aa r X i v : . [ m a t h . A T ] J a n Enriched Lie algebras in topology, I
Yves F´elix and Steve HalperinJanuary 5, 2021
To each path connected space X the Sullivan theory of minimal models associates acommutative differential graded algebra, its minimal model ( ∧ V, d ), and with it a gradedLie algebra L X that is the homotopy Lie algebra of its geometric realization < ∧ V, d > .When X is a simply connected space with finite Betti numbers, then L X is isomorphic tothe Lie algebra of X , π ∗ (Ω X ) ⊗ Q , equipped with the Whitehead bracket.In the general case, L X is a complete enriched Lie algebra: An enriched Lie algebra isa graded Lie algebra L = L ≥ together with a family of ideals I α indexed by a partiallyordered set with α ≥ β if and only if I α ⊂ I β , and such that each quotient L/I α is afinite dimensional nilpotent Lie algebra. The enriched Lie algebra ( L, { I α } ) is complete if L = lim ←− α L/I α .Enriched Lie algebras are the natural extension of graded Lie algebras for path con-nected spaces, because each complete enriched Lie algebra L is the homotopy Lie algebraof a path connected space X .This text is the first part of a general study of complete enriched Lie algebras. Appli-cations will be contained in a second part. 1 ontents I Enriched Lie algebras 4 ( L, I ) ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 U L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Quadratic ∧ V -modules and holonomy representations . . . . . . . . . . . . 235.5 Acyclic closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Adjoint representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.7 Profree L -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.8 Closed ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 II Profree Lie algebras 29
III Topological spaces, Sullivan models, and their homotopy Lie al-gebras 40 art I Enriched Lie algebras
By an enriched Lie algebra we mean a graded Lie algebra, L = L ≥ , equipped with afamily of ideals I = { I α } satisfying the following conditions(i) Each L α = L/I α is a finite dimensional nilpotent Lie algebra. In particular, for some n α , L n α = 0, L k denoting the linear span of iterated Lie brackets of length k .(ii) The index set, { α } is a directed set under the partial order given by α ≥ β ⇐⇒ I α ⊂ I β :i.e. for each α, β there is a γ such that I γ ⊂ I α ∩ I β .(iii) ∩ α I α = 0.The corresponding quotient maps will be denoted ρ α : L → L/I α := L α . Moreover, we shall use I to denote both the family { I α } of ideals, and the index set { α } . Example.
In any graded Lie algebra L = L ≥ the ideals J α ⊂ L for which L/J α is finitedimensional and nilpotent form a directed system as above. The intersection of theseideals consists of the elements x ∈ L which, for every n , can be expressed as a linearcombination of commutators of length n . If this intersection is zero, the ideals J α definean enriched structure, and so this condition is necessary and sufficient for L to admit anenriched structure. However, in principle, L may admit two distinct such structures.Finally, as we shall show in §
8, a minimal Sullivan algebra induces a natural enrichedstructure in its homotopy Lie algebra.The completion of an enriched Lie algebra ( L, I ) is the inclusion λ L : L → lim ←− α L/I α := L, and L is complete if λ is an isomorphism. In particular, the enriched structure in thehomotopy Lie algebra of a minimal Sullivan algebra is complete. Now, in general, theinverse limit provides surjections ρ α : L → L/I α , and their kernels make L into a completeenriched Lie algebra, ( L, I ).Note that if β > α then ρ α factors to yield a surjection ρ αβ : L β → L α . Thus L consistsof the coherent families ( x α ), α ∈ I , i.e. those satisfying ρ αβ x β = x α . In particular, L ∩ ker ρ α = ker ρ α . (1)4 efinition . Suppose R ⊂ E and T ⊂ F are subspaces of enriched Lie algebras ( E, J ) and( F, I ). A linear map f : R → T is coherent if deg f = 0 and for each α ∈ I there is a β ∈ J such that J β ∩ R ⊂ f − ( I α ∩ T ) . In particular, a morphism of enriched Lie algebras is a coherent homomorphism ϕ :( E, J ) → ( F, I ) of graded Lie algebras. In other words, for each α ∈ I there is a β ( α ) ∈ G and a morphism ϕ α : E β ( α ) → F α such that ρ α ◦ ϕ = ϕ α ◦ ρ β ( α ) . Thus a morphism ofcomplete enriched Lie algebras ϕ : E → F is a morphism of limits E = lim ←− β E β → lim ←− α E β ( α ) → lim ←− α F α = F. It is therefore straightforward to verify that a morphism ϕ : ( E, G ) → ( F, I ) extendsuniquely to a morphism ϕ : ( E, G ) → ( F , I ) satisfying ϕ ◦ λ E = λ F ◦ ϕ. Next, if I and I ′ are families of ideals in L satisfying the conditions above, we say that I and I ′ are equivalent ( I ∼ I ′ ) if the identities, ( L, I ) → ( L, I ′ ) and ( L, I ′ ) → ( L, I ) aremorphisms. This is an equivalence relation, and if ϕ : ( F, G ) → ( E, I ) is a morphism, then ϕ : ( F, G ′ ) → ( E, I ′ ) is also a morphism whenever G ∼ G ′ and I ∼ I ′ . In particular, if I and I ′ are equivalent families of ideals, then the identity induces an isomorphismlim ←− I α ∈ I L/I α ∼ = −→ lim ←− J β ∈ I ′ L/J β . Thus the completion, λ L : L → L is independent of the choice of I in its equivalence class. Examples. (1). Let ( L, { I α } ) be an enriched Lie algebra and let ( I β ) be a subfamily suchthat for each α there is a β with I β ⊂ I α . Then ( I α ) and ( I β ) are equivalent families ofideals.(2). Full enriched Lie algebras . An enriched Lie algebra ( L, I ) = { I α } ) is full ifwhenever an ideal I contains some I β then I = I α for some α . If ( L, J ) = { J β } ) is anyenriched Lie algebra then J extends to a unique full enriched family I consisting of allthe ideals containing some J β . Evidently I ∼ J . Moreover if I and J are equivalent fullenriched structure, then they coincide.(3). Suppose that L is the projective limit of a directed system of nilpotent finitedimensional Lie algebras Q α , L = lim ←− α Q α . Denote ρ α : L → Q α the correspondingprojections. Then ( L, { ker ρ α } ) is a complete enriched Lie algebra with L α = L/ ker ρ α =Im ρ α ⊂ Q α , L = lim ←− α L α . (4). Free Lie algebras.
The Lie algebra freely generated by a subspaces S is denotedby L S and, in particular, is the direct sum of the subspaces L S ( k ) = [ S, . . . , S ] ( k factors . )5ts classical completion , b L S is then the inverse limit lim ←− n L S / ⊕ k>n L S ( k ) . When S is a graded vector space of finite type the ideals ⊕ k>n L S ( k ) define an enriched in L S , and b L S = L S , the corresponding completion.(5). When the space [ L, L ] of commutators in a graded Lie algebra L has finite codi-mension then (see Proposition 4) its completion L has a unique structure as an enrichedLie algebra, namely that provided by its lower central series. Remark.
Let f : ( L, { J β } ) → ( L ′ , { I α } ) be a coherent morphism of complete enriched Liealgebras. For each α , let β ( α ) an index such that J β ( α ) ⊂ f − ( I α ) and form the index set K = { ( α, β ) | β ≥ β ( α ) } . Then for each ( α, β ) ∈ K , let I ( α,β ) = I α and J ( α,β ) = J β . Thisgives for each ( α, β ) ∈ K a morphism f ( α,β ) = L ( α,β ) → L ′ ( α,β ) , and f = lim ←− ( α,β ) f ( α,β ) .Now we define the category, C , of enriched Lie algebras as follows: • The objects of C are the pairs ( L, { I } ) in which L = L ≥ is a graded Lie algebra and { I } is an equivalence class of ideals satisfying the conditions above. • The morphism of C are the morphisms ϕ : ( F, G ) → ( E, I ).In particular L L is a functor from C to the sub category of complete enriched Liealgebras, the inclusion L → L is a morphism, and any morphism ( E, J ) → ( F, I ) extendsuniquely to a morphism E → F .A key aspect of an enriched Lie algebra ( L, I ) is that I makes the Lie algebra accessibleto finite dimensional arguments, even when L is not a graded vector space of finite type.In particular, enriched Lie algebras behave well with respect to inverse limits, as followsfrom the following Theorem ([1]): Suppose → A α → B α → C α → is a directed system of short exact sequences ofgraded vector spaces of finite type. Then → lim ←− α A α → lim ←− α B α → lim ←− α C α → is also short exact. In particular, for chain complexes of finite type, homology commuteswith inverse limits. Complete enriched Lie algebras arise naturally in Sullivan’s approach to rational ho-motopy theory : the homotopy Lie algebra of a Sullivan model is naturally a completeenriched Lie algebra and this is an essential aspect when the theory is extended to all con-nected CW complexes. Thus our objective here is to develop the properties of enrichedLie algebras, in particular with a view to future topological applications. In particular,while many of the basic properties of C are parallel those for the category of all gradedLie algebras, the enrichment gives the objects in C a distinct ”homotopy flavour”. In par-ticular each enriched Lie algebra ( L, I ) determines a quadratic Sullivan algebra ∧ Z and asimplicial set h∧ Z i . 6 Subspaces, sub Lie algebras, ideals, and universal envelop-ing algebras
Throughout this entire subsection ( L, I ) denotes a fixed complete enriched Lie algebra, withquotient maps ρ α : L → L/I α := L α . Any subspace S ⊂ L determines the inclusions, S → lim ←− α S/S ∩ I α = lim ←− α ρ α S ⊂ L, and the subspace lim ←− α ρ α S ⊂ L is called the closure of S in L . In particular, the closure ofa sub Lie algebra E ⊂ L coincides with its completion, E , and so we use the same notationand write S = lim ←− α ρ α S. Since ρ β S → ρ α S is surjective for β ≥ α it follows that ρ α S = ρ α S, α ∈ I . Remark.
Suppose ( x k ) k ≥ is a sequence of elements in S . If for each α ∈ I there is some r ( α ) such that m X k = r ( α )+1 x k ∈ I α , m > r ( α ) , then the elements ρ α ( P r ( α ) k =1 x k ) ∈ ρ α ( S ) define a single element y ∈ S , and we write y = P k x k . Lemma 1. If S ⊂ T ⊂ L are subspaces, then T /S = lim ←− α ρ α T /ρ α S. proof: Since ρ α S ⊂ ρ α T , 0 → ρ α S → ρ α T → ρ α T /ρ α S → (cid:3) Lemma 2. (i) A finite sum of closed subspaces of L is closed.(ii) An arbitrary intersection of closed subspaces of L is closed.(iii) For any S ⊂ L , S = ∩ α ( S + I α ) .(iv) If S ⊂ L is a graded space of finite type, then S and [ S, L ] are closed. roof. (i) It is sufficient to show that if S and T are closed then S + T is closed. ByLemma 1, S + T /S = lim ←− α ρ α ( S + T ) /ρ α ( S ) = lim ←− α ρ α ( S ) + ρ α ( T ) ρ α ( S )= lim ←− α ρ α ( T ) ρ α ( S ) ∩ ρ α ( T )= T / lim ←− α ρ α ( S ) ∩ ρ α ( T )In particular, it follows that the inclusion of S and T in S + T define a surjection S + T → S + T . Since S and T are closed, S + T → S + T is an isomorphism.(ii) Suppose x ∈ ∩ σ S σ , where each S σ is a closed subspace of L . Then ρ α x ∈ ρ α ( ∩ S σ ) ⊂ ∩ σ ρ α ( S σ ) . Therefore ρ α x ∈ ρ α ( S σ ) for each σ . Thus x = ( ρ α x ) ∈ lim ←− α ρ α ( S σ ) = S σ = S σ . Thus x ∈ ∩ σ S σ and ∩ σ S σ = ∩ σ S σ .(iii) If x ∈ S then ρ α x ∈ ρ α ( S ). Thus for some x α ∈ S , ρ α ( x − x α ) = 0and so x − x α ∈ I α . Therefore x ∈ S + I α for all α .In the reverse direction, suppose x ∈ ∩ α ( S + I α ). Then ρ α x ∈ ρ α S. Thus the coherent family x = ( ρ α x ) ∈ S .(iv) In view of (i) it is sufficient to prove that for any x ∈ L , [ x, L ] is closed. Let C α = { y α ∈ L α | [ ρ α x, y α ] = 0 } . It is immediate from the definition that { C α } is an inversesystem, and we set C = lim ←− α C α . Now we show that C = { y ∈ L | [ x, y ] = 0 } . Indeed, if y ∈ C , then for all α , ρ α [ x, y ] = [ ρ α x, ρ α y ] ∈ [ ρ α x, C α ] = 0 , and hence [ x, y ] = 0. On the other hand, if [ x, y ] = 0 then [ ρ α x, ρ α y ] = ρ α [ x, y ] = 0 andso each ρ α y ∈ C α . Thus y ∈ C .Finally, apply Lemma 1 to obtain the commutative diagram L/C = lim ←− α L α / lim ←− α C α ∼ = [ x, − ] (cid:15) (cid:15) = / / lim ←− α L α /C α ∼ = [ x, − ] (cid:15) (cid:15) [ x, L ] = / / lim ←− α [ ρ α x, L α ]Since lim ←− α [ ρ α x, L α ] = [ x, L ], (iv) is established. (cid:3) efinition. (i). If E ⊂ L is a sub Lie algebra then the corresponding sub enriched Liealgebra ( E, I E ) is defined by I E = { E ∩ I α } . (ii). If I ⊂ L is a closed ideal then the corresponding quotient enriched Lie algebra,( L/I, I L/I ), defined by I L/I = { ( I + I α ) /I } , is complete.Straightforward arguments from the definitions then establish the Remarks.
