Sullivan homotopy in the category of dglas: twodgla maps f, g : g → h are Sullivan homotopic if there is a dgla map g → h [ z, dz ] restricting to f and g at z = 0 and z = 1 respectively. Furthermore, the dgla h [ z, dz ] is a path object for h and thus, the Sullivan homotopy for dglas is an instance of the closed model theoretic notion of right homotopy . Similarly, one would like to be able to view B [ z, dz ] as a path object for B anda Sullivan homotopy as a right homotopy in the closed model category of formal cdgas. Suchan interpretation is not available since k [ z, dz ] is not a formal cdga. Fortunately, the followingresult shows that for all practical purposes one can treat B [ z, dz ] as if it were a path object for B . Theorem 3.6.
Let A be a cofibrant formal cdga, B be an arbitrary formal cdga and f, g : A → B be two maps in F Alg . Then f and g are Sullivan homotopic if and only if they are homotopic,i.e. represent the same maps in the homotopy category of F Alg . Thus, the set of equivalenceclasses of Sullivan homotopic maps A → B is bijective with the set of maps from A into B inthe homotopy category of F Alg . This set will be denoted by [ A, B ] ∗ .Proof. Let h : A → B [ z, dz ] be a homotopy between f and g . This homotopy can be viewed asa k [ z, dz ]-linear map A [ z, dz ] → B [ z, dz ], i.e. as a map of k [ z, dz ]-linear formal cdgas. Applyingthe functor L k [ z,dz ] to it we obtain a k [ z, dz ]-linear map L ( B )[ z, dz ] ∼ = L k [ z,dz ] ( B [ z, dz ]) → L k [ z,dz ] ( A [ z, dz ]) ∼ = L ( A )[ z, dz ]which is the same as a dgla homotopy L ( B ) → L ( A )[ z, dz ] . Clearly at z = 0 , L ( f ) , L ( g ) : L ( B ) → L ( A ) respectively. It follows that the maps L ( f ) and L ( g ) are homotopic and thus, f and g are homotopic as well.Conversely, suppose that f and g are homotopic. This implies that the dgla maps L ( f ) and L ( g ) : L ( B ) → L ( A ) are homotopic and thus (since L ( B ) is a cofibrant dgla) there exists aSullivan dgla homotopy L ( B ) → L ( A )[ z, dz ] which restricts to L ( f ) and L ( g ) at z = 0 , k [ z, dz ]-map and applying the functor L k [ z,dz ] we obtain a map f formal k [ z, dz ]-linear cdgas C L ( A )[ z, dz ] → C L ( B )[ z, dz ] which is the same as a Sullivanhomotopy s : C L ( A ) → C L ( B )[ z, dz ]which specializes to C L ( f ) and C L ( g ) at z = 0 and z = 1 respectively. Consider the followingdiagram C L ( A ) s / / q ( A ) (cid:15) (cid:15) C L ( B )[ z, dz ] q ( B [ z,dz ]) (cid:15) (cid:15) A i T T ✖✤✭ B [ z, dz ]Here the dotted map i is a section of q ( A ) (which exists because A is a cofibrant formal cdga).Now the required homotopy h : A → B [ z, dz ] is defined by the formula h = q ( B [ z, dz ]) ◦ s ◦ i . (cid:3) Corollary 3.7.
Let A be a cofibrant formal cdga and B be an arbitrary formal cdga: (1) The relation of Sullivan homotopy on the set of maps A → B is an equivalence relation. (2) If A ′ is a cofibrant formal cdga weakly equivalent to A and B ′ is weakly equivalent to B then there is a bijective correspondence [ A, B ] ∗ ∼ = [ A ′ , B ′ ] ∗ . (3) Two cofibrant formal cdgas are Sullivan homotopy equivalent if and only if they areweakly equivalent.Proof.
All these statements follow from Theorem 3.6 in a straightforward fashion. For example,if f : A → B is a weak equivalence between two cofibrant formal cdgas then f is invertible inthe homotopy category of F Alg ; the inverse map is then represented by a map g : B → A since B is cofibrant and all formal cdgas are fibrant. We have f ◦ g is homotopic to id B and so f ◦ g is Sullivan homotopic to id B ; similarly g ◦ f is Sullivan homotopic to id A . Other claimsare equally obvious. (cid:3) From now on we will refer to Sullivan homotopy as simply homotopy (assuming that thesource is cofibrant as it will always be).3.1.
Free homotopy.
We will now discuss the notion of a free homotopy in the category F Alg .Unfortunately, this notion does appear slightly ad hoc and from the abstract point of view itseems unjustified. The reason for introducing it is that it corresponds to the notion of a notnecessarily basepoint-preserving homotopy between connected spaces and this correspondencewill be made precise later on in the paper. Recall that for a formal cdga A we denoted by ˜ A the algebra obtained from A by adjoining a unit. Definition 3.8.
Let f, g : A → B be a map between two formal cdgas and ˜ f , ˜ g : ˜ A → ˜ B are the corresponding maps between the unital algebras ˜ A and ˜ B . We say that f and g are freely homotopic if there exists a (continuous) map h : ˜ A → ˜ B [ z, dz ] such that h | z =0 = ˜ f and h | z =1 = ˜ g . The set of equivalence classes generated by the relation of free homotopy will bedenoted by [ A, B ] . Remark 3.9.
It is clear that for two formal cdgas A and B , any map ˜ A → ˜ B is induced bya map A → B . Furthermore, a homotopy between two maps f, g : A → B gives rise to a freehomotopy, but not vice-versa: there may be maps in F Alg which are freely homotopic but not homotopic.
The following proposition shows that the for two formal cdgas
A, B , the set [
A, B ] has ahomotopy invariant meaning.
Proposition 3.10.