1. If E ⊂ F ⊂ L are sub Lie algebras, then ( E, I E ) → ( F, I F ) is a morphism. Moreover,the closure and completion of E coincide as complete enriched Lie algebras.2. If I is an ideal in L then I is an ideal.3. If I is a closed ideal in L then ( L/I, I L/I ) is complete. (This requires Lemma 1.)4. Suppose ( E, G ) ϕ → ( L, I ) is a morphism of complete enriched Lie algebras. Thenker ϕ is closed in E , and so (ker ϕ, I ker ϕ ) is a complete enriched Lie algebra. Theinduced map E/ ker ϕ → L is a morphism of enriched Lie algebras.5. If L = lim −→ σ L ( σ ) is the direct limit of closed sub Lie algebras L ( σ ) then for each α there is some σ for which I α = L ( σ ), and C ( L, − ) = lim −→ σ C ( L ( σ ) , − ) . A weight decomposition in a graded Lie algebra is a decomposition L = ⊕ k ≥ L ( k )in which [ L ( k ) , L ( ℓ )] ⊂ L ( k + ℓ ). A weighted subspace of L is a subspace S ⊂ L such that S = ⊕ k S ∩ L ( k ). We denote S ( k ) = S ∩ L ( k ).A weighted enriched Lie algebra is a weighted Lie algebra with a defining set of idealsof the form I α = ⊕ k I α ( k ). Henceforth in this example, L = ⊕ k ≥ L ( k ) denotes a fixed weighted enriched Lie alge-bra with a defining set of weighted ideals I α . There follows the
Proposition 1.
With the hypotheses and notation above,(i) L = Q k L ( k ) . (ii) If S ⊂ L is a weighted subspace then S = Q k S ( k ) . In particular, S is closed if andonly if each S ( k ) is closed, in which case any subspace of the form Q i S ( k i ) is closed. iii) If each L ( k ) is finite dimensional, then L = lim ←− r L/ Q k>r L ( k ) . proof. (i) Denote as usual by ρ α : L → L α = L/I α the projection on the finite dimensionalLie algebra L α . Then L α = ⊕ k ρ α ( L ( k )) = Y k ρ α ( L ( k )) , and L = lim ←− α L α = Y k lim α ρ α ( L ( k )) = Y k lim ←− α L ( k ) /I α ( k ) = Y k L ( k ) . (ii) Similar proof than the one for (i), and (iii) is a direct consequence. (cid:3) Recall that the classical completion of the universal enveloping algebra
U E of a gradedLie algebra E is the inverse limit d U E = lim ←− n U E/J n ,J n denoting the n th power of the augmentation ideal. In particular d U E = Q ⊕ b J , and b J := lim ←− n J/J n is the augmentation ideal for d U E . Definition.
The
Sullivan completion
U L of an enriched Lie algebra is defined by
U L := lim ←− α [ U L α = lim ←− n,α U L α /J nα = lim ←− γ U L/K γ , where K γ runs over all finite codimensional ideals of U L .Passing to inverse limits shows that the inclusions L α → L α /J nα define an inclusion L → U L.
Note also that
U L = Q ⊕ J , where J = lim ←− α,n J α /J nα ; J is the augmentation ideal for U L . It is also immediate that amorphism ϕ : E → L of enriched Lie algebras extends to a morphism U ϕ : U E → U L .Finally, in analogy with the filtration of
U L by the ideals J n we filter U L by the ideals J ( n ) := lim ←− k,α J nα /J k + nα ⊂ J = J (1) . Then
U L is complete with respect to this filtration:
U L = lim ←− n U L/J ( n ) . (3)10n fact, (2) yields U L = lim ←− n,α U L α /J nα = lim ←− k lim ←− n,α ( U L α /J n + kα ) / ( J nα /J n + kα )= lim ←− n lim ←− k,α ( U L α /J n + kα ) / lim ←− k,α ( J nα /J n + kα )= lim ←− n U L/J ( n ) . Examples.
1. When dim L/ [ L, L ] < ∞ , then U L = d U L .2. Let L = b L ( V ) be the completion of the free graded Lie algebra on a finite dimensiongraded vector space V . Then U L is isomorphic to Q n T n V . Moreover the shortexact sequence of U L -modules0 → U L ⊗ sV d → U L ε → Q → , with d ( sx ) = x shows that U L is an algebra of global dimension one.
Throughout this entire section ( L, I ) denotes a fixed complete enriched Lie algebra, withquotient maps ρ α : L → L/I α := L α . A key invariant for any graded Lie algebra E is its lower central series E = E ⊃ · · · ⊃ E k ⊃ . . . of ideals in which E k is the linear span of iterated commutators of length k of elements in E . The classical completion of E is the inverse limit b E = lim ←− n E/E n , and E is pronilpotent if E ∼ = → b E .The analogue of the lower central series for L is the sequence of ideals L = L (1) ⊃ · · · ⊃ L ( k ) ⊃ . . . defined by L ( k ) = L k . It is immediate that L L ( k ) is a functor. Moreover, since L k ⊂ L ( k ) , each L/L ( k ) is a nilpotent Lie algebra : (cid:0) L/L ( k ) (cid:1) k = 0 . Lemma 3.
Let L be a complete enriched Lie algebra.(i) L ( k ) = lim ←− ℓ L ( k ) /L ( k + ℓ ) . In particular L = lim ←− k L/L ( k ) . (ii) [ L ( k ) , L ( ℓ ) ] ⊂ L ( k + ℓ ) . (iii) L is a retract of b L = lim ←− n L/L n . roof. (i) Since each L α is nilpotent, for some ℓ = ℓ ( α ), L k + ℓα = 0. Thus by Lemma 1,lim ←− ℓ L k /L k + ℓ = lim ←− ℓ,α L kα /L k + ℓα = lim ←− α L kα = L k . (ii) From (2) we obtain ρ α ( L ( k ) ) = ρ α (lim β ≥ α ρ β ( L k )) = ρ α ( L k ) = L kα . It follows that ρ α [ L ( k ) , L ( ℓ ) ] ⊂ [ L kα , L ℓα ] ⊂ L k + ℓα , and so [ L ( k ) , L ( ℓ ) ] ⊂ [ L ( k ) , L ( ℓ ) ] ⊂ L ( k + ℓ ) . (iii) For each n and α , the surjection L → L α /L nα (induced from ρ α ) factors as L → L/L n → L α /L nα . Thus we obtain L → lim ←− n L/L n → lim ←− n,α L α /L nα = lim ←− α L α = L, which decomposes id L as L ϕ → b L ψ → L. (cid:3) Example.
The completion of the free Lie algebra on an infinite number of generatorsmay not be pronilpotent.Let W be the vector space whose basis is given by the elements v i and w i , i ≥ v i = deg w i = 1. We equip W with trivial multiplication W · W = 0 and we denote by( ∧ V, d ) the minimal model of ( Q ⊕ W, V = V = ⊕ n ≥ V n with V = V ∩ ker d andfor n > d : V n → ( ∧ V ) n − . In particular V = W . Then, for i ≥ c i anelement of V with dc i = v i w i , and by induction for n >
1. Let c in be an element of V n with d ( c in ) = v i c i,n − .Let L be the homotopy Lie algebra of ( ∧ V, d ). We denote by x i and y i the elementsof L defined by < v i , sx j > = δ ij , < w i , sx j > = 0 , and < V ≥ , sx j > = 0 ,< w i , sy j > = δ ij , < v i , sy j > = 0 , and < V ≥ , sy j > = 0 . It follows that < c j , s [ x i , y i ] > = − < v j ∧ w j , sx i , sy i > = − δ ij . In the same way, < c jn , s ad px i ( y i ) > = − < v j ∧ c j,n − , sx i , s ad p − x i ( y i ) > = ( − n δ ij δ pn . Now let U ⊂ V be the subspace formed by the elements z such that < z, sx i > =
1. Then L is the completionof E . In particular the series ω = X n ≥ ad nx n ( y n ) = [ x , y ] + [ x , [ x , y ]] + . . . is a well defined element in L .We first suppose that ω belongs to L and will arrive to a contradiction. This will implythat ω is an indecomposable element in L . So suppose ω = P Ni =1 [ ω i , ω ′ i ]. Denote by K the intersection K = ∩ Ni =1 ker ω i ∩ ∩ Ni =1 ker ω ′ i . Then K is a finite codimensional subspaceof V . Since the v i generate a subspace of infinite dimension, some linear combination a = P mi = p α i v i , α i ∈ Q , belongs to K , and we can suppose that the first term α p = 1. Weconsider then the sequence of elements a r ∈ V r , r ≥
1, defined by da = ay p and for r ≥ da r = a · a r − . Then < a r , sω > = − N X i =1 < da r , sω i , sω ′ i > = 0 . On the other hand, a = c p + u with u ∈ U , and by induction a k = c pk + u k with u k ∈ U . Now we compute < a p , sω > using the decomposition of ω as a series. Since a p ∈ V p , for each iterated Lie bracket α in the x i , y i of length different of p , we have < a p , α > = 0. Therefore < a p , sω > = < a p , s ad px p ( y p ) > = < c pp , s ad px p ( y p ) > = − , which gives the required contradiction.Now write α n = P nk =1 ad kx k ( y k ). Then α n +1 − α n ∈ L n +1 and the sequence ( α n ) is acoherent sequence of elements in the tower . . . / / L/L n +1 q n +1 ,n / / L/L n q n,n − / / L/L n − / / . . . This sequence defines an element α ∈ b L = lim ←− n L/L n .We use the notation of Lemma 3 for the retraction L ϕ → b L ψ → L . Since the naturalprojections q n : L/L n → L/L n maps α n to α n , at the limit we have ψ ( α ) = ω . Now if ϕ is an isomorphism, because ψ ◦ ϕ is the identity, we would have ϕ ( ω ) = α , which is notthe case because ω being indecomposable, in L/L we have p ( ω ) = ω = p ( α ) = α . Remark.
While L may not be pronilpotent, Lemma 3 identifies the sequence L ⊃ · · · ⊃ L ( k ) ⊃ . . . as an N -suite as defined by Lazard in [11]. Example.
Let E be an enriched Lie algebra. Then E and b E can be very different. Let E be the abelian Lie algebra with basis the countably infinite family x , x , . . . . We equip13 with the enriched structure given by the ideals I n generated by the x i , i ≥ n . Thecompletion E is the vector space of series P i ≥ α i x i , α i ∈ Q . Therefore the injection b E → E is not an isomorphism, even when E is nilpotent. Lemma 4.
Suppose E ⊂ L is a sub Lie algebra.(i) If E + L (2) = L , then E = L .(ii) E n = E ( n ) , n ≥ . proof. (i) By hypothesis ρ α L = ρ α E + ρ α L (2) = ρ α E + [ ρ α L, ρ α L ]. Since ρ α L is nilpotentthis gives ρ α E = ρ α L and E = L .(ii) Since E ⊂ E , ρ α ( E ) = ρ α ( E ). Therefore E n = lim ←− α ρ α ( E n ) = lim ←− α ( ρ α ( E )) n = lim ←− α ( ρ α ( E ) n = lim ←− α ρ α ( E n ) = E ( n ) . (cid:3) Proposition 2.
Suppose L is a complete enriched Lie algebra. If dim L/L (2) < ∞ , then(i) For any integer k there is an α such that the projection L/L ( k ) → L α /L kα is anisomorphism.(ii) L k = L ( k ) , for k ≥ . proof. (i). Since L/L (2) is finite dimensional, by Lemma 1 there is an α , such that foreach β ≥ α the projection p β : L/L (2) → L β /L β is an isomorphism. This shows that foreach α the dimension of L α /L α is bounded by the integer N = dim L α /L α . It followsthat for any r ≥
1, and any α the dimension of L rα /L r +1 α is bounded by N r . Thus itfollows from Lemma 1 that L ( r ) /L ( r +1) = lim ←− α L rα /L r +1 α is finite dimensional. Thereforefor any integer k there is an index α k such that the projection L/L ( k ) → L α k /L kα k is an isomorphism.(ii) It follows from Lemma 1 that for some α the Lie bracket in LL/L (2) ⊗ L ( r ) /L ( r +1) → L ( r +1) /L ( r +2) , r ≥ , (4)may be identified with the corresponding surjection L α /L α ⊗ L rα /L r +1 α → L r +1 α /L r +2 α . This implies that the Lie bracket (4) is surjective.Now let x , . . . , x m ∈ L represent a basis of, L/L (2) . Then for any y ∈ L ( r +1) it followsthat there are elements y i ∈ L ( r ) such that y − r X i =1 [ x i , y i ] ∈ L ( r +2) . x ∈ L ( n ) , this yields an inductive construction of elements y i ( ℓ ) ∈ L ( ℓ − , ℓ ≥ k ,such that x − m X i =1 r X ℓ = k − [ x i , y i ( ℓ )] ∈ L ( r ) . Set y i = P ℓ y i ( ℓ ). Then by construction x = m X i =1 [ x i , y i ] . This shows that L ( k ) ⊂ [ L, L ( k − ]. Induction on k gives L ( k ) ⊂ L k , and the reverse inclusion is obvious. (cid:3) Corollary 1.