Let
A, A ′ , B and B ′ be cofibrant formal cdgas with A weakly equivalent to A ′ and B weakly equivalent to B ′ . Then there is a bijective correspondence [ A, B ] → [ A ′ , B ′ ] .Proof. Since A and A ′ are cofibrant and weakly equivalent they must be Sullivan homotopyequivalent, so that there are maps A → A ′ and A ′ → A which are homotopy inverse. Thisdetermines two mutually inverse maps [ A, B ] ⇆ [ A ′ , B ]. Similarly there is a bijective correspon-dence [ A ′ , B ] ⇆ [ A ′ , B ′ ]. (cid:3) emark 3.11. It is possible that the statement of the above proposition holds without thecondition that A and A ′ be cofibrant (compare Corollary 3.7, (2)). To address this questionproperly one would have to go beyond the category F Alg , allowing cdgas with more than onemaximal ideal (equivalently considering not necessarily cocomplete dg coalgebras). The versionstated here is sufficient for most applications, however. The Quillen-Neisendorfer closed model category structure
There is another, apparently, more natural, notion of a weak equivalence on the categoryof formal cdgas due to Quillen [22] and Neisendorfer [21] (in fact, Quillen and Neisendorferworked with coalgebras but we saw that this difference is immaterial). Namely, we say that twoformal cdgas A and B are homologically equivalent if there is a (continuous) morphism A → B inducing an isomorphism on cohomology. In other words, our new weak equivalences are now(continuous) quasi-isomorphisms.It is clear that weakly equivalent formal cdgas are homologically equivalent. Indeed, if f : A → B is a weak equivalence, then (by definition) the map of dglas L ( f ) : L ( B ) → L ( A )is a quasi-isomorphism and then the map of formal cgdas C L ( f ) : C L ( A ) → C L ( B ) willbe a quasi-isomorphism from which it follows that f was a quasi-isomorphism to begin with.On the other hand, a homology equivalence in F Alg need not be a weak equivalence; e.g. forany semisimple Lie algebra g the (formal) cdga C ( g ) is quasi-isomorphic to its cohomology ring(which is the symmetric algebra on a collection of odd generators) but this quasi-isomorphism isnot a weak equivalence because otherwise g would be quasi-isomorphic, and hence, isomorphic,to an abelian Lie algebra.It makes sense, therefore, to consider the Bousfield localization of Hinich’s closed modelcategory F Alg with respect to homology equivalences. We will not treat this problem in fullgenerality but note that the solution for the subcategory F Alg + consisting of connected (i.e.positively-graded) formal cdgas was given in [21], generalizing the previous work of Quillen,[22]. Note that the functor L : F Alg L ie restricts to the functor (which we denote bythe same symbol) L : F Alg + L ie + where L ie + stands for the category of non-negativelygraded dglas. Similarly the functor C : L ie F Alg restricts to C : L ie + F Alg + . Theorem 4.1. (1)
The category F Alg + admits a closed model category structure such that fibrations aresurjective maps and weak equivalences are homology equivalences of connected formalcdgas. (2) The adjoint functors C and L induce an equivalence between the homotopy category of F Alg + and the full subcategory of the homotopy category of L ie + consisting of dglaswhose homology are pro-nilpotent graded Lie algebras.Proof. The statement of the theorem is just a reformulation of Proposition 5.2 and Proposition7.2 of [21]. (cid:3)
Since a weak equivalence of formal cdgas is stronger than a homology equivalence there arefewer cofibrant objects in the Quillen-Neisendorfer structure on F Alg + compared to Hinich’sone. For example, the formal cdga C ( g ) will not be Quillen-Neisendorfer cofibrant unless g haspro-nilpotent homology Lie algebra; more generally a formal cdga ˆ S Σ − V ∗ representing an L ∞ algebra V will not be cofibrant unless a suitable nilpotency condition is satisfied. We restrictourselves by describing a special kind of Quillen-Neisendorfer cofibrant objects coming fromdglas with nilpotent homology; a more general result is contained in [21]. Before formulating itrecall the notion of a Sullivan minimal cdga. Definition 4.2.
A Sullivan minimal cdga is a cdga of the form ( S ( W ) , d ) where W is a positivelygraded dg vector space which is a union of its subspaces W = S W i , i = 1 , , . . . with d ( W i ) ⊂ S ( W i − ) and d is decomposable: d ( W ) ⊂ S ( W ) + · S ( W ) + . emark 4.3. Note that a Sullivan minimal cdga can always be completed and thus considered asa formal cdga (which represents a certain L ∞ algebra). We will be mostly interested in the casewhen W is of finite type; in this case we will refer to ( S ( W ) , d ) as a Sullivan cdga of finite type.Note that in that case there is no difference between completed and uncompleted symmetricalgebra. This is the most pleasant case: a Sullivan minimal cdga of finite type represents acofibrant object both in the category F Alg and F Alg + . Proposition 4.4.
Let g be a non-negatively graded dgla with H ( g ) nilpotent and of finite type.Then the formal cdga C ( g ) = ˆ S Σ − g ∗ is Quillen-Neisendorfer cofibrant. Moreover, there is anisomorphism in F Alg + : C ( g ) ∼ = A ⊗ B where A is a Sullivan minimal cdga and B = ˆ S Σ − V ∗ represents a linear contractible L ∞ algebra V .Proof. This is essentially Proposition 3.12 of [21] and we outline how its proof is adopted tothe present context. Consider C ( g ) as a formal cdga representing an L ∞ algebra; then by thedecomposition theorem there is an isomorphism of formal cdgas C ( g ) ∼ = ˆ S Σ − [ H ( g )] ∗ ⊗ B where B represents a linear contractible L ∞ algebra and ˆ S Σ − [ H ( g )] ∗ represents the L ∞ minimalmodel of the dgla g . Note that since H = H ( g ) is a graded Lie algebra of finite type thecompleted symmetric algebra on Σ − [ H ( g )] ∗ is the same as the uncompleted one.Set H ( m,
0) := L mi =0 H i , the m ’th ‘Postnikov stage’ of H . Consider further H ( m,
1) :=[ H , H ( m, H ( m,
2) = [ H , H ( m, H ( m, ⊃ H ( m ) ⊃ . . . ⊃ H ( m, n ) is afinite filtration on H ( m,
0) for every m . Then we have the refined filtration on H (which corre-sponds to the principal refinement of the Postnikov tower of the rational space correspondingto H ): H (0 , n ) ⊂ . . . ⊂ H (0 ,
0) = H ⊂ H (1 , n ) ⊂ . . . ⊂ H (1 ,
0) = H ⊕ H ⊂ . . . and the induced filtration on S Σ − H ∗ will satisfy the second condition of the Sullivan minimalmodel by the nilpotency of H . (cid:3) Remark 4.5.
In the proposition above it was essential that g (and so H = H ( g ) ) was non-negatively graded. Without this assumption the nilpotency of H alone does not even guaranteethat the formal cdga representing an L ∞ minimal model of g would be a polynomial algebra, letalone that it would be a minimal Sullivan cdga. The MC moduli set and its homotopy invariance
We will now describe the notion of an MC moduli set in the context of L ∞ algebras andformal cdgas. Definition 5.1.
Let ( V, m ) be an L ∞ algebra and A be a formal cdga. The formal cdga A ⊗ ˆ S Σ − V ∗ represents an A -linear L ∞ algebra by A -linear extension of scalars; the corresponding L ∞ structure will be denoted by m A . Then an element ξ ∈ ( A ⊗ Σ V ) is Maurer-Cartan (MCfor short) if ( d A ⊗ Σ V )( ξ ) + P ∞ i =2 1 i ! m Ai ( ξ ⊗ i ) = 0 . The set of Maurer-Cartan elements in A ⊗ Σ V will be denoted by MC(
V, A ) . The correspondence (
V, A ) MC(
V, A ) is clearly functorial in A and V . It is straightforwardto check and well-known (see, e.g. [10]) that as a functor of the second argument MC( V, A ) isrepresentable by the formal cdga ˆ S Σ − V ∗ . In other words, there is a natural bijective corre-spondence MC( V, A ) ∼ = Hom F Alg ( ˆ S Σ − V ∗ , A ) . Furthermore, if V is a dgla then we also have the following natural bijection, [22]:MC( V, A ) ∼ = Hom L ie ( L ( A ) , V ) . If ξ ∈ MC(
V, A ) then we will abuse the notation and denote the corresponding map of formalcdgas ˆ S Σ − V ∗ → A by the same symbol ξ . efinition 5.2. Two MC elements ξ, η ∈ MC(
V, A ) are called homotopic if there exists anMC-element h ∈ ( V, A [ z, dz ]) such that h | z =0 = ξ and h | z =1 = η . Remark 5.3.
Note that there is a slight imprecision built into this definition: an MC elementas we defined it has coefficients in a formal cdga, but A [ z, dz ] is not formal. Nevertheless,the definition of an MC element does make sense for such a coefficient cdga and this will notcause us any further problems. This issue has already come up in the definition of a Sullivanhomotopy for maps between formal cdgas. It is likely that this minor nuisance can be fixed eitherby extending the category of formal cdgas suitably or by modifying the notion of the Sullivanhomotopy, however we will refrain from elaborating on this further. Proposition 5.4.
The relation of homotopy on
MC(
V, A ) is an equivalence relation. The setof equivalence classes under this relation is called the MC moduli set of V with coefficients in A and it will be denoted by M C ( V, A ) .Proof. One only has to note that a homotopy between two MC elements ξ, η ∈ MC(
V, A ) is thesame as a Sullivan homotopy between the maps ξ, η : ˆ S Σ − V ∗ → A ; the statement then followsfrom Corollary 3.7, (1). (cid:3) The following is one of the main results about the MC moduli set. This result has a longhistory; a version of it is contained in the unpublished manuscript of Schlessinger and Stasheff[24]; it was further elaborated in [13]. Kontsevich formulated it using L ∞ algebras in [19] andKeller’s method [17] is essentially the same as ours. We believe that our formulation is the mostgeneral among those currently in existence; its obvious Z / Theorem 5.5.
Let V be an L ∞ algebra and A be a formal cdga. Then for any L ∞ algebra U which is L ∞ quasi-isomorphic to V and any formal cdga B which is weakly equivalent to A there is a natural bijective correspondence M C ( V, A ) ∼ = M C ( U, B ) . Proof.
Let ˆ S Σ − V ∗ and ˆ S Σ − U ∗ be the formal cdgas which represent the L ∞ algebras V and U respectively. Then we have M C ( V, A ) ∼ = [ ˆ S Σ − V ∗ , A ] ∗ and M C ( U, A ) ∼ = [ ˆ S Σ − U ∗ , B ] ∗ sothe statement follows from Corollary 3.7, (2). (cid:3) Remark 5.6.
In applications it often happens that the L ∞ algebra V that figures in Theorem5.5 is in fact a pronilpotent dgla in which case the two MC elements are equivalent if and onlyif they are gauge equivalent , an important result due to Schlessinger-Stasheff [24] , cf. also [9] for a discussion of the notion of a gauge equivalence and a short proof. Sometimes one wants to consider MC elements in V with coefficients in a not necessarilyformal cdga A (such as the ground field k , for instance). In such a situation the immediateproblem is that the series ( d A ⊗ Σ V )( ξ ) + P ∞ i =2 1 i ! m Ai ( ξ ⊗ i ) = 0 defining the MC element ξ neednot converge. One important case when this problem goes away is when V is a strict dgla. Inthat case the MC equation becomes the familiar equation of a flat connection: d ( ξ ) + 12 [ ξ, ξ ] = 0 . One can still call the elements ξ ∈ A ⊗ Σ V satisfying the above equation the MC elements of V with coefficients in A and denote the corresponding set by MC( V, A ). Furthermore, the notionof a homotopy (or a gauge equivalence) makes perfect sense and we are entitled to form themoduli set
M C ( V, A ) just as above; in the case when V ⊗ A is not a nilpotent dgla the homotopymay not be transitive and we have to take its transitive closure. However the issue of homotopyinvariance of M C ( V, A ) is more subtle; e.g. it is no longer true that an L ∞ quasi-isomorphismin V induces a bijection on the MC moduli set, even when V is nilpotent. Here is a simple (insome sense simplest) example. Let V be the 2-dimensional dgla generated by one element x indegree − d ( x ) = − [ x, x ]. Clearly V is acyclic, that is to say it is quasi-isomorphic tothe zero Lie algebra. However it is easy to see that M C ( V, k ) consists of two elements – 0 and . This demonstrates that one cannot simply remove the condition that the coefficient cdga A be formal.In our applications we will not consider MC elements with coefficients in a non-formal cdga,however we will need a version of the MC moduli set which could be viewed as a small stepaway from the formal situation. As far as we are aware this notion has not been consideredbefore. Definition 5.7.
Let ( V, m ) be an L ∞ algebra with representing formal cdga ˆ S Σ − V ∗ and A bea formal cdga. Then two M C elements ξ, η ∈ MC(
V, A ) are called freely homotopic if they arefreely homotopic as maps of formal cdgas ˆ S Σ − V ∗ → A . Denote the transitive closure of therelation of being freely homotopic by ∼ ; then the reduced MC moduli space of V with coefficientsin A is defined as ] M C ( V, A ) := MC(
V, A ) / ∼ . The following result is a reformulation of Proposition 3.10.
Proposition 5.8.
Let V be an L ∞ algebra and A be cofibrant formal cdga. Then the set ] M C ( V, A ) is homotopy invariant in the sense that for any L ∞ algebra V ′ quasi-isomorphic to V and a cofibrant formal cdga A ′ quasi-isomorphic to A there is a bijective correspondence ] M C ( V, A ) ∼ = ] M C ( V ′ , A ′ ) . (cid:3) Remark 5.9.
We see that the relation on the set of MC elements of being freely homo-topic is stronger then the usual homotopy relation. It follows that there is a surjective map
M C ( V, A ) → ] M C ( V, A ) . A more precise relationship between these two moduli sets can some-times be established; in fact it follows from our topological interpretation given in the last sectionthat in the case when the representing formal cdga of V is a Sullivan model of a rational space X there is an action of the group π ( X ) on M C ( V, A ) so that the quotient by this action is ] M C ( V, A ) . In the context of deformation theory we can offer the following analogy: given anobject O and its universal deformation with a (perhaps dg) base X there is a ‘residual’ actionof the group Aut( O ) of automorphisms of O on X . In favorable cases one expects the quotientto be the germ of the moduli space at O . Connected covers of dglas and MC moduli.
For the needs of rational homotopytheory we need to consider connected formal cdgas (such a cdga A is required to satisfy A i = 0for i ≤
0) and connected L ∞ algebras (such an L ∞ algebra V is required to satisfy V i = 0 for i < Definition 5.10.