Let E be an enriched Lie algebra with dim E/E < ∞ . Then dim E/E k < ∞ , k ≥
1, and the lower central series for E satisfies E ( k ) = E k = E k . proof. There is an α such that for each α ≥ α , the projection E α /E α → E α /E α isan isomorphism. Thus by Lemma 1 E/E (2) is isomorphic to E α /E α , and so is finitedimensional. Proposition 2 gives then an isomorphism E/E (2) = E/E = E α /E α Inparticular, E + E (2) = E . The result is then a direct consequence of Proposition 2(ii) andLemma 4(ii). (cid:3) Corollary 2.
An complete enriched Lie algebra L is the direct limit of closed sub Liealgebras, E , satisfying dim E/E < ∞ . proof. Any Lie algebra is the direct limit of its finitely generated Lie algebras, F . Thus L is the direct limit of the completions, F . By Proposition 2 and Corollary 1, each F satisfies dim F /F < ∞ . (cid:3) Proposition 2 has the following analogue:
Proposition 3. If E ⊂ L is a sub Lie algebra and E/E is a graded vector space of finitetype then(i) Each E/E k has finite type.(ii) E k = E k = E ( k ) . proof: (i) In the proof of Proposition 2(i), replace L/L k by any E n / ( E k ) n to obtain that( E ) n / ( E k ) n is finite dimensional.(ii) This follows directly by the same proof as in Corollary 3.. (cid:3) roposition 4. Suppose L is a complete enriched Lie algebra such that L/L has finite type,then L has a unique (up to equivalence) structure as an enriched Lie algebra ( L, I ) , namelythat given by I = { I k } k ≥ with I k = L k + L ≥ k . proof: Suppose a second family J = { J β } of ideals in L also makes L into an enriched Liealgebra. Since L/J β is finite dimensional and nilpotent, it is also immediate that J β ⊃ I k for some k .In the reverse direction, apply Proposition 2(i) to conclude that for fixed k and n thereis some β = β ( k, n ) such that ρ β : L n / ( L k ) n ∼ = −→ ( L β ) n / ( L kβ ) n . It follows that ( L k ) n = ρ − β ( L kβ ) n ⊃ ( J β ) n . Choose γ ∈ J so that J γ ⊂ ∩ kn =1 J β ( k,n ) . Then ( L k ) ≤ n ⊃ ( J γ ) ≤ n , and so I k ⊃ J γ . (cid:3) Corollary.
Let L be a complete enriched Lie algebra and E ⊂ L be a subalgebra suchthat E/E has finite type. Then E is closed if and only if E is pronilpotent. proof. By Proposition 4, the structure systems ρ α E and E/E n are equivalent. This impliesthe result. (cid:3) Example.
Suppose a graded Lie algebra E satisfies ∩ k E k = 0 , dim E/E < ∞ , and dim E = ∞ . Let z ∈ E satisfy E ⊕ Q z ⊂ E and let F be the sub Lie algebra generated by E and Q z .Then E ⊂ F ⊂ E , and it follows that E = F . However the dimension of F/F dependson the choice of z .For instance, suppose E is the completion of the free Lie algebra L ( x, y ) generated by x, y , and with the unique enriched structure described in Proposition 4. Let F be the subLie algebra generated by x , y and z = e ad x ( y ). Denote by t the operator ad x . The series t and e t are algebraically independent in Q [[ t ]], i.e., if P ( a, b ) is a polynomial in 2 variables a and b and P ( t, e t ) = 0 then P is identically zero. It follows that if we have a polynomialrelation of the form X n α n ad nx ( y ) + X n β n ad nx e ad x ( y ) = 0then ( P α n t n + P β n t n e t )( y ) = 0, and α n = β n = 0 for all n . Therefore the classes of x, y and z are linearly independent in F/F , and F/F has dimension 3.On the other hand, with the same E , if z = P n ≥ ad nx ( y ), then z − y = [ x, z ] and F/F = E/E . 16 The quadratic model of ( L, I ) First recall that a quadratic Sullivan algebra is a Sullivan algebra ( ∧ V, d ) in which d : V →∧ V . This endows H ( ∧ V ) with the decomposition H ( ∧ V ) = ⊕ k H [ k ] ( ∧ V )in which H [ k ] ( ∧ V ) is the subspace represented by ∧ k V ∩ ker d . Moreover, V = lim −→ n V n where V = V ∩ ker d and V n +1 = V ∩ d − ( ∧ V n ). In particular a morphism ϕ : ∧ V → ∧ W of quadratic Sullivan algebras restricts to linear maps ϕ n : V n → W n . On the other hand, recall that the classical functor L C ∗ ( L ) from graded Lie algebrasto cocommutative chain coalgebras is given by C ∗ ( L ) = ∧ sL with differential determinedby the condition ∂ ( sx ∧ sy ) = ( − deg x +1 s [ x, y ] . The dual, C ∗ ( L ) = Hom( C ∗ ( L ) , Q ) is, with the differential forms on a manifold, one ofthe earliest examples of a commutative differential graded algebra (cdga). In particular,if dim L < ∞ then C ∗ ( L ) = ∧ ( sL ) ∨ , and the differential is determined by the condition < dv, sx, sy > = ( − deg y +1 < v, s [ x, y ] >, v ∈ L ∨ . (5)Now suppose ( L, I ) is an enriched graded Lie algebra. Then L = lim ←− α ∈ I L α , andsince L α is finite dimensional, C ∗ ( L α ) = ∧ ( sL α ) ∨ . Set V α = ( sL α ) ∨ and note that then sL α = V ∨ α . In particular, ( ∧ V, d ) := lim −→ α ∧ V α = lim −→ α C ∗ ( L α )is a cdga, and d : V → ∧ V . Moreover, because each L α is nilpotent, each ∧ V α is a(quadratic) Sullivan algebra, and therefore ( ∧ V, d ) is a quadratic Sullivan algebra. Inview of (5), which then holds for all v ∈ V and x, y ∈ L , L is the homotopy Lie algebra of ∧ V , as defined for example in [8, § Definition. ∧ V is the quadratic model of ( L, I ). Remark.
Note that the natural morphism ∧ V = lim −→ C ∗ ( L α ) → C ∗ ( L ) will not, in general,be a quasi-isomorphism. For instance, if V = V has zero differential and countably infinitedimension then ∧ V is countable but L and C ∗ ( L ) are uncountable, and since L is abelian, C ∗ ( L ) = H ( C ∗ ( L )).With the notation at the start of the Introduction note that a morphism of completeenriched Lie algebras, ψ : E → L , induces morphisms E = lim ←− β E β −→ lim ←− α E β ( α ) −→ lim ←− α L α = L. The morphism E β ( α ) → E α dualizes to a morphism of quadratic models, and the compos-ites define a unique morphism ϕ : ∧ V L → ∧ V E ψ = ( ϕ | V L ) ∨ . This identifies the correspondence ( L, I )
7→ ∧ V L as a functor fromthe category of complete enriched Lie algebras to the category C ′ of quadratic Sullivanalgebras in which the morphisms ∧ V → ∧ W map V into W .More generally, and in the same way, a coherent linear map f : E → L induces a linearmap ϕ : V L → V E for which ϕ ∨ = f . Proposition 5.
The correspondence ( L, I ) ∧ V is a contravariant isomorphism fromthe category of complete enriched Lie algebras to the category C ′ of quadratic Sullivanalgebras. proof. Each quadratic Sullivan algebra, ∧ V , determines an enriched Lie algebra ( L, I ).Namely, L is the homotopy Lie algebra of ∧ V , given by L = ( sV ) ∨ and < v, s [ x, y ] > = ( − deg y +1 < dv, sx ∧ sy > . Moreover V = lim −→ β V β where the V β are the finite dimensional subspaces for which ∧ V β is preserved by d . The inclusions ∧ V β → ∧ V dualize to surjections ρ β : L → L β betweenthe homotopy Lie algebras, and these induce an isomorphism L ∼ = −→ lim ←− β L β . This then endows L with the enriched structure ( L, I ) with I = { ker ρ β } . If ϕ : ∧ V → ∧ W is a morphism then ( ϕ | V ) ∨ is the corresponding morphism L W → L V . (cid:3) Remark.
Proposition 5 couples the categories of complete enriched Lie algebras, L , andquadratic Sullivan algebras, ∧ V , as pairs ( L, ∧ V ) where ∧ V is the quadratic Sullivanmodel of L and, equivalently, L is the homotopy Lie algebra of ∧ V . Example.
Inverse limits
Suppose { L ( σ ) , ϕ σ,τ : L ( τ ) → L ( σ ) } is an inverse system of morphisms of completeenriched Lie algebras. Then L := lim ←− σ L ( σ )is naturally a complete enriched Lie algebra.In fact under the correspondence of Proposition 5, this inverse system is the dual ofthe directed system {∧ V ( σ ) } of the quadratic Sullivan models of the L ( σ ). But then ∧ V := lim −→ σ ∧ V ( σ ) is a quadratic Sullivan algebra, and L is the homotopy lie algebra of ∧ V . It is immediate that L is the inverse limit in the category of complete enriched Liealgebras.The correspondence of Proposition 5 is reflected in the next Lemma. Lemma 5.
The degree 1 identification L ∼ = → V ∨ induces isomorphisms L/L ( n +2) ∼ = −→ V ∨ n , n ≥ , and therefore identifies L ( n +2) = { x ∈ L | < V p , sx > = 0 } . roof. This is straightforward when L and V are replaced by L α and V α . In the generalcase by Lemma 1 we have the following sequence of isomorphisms L/L ( n +2) = L/L n +2 = lim ←− α ρ α ( L ) /ρ α ( L n +2 ) = lim ←− α L α /L n +2 α = lim ←− α ( V α ) n ∨ = lim −→ α ( V α ) n ! ∨ = V ∨ n . (cid:3) Remark.
A morphism ψ : E → L of complete enriched Lie algebras induces morphisms ψ ( n ) : E/E ( n ) → L/L ( n ) , n ≥
1. If ϕ : ∧ V L → ∧ V E is the corresponding morphism ofquadratic models then ψ ( n + 2) is dual to to the linear map ϕ n : ( V L ) n → ( V E ) n .Finally, note that a surjection ξ : V → P dualizes to an inclusion ξ ∨ : ( sP ) ∨ → L . Lemma 6. (i) If S ⊂ L , let ξ : V → P = V /K be the surjection given by K = { v ∈ V | < v, sS > = 0 } . Then S = Im ξ ∨ . (ii) In particular, the closed subspaces of L are precisely the subspaces of the form Im ξ ∨ as ξ runs through the surjections V → P .(iii) Any closed subspace S ⊂ L has a closed direct summand. proof. (i) Use ξ ∨ to identify ( sP ) ∨ with a subspace of L . Since L = lim ←− α L α and L α = V ∨ α ,for x ∈ L and v ∈ V α we have < v, sx > = < v, sρ α x > . Thus V α ∩ K = { v ∈ V α | < v, sρ α S > = 0 } . Set P α = V α /V α ∩ K . Then, because V α and L α are finite dimensional, this gives ρ α S = { x ∈ L α | < V α ∩ K, sx > = 0 } = ( sP α ) ∨ . Thus S = lim ←− α ρ α S = lim ←− α ( sP α ) ∨ = (lim −→ α sP α ) ∨ = ( sP ) ∨ . (ii) If follows exactly as in (i) that each ( sP ) ∨ ⊂ L is closed.(iii) The inclusion S → L is the dual of a surjection V → V /K . Dualizing the inclusion K → V provides a surjection L → ( sK ) ∨ onto a closed direct summand of S . (cid:3) .1 Closed subalgebras and ideals Suppose E ⊂ L is a closed subalgebra of a complete enriched Lie algebra. Then theinclusion is the dual of a surjection ∧ V ρ / / ∧ Z of the respective quadratic models, andby Lemma 6, E = { x ∈ L | < ker ρ | V , sx > = 0 } . Moreover ([8, Lemma 10.4]), d : ker ρ | V → ker ρ | V ∧ V . Recall also that every such surjec-tion induces an inclusion of a closed sub algebra of L .Now suppose I ⊂ L is a closed ideal. Let ρ : ∧ V → ∧ Z be the corresponding surjection,and denote W = ker ρ | V . In this case ([8, Lemma 10.4]) the condition that I be an ideal isequivalent to the condition d : W → ∧ W . Thus ∧ W is a sub quadratic Sullivan algebraand I decomposes ∧ V as the Sullivan extension ∧ W λ / / ∧ W ⊗ ∧ Z = ∧ V ρ / / ∧ V ⊗ ∧ W Q = ∧ Z, which dualizes to L/I ← L ← I. In particular this identifies
L/I as the homotopy Lie algebra of ∧ W . Proposition 6.