Let ( V, m V ) be an L ∞ algebra; then its connected cover is the L ∞ algebra V h i defined by the formula V h i i = V i , if i > { d : V → V − } if i = 00 , if i < . The L ∞ structure m V h i on V h i is the obvious restriction of m V . It is clear that there is a natural (strict) L ∞ map V h i → V , moreover for two L ∞ quasi-isomorphic L ∞ algebras V and U the corresponding connected covers V h i and U h i are L ∞ quasi-isomorphic. Proposition 5.11.
For any connected formal cdga A and an L ∞ algebra V there is a naturalbijection M C ( V, A ) ∼ = M C ( V h i , A ) . Proof.
Without loss of generality we can assume that V is a minimal L ∞ algebra. Choose ahomogeneous basis { x i } , { y i } in V for which | x i | ≥ | y i | <
0. Passing to the dual wesee that the L ∞ algebra V is represented by a formal cdga ˆ S { Σ − x ∗ i , Σ − y ∗ i } whereas the L ∞ lgebra V h i is represented by its quotient formal cdga ˆ S { Σ − x ∗ i } . Since MC elements in V and V h i with coefficients in A are represented by maps from ˆ S { Σ − x ∗ i , Σ − y ∗ i } and ˆ S { Σ − x ∗ i } into A the degree considerations give a bijection MC( V, A ) ∼ = MC( V h i , A ). It is likewise clearthat any homotopy of elements in MC( V, A ) lifts to a homotopy in g M C ( V, A ) from which thedesired statement follows. (cid:3) Maurer-Cartan twisting
We now recall the procedure of twisting in L ∞ algebras by MC elements, following [9]. Let( V, m ) be an L ∞ algebra, A be a formal cdga and ξ ∈ MC(
V, A ). We may regard ξ as a (formal) A -linear derivation of the formal cdga A ⊗ ˆ S Σ − V ∗ ; indeed ξ could be viewed as a linear functionon Σ − V ∗ ∼ = 1 ⊗ Σ − V ∗ ⊂ A ⊗ ˆ S Σ − V ∗ and we extend it to the whole of A ⊗ ˆ S Σ − V ∗ by the A -linear Leibniz rule.Exponentiating the derivation ξ we obtain an automorphism of A ⊗ ˆ S Σ − V ∗ so that e ξ :=id + ξ + ξ + . . . ; the convergence of this formal power series is ensured by the formality of A .Then it is straightforward to see that the MC condition on ξ implies (in fact, is equivalent to)that the formal derivation e ξ ( d A ⊗ Σ V + m A ) e − ξ = e ξ m A e − ξ − d A ⊗ Σ V ( ξ ) is an L ∞ structure on A ⊗ ˆ S Σ − V ∗ , in other words, that it has no constant term. We will write this new A -linear L ∞ structure on V as (( A ⊗ V ) ξ , m ξ ) and call it the twisting of m by ξ . Note that the formalderivation e ξ m A e − ξ − d A ⊗ Σ V ( ξ ) is also defined on ˜ A ⊗ ˆ S Σ − V ∗ and thus, we can define theunital version of the MC twisting: ( ˜ A ⊗ V ) ξ . The following result shows that homotopic M C elements determine weakly equivalent twistings.
Proposition 6.1.
Let ξ, η be two homotopic elements in
MC(
V, A ) where ( V, m ) is an L ∞ algebra and A is a formal cdga. Then: (1) The L ∞ algebras ( ˜ A ⊗ V ) ξ and ( ˜ A ⊗ V ) η are L ∞ isomorphic. (2) The L ∞ algebras ( A ⊗ V ) ξ and ( A ⊗ V ) η are L ∞ isomorphic.Proof. Let us prove the first statement; the proof of the second statement is similar. A homotopybetween ξ and η is an MC element h ∈ MC( ˜ A ⊗ V [ z, dz ]) that specializes to ξ and η at z = 0and z = 1 respectively. Consider the L ∞ algebra ( ˜ A ⊗ V [ z, dz ]) h , the twisting of ˜ A ⊗ V [ z, dz ]by h . This L ∞ algebra structure can be viewed as an element in MC( g , A [ z, dz ]) where g isthe Lie algebra of formal derivations of ˆ S Σ − V ∗ having no constant and linear terms. In otherwords, h is a homotopy between two elements in MC( g , ˜ A ). Such a homotopy implies gaugeequivalence (see Remark 5.6) and it follows that the L ∞ algebras ( A ⊗ V ) ξ and ( A ⊗ V ) η are L ∞ isomorphic as desired. (cid:3) Remark 6.2.
Let V be a dgla (as opposed to an L ∞ algebra), not necessarily nilpotent. Thenone can define its twisting without tensoring with a formal cdga; namely for ξ ∈ MC( V ) :=MC( V, k ) we get a twisted dgla V ξ whose Lie bracket is the same as that of V and the differentialis given as d ξ ( v ) = d ( v ) + [ ξ, v ] for v ∈ V . Then it is easy to see that in this situation the proofof Proposition 6.1 still implies that two homotopic MC elements give rise to L ∞ isomorphictwisted dglas. However the next result uses the formal cdga variable in an essential way. Proposition 6.3. (1)
Let f : ( V, m V ) → ( U, m U ) be an L ∞ quasi-isomorphism, A be a formal cdga, ξ ∈ MC(
V, A ) and f ∗ ( ξ ) be the element in MC(
U, A ) corresponding to ξ under the map f ∗ :MC( V, A ) → MC(
U, A ) induced by f . Then there are natural L ∞ quasi-isomorphisms f ξ : ( A ⊗ V ) ξ → ( A ⊗ U ) f ∗ ( ξ ) and ˜ f ξ : ( ˜ A ⊗ V ) ξ → ( ˜ A ⊗ U ) f ∗ ( ξ ) (2) Let ( V, m ) be an L ∞ algebra, g : A → B be a weak equivalence between two formalcdgas A and B , ξ ∈ MC(
V, A ) and g ∗ ( ξ ) be the element in MC(
V, B ) corresponding to ξ under the map g ∗ : MC( V, A ) → MC(
V, B ) induced by g . Then there are natural L ∞ quasi-isomorphisms g ξ : ( A ⊗ V ) ξ → ( B ⊗ V ) g ∗ ( ξ ) and ˜ g ξ : ( ˜ A ⊗ V ) ξ → ( ˜ B ⊗ V ) g ∗ ( ξ ) . roof. We restrict ourselves with proving the statements involving f ξ and g ξ ; the proofs for ˜ f ξ and ˜ g ξ are completely analogous. For (1) set ˜ m V := d A ⊗ Σ V + m AV and denote by ˜ m ξV the formal A -linear derivation of A ⊗ ˆ S Σ − V ∗ obtained by twisting ˜ m V with ξ so that ˜ m ξV = e ξ ˜ m AV e − ξ ;similarly set ˜ m U := d A ⊗ Σ U + m AU and ˜ m ξU = e f ∗ ( ξ ) ˜ m AU e − f ∗ ( ξ ) . Then the map f ξ := e f ∗ ( ξ ) f e − ξ :ˆ S Σ − U ∗ → ˆ S Σ − V ∗ determines an L ∞ map ( A ⊗ V ) ξ → ( A ⊗ U ) f ∗ ( ξ ) . Indeed, f ξ ˜ m ξV =[ e f ∗ ( ξ ) f e − ξ ][ e ξ ˜ m AV e − ξ ]= e f ∗ ( ξ ) f ˜ m AV e − ξ = e f ∗ ( ξ ) ˜ m AU f e − ξ =[ e f ∗ ( ξ ) ˜ m AU e − f ∗ ( ξ ) ][ e f ∗ ( ξ ) f e − ξ ]= ˜ m f ∗ ( ξ ) U f ξ . Here we used the equality f ˜ m AV = ˜ m AU f which holds since f is an L ∞ map.Further, the 1-component of the map f ξ is a map between filtered dg vector spaces f ξ : A ⊗ V → A ⊗ U where the filtration is induced by the canonical multiplicative filtration on the formal cdga A . Itis clear that on the level of the associated graded the map f ξ reduces to f , the first componentof the original L ∞ map between V and U . Since the latter is a quasi-isomorphism we concludethat f ξ is likewise a quasi-isomorphism as desired.Now let us prove (2). Using (1) and the fact that any L ∞ algebra is L ∞ quasi-isomorphic toa strict dgla we reduce the statement to the case when V is a strict dgla. Furthermore, withoutloss of generality we can assume that A → B is a filtered map (with respect to some admissiblefiltrations on A and B ), cf. Remark 3.4. Let ξ ∈ MC(
V, A ); the twisted differential on ( A ⊗ V ) ξ will have the form d ξ = d V ⊗ A + [ ξ, ?] where d V ⊗ A is the untwisted differential on A ⊗ V . Itfollows that there is an isomorphism of associated graded dglas:Gr( A ⊗ V ) ξ ∼ = Gr( A ⊗ V )since the ‘twisted’ part [? , ξ ] of the differential d ξ vanishes upon passing to the associated gradeddglas. Similarly Gr( B ⊗ V ) g ∗ ξ ∼ = Gr( B ⊗ V )It follows that the map g ⊗ id : A ⊗ V → B ⊗ V induces a quasi-isomorphismGr( A ⊗ V ) ξ → Gr( B ⊗ V ) g ∗ ξ and so g ⊗ id must be a quasi-isomorphism. (cid:3) Remark 6.4.
If an L ∞ map f : ( V, m V ) → ( U, m U ) is not a quasi-isomorphism and ξ ∈ MC(
V, A ) then there is still an L ∞ map f ξ : ( A ⊗ V ) ξ → ( A ⊗ U ) f ∗ ( ξ ) given by the same formula f ξ := e f ∗ ( ξ ) f e − ξ ; of course it need not be an L ∞ quasi-isomorphism. It is straightforward towrite f ξ in components; one has for x i ∈ Σ − V, i = 1 , . . . : f ξn ( x , . . . , x n ) = ∞ X i =0 i ! f n + i ( ξ, . . . , ξ, x , . . . , x n ) . We will not need this explicit formula.
Combining Propositions 6.1 and 6.3 we obtain the following result.
Corollary 6.5.
Let f : ( V, m V ) → ( U, m U ) be an L ∞ quasi-isomorphism and A → B be a weakequivalence between formal cdgas A and B . For ξ ∈ M C ( V, A ) denote by ˜ ξ ∈ M C ( U, A ) theequivalence class corresponding to ξ under the bijection M C ( V, A ) ∼ = M C ( U, B ) induced by f and g . Then the L ∞ algebras ( A ⊗ V ) ξ and ( B ⊗ U ) ˜ ξ are L ∞ quasi-isomorphic. (cid:3) . Examples of twisting: Chevalley-Eilenberg and Harrison cohomology
In this section we explain how to treat Chevalley-Eilenberg cohomology of dglas and Harrison-Andr´e-Quillen cohomology of cdgas as instances of MC twisting. First let us recall the standarddefinitions of Chevalley-Eilenberg and Harrison cohomology, cf. for example [20, 2].
Definition 7.1. • Let V be a Lie algebra and M be a V -module. Then the Chevalley-Eilenberg complex of V with coefficients in M is defined as C n CE ( V, M ) ⊂ Hom( V ⊗ n , M ) consisting of skew-symmetric multilinear functions on V with values in M . The differential C n CE ( V, M ) → C n +1CE ( V, M ) is defined as follows: ( df )( v , . . . , v n +1 ) = X ≤ i The reader will note the apparent asymmetry in our exposition of Chevalley -Eilenberg and Harrison cohomologies; indeed it is possible to treat the latter for not necessarilyformal cdgas or even C ∞ algebras, cf. [16] for a detailed discussion of C ∞ algebras. The reasonfor this asymmetry is that the present framework is adequate for the needs of rational homotopytheory; additionally C ∞ algebras are more difficult to work with than L ∞ algebras; particularlythe notion of an MC element of a dgla with values in a C ∞ algebra is combinatorially rathermessy and we do not have clear applications justifying this added complexity. roposition 7.4. If V and U are Lie algebras and ξ : V → U is a Lie algebra map thenDefinition 7.1 (1) is equivalent to Definition 7.2 (1) so that there is an isomorphism C ξ CE ( V, U ) ∼ = C CE ( V, U ) . Similarly if A and B are (non-unital) finite dimensional nilpotent algebras and A → B is analgebra map then Definition 7.1 (2) is equivalent to Definition 7.2 (2) in the following sense C ξ Harr ( A, B ) ∼ = C Harr ( A, B ) . Proof. Let us prove the first statement. Consider a map f : Hom([ T n Σ − V ] , U ) S n → Hom( T n Σ − V, U ) S n ∼ = [ ˆ T n Σ − V ∗ ] S n ⊗ U ∼ = ˆ S n Σ − V ∗ ⊗ U, the natural isomorphism from S n -invariants to S n -coinvariants. The map f identifies the n thcochains of the Lie algebra V with coefficients in the Lie algebra U in the sense of Definition7.1 with ( ˆ S n Σ − V ∗ ⊗ U ) ξ . Recall that the differential in the dgla ( ˆ S n Σ − V ∗ ⊗ U ) ξ has the form[ m V , ?] + [ ξ, ?]; it is then straightforward to check that [ m V , ?] and [ ξ, ?] correspond to the firstand second terms in the formula (7.1) respectively.For the second statement consider a map g : Hom([ T n Σ − A ] , B ) → ˆ T n Σ − A ∗ ⊗ B. Then the restriction of g onto the space of those multilinear maps which vanish on shuffle prod-ucts can be identified with (Prim ˆ T Σ − A ∗ ) T ( ˆ T n Σ − A ∗ ) ⊗ B where Prim ˆ T Σ − A ∗ is the spaceof primitive elements in the Hopf algebra ˆ T Σ − A ∗ (supplied with the standard cocommutativecoproduct). This space