Let f : L → E be a morphism of complete enriched Lie algebras. If L/L (2) → E/E (2) is surjective, then f is surjective. proof. Denote by ϕ : ∧ V E → ∧ V L the corresponding morphism given by Proposition5. By Lemma 5, ( V E ∩ ker d ) ∨ ∼ = L/L (2) and ( V L ∩ ker d ) ∨ ∼ = E/E (2) . It follows that ϕ : V E ∩ ker d → V L ∩ ker d is injective. We suppose that for some integer n , ϕ : ( V E ) n → ( V L ) n is injective. Then ϕ : ∧ ( V E ) n → ∧ ( V L ) n is injective. Suppose that v ∈ ( V E ) n +1 isin the kernel of ϕ , then ϕ ( dv ) = 0, and so dv = 0. Therefore v ∈ ( V E ) which implies that v = 0. It follows that the dual map f is surjective. (cid:3) Proposition 7. If f : L → L ′ is a morphism of complete enriched Lie algebras, thenKer f is a closed ideal in L and Im f is a closed subalgebra of L ′ . proof. Let ϕ : ∧ V ′ → ∧ V be the morphism associated to f . We denote by E the idealgenerated by V ′ ∩ Ker ϕ in ∧ V ′ and we denote by ∧ W = ∧ V ′ /E the quotient quadraticSullivan algebra. Then ϕ factorizes through an injection ϕ ′ : ∧ W → ∧ V . We extend then ϕ ′ to an isomorphism ψ : ∧ W ⊗ ∧ T ∼ = → ∧ V from a Λ-extension. Then ∧ W is the quadraticmodel of Im f and the quotient cdga ( ∧ T, d ) is the quadratic model of Ker f .The surjection ∧ V ′ → ∧ W shows that Im f is a closed subalgebra of L ′ , and themorphism ∧ W ⊗ ∧ T → ∧ T that I is a closed ideal in L . (cid:3) Corollary.
Let f : L → W a surjective coherent morphism between complete enrichedabelian Lie algebras. Then f admits a coherent section, σ . proof. By Proposition 7, Ker f is a closed subspace. Now by Lemma 6(iii), Ker f ⊂ L admits a closed direct summand S . It follows that f | S : S → Im f = W is an isomorphism,and we define σ = ( f | S ) − . (cid:3) xample. Weighted Lie algebras. Let E = ⊕ k E ( k ) be a weighted enriched Liealgebra with defining ideals I α = ⊕ k I α ( k ). The weighting then induces another gradation V = ⊕ k V ( k ) in the generating space of the corresponding quadratic model: V ( k ) = lim −→ α s ( E ( k ) /I α ( k )) ∨ . It is immediate that the differential preserves the induced gradation in ∧ V . Remark.
ZFC - set theory alone does not permit us to extend Proposition 5 to all gradedLie algebras, since it is consistent with the ZFC-axioms that the same graded Lie algebracan support two inequivalent enriched structures.In fact, let L = L be a vector space and suppose (consistent with the ZFC axioms)that there are isomorphisms V ∨ ∼ = sL ∼ = W ∨ in which card V = card W . Let ∧ V and ∧ W be the quadratic Sullivan algebras with zerodifferential. Then L , regarded as an abelian Lie algebra is the homotopy Lie algebra ofboth ∧ V and ∧ W , but since ∧ V = ∧ W the corresponding enriched structures in L arenot equivalent. Throughout this entire section ( L, I ) denotes a fixed complete enriched Lie algebra, withquotient maps ρ α : L → L/I α := L α and quadratic model ∧ V . Definition. (i) An elementary L -module is a finite dimensional nilpotent L -module, Q ,for which some I α · Q = 0.(ii) An enriched L -module is an L -module, N , together with a decomposition, N = lim ←− τ N τ , of N as an inverse limit of elementary L -modules.(iii) A coherent L -module is an L -module, M , together with a decomposition, M = lim −→ σ M σ , of M as a direct limit of elementary L -modules.(iv) A morphism of enriched (resp. coherent) L -modules is a morphism of L -moduleswhich is the inverse limit (resp. direct limit) of morphisms of elementary L -modules.Note that an enriched L -module N is the dual of the coherent L -module M definedby M = lim −→ σ N ∨ σ . Example.
Let L be the free Lie algebra on one generator x in degree 0. The space M of finite sequences of rational numbers ( a , . . . , a n ) equipped with the L -structure definedby x · ( a , . . . , a n ) = ( a , . . . a n ) is a coherent L -module. On the other hand the space N of infinite sequences ( a , a , . . . ) with the L -structure defined by ( a , a , . . . ) · x =(0 , a , a , . . . ) is an enriched L -module. 21 emarks.
1. When dim L/ [ L, L ] < ∞ , any finite dimensional nilpotent L -module is elementary.This however is not true in general. In fact, if L = L is an infinite dimensional abelian Liealgebra with basis { x i } , the subspaces I ( k ) spanned by the x i , i ≥ k , define an enrichedstructure in L . Then, consider the finite dimensional nilpotent L -module M = Q a ⊕ Q b defined by x i · a = b and x i · b = 0 . Since dim L = ∞ , M cannnot be an elementary L -module.2. The dual, M ∨ , of a coherent L -module M = lim −→ σ M σ inherits the enriched L -modulestructure given by M ∨ = lim ←− σ M ∨ σ . Moreover, each enriched L -module is the dual of aunique coherent L -module.3. The dual, ϕ ∨ , of a morphism, ϕ , of coherent L -modules is a morphism of the dualenriched L -modules. Moreover, every morphism of the dual modules is the dual of a uniquemorphism of the corresponding coherent modules.4. Since for each elementary L -module Q there is an ideal I α with I α · Q = 0, the module Q is naturally a [ U L α -module for some α . Thus the representations of L in enriched andcoherent L -modules naturally extend to representations of the algebra U L .5. Let N be a right enriched L -module. The space of decomposable elements of N is by definition N · L = lim ←− α N α · L . For instance each complete Lie algebra L is a rightenriched module over itself for the adjoint action and N · L = L (2) . An elementary L -module N τ determines the family I α ⊂ I of ideals satisfying N τ · I α = 0.However, the trivial L -modules, e N τ := N τ ⊗ [ UL α Q = N τ /N τ · L are independent of thechoice of α . Thus e N = lim ←− τ f N τ is a trivial enriched L -module and the induced morphism ε N : N → e N is a morphism of enriched L -modules: ε N is the augmentation for N . By construction, ε N : N → e N depends naturally on N .Let M and f M be the coherent L -modules defined by M ∨ = N and f M ∨ = e N . Then f M = { m ∈ M | L · m = 0 } . ⊗ Suppose N , M , and M ′ are respectively a right enriched L -module and two left coherent L -modules. Then, Hom( M, N ) = lim ←− σ,τ Hom( M σ , N τ )and M ⊗ M ′ = lim −→ σ,σ ′ M σ ⊗ M ′ σ ′ identify Hom( M, N ) and M ⊗ M ′ respectively as a right enriched and a left coherent L -module. 22 .3 U L
The surjections
U L → [ U L α /J nα identify U L as a right enriched L -module with the repre-sentation given by right multiplication. ∧ V -modules and holonomy representations Quadratic ∧ V -modules are examples of the semifree ∧ V -modules recalled in the Appendix: Definition. (i) A quadratic ∧ V -module is a ∧ V -module of the form ∧ V ⊗ M in which d : 1 ⊗ M → V ⊗ M and M = ∪ k ≥ M k , where M = M ∩ d − ( V ) and M k +1 = M ∩ d − ( V ⊗ M k ) . (ii) The holonomy representation for a quadratic ∧ V -module ∧ V ⊗ M is the represen-tation of L in M given by x · m = − X < v i , sx > m i , where d (1 ⊗ m ) = P v i ⊗ m i . It is immediate that the holonomy operation makes M into a coherent left L -module.In the reverse direction, suppose M = lim −→ σ M σ is a coherent left L -module. The classicalCatan-Eilenberg-Serre construction then has the form C ∗ ( L α ( σ ) , M σ ) = ∧ ( sL α ( σ ) ) ∨ ⊗ M σ = ∧ V α ( σ ) ⊗ M σ in which d : M σ → V α ( σ ) ⊗ M σ . Passing to direct limits constructs the quadratic ∧ V -module ∧ V ⊗ M = lim −→ σ ∧ V α ( σ ) ⊗ M σ . This establishes a bijection between quadratic ∧ V -modules and coherent left L -modules.Observe as well that since quadratic ∧ V -modules are semifree a morphism from aquadratic ∧ V -module lifts through any surjective quasi-isomorphism of ∧ V -modules [6, § ∧ V ⊗ M ≃ → ∧ V ⊗ M ′ of quadratic ∧ V -modules hasa homotopy inverse. It follows that it induces an isomorphism M ∼ = → M ′ of coherent left L -modules.Finally, as recalled in §
4, associated with L is the differential coalgebra ( ∧ sL, ∂ ) inwhich ∂ ( sx ∧ sy ) = ( − deg sx s [ x, y ]. More generally, (eg. [8, Chap 2]) associated with M is the differential ( ∧ sL, ∂ )-comodule, ( ∧ sL ⊗ M, ∂ ), characterized by ∂ ( sx ⊗ m ) = ( − deg sx x · m. As recalled in [8, Chap 10], there are natural isomorphismsTor
ULp ( Q , M ) ∼ = H [ p ] ( ∧ sL ⊗ M ) , p ≥ , where H [ p ] denotes the subspace of H ( ∧ sL ⊗ M ) represented by cycles in ∧ p sL ⊗ M .23 .5 Acyclic closures Recall ([8, Chap 3]) that the acyclic closure is the special case, ∧ V ⊗ ∧ U , of a Λ-extensionconstructed inductively as follows. Write V = ∪ n V n with V = V ∩ ker d and V n +1 = V ∩ d − ( ∧ V n ). Then U = ∪ n ≥ U n and there is a degree 1 isomorphism p : U ∼ = → V restricting to isomorphisms U n ∼ = → V n . Thus this identifies L = U ∨ . Finally, the differentialis determined by the conditions du = pu, u ∈ U and ( d − p ) : U n +1 → V n ⊗ ∧ U n . In particular the augmentation ε U : ∧ U → Q , U → ∧ V define a quasi-isomorphism, ∧ V ⊗ ∧ U ≃ −→ Q . This construction identifies ∧ V ⊗ ∧ U as a quadratic ∧ V -module. In this case theholonomy representation is a representation θ : L → Der( ∧ U ) of L by derivations in ∧ U ,and if x ∈ L and u ∈ U n then θ ( x ) u + < pu, x > ∈ V n − ⊗ ∧ U n − . (6)Now denote by η L the morphism of enriched L -modules, η L : U L −→ ( ∧ U ) ∨ , defined by η L ( a )(Φ) = ε U ( a · Φ) , a ∈ U L, Φ ∈ ∧ U. By [8, Theorem 6.1], η α : [ U L α → ( ∧ U α ) ∨ is an isomorphism, and since η L is a morphismof inverse limits, we have Proposition 8.
The morphism η L : U L ∼ = −→ ( ∧ U ) ∨ is an isomorphism of U L -modules.
The right adjoint representation of L α in L α makes L α into an L α -module. This exhibits L = lim ←− α L α with the right adjoint representation of L as a right enriched L -module. Since sL α = V ∨ α the corresponding left coherent L -module is a representation of U L in V . It isgiven explicitly by < x · v, sy > = − < dv, sx, sy >, v ∈ V, x, x ∈ L. Lemma 7. If S ⊂ L is any subspace then S · U L is an ideal. If S is a graded space offinite type, then S · U L is closed. roof. Observe that [ x · Φ , y ] = x · Φ y , x, y ∈ L, Φ ∈ U L, and so S · U L is an ideal. Since a graded space is closed if and only if each component ina given degree is closed, if S has finite type it is sufficient to show that each S k · U L isclosed. Since a finite sum of closed subspaces is closed we have only to show that x · U L is closed for each x ∈ L .Denote ρ α x by x α . Then the closure of x · U L is the inverse limitlim ←− α ρ α ( x · U L ) = lim ←− α x α · [ U L α = Q x α ⊕ lim ←− α x α · c J α = Q x α ⊕ lim ←− n,α ( x α · J α ) / ( x α · J nα ) . On the other hand, because each J α /J nα is finite dimensional, the surjections J α /J nα → ( x α · J α ) /x α · J nα ) induce a surjection J → lim ←− n,α x α · J α / ( x α · J nα ) . This factors through the map J → x · J , and therefore shows that x · J → lim ←− α ρ α ( x · J )is surjective. But this is the inclusion of x · J in its closure, and so x · J is closed. (cid:3) L -modules Definition. A profree L -module is an enriched L -module of the form Hom( S, U L ) where S is a trivial coherent L -module. Remarks.
1. It is immediate from the definition that the augmentation in Hom(
S, U L ) is themorphism Hom(
S, U L ) → S ∨ given by γ ε ◦ γ , where ε is the augmentation U L → Q with kernel J .2. If S is a graded vector space of finite type and S = S ≥ then it is immediate thatHom( S, U L ) = S ∨ ⊗ U L is a free
U L -module.
Now fix a left coherent L -module, M , and define a left coherent L -module S by setting S = { a ∈ M | L · a = 0 } . It follows from § M ∨ → S ∨ is the augmentation for S ∨ .As usual we denote by ∧ V the quadratic Sullivan model of L and by ( ∧ V ⊗ M, d ) theholonomy representation defined in § d (1 ⊗ S ) = 0.25 roposition 9. With the hypotheses and notation above the following conditions are equiv-alent:(i) The inclusion S → ⊗ S ⊂ ∧ V ⊗ M induces an isomorphism S ∼ = −→ H ( ∧ V ⊗ M ) . (ii) M ∼ = ∧ U ⊗ S as left coherent L -modules.(iii) M ∨ is a profree L -module with augmentation M ∨ → S ∨ . proof. Choose any linear retraction ρ : M → S . Tensored with the augmentation of ∧ V this provides a morphism ∧ V ⊗ M → S of ∧ V -modules. Lifting this through the quasi-isomorphism ∧ V ⊗ ∧ U ⊗ S → S provided by the augmentation of ∧ V ⊗ ∧ U , we obtain amorphism, ψ : ∧ V ⊗ M → ∧ V ⊗ ∧ U ⊗ S, of ∧ V -modules. Then division by ∧ ≥ V yields a morphism ϕ : M → ∧ U ⊗ S of coherentleft L -modules.To complete the proof we show first that (i) ⇔ (ii). In fact it is immediate that (ii) ⇒ (i). On the other hand, if (i) holds then ∧ V ⊗ M → S is a quasi-isomorphism, and henceso is ψ . It follows that ϕ is an isomorphism.Finally, to show (ii) ⇔ (iii) note that since ( ∧ U ) ∨ = U L , (ii) ⇒ (iii) is immediatefrom the definition of profree. In the reverse direction, suppose M ∨ is profree with aug-mentation M ∨ → S ∨ . Then, since augmentations are preserved by morphisms, there is acommutative diagram M ∨ χ ∼ = / / ! ! ❈❈❈❈❈❈❈❈ Hom(
S, U L ) y y sssssssssss S ∨ of right enriched L -modules.On the other hand, the dual of the morphism ϕ is a morphism ϕ ∨ : ( ∧ U ⊗ S ) ∨ → M ∨ of enriched L -modules. Since the representation of L in ∧ U ⊗ S is just the holonomyrepresentation in ∧ U and since ( ∧ U ) ∨ ∼ = U L , ϕ ∨ is a morphism from Hom( S, U L ). Thusthe diagram Hom(
S, U L ) ϕ ∨ / / ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ M ∨ (cid:15) (cid:15) χ ∼ = / / Hom(
S, U L ) v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ S ∨ commutes.To establish (ii) we will show that ϕ is an isomorphism, and of course it is sufficientto show that ϕ ∨ is an isomorphism or, equivalently, that γ := χ ◦ ϕ ∨ is an isomorphism.But the commutative diagram above shows that γ − id : Hom( S, U L ) → Hom(
S, J ) . γ is a morphism of enriched L -modules it follows that( γ − id ) : Hom( S, J ( n ) ) → Hom(
S, J ( n +1) ) . Finally, because
U L = lim ←− n U L/J ( n ) , it follows thatHom( S, U L ) = lim ←− n Hom(
S, U L/J ( n ) )and therefore γ is an isomorphism. (cid:3) Suppose I ⊂ L is a closed ideal and denote by ∧ W λ / / ∧ W ⊗ ∧ Z = ∧ V ρ / / ∧ V ⊗ ∧ W Q = ∧ Z, the Λ-extension in which λ and ρ dualize to the short exact sequence L/I ← L ← I. Next let ( ∧ W ⊗∧ U W , d W ) and ( ∧ Z ⊗∧ U Z , d Z ) be the respective acyclic closures. Since ∧ W ⊗ ∧ U W and ∧ W ⊗ ∧ Z are Λ-extensions we can form the Λ-extension ∧ W ⊗ ∧ U W ⊗ ∧ Z := ( ∧ W ⊗ ∧ U W ) ⊗ ∧ W ( ∧ W ⊗ ∧ Z ) . Now it follows from [5, Proposition 1] that the acyclic closure of ∧ V is a Λ-extension of ∧ W ⊗ ∧ U W ⊗ ∧ Z of the form( ∧ W ⊗ ∧ U W ⊗ ∧ Z, d ) → ( ∧ W ⊗ ∧ U W ⊗ ∧ Z ⊗ ∧ U Z , d )in which ( d − d Z ) : ∧ U Z → W ∧ ( ∧ W ⊗ ∧ U W ⊗ ∧ Z ⊗ ∧ U Z ) . (7)Since Λ-extensions are semifree modules with respect to the underlying quadratic Sullivanalgebras it follows that ∧ U Z is the union of subspaces M (0) ⊂ · · · ⊂ M ( k ) ⊂ M ( k + 1) ⊂ . . . satisfying d : M (0) → ∧ W ⊗ ∧ U W ⊗ ∧ Z and d : M ( k + 1) → ( ∧ W ⊗ ∧ U W ⊗ ∧ Z ) ⊗ M ( k ) . (8)On the other hand, since [ I, L ] ⊂ I , I is an L -submodule with respect to the rightadjoint representation. Moreover, this representation factors to yield a representation of L/I in I/I . Proposition 10.
With the hypotheses above:(i) I is an enriched sub-module of L for the adjoint representation.(ii) The adjoint representation of L/I in I/I factors over the surjection I/I → I/I (2) to identify
I/I (2) as an enriched
L/I -module. roof. (i). Each ρ α I ⊂ L α is an L α -module: [ ρ α I, L α ] ⊂ ρ α I . This exhibits I = lim ←− α ρ α I as an enriched L -sub module of L .(ii). By definition, I (2) = lim ←− α [ ρ α I, ρ α I ] is also a closed ideal. Now the right adjointrepresentation of L α factors to give a representation of L α /ρ α I in ρ α I/ [ ρ α I, ρ α I ]. But by §
2, lim ←− α L α /ρ α I = L/I and lim ←− α ρ α / [ ρ α I, ρ α I ] = I/I (2) . (cid:3) Now, since ∧ V = ∧ W ⊗ ∧ Z is a Λ-extension of ∧ W it is a semifree ∧ W -module. Inparticular from the equation d = 0 n ∧ V it follows [8, Chap. 4] that the holonomyrepresentation is a representation of L W in ( ∧ Z, d Z ).Denote by θ the induced representation in H ( ∧ Z, d Z ), and note that it follows immedi-ately from the construction that θ restricts to a representation in the subspaces H [ p ] ( ∧ Z )represented by the cycles in ∧ p Z . Moreover, filtering ∧ W ⊗ ∧ Z by the ideals ∧ ≥ r W ⊗ ∧ Z induces a spectral sequence whose E -term ∧ W ⊗ H ( ∧ Z ) is a semifree ∧ W -module. It isimmediate from the construction that θ is the resulting holonomy representation. (cid:3) Proposition 11.
Let I be a closed ideal in a profree Lie algebra L with associated quadraticalgebra ∧ Z . Then there is a natural isomorphism H [1] ( ∧ Z ) ∨ ∼ = s I/I (2) of enriched L/I -modules, where
L/I acts in H [1] ( ∧ Z ) ∨ by the dual of the holonomy rep-resentation and in s I/I (2) by the right adjoint representation. proof. First observe that Lemma 5 provides an isomorphism H [1] ( ∧ Z ) ∨ = ( Z ∩ ker d ) ∨ ∼ = s I/I (2) . It remains to show that this is an isomorphism of enriched
L/I -modules. A limit argumentreduces this to the case dim
V < ∞ , where it is sufficient to show that it is an isomorphismof L -modules.Let x ∈ I , y ∈ L and z ∈ ker d . From (5) we have < d (1 ⊗ z ) , sx, sy > = ( − deg y +1 < z, s [ x, y ] > = ( − deg x < s [ x, y ] , z > . On the other hand, since < V, sx > = 0, we have from (6) that < d (1 ⊗ z ) , sx, sy > = P < v i , sy > < z i , sx > = − < θ ( y ) z, sx > = ( − deg x < sx, θ ( y ) z > . (cid:3) art II Profree Lie algebras
Free graded Lie algebras freely generated by a space V , and which we denote by L V , playa key role in Lie algebra theory. Profree Lie algebras are the analogue of free Lie algebrasin the category of complete enriched Lie algebras. More precisely, let L be a completeenriched Lie algebra, and recall from Lemma 6 that L admits direct decompositions L = L (2) ⊕ T in which T is a closed subspace. Moreover, as we shall show in Proposition 13,if L is profree then T freely generates a free sub Lie algebra of L . Definition
1. The complete enriched Lie algebra, L , is profree if, for some decomposition L = L (2) ⊕ T with T closed, any coherent linear map f : T → E into a complete enriched Lie algebra extends to a morphism ϕ : L → E.
2. A decomposition L = L (2) ⊕ T with T closed is extendable if it satisfies this condition. Remarks
1. If T is a closed direct summand of L (2) in a complete enriched Lie algebra L then(Lemma 4) L is the closure of the sub Lie algebra generated by T . It follows that ϕ is uniquely determined by f .2. Suppose ∧ V is the quadratic model of a complete enriched Lie algebra L . As in § V := V ∩ ker d. If L = L (2) ⊕ T with T closed then by Lemma 5, division by L (2) induces an isomor-phism T ∼ = −→ V ∨ . Theorem 1.
A complete enriched Lie algebra, L , is profree if and only if its quadraticmodel ∧ V satisfies H ( ∧ V ) = Q ⊕ ( V ∩ ker d ) . In this case each direct decomposition L = L (2) ⊕ T with T closed is extendable. The proof of Theorem 1 will be carried out in the next two subsections.29 .1 Quadratic Sullivan algebras
For any quadratic Sullivan algebra, ∧ V , we write H ( ∧ V ) = ⊕ k H [ k ] ( ∧ V ) , where H [ k ] ( ∧ V ) is the image in H ( ∧ V ) of the space of cycles in ∧ k V . Our objective hereis to prove (Lemma 9) that H [2] ( ∧ V ) = 0 ⇐⇒ H ( ∧ V ) = Q ⊕ ( V ∩ ker d ) . Lemma 8.
Suppose ψ : E → L is a morphism of complete enriched Lie algebras.(i) ψ is surjective if and only if ψ (2) : E/E (2) → L/L (2) is surjective.(ii) Suppose the quadratic model, ∧ V , of L satisfies H [2] ( ∧ V ) = 0 . Then the followingassertions are equivalent:(a) ψ (2) is an isomorphism,(b) ψ ( n ) is an isomorphism, n ≥ ,(c) ψ is an isomorphism. proof. First recall that ψ is the dual of ϕ : V → W where ϕ : ∧ V → ∧ W is thecorresponding morphism between the quadratic models of L and E . In view of Lemma 5, ψ ( n + 2) : E/E ( n +2) → L/L ( n +2) is the dual of ϕ n : V n → W n .(i) We have only to show that if ϕ is injective then ϕ is injective. Assume by inductionthat ϕ n is injective. if v ∈ V n +1 and ϕv = 0 then since dv ∈ ∧ V n and ϕ n is injective itfollows that dv = 0. Thus v ∈ V and since ϕ is injective v = 0.(ii) Suppose (a) is satisfied and assume by induction that ϕ n : V n ∼ = → W n . If w ∈ W n +1 then dw is a cycle in ∧ W n and so dw = ϕ Φ where Φ is a cycle in ∧ V n . Because H [2] ( ∧ V ) = 0, it follows that Φ = dv for some v ∈ V n +1 . Thus d ( w − ϕv ) = 0 and so w − ϕ ( v ) ∈ W . By hypothesis, w − ϕv = ϕv for some v ∈ V , and so W n +1 ⊂ ϕ ( V n +1 ).On the other hand, by (i), ϕ n +1 : V n +1 → W n +1 is injective. This proves that (a) ⇒ (b).But then ϕ = lim −→ ϕ n and so ϕ is an isomorphism. Thus (b) ⇒ (c). Finally, it isimmediate that if ϕ is an isomorphism then so is ϕ . (cid:3) Lemma 9.
If a morphism ρ : ∧ V → ∧ Z of quadratic Sullivan algebras restricts to asurjection ρ : V → Z , and if H [2] ( ∧ V ) = 0 , then H [ k ] ( ∧ Z ) = 0 , k ≥ . proof. The proof is in two Steps.