of primitive elements is further identified with the free Lie algebra onΣ − A ∗ ; therefore C n Harr ( A, B ), the space of n th cochains of the algebra A V with coefficients inthe Lie algebra U in the sense of Definition 7.1 is isomorphic to B ⊗ L ie ( A ). A simple checkshows that the differential (7.2) agrees with the differential in [ B ⊗ L ie ( A )] ξ . (cid:3) Remark 7.5. • We see that in the situation of Definition 7.2 the Chevalley-Eilenberg complex C CE ( V, U ) has the structure of an L ∞ algebra and the Harrison complex C Harr ( A, B ) has the struc-ture of a dgla. According to our convention we should, therefore, view them as homo-logically graded dg spaces; however this would contradict with the traditional usage andso we retain the cohomological grading for these complexes. • A natural question is whether one can extend our definitions to the case of an arbitrarymodule of coefficients . We will now outline how this can be done in the Chevalley-Eilenberg case; the Harrison case could be dealt with similarly.Suppose that V is an L ∞ algebra with the representing formal cdga ˆ S Σ − V ∗ and M is an L ∞ module over V ; that means that there is an L ∞ map f : V → End( M ) . Thenwe have the induced map f ∗ : MC( V, ˆ S Σ − V ∗ ) → MC(End( M ) , ˆ S Σ − V ∗ ) . Consider the element f ∗ ( ξ ) ∈ MC(End( M ) , ˆ S Σ − V ∗ ) where ξ ∈ MC( V, ˆ S Σ − V ∗ ) is thecanonical MC element. The element f ∗ ( ξ ) can be viewed as an ˆ S Σ − V ∗ -linear endo-morphism of ˆ S Σ − V ∗ ⊗ M and we define the Chevalley-Eilenberg complex of V withcoefficients in M as C CE ( V, M ) = ˆ S Σ − V ∗ ⊗ M supplied with the twisted differential d f ∗ ( ξ ) ( a ⊗ m ) = d ( a ⊗ m ) + f ( a ⊗ m ) for a ∈ ˆ S Σ − V ∗ , m ∈ M . From now on we will omit the superscript ξ in the symbols for the Harrison and Chevalley-Eilenberg cohomology whenever the choice of the corresponding MC element is clear from thecontext. Additionally, we introduce the notion of truncated Chevalley-Eilenberg and Harrisoncohomology. efinition 7.6. • Let ( V, m V ) be an L ∞ algebra representable by the formal cdga ˆ S Σ − V ∗ , ( U, m U ) beanother L ∞ algebra and ξ ∈ MC( U, ˆ S Σ − V ∗ ) be the MC element corresponding to an L ∞ map V → U . Then the truncated Chevalley-Eilenberg complex of V with coefficientsin U is defined as C CE ( V, U ) := ([ ˆ S Σ − V ∗ ] + ⊗ U ) ξ . • Let A and B be formal cdgas and ξ ∈ M C ( L ( A ) , B ) be the MC element correspondingto a map of formal cdgas A → B . Then the truncated Harrison complex of A withcoefficients in B is defined as C Harr ( A, B ) := ( B ⊗ L ( A )) ξ . It is clear that there C CE ( V, U ) and C Harr ( A, B ) are sub dglas of C CE ( V, U ) and C Harr ( A, B );moreover there are the following short exact sequences of dg vector spaces: C CE ( V, U ) → C CE ( V, U ) → U ; C Harr ( A, B ) → C Harr ( A, B ) → B. It turns out that and Chevalley-Eilenberg cohomology of L ∞ algebras reduce to Harrisoncohomology of formal cdgas. Proposition 7.7. Let ( V, m V ) and ( U, m U ) be L ∞ algebras with representing formal cdgas B = ˆ S Σ − V ∗ and A = ˆ S Σ − U ∗ respectively; let ˆ S Σ − U ∗ → ˆ S Σ − V ∗ be the map of formalcdgas representing an L ∞ map V → U . Then there are natural L ∞ quasi-isomorphisms: C CE ( V, U ) ∼ = C Harr ( A, B ); C CE ( V, U ) ∼ = C Harr ( A, B ); Proof. We will restrict ourselves with proving the first isomorphism; the proof of the secondone is similar. Denote by ξ the MC element in ˆ S Σ − V ∗ ⊗ U corresponding to the map ξ : A =ˆ S Σ − U ∗ → B = ˆ S Σ − V ∗ ; similarly denote by ξ the MC element in B ⊗ L ( A ) corresponding to ξ . Recall that C CE ( V, U ) = ( ˆ S Σ − V ∗ ⊗ U ) ξ = ( B ⊗ U ) ξ and that C Harr ( A, B ) = [ B ⊗ L ( A )] ξ .The adjunction morphism C L ( A ) → A is a weak equivalence of formal cdgas by Theorem 2.7from which it follows that L ( A ) and U are L ∞ quasi-isomorphic. Note also that ξ correspondsto ξ under this L ∞ quasi-isomorphism. Now the desired statement follows from Proposition6.3. (cid:3) The Chevalley-Eilenberg and Harrison cohomology are homotopy invariant in the followingsense. Proposition 7.8. (1) Let ( V, m V ) , ( U, m U ) be two L ∞ algebras with representing formal cdgas ˆ S Σ − V ∗ and ˆ S Σ − V ∗ respectively. Let ξ, η : ˆ S Σ − U ∗ → ˆ S Σ − V ∗ be two maps representing twohomotopic L ∞ maps V → U . Then the L ∞ algebras C ξ CE ( V, U ) and C ηCE ( V, U ) are L ∞ isomorphic. Similarly the L ∞ algebras C ξ CE ( V, U ) and C ηCE ( V, U ) are L ∞ isomorphic. (2) Let A, B be two formal cdgas and ξ, η : A → B be two homotopic maps between them.Then the dglas C ξ Harr ( A, B ) and C η Harr ( A, B ) are L ∞ isomorphic. Similarly the dglas C ξ Harr ( A, B ) and C η Harr ( A, B ) are L ∞ isomorphic.Proof. We restrict ourselves with proving the corresponding statements for the untruncatedcomplexes; the proofs for the truncated analogues are completely parallel. For (1) considerthe MC elements ξ, η ∈ MC( U, ˆ S Σ − V ∗ ) associated with the corresponding L ∞ maps. TheseMC elements are homotopic and thus, by Proposition 6.1 the L ∞ algebras ( ˆ S Σ − V ∗ ⊗ U ) ξ and ( ˆ S Σ − V ∗ ⊗ U ) η are L ∞ isomorphic as required. For (2) viewing ξ and η as elements inMC( L ( A ) , B ) and noting that these elements are homotopic, we conclude similarly that thedglas [ B ⊗ L ( B )] ξ and [ B ⊗ L ( B )] η are L ∞ isomorphic. (cid:3) he following result is a direct