Step One. If, in addition, ρ is surjective then H [ k ] ( ∧ Z ) = 0 , k ≥ . Here we first show that H [2] ( ∧ Z ) = 0. Since ρ is surjective, by [8, Cor.3.4] ρ extendsto a quasi-isomorphism ρ : ∧ V ⊗ ∧ U ≃ → ∧ Z
30n which ∧ V ⊗ ∧ U is a Λ-extension of ∧ V and d : ∧ U → V ⊗ ∧ U . Since ρ is surjective thereis a quasi-isomorphism σ : ∧ Z → ∧ V ⊗∧ U satisfying ρ ◦ σ = id . Because d : ∧ U → V ⊗∧ U and ∧ V ⊗ ∧ U is a Λ-extension it is straightforward to verify that σ may be constructedso that σ : Z → V ⊗ ∧ U . In particular, the decomposition ∧ V ⊗ ∧ U = ⊕ k ∧ k V ⊗ ∧ U induces a decomposition of its homology, and σ : H [ k ] ( ∧ Z ) ∼ = → H [ k ] ( ∧ V ⊗ ∧ U ) , k ≥ . On the other hand, because ∧ V ⊗ ∧ U is a Λ-extension, ∧ U is the increasing union ofthe subspaces ( ∧ U ) q given by( ∧ U ) = ∧ U ∩ d − ( ∧ V ⊗
1) and ( ∧ U ) q +1 = ∧ U ∩ d − ( ∧ V ⊗ ( ∧ U ) q ) . Thus H [ k ] ( ∧ V ⊗ ∧ U ) = lim −→ q H [ k ] ( ∧ V ⊗ ( ∧ U ) q ) . But the differential in the quotients ∧ V ⊗ (( ∧ U ) q +1 / ( ∧ U ) q ) is just d ⊗ id , and it followsthat H [2] ( ∧ V ⊗ ∧ U ) = 0.It remains to show that H [ k ] ( ∧ Z ) = 0, k ≥
3. Suppose Φ ∈ ∧ k Z is a cycle. Thereis then a sequence z , . . . , z r of elements in Z such that dz = 0, dz i +1 ∈ ∧ ( z , . . . , z i ),Φ ∈ ∧ ( z , . . . , z r ) and such that division by z , . . . , z r maps Φ to zero. When r = 1, Φ = z k is a boundary. We use induction on r and on k to show that Φ is a boundary.Observe first that division by z gives a quadratic Sullivan algebra ( ∧ Z ′ , d ′ ). By whatwe just proved, H [2] ( ∧ Z ′ , d ′ ) = 0. Thus by induction on r , the image of Φ in ∧ Z ′ is aboundary. It follows that for some Φ ′ ∈ ∧ k − Z ,Φ − d (1 ⊗ Φ ′ ) = z ⊗ Φ ′′ , with Φ ′′ ∈ ∧ Z . In particular, in ∧ Z ′ , Φ ′′ is a d ′ -cycle. By induction on k in ∧ Z ′ , Φ ′′ = d ′ Ωfor some Ω ∈ ∧ k − Z ′ . Therefore Φ ′′ = d Ω + z ⊗ Ψ for some Ψ.If deg z is odd then z ⊗ Φ ′′ = d ( − z ⊗ Ω), and so Φ is a boundary. If deg z is eventhen, since H [2] ( ∧ Z ) = 0, we may choose z so dz = z . Division by z and z then givesa quasi-isomorphism ∧ Z ∼ = −→ (cid:0) ( ∧ z ) /z (cid:1) ⊗ ∧ Z ′′ and the same argument as above showsthat Φ is a boundary. Step Two. Completion of the proof of Lemma 9.
We define a sequence of surjective morphisms ∧ V = ∧ V (1) ։ ∧ V (2) ։ · · · ։ ∧ V ( p ) ։ . . . such that ρ factors through each to yield morphisms ρ ( p ) : ∧ V ( p ) ։ ∧ Z. In fact, if ρ ( p ) has been defined let ϕ p : ∧ V ( p ) ։ ∧ V ( p + 1) be obtained by division by V ( p ) ∩ ker ρ ( p ) . Then the kernels of the surjections ∧ V → ∧ V ( p ) form an increasing sequence of sub-spaces K ( p ) ⊂ V . Set K = ∪ p K ( p ) and let ϕ : ∧ V → ∧ W
31e the surjection obtained by dividing V by K . Then ϕ = lim −→ p ϕ p , and so ϕ and ϕ aresurjective. Thus by Step One H [ k ] ( ∧ W ) = 0, k ≥ ρ factors as ∧ V ϕ → ∧ W γ → ∧ Z. Moreover, ∧ W = lim −→ p ∧ V ( p ) and so W = lim −→ p V ( p ) . Let I ( p ) be the image of V ( p ) in V ( p + 1) . Then also W = lim −→ p I ( p ) . Thus by construction, γ : W ∼ = → Z . Now Lemma 8(ii) asserts that γ is an isomorphism,and hence H [ k ] ( ∧ Z ) = H [ k ] ( ∧ W ) = 0 , k ≥ . (cid:3) This is in two steps.
Step One. If L = L (2) ⊕ T is extendable then H ( ∧ V ) = Q ⊕ ( V ∩ ker d ).In this case, by adjoining additional variables construct an inclusion λ : ∧ V → ∧ W ofquadratic Sullivan algebras for which V ∼ = −→ W and H [2] ( ∧ W ) = 0 . Dualizing V → W gives a surjection L ρ ← L W of complete enriched Lie algebras. Moreover,since V ∼ = → W , L W = L (2) W ⊕ T W where T W is closed and ρ T : T ∼ = ←− T W . Here both ρ T and its inverse, σ , are coherent and so σ extends to a morphism ϕ : L → L W . Since ρ ◦ ϕ | T = id | T it follows that ρ ◦ ϕ = id L . Now dualizing, ϕ , gives a morphism ψ : ∧ V ← ∧ W such that ψ ◦ λ = id .On the other hand, the inclusion λ defines a Λ-extension ∧ V ⊗ ∧ Z ∼ = −→ ∧ W, in which, if Z = 0 then some z ∈ Z satisfies dz ∈ ∧ V . Thus dψz = ψdz = dz. Hence ψz − z is a cycle in W . This gives ψz − z ∈ W = V z ∈ V . This contradicts z ∈ Z and it follows that Z = 0 and ψ is an isomorphisminverse to λ . In particular H [2] ( ∧ V ) = 0 and now Lemma 9 implies that H ( ∧ V ) = Q ⊕ ( V ∩ ker d ) . Step Two. If H ( ∧ V ) = Q ⊕ ( V ∩ ker d ) then any decomposition L = L (2) ⊕ T with T closedis extendable.Suppose f : T → E is a coherent linear map into a complete enriched Lie algebra.Since f is coherent by Lemma 6 it is the dual of a linear map g : V ← W, where ∧ W is the quadratic model of E . Now recall from Lemma 6 that since T is closed, V = V ⊕ C with C = { v ∈ V | < v, sT > = 0 } . We shall construct a morphism ϕ : ∧ V ← ∧ W of quadratic Sullivan algebras such that ϕw − gw ∈ C .In fact, extend g to a morphism ∧ V ← ∧ W and then recall that W = ∪ k W k with W k +1 = W ∩ d − ( ∧ W k ). Suppose that ϕ has been constructed in ∧ W k . If w ∈ W k +1 then ϕdw is a cycle in ∧ V and therefore there is a unique v ∈ C such that dv = ϕdw .Extend ϕ to W k +1 by setting ϕw = v + g ( w ) . The uniqueness of v implies that this coincides with the construction of ϕ in W k , and sothis inductive process constructs the morphism ϕ .Finally, let γ : L → L W be the morphism determined by ϕ . Then for x ∈ T and w ∈ W , < w, s ( γx ) > = < ϕw, sx > = < v + gw, sx > . Since < C, sT > = 0 and v ∈ V we have < w, s ( γx ) > = < gw, sx > = < w, sf ( x ) > . Thus γ extends f .This completes the proof of Step Two, and with it of Theorem 1. (cid:3) In analogy with properties of the free Lie algebras, we have from Theorem 1:
Corollary 1.
A complete enriched Lie algebra L admits a surjective morphism of enrichedLie algebras E ։ L in which E is profree. proof. Let ∧ V be the quadratic model of L , and adjoin variables to V to obtain an inclusion ∧ V → ∧ W of quadratic Sullivan algebras such that V ∼ = → W and H [2] ( ∧ W ) = 0. Then33ualize to obtain a surjective morphism E ։ L from the homotopy Lie algebra of W ,which, by Lemma 9 and Theorem 1, is profree. (cid:3) Corollary 2. If ψ : ( L, I ) → ( F, G ) is a surjective morphism from a complete enriched Liealgebra to a profree Lie algebra then there is a morphism σ : ( F, G ) → ( L, I )such that ψ ◦ σ = id F . proof. Let ϕ : ∧ V F → ∧ V be the injective morphism of the corresponding quadratic modelswhich dualizes to ψ . Write V = ϕ ( V F ) ⊕ S . Inverting ϕ then defines an isomorphism γ from the sub quadratic algebra ∧ ϕ ( F ) to ∧ V F and clearly γ ◦ ϕ = id . It remains to extend γ to S so that γ ( dw ) = dγ ( w ), w ∈ S .For this set γ = 0 in S ∩ V . Now suppose γ has been defined in S ∩ V n and let { z i } bea basis of a direct summand of S ∩ V n in S ∩ V n +1 . Then γ ( dz i ) is a cycle in ∧ V F . Hence γ ( dz i ) = dw i for some w i ∈ V F . Extend γ by setting γz i = w i . (cid:3) Now recall from [8, Chap 9] that the category of a minimal Sullivan algebra, ∧ V , isthe least p (or ∞ ) such that ∧ V is a homotopy retract of ∧ V / ∧ >p V . Proposition 12. (i) A complete enriched Lie algebra is profree if and only if its quadraticmodel satisfies cat ( ∧ V ) = 1 .(ii) Any closed sub Lie algebra of a profree Lie algebra is profree. proof. (i) The condition cat( ∧ V ) = 1 for a quadratic Sullivan algebra is equivalent to thecondition Q ⊕ ( V ∩ ker d ) ≃ → ∧ V . Thus (i) follows from Theorem 1.(ii) Suppose ∧ V → ∧ Z is the morphism of quadratic models corresponding to aninclusion E → L of a closed sub Lie algebra in a complete enriched Lie algebra. Then [8,Theorem 9.3] gives cat( ∧ Z ) ≤ cat( ∧ V ). Thus (ii) follows from (i). (cid:3) Proposition 13. (i) Suppose E = R ⊕ E is a free graded Lie algebra. If dim R < ∞ and E is equipped with the unique (Proposition 4) enriched structure then E is profree.(ii) If L = T ⊕ L (2) is a decomposition of a profree Lie algebra in which T is closedthen the sub Lie algebra generated by T is a free Lie algebra. proof. (i) By Proposition 2, E = R ⊕ E (2) . Moreover any linear map from R into a completeenriched Lie algebra, F , is coherent, and therefore extends to a morphism E → F . Thistherefore extends to a morphism E → F , and so by definition, E is profree.(ii) It is sufficient to show that any finite dimensional subspace R of T generates a freeLie algebra. It follows from Lemma 6 that T = R ⊕ Q in which Q is also closed. Thendivision by Q is a surjection T → R and this is a coherent linear map.Now let E be the completion of the free Lie algebra generated by R with respect toits unique enriched structure. The surjection T → R extends to a morphism L → E which maps the sub Lie algebra generated by R in L onto the free Lie algebra generatedby R in E . It follows that the sub Lie algebra generated by R in L is free. (cid:3) roposition 14. Suppose L = T ⊕ L (2) is a profree Lie algebra. Then(i) If S ⊂ T is any closed subspace then the closure E of the sub Lie algebra E generatedby S satisfies E = S ⊕ E (2) , and E (2) = E ∩ L (2) .(ii) If = E is a solvable sub Lie algebra of the profree Lie algebra, L , then E is a freeLie algebra on a single generator. proof. (i) Since the Lie algebra E T generated by T is a free Lie algebra and E is a retractof E T , E is also a free Lie algebra. Choose a decomposition T = S ⊕ S ′ in which S ′ isalso closed. This gives a coherent retraction E T → E which then extends uniquely to aretraction ρ : L → E . In particular the identity of E (2) factors as E (2) → E ∩ L (2) ρ → E (2) , because ρ : L (2) → E (2) . Since ρ | E is injective, ρ : E ∩ L (2) → E (2) is injective, and itfollows that E (2) = E ∩ L (2) . This gives E = S ⊕ E (2) .(ii) Denote by E ⊃ · · · ⊃ E [ k ] ⊃ . . . the sequence of ideals defined by E [ k +1] =[ E [ k ] , E [ k ] ]. By hypotheses, some E [ n +1] = 0. Then, by induction on n , we may assume[ E, E ] is either zero or a free Lie algebra on a single generator. It follows that the closure, E ⊂ L satisfies dim E (2) ≤
2, since (Lemma 4) E (2) = E .But since E = 0, Proposition 13 gives that E = S ⊕ E (2) and S generates a free Liealgebra. This implies that E is the free Lie algebra generated by a single element. Inparticular dim E ≤ E = E . (cid:3) Remark.
Recall that any sub Lie algebra of a free Lie algebra is free ([16]). By Proposition12(i) the analogous statement for complete Lie algebras and closed sub algebras is alsotrue.
Proposition 15.
Let L be a profree Lie algebra.(i) When dim L/ [ L, L ] < ∞ , there is a graded vector space T , and L ∼ = b L ( T ) = lim ←− n L T / L nT . (ii) In the general case, L = lim ←− σ L σ , where the L σ are profree Lie algebras satisfyingdim L σ / [ L σ , L σ ] < ∞ . roof. (i). Since L/ [ L, L ] is finite dimensional, L ( k ) = L k and L ∼ = b L ( sV ) ∨ .(ii) The quadratic model, ∧ V , of a profree Lie algebra, L , decomposes as the directlimit ∧ V = lim −→ σ ∧ V ( σ )where the sub quadratic algebras ∧ V ( σ ) are characterized by the two conditionsdim V ( σ ) ∩ ker d < ∞ , and V ( σ ) ∩ ker d ∼ = −→ H ≥ ( ∧ V ( σ )) . This in turn yields the isomorphism L ≃ −→ lim ←− σ L ( σ )of the corresponding homotopy Lie algebras. (cid:3) With the notations of the proof of Proposition 15, denote by ρ σ : L → L ( σ ) the dualof the inclusion V ( σ ) → V . Lemma 10. If S ⊂ L is any subspace then S = lim ←− σ ρ σ ( S ) . proof. Set K ( σ ) = { v ∈ V ( σ ) | < v, sρ σ ( S ) > = 0 } , and K = lim −→ σ K ( σ ). It is straightfor-ward to check that K = { v ∈ V | < v, sS > = 0 } . It follows from Lemma 6 that S = image ( ( V /K ) ∨ → L ) = lim ←− σ image ( ( V ( σ ) /K ( σ )) ∨ → L ( σ ))= lim ←− σ ρ σ ( S ) . (cid:3) Definition.
A vector space S is called profinite if S is the projective limit S = lim ←− α S α of a family of finite dimensional vector spaces indexed by a directed set. Associate to S the enriched Lie algebra L S = lim ←− α,n L ( S α ) / L >n ( S α ) . We denote then by I α the kernel of the projection S → S α . Proposition 16.
The enriched Lie algebra L S is a profree Lie algebra. Conversely, if L = T ⊕ L (2) is a profree Lie algebra and T is closed, then L = L T . proof. By construction L S = S ⊕ L (2) S . Now let E = lim ←− β E β be a complete enrichedLie algebra and f : S → E be a coherent linear map. Denote by J β the kernel of theprojection p β : E → E β . Since f is coherent, for each β there is α with f ( I α ) ⊂ J β . Then p β ◦ f factors through S α , and since L β is nilpotent, we get a map L ( S α ) / L >n ( S α ) → E β for some integer n . By composition with the projection L S → L ( S α ) / L >n ( S α ) this givesa map f β : L S → E β . Finally, together the f β define a map f : L S → E .36he converse is Proposition 15(ii). (cid:3) Again, suppose that ∧ V is the quadratic model of a profree Lie algebra, L . Recallthat for some closed subspace T ⊂ L we have L = T ⊕ L (2) , so that T generates a free Liealgebra, E , with E ⊂ L (2) . In particular E = ⊕ k ≥ T ( k ) , where T ( k ) is the linear span of the iterated Lie brackets of length k in elements of T .In general, however the subspaces T ( k ) may not be closed. Nevertheless we do have ananalogous structure for L . Proposition 17. If L is a profree Lie algebra then, with the notation above and of Propo-sition 15(ii),(i) L ( k ) = lim ←− σ L ( σ ) k = T ( k ) ⊕ L ( k +1) , k ≥ .(ii) For each k, ℓ , [ T ( k ) , T ( ℓ )] ⊂ T ( k + ℓ ) . In particular F := ⊕ k T ( k ) is a weighted sub Lie algebra of L .(iii) L ∼ = −→ lim ←− n L/J ( n ) , where J ( n ) = Q k ≥ n T ( k ) .(iv) L ( k ) /L ( k +1) ∼ = T ( k ) ∼ = lim ←− σ L ( σ ) k /L ( σ ) k +1 . proof. (i) The inclusions V ( σ ) ⊂ V dualize to surjections ρ σ : L → L ( σ ) and, as observedabove, since V = lim −→ σ V ( σ ), these induce an isomorphism L ∼ = −→ lim ←− σ L ( σ ) . Moreover, for each n , V n = lim −→ σ V ( σ ) n , and therefore for each n , V /V n = lim −→ σ V ( σ ) /V ( σ ) n . It follows from Lemma 6 that ρ σ : L ( n ) → L ( σ ) ( n ) is also surjective and that these definean isomorphism L ( n ) ∼ = −→ lim ←− σ L ( σ ) ( n ) . Next observe that since T is closed the inclusion of T in L is (Lemma 6) dual toa surjection of V onto a direct summand V ′ of V ∩ ker d in V . This then restricts tosurjections of V ( σ ) onto direct summand of V ( σ ) ∩ ker d in V ( σ ). These in turn dualizeto inclusions T ( σ ) → L ( σ ) into direct summands of L ( σ ) (2) in L ( σ ). By construction each ρ ( σ ) : T → T ( σ ) is surjective and these define an isomorphism: T ∼ = −→ lim ←− σ T ( σ ) .
37n the other hand, let T ( σ, k ) be the linear span of the iterated Lie brackets of length k in elements of T ( σ ). Since L ( σ ) is the closure of the free Lie algebra E ( σ ) generated by T ( σ ), the fact that dim T ( σ ) < ∞ implies (Proposition 2) that for each n , L ( σ ) ( n ) = lim ←− k E ( σ ) n /E ( σ ) k + n = T ( σ, n ) ⊕ L ( σ ) ( n +1) . Since ρ σ : T → T σ is surjective each ρ σ : T ( k ) → T ( σ, k ) is also surjective. Thus Lemma10 gives T ( n ) ∼ = −→ lim ←− σ T ( n, σ ) . Finally, it follows from Lemma 6 that L ( n ) /L ( n +1) → lim ←− σ L ( σ ) ( n ) /L ( σ ) ( n +1) is the dual of the isomorphism V n +1 /V n ∼ = lim −→ σ V ( σ ) n +1 /V ( σ ) n . Thus the commutativediagram T ( n ) (cid:15) (cid:15) ∼ = / / lim ←− σ T ( σ, n ) ∼ = (cid:15) (cid:15) L ( n ) /L ( n +1) ∼ = / / lim ←− σ L ( σ ) n ) /L ( σ ) ( n +1) implies that T ( n ) ⊕ L ( n +1) = L ( n ) .(ii) This is immediate from the fact that [ T ( k ) , T ( ℓ )] ⊂ T ( k + ℓ ) and the fact that theLie bracket preserves closures.(iii) This is immediate from the relation J ( n ) = lim ←− σ,k ≥ n T ( σ, k ) = lim ←− σ L ( σ ) ( n ) = L ( n ) where the first equality again follows from Lemma 6.(iv) The first isomorphism is a consequence of (i). The second isomorphism followsfrom Lemma 10. (cid:3) The next Proposition explains the relation between L and L (2) for general profree Liealgebras. Proposition 18. If L = L is a profree Lie algebra and dim L/ [ L, L ] = ∞ , then L = L (2) . proof. Indeed, let ( ∧ W, d ) be the quadratic model of L . We can decompose W as an union W = ∪ n W n with W = W ∩ ker d , and for n > W n = d − ( ∧ W n − ). We denote by Z n a direct summand of W n − in W n . Then by hypothesis W is infinite dimensional andthere is a quasi-isomorphism ϕ : ( ∧ W, d ) → ( Q ⊕ W ,
0) that is the identity on W andthat maps each Z n to 0.For sake of simplicity, we write V = W and Z = Z . By construction the isomorphism L = ( sW ) ∨ induces isomorphisms L/L (2) ∼ = ( sV ) ∨ , and L (2) /L (3) ∼ = ( sZ ) ∨ . W countable and denote by w , w , . . . a basis of V . Since d = 0 on V , d : Z → ∧ V is an isomorphism. We denote by w ij , i < j the basis of Z defined by d ( w ij ) = w i ∧ w j . Denote by E teh vector space of column matrices X = ( x i ) with only afinite number of nonzero a i . Then the map ( a i ) P a i w i defines an isomorphism E ∼ = → V .Let represent an element ϕ ∈ Z ∨ by the infinite dimensional antisymmetric matrix M ϕ , ( M ϕ ) ij = ϕ ( w ij ) . In a similar process, an element f ∈ V ∨ can be represented by a column matrix A f , with( A f ) i = f ( w i ). The vector space ker f can then be identified with the sub vector spaceof E formed by the column matrices X satisfying A tf · X = 0. (Here A t denotes the linematrix transposition of a column matrix A .)Note that when we have two column matrices A and B , we can form the antisymmetricmatrix A · B t − B · A t . Now remark that for f, g ∈ L , we have M [ f,g ] = A f · B tg − B g · A tf . Let ϕ ∈ Z ∨ be the particular element defined for i < j by ϕ ( w ij ) = (cid:26) w ij = w k +1 , k +2 , for some k ϕ can be extended to all of W , by ϕ ( V ) = 0 and ϕ ( Z n ) = 0, for n > ϕ ∈ L (2) . The associated matrix is M = B . . . B . . . B . . .. . . . . . . . . . . . with B = (cid:18) − (cid:19) . Now consider a finite sum P ni =1 [ f i , g i ], with f i and g i ∈ V ∨ . The associated matrix is P ni =1 M [ f i ,g i ] . Then K = ( ∩ ni =1 ker f i ) ∩ ( ∩ ni =1 ker g i )is infinite dimensional, and so for some non-zero X ∈ Z ∨ , ( P M [ f i ,g i ] ) · X = 0. Since M · X = 0, it follows that L ⊂ = L (2) .In the general case, let E ⊂ V be a countable subvector space and let ∧ T be theminimal model of ( Q ⊕ E, L T is a retract of L , L T j ) ) L. ρ j j Let ϕ ∈ L (2) T , not in L T .Then j ( ϕ ) ∈ L (2) and not in L because otherwise ϕ = ρ ( ϕ )would belong to L T . (cid:3) art III Topological spaces, Sullivan models, andtheir homotopy Lie algebras
Enriched Lie algebras effectively describe the homotopy groups, π ∗ ( X Q ) and π ∗ < ∧ V > for any connected space, X , and minimal Sullivan algebra, ∧ V . (Recall that if ∧ V is theminimal Sullivan model of X , then X Q = < ∧ V > . Recall also that for simplicity we write H ( X ) to mean H ∗ ( X ; Q ))To make this description explicit, we fix a minimal Sullivan algebra, ( ∧ V, d ) . Recallfrom [8, Chap 2] that the homotopy Lie algebra of ( ∧ V, d ) is the graded Lie algebra, L = L ≥ , defined by: sL = V ∨ and < v, s [ x, y ] > = ( − deg y +1 < d v, sx, sy >, where d v is the component of dv in ∧ V . Now observe that ( ∧ V, d ) determines a naturalenriched structure in L : it is immediate from the defining condition for d that V isthe union of the finite dimensional subspaces V α for which ∧ V α is preserved by d . Theinclusions V α → V then dualize to surjections L → L/I α = L α onto finite dimensional nilpotent Lie algebras, and this yields an enriched structure ( L, { I α } )for L .On the other hand, the linear map d : V → ∧ V extends to a derivation in ∧ V ,and ( ∧ V, d ) is a quadratic Sullivan algebra: the quadratic Sullivan algebra associated with ( ∧ V, d ). It is immediate that ( L, { I α } ) is the homotopy Lie algebra for ( ∧ V, d ), introducedin §
4. In particular, it is complete.
Remark.
The choice of generating space V can be modified by a map, v v + σ ( v )in which σ : V → ∧ ≥ V , without changing the associated quadratic Sullivan algebra( ∧ V, d ) or the enriched Lie algebra ( L, { I α } ), although it may replace the ideals I α by anequivalent set of ideals.Now consider the adjoint bijectionsCdga( ∧ V, A
P L ( X )) = Simpl( X, < ∧ V > ) (9)for any connected space X . In particular, adjoint to id < ∧ V > is a morphism ϕ ∧ V : ∧ V → A P L < ∧ V > .
Then adjoint to any morphism ϕ : ∧ V → A P L ( X ) from a minimal Sullivan algebra ∧ V isa simplicial map < ϕ > : X → < ∧ V > .
40 straightforward check from the definitions then shows that ϕ decomposes as the com-posite ϕ : ∧ V ϕ ∧ V / / A P L < ∧ V > A PL <ϕ> / / A P L ( X ) . (10)There follows Lemma 11.