consequence of Proposition 6.3: Proposition 7.9. (1) Let ( V, m V ) , ( V ′ , m V ′ ) , ( U, m U ) and ( U ′ , m ′ U ) be L ∞ algebras, ( V, m V ) → ( V ′ , m V ′ ) and ( U ′ , m U ′ ) → ( U, m U ) be L ∞ quasi-isomorphisms and ( V ′ , m V ′ ) → ( U ′ , m ′ U ) be an L ∞ map. Then there are natural L ∞ quasi-isomorphisms C CE ( V, U ) → C CE ( V ′ , U ′ ); C CE ( V, U ) → C CE ( V ′ , U ′ ) where C CE ( V, U ) is formed using the composition L ∞ map ( V, m V ) → ( V ′ , m V ′ ) → ( U ′ , m ′ U ) → ( U, m U ) . (2) Let A, A ′ , B and B ′ be formal cdgas, A → A ′ and B ′ → B be weak equivalences and A ′ → B ′ be a map of formal cdgas. Then there are natural L ∞ quasi-isomorphisms C Harr ( A, B ) → C Harr ( A ′ , B ); C Harr ( A, B ) → C Harr ( A ′ , B ) . where C Harr ( A, B ) is formed using the composition A → A ′ → B ′ → B . (cid:3) Remark 7.10. As an aside we mention that there is an analogue of the dgla structure on C Harr or C CE in the context of the Hochschild complex of an A ∞ algebra; for a detailed discussionof the latter concept see [18] or [16] . Namely, suppose that ( V, m V ) , ( U, m U ) are two A ∞ al-gebras with representing formal dgas ˆ T Σ − V ∗ and ˆ T Σ − U ∗ respectively; suppose further that ξ : ˆ T Σ − U ∗ → ˆ T Σ − V ∗ is a map representing an A ∞ morphism V → U . The latter morphismcorresponds to an A ∞ MC element ξ ∈ MC( U, ˆ T Σ − V ∗ ) , cf. [10] concerning A ∞ MC elementsand associated twistings. Then C Hoch ( V, U ) , the Hochschild complex of V with coefficients in U can be naturally identified with ( ˆ T Σ − V ∗ ⊗ U ) ξ . We conclude that C Hoch ( V, U ) has the structureof an A ∞ algebra. This structure has been studied in some detail in the case when V = U and ξ is the identity morphism, [12] cf. in which case it is homotopy abelian as part of a richerstructure of a G ∞ algebra. In general this A ∞ structure may not be homotopy abelian. Rational homotopy of function spaces In this section we will use the developed technology of MC twistings to construct explicitrational models for function spaces. Let X and Y be two connected nilpotent rational CW complexes of finite type. Additionally, we assume that either X is a finite CW complex or Y has a finite Postnikov tower; this condition ensures that the spaces of maps between X and Y are homotopically equivalent to finite type complexes. We remark that the latter condition isnot necessary, at least in the philosophical sense; we impose it only because the current stateof rational homotopy theory does not provide a Lie-Quillen model for a nilpotent space thatis simultaneously not simply-connected and not of finite type. We believe that such a modelshould exist. Denote by F ( X, Y ) ( F ∗ ( X, Y )) the spaces of continuous maps (based continuousmaps) between X and Y . We denote by A ( X ) a Sullivan minimal model of X , as describedin [3] and by L ( Y ) = L ( A ( X ) + ) its Lie-Quillen model of Y , [21]. Note that A ( X ) + can beviewed as a formal cdga since it is a symmetric algebra on finitely many generators in positivedegrees. Then we have the following result. Theorem 8.1. (1) (a) There is a bijection π F ( X, Y ) ∼ = ] M C ( L ( Y ) , A ( X ) + ) . (b) There is a bijection π F ∗ ( X, Y ) ∼ = M C ( L ( Y ) , A ( X ) + ) . (2) Let ξ : X → Y denote both a base point in F ( X, Y ) and a representative of the cor-responding element in MC( L ( Y ) , A ( X ) + ) . Further, denote by F ξ ( X, Y ) and F ξ ∗ ( X, Y ) the connected component of F ( X, Y ) and of F ∗ ( X, Y ) containing ξ . (a) The dgla [ A ( X ) ⊗ L ( Y )] ξ h i is a Lie-Quillen model of F ξ ( X, Y ) . (b) The dgla [ A ( X ) + ⊗ L ( Y )] ξ h i is a Lie-Quillen model of F ξ ∗ ( X, Y ) . roof. We have a natural bijection π F ( X, Y ) ∼ = [ A ( Y ) , A ( X )] ∼ = ] M C ( L ( Y ) , A ( X ) + ) . Similarly, π F ∗ ( X, Y ) ∼ = [ A ( Y ) , A ( X )] ∗ ∼ = M C ( L ( Y ) , A ( X ) + )which proves (1). Let us now prove (2) starting with part (a). Let Z be any nilpotent rationalCW complex Z of finite type. We will prove the following natural bijection of sets:(8.1) [ Z, F ξ ( X, Y )] ∗ ∼ = [ L ( Z ) , [ A ( X ) ⊗ L ( Y )] ξ h i ] . Here the left hand side of (8.1) is the set of pointed homotopy classes of maps of spaces whereasthe right hand side is the set of homotopy classes of dgla maps. By the Yoneda lemma thiswould imply the desired statement. Note first that [ Z, F ξ ( X, Y )] ∗ can be identified with thefiber over ξ ∈ [ X, Y ] of the map (induced by the inclusion of the base point into Z ):[ Z × X, Y ] → [ X, Y ] . By part (1) this is the same as the fiber over ξ ∈ ] M C ( L ( Y ) , A ( X )) of the map ] M C ( L ( Y ) , A ( Z ) ⊗ A ( X )) → ] M C ( L ( Y ) , A ( X )) . We have: [ L ( Z ) , [ A ( X ) ⊗ L ( Y )] ξ h i ] ∼ = [ L A ( Z ) + , [ A ( X ) ⊗ L ( Y )] ξ h i ] ∼ = M C ([ A ( X ) ⊗ L ( Y )] ξ h i , A ( Z ) + ) ∼ = M C ( A ( X ) ⊗ L ( Y ) ξ , A ( Z ) + )(8.2)where in the last equality we have used Proposition 5.11.Next consider the following sequence of maps of sets:(8.3) M C ( A ( X ) ⊗ L ( Y ) ξ , A + ( Z )) → ] M C ( A ( X ) ⊗ L ( Y ) ξ , A ( Z )) → ] M C ( L ( Y ) , A ( Z ))where the first map is induced by the natural inclusion A + ( Z ) → A ( Z ) and the second – bythe projection A ( Z ) → Q . This sequence is exact in the sense that second map is onto whereasthe first term is its fiber over ξ ∈ ] M C ( L ( Y ) , A ( X )). Next we have natural bijections(8.4) ] M C ( A ( X ) ⊗ L ( Y ) ξ , A ( Z )) ∼ = ] M C ( A ( X ) ⊗ L ( Y ) , A ( Z )) ∼ = ] M C ( L ( Y ) , A ( Z ) ⊗ A ( X )) . Here the first bijection is given for η ∈ A ( X ) ⊗ L ( Y ) ⊗ A ( Z ) by η η + ξ ⊗ L ( Z ) , [ A ( X ) ⊗ L ( Y )] ξ h i ] isbijective with the fiber of ] M C ( L ( Y ) , A ( Z ) ⊗ A ( X )) → ] M C ( L ( Y ) , A ( Z ))over ξ ∈ ] M C ( L ( Y ) , A ( Z )) and so the bijection (8.1) is proved.The proof of the based version (b) follows from the unbased version (a). Note, first of all,that there is a homotopy fiber sequence of spaces F ξ ∗ ( X, Y ) → F ξ ( X, Y ) → Y where the last map is induced by the inclusion of the base point into X . Next, under the functor? L (?) the map F ξ ( X, Y ) → corresponds to[ A ( X ) ⊗ L ( Y )] ξ h i → L ( Y )induced by the augmentation A ( X ) → Q . Clearly the homotopy fiber of the latter map (whichcoincides in this case with the actual fiber) is isomorphic to [ A ( X ) ⊗ L ( Y )] ξ h i . (cid:3) Remark 8.2. Note the manifest homotopy invariant nature of Theorem 8.1: the Quillen-Liemodels of F ( X, Y ) and F ∗ ( X, Y ) do not depend (up to quasi-isomorphism) on the choice ofmodels L ( Y ) and A ( X ) and also on the choice of the MC element ξ within its homotopy class. ote that there is a weak equivalence between formal cdgas: A ( X ) + ∼ = C ( L ( X )) . By Proposition 6.3 it follows that there are L ∞ quasi-isomorphisms( A ( X ) + ⊗ L ( Y )) ξ ∼ = ( C ( L ( X )) ⊗ L ( Y )) ˜ ξ ;here ˜ ξ is the MC element in ( C ( L ( X )) ⊗ L ( Y )) corresponding to a given map X → Y . Bydefinition ( C ( L ( X )) ⊗ L ( Y )) ˜ ξ ∼ = C CE ( L ( X ) , L ( Y )). Similarly we obtain the following naturalisomorphism of dglas: ( A ( X ) ⊗ L ( Y )) ξ ∼ = C CE ( L ( X ) , L ( Y )) . This gives an interpretation of the Quillen-Lie algebras of F ( X, Y ) and F ∗ ( X, Y ) in terms ofChevalley-Eilenberg complexes. By Proposition 7.7 we can further interpret them in terms ofHarrison complexes. Putting everything together we obtain the following result. Corollary 8.3. (1) The dglas C Harr ( A ( Y ) , A ( X )) h i and C CE ( L ( X ) , L ( Y )) h rangle are Lie-Quillen modelsfor a connected component of F ( X, Y ) . (2) The dglas C Harr ( A ( Y ) , A ( X )) h i and C CE ( L ( X ) , L ( Y )) h i are Lie-Quillen models for aconnected component of F ∗ ( X, Y ) . (cid:3) One further consequence is that the homotopy groups of function spaces can be expressed interms of Harrison or Chevalley-Eilenberg cohomology; this was already known [2]: Corollary 8.4. We have the following isomorphisms for i > : (1) π n F ( X, Y ) ∼ = H − n CE ( L ( X ) , L ( Y )) ∼ = H − n Harr ( A ( Y ) , A ( X )) . (2) π n F ( X, Y ) ∼ = H − n CE ( L ( X ) , L ( Y )) ∼ = H − n Harr ( A ( Y ) , A ( X )) . (cid:3) Theorem 8.1 can also be used to obtain Sullivan models of mapping spaces: Corollary 8.5. (1) The cdga C CE ([ A ( X ) ⊗ L ( Y )] ξ ) h i is a Sullivan model for F ( X, Y ) . (2) The cdga C CE ([ A ( X ) ⊗ L ( Y )] ξ h i ) is a Sullivan model for F ∗ ( X, Y ) . (cid:3) Corollary 8.6. Let X be a formal space (i.e. A ( X ) is quasi-isomorphic to H ( X ) ), Y bea coformal space (i.e. L ( Y ) is quasi-isomorphic to π Q ( Y ) , the Whitehead Lie algebra of Y )and X → Y be the constant map corresponding to ∈ MC( L ( Y ) , A ( X )) . Then the functionspaces F ( X, Y ) and F ∗ ( X, Y ) are both coformal; their Whitehead Lie algebras are isomorphicrespectively to H ∗ ( X ) ⊗ π Q ( Y ) and H ∗ ( X ) + ⊗ π Q ( Y ) . (cid:3) Remark 8.7. Note that the Sullivan models described in the above corollary will not, in general,be minimal. However if A ( X ) is formal (in the sense of being quasi-isomorphic to its cohomol-ogy) a minimal model for the components of F ( X, Y ) and F ∗ ( X, Y ) containing the constantmap can be constructed. Indeed, consider instead of a dgla L ( Y ) a minimal L ∞ algebra V equiv-alent to it (effectively, a Sullivan minimal model of X ). Then ( H ( X ) ⊗ V ) h i is itself a minimal L ∞ algebra whose representing formal cdga ˆ S Σ − [( H ( X ) ⊗ V ) h i ] ∗ is a Sullivan minimal modelof F ( X, Y ) ; a similar argument applies to construct a Sullivan minimal model of F ∗ ( X, Y ) . xample 8.8. • Consider the component of the constant map in F ( S n , X ) with n ≥ . It follows fromTheorem 8.1 that a Lie model of F ( X, Y ) is the dgla L ( X ) ⋉ Σ − n L ( X ) h i , the square-zero extension of L ( X ) by Σ − n L ( X ) h i . It further follows from Corollary 8.5 that aSullivan model of F ( S n , X ) is the cdga C CE ( L ( X ) ⋉ Σ − n L ( X ) h i ) ∼ = C CE ( L ( X ) , ˆ S Σ − [Σ − n L ( X ) h i ] ∗ ) , the Chevalley-Eilenberg complex of L ( X ) with coefficients in the (completed) symmetricalgebra of Σ − [Σ − n L ( X ) h i ] ∗ . In the case X is n -connected we have [Σ n L ( X ) h i ] ∗ =[Σ n L ( X )] ∗ and so we get C CE ( L ( X ) , ˆ S [Σ n − L ( X ) ∗ ]) as a Sullivan model of F ( S n , Y ) .Specializing further to the case n = 1 we conclude that the cdga C CE ( L ( X ) , ˆ S [ L ( X ) ∗ ]) isa Sullivan model of the free loop space of X . Note that the universal enveloping algebraof L ( X ) is quasi-isomorphic to C ∗ (Ω( X )) , the chain algebra of the based loop spaceof X whereas ˆ S [ L ( X ) ∗ ] is quasi-isomorphic to its dual C ∗ (Ω( X )) and so we obtain aquasi-isomorphism C CE ( L ( X ) , S [ L ( X ) ∗ ]) ∼ = C ∗ Hoch ( C ∗ Ω( X ) , C ∗ Ω( X )) . 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