Suppose ϕ : ∧ V → A P L ( X ) is a morphism from a minimal Sullivan algebra.If A P L < ϕ > is a quasi-isomorphism then ϕ ∧ V is a quasi-isomorphism if and only if ϕ is the minimal Sullivan model of X . Next observe that (9) provides bijections, linear for n ≥ π n < ∧ V > = [ ∧ V, A
P L ( S n )] = ( V n ) ∨ , n ≥ . Since ∧ V is minimal, the retraction ξ : ∧ ≥ V → V with kernel ∧ ≥ V induces a linear map H ( ξ ) : H ≥ ( ∧ V ) → V . It is immediate from the definitions that for x ∈ V ∨ = π ∗ < ∧ V > and [Φ] ∈ H ( ∧ V ), we have < H ( ξ )[Φ] , x > = < H ( ϕ ∧ V )[Φ] , hur x > (11)where hur: π ∗ < ∧ V > → H ∗ < ∧ V > denotes the Hurewicz homomorphism.Since ( V n ) ∨ = L n − , the bijection π n < ∧ V > = ( V n ) ∨ also identifies π ∗ < ∧ V > asthe suspension sL of L . With this identification, the enriched Lie algebra structure of L provides explicit formulas for the product in π < ∧ V > , the action of this group on π n < ∧ V >, n ≥
2, and for the Whitehead products.The formulas involving π depend on a functor L G L ⊂ U L from complete enrichedLie algebras to groups, together with a natural bijectionexp : L ∼ = −→ G L . These are defined as follows. Since each L α is finite dimensional, it is a classical fact ([15],[14], [8, Chap 2]) that the standard power series is a bijection,exp : ( L α ) ∼ = −→ G L α onto the group of group-like elements in [ U L α .Passing to inverse limits yields a functor L G L ⊂ U L from complete enriched Liealgebras to groups, and the natural bijectionexp : L ∼ = −→ G L . Moreover, the Baker-Campbell-Hausdorff series in L is convergent, and provides an ex-plicit formula for the product.Next recall (Corollary 2 to Proposition 2) that L is the direct limit of pronilpotentsub Lie algebras L ( σ ) satisfying dim L ( σ ) /L ( σ ) < ∞ . It follows that G L is the directlimit of the corresponding groups G ( σ ), and that exp is the direct limit of the bijectionexp( σ ). 41ow denote by G n the subgroup of a group G generated by iterated commutators oflength n . The direct limits above then induce bijectionslim −→ σ L ( σ ) n ∼ = −→ L n and lim −→ σ G ( σ ) n ∼ = −→ G n ( σ ) . Moreover [8, Sec. 2.4] shows that exp restricts to a family of bijections L ( σ ) n ∼ = −→ G ( σ ) n .Therefore exp restricts to bijectionsexp : L n ∼ = −→ G nL , n ≥ . Moreover, again by [8, Sec. 2.4] for each σ the bijections L ( σ ) n ∼ = → G ( σ ) n factor to givebijections, L ( σ ) n /L ( σ ) n + k ∼ = −→ G ( σ ) n /G ( σ ) n + k , k ≥ , which are linear isomorphisms when k = 1.Since the direct limit of quotients is the quotient of the direct limits these definebijections, which are linear when k = 1, L n /L n + k ∼ = −→ G nL /G n + kL , k ≥ . In particular each G nL /G n +1 L is a rational vector space. Finally, and in the same way, thediagrams L q /L q +10 ⊗ L p /L p +10 / / ∼ = (cid:15) (cid:15) L p + q /L p + q +10 ∼ = (cid:15) (cid:15) G qL /G q +1 L ⊗ G pL /G p +1 L / / G p + qL /G p + q +1 L , induced by the respective adjoint actions, commute.Moreover, all these relations translate immediately to π < ∧ V > as follows: Theorem2.4 in [8] provides natural isomorphisms G L α ∼ = π ( ∧ V α ) of groups. Passing to inverselimits yields the natural isomorphism G L ∼ = π < ∧ V > .
Then all the other relations translate simply by replacing G L by π < ∧ V > .Thus Theorem 2.5 in [8] shows that the right action of π < ∧ V > on π n < ∧ V > , n ≥
2, is given by β · α = [ β, α ] , α ∈ L , β ∈ L n , n ≥ , while, for k, ℓ ≥
2, the Whitehead products π k < ∧ V > × π ℓ < ∧ V > → π k + ℓ − < ∧ V > also translate the Lie bracket in L .Finally, suppose ϕ : ∧ V → ∧ W is a morphism of minimal Sullivan algebras. Filteringby wedge degree yields a morphism ϕ : ( ∧ W, d ) → ( ∧ V, d ) of the associated quadraticalgebras, whose restriction to W dualizes to a morphism L ϕ : L V ← L W π , its action on π n , n ≥ Lemma 12.
Suppose ϕ, ψ : ∧ V → ∧ W are two morphisms. Then the following conditionsare equivalent:(i) ϕ and ψ are based homotopic.(ii) ϕ = ψ .(iii) L ϕ = L ψ . Let ϕ X : ∧ V ≃ → A P L ( X ) be a minimal Sullivan model for a connected space, X , and let L X denote the homotopy Lie algebra of ∧ V . The adjoint map < ϕ X > : X → < ∧ V > := X Q is called the Sullivan completion of X . The discussion in § π ∗ ( X Q ) = sL X and shows how the Lie bracket and enriched structure in L X determine the product in π ( X Q ), its action in π ≥ ( X Q ), and the higher order Whitehead products.Thus while H ( X ) and π ∗ ( X Q ) are directly computable from ∧ V , H ( X Q ) may not besuh a simple invariant of the model. For example, H (( S ∨ S ) Q ) is uncountably infinite([10]). This leads to the Definition.
If a Sullivan completion < ϕ X > : X → X Q satisfies H ( < ϕ X > ) : H ( X Q ) ∼ = −→ H ( X )then < ϕ X > is a Sullivan rationalization of X .If < ϕ X > is a homotopy equivalence, then X is Sullivan rational . Remarks. 1.
The condition that < ϕ X > be a Sullivan rationalization implies that theminimal Sullivan model of X directly computes both H ( X Q ) and π ∗ ( X Q ). If X Q is Sullivan rational then clearly A P L < ϕ X > is a quasi-isomorphism and < ϕ X > is a Sullivan rationalization. Conversely, if < ϕ X > is a Sullivan rationalizationthen by Lemma 11 it identifies the minimal Sullivan model ∧ V of X with a minimal modelof < ∧ V > = X Q . Thus X Q ≃ ( X Q ) Q and X Q is Sullivan rational.Recall that([14]) Quillen’s rationalization for simply connected spaces assigns to eachsuch space a map Y → Q ( Y ) which induces isomorphisms H ≥ ( Y ) ⊗ Q ∼ = → H ≥ ( Q ( Y )) and π ∗ ( Y ) ⊗ Q ∼ = → π ∗ ( Q ( Y )) . In particular, Q ( Q Y ) = Q ( Y ). 43ullivan’s completion, X Q , is analogous to Quillen’s rationalization, and also to theBousfield-Kan completion Q ∞ ( X ) - cf [3]. Moreover, if H ( X ) has finite type then Q ∞ ( X )and X Q are homotopy equivalent ([2]). But it is not always true that every X Q is Sullivanrational. For instance ([10]), H (( S ∨ S ) Q ) has uncountable dimension, where as if X Q is Sullivan rational by the Corollary to Theorem 2 below, H ( X Q ) has finite type. Remarks 1. If ϕ X : ∧ V → A P L ( X ) is a minimal Sullivan model then X is Sullivanrational if and only if each π k < ϕ X > : ( V k ) ∨ ∼ = −→ π k ( X ) . Let Y be a Sullivan rational space and let X be a connected space. Since f Q ◦ ϕ X = ϕ Y ◦ f , and ϕ Y is a homotopy equivalence, it follows that f f Q yields an injection[ X, Y ] → [ X Q , Y Q ] . The next Theorem extends results in [8, Chap.7].
Theorem 2.
Suppose X is a connected space with fundamental group G X and universalcovering space e X . Then the following conditions are equivalent:(i) X is Sullivan rational.(ii) BG X and e X are Sullivan rational, and G X acts locally nilpotently in H ( e X ) . Corollary.
If a connected space X is Sullivan rational, then(i) Each π ( X ) n /π ( X ) n +1 is a finite dimensional rational vector space.(ii) π ( X ) = lim ←− n π ( X ) /π ( X ) n .(iii) Each π k ( X ) , k ≥ , is a rational vector space and a finite dimensional nilpotent π ( X ) -module.(iv) H ( X ) is a graded vector space of finite type. Before proceeding to the proof of Theorem 2 and of its Corollary, we establish twoLemmas.
Lemma 13.
Suppose for some minimal Sullivan algebra, ∧ V , that ϕ ∧ V : ∧ V → A P L < ∧ V > is a quasi-isomorphism. Then(i) < ∧ V > is Sullivan rational.(ii) A basis of V is at most countable.(iii) H ( ∧ V ) is a graded vector space of finite type. proof. (i) Since < ϕ ∧ V > = id < ∧ V > , < ϕ ∧ V > is a homotopy equivalence and < ∧ V > isSullivan rational by definition. 44ii) Formula (11) identifies the surjection V ∨ → ( V ∩ ker d ) ∨ as a composite V ∨ hur / / H ∗ < ∧ V > / / ( V ∩ ker d ) ∨ . Thus H ∗ < ∧ V > → ( V ∩ ker d ) ∨ is surjective. Dualizing gives injections ( V n ∩ ker d ) ∨∨ → H n ( ∧ V ) . Now assume by induction that V
Suppose ∧ V is a minimal Sullivan algebra. If V = 0 then the followingconditions are equivalent:(i) ϕ ∧ V : ∧ V → A P L < ∧ V > is a quasi-isomorphism.(ii) Each dim H k ( ∧ V ) < ∞ . (iii) Each dim V k < ∞ .(iv) Each dim π k < ∧ V >< ∞ . proof. (i) ⇒ (ii). This is established in Lemma 13.(ii) ⇒ (iii). This is immediate because, since V = 0, d : V k → ∧ ≥ V
2. Thus the hypotheses of [8, Theorem 7.8(i)] are satisfied. Butthe proof of that Theorem shows that ϕ B : ∧ V → A P L < ∧ V > is a quasi-isomorphism. Since < ϕ B > is also a homotopy equivalence it follows that BG X is Sullivan rational.Finally [8, Corollary 4.3] implies that the holonomy representation of BG X in H ( e X )is nilpotent.(ii) ⇒ (i). Again consider diagram (12). As observed in the proof of [8, Theorem 7.1], ϕ B extends to a minimal Sullivan model of the form ∧ V ⊗∧ Z ≥ ≃ −→ A P L ( BG X ). Since BG X is Sullivan rational it follows that ( V ⊗ Z ≥ ) ∨ ∼ = π ( BG X ), and so Z ≥ = 0. Therefore46 B : ∧ V ∼ = −→ A P L ( BG X ). This, together with the hypothesis on the action of G X on H ( e X ) allows us to apply [8, Theorem 5.1] and conclude that e ϕ is a quasi-isomorphism.But by hypothesis, e X is Sullivan rational. Therefore < e ϕ > is a homotopy equivalence.This in turn implies that < ϕ X > is a homotopy equivalence and so X is Sullivan rational. (cid:3) proof of the Corollary to Theorem 2. Let L be the homotopy Lie algebra of ∧ V . Then § L / [ L , L ] ∼ = G X / [ G X , G X ] = H ( BG X ; Z ) . In particular, H ( G X ; Z ) is a rational vector space.On the other hand, Lemma 13 implies that H ( ∧ V ) is a finite dimensional vectorspace and therefore that dim H ( BG X ) = dim L / [ L , L ] < ∞ .In particular, we obtain that dim L /L (2)0 < ∞ . It follows that L is pronilpotent and L ( n )0 = L n , n ≥
1. Thus we may combine Lemma 5 with § G X /G n +2 X ∼ = L /L ( n +2)0 ∼ = ( V n ) ∨ where V = V ∩ ker d and V n +1 = d − ( ∧ V n ). It follows that G X = ( V ) ∨ = lim ←− n ( V n ) ∨ = lim ←− n G X /G n +2 X , and that each G kX /G k +1 X is a finite dimensional rational vector space.This proves (i) and (ii), while (iii) is established directly in the Theorem. Moreover, if X is Sullivan rational then by definition X ≃ < ∧ V > , where ∧ V is its minimal Sullivanmodel. Thus (iv) follows from Lemma 13. (cid:3)
10 Towers of Lie algebras
Proposition 19.
Let L be the homotopy Lie algebra of a minimal algebra ∧ V . Then thefollowing are equivalent :(i) L is the possibly finite inverse limit of a tower of finite dimensional nilpotent Liealgebras L = lim ←− . . . L ( n ) → L ( n − → . . . L (0) → (ii) dim V is finite or countably infinite(iii) the dimension of H ( ∧ V ) is finite or countably infinite. proof. Denote by d the quadratic part of the differential d in ∧ V , d : V → ∧ V . Then,( ∧ V, d ) is the quadratic model of L .(i) = ⇒ (ii). Denote by ∧ V n the quadratic model of L ( n ). Then the quadratic model( ∧ V, d ) of L satisfies ∧ V = ∪ n ∧ V n . Since each V n is finite dimensional, dim V is finiteor countably infinite. 47ii) = ⇒ (iii). This follows because in that case ∧ V is countably infinite and, as vectorspaces we have an injection H ( ∧ V ) ⊂ ∧ V .(iii) = ⇒ (i). Write V as the increasing union of finite dimensional subspaces W n . Thendefine subspaces V n ⊂ W n inductively by setting V n = { v ∈ W n | dv ∈ ∧ V n − } . A straightforward argument shows that V = ∪ V n . Evidently the homotopy Lie algebras L ( n ) of ∧ V n satisfy (i). (cid:3) Corollary.
Let X be a path connected space. Then L X is the projective limit of a towerof finite dimensional nilpotent Lie algebras if and only if H ∗ ( X ) is a finite type gradedvector space. proof. Recall that for any integer n , either H n ( X ) is finite dimensional or else uncountablyinfinite. (cid:3)
11 Wedge of spheres and rationally wedge-like spaces
Suppose { S n α } α ∈ S is a collection of spheres in which S is a linearly ordered set and each n α ≥
1. For each finite subset σ ⊂ S we write σ = { σ , . . . , σ r } with σ < · · · < σ r , andset | σ | = r . If σ ⊂ τ , so that | σ | ≤ | τ | = q , the inclusion defines an inclusion j σ,τ : S n σ ∨ · · · ∨ S n σr −→ S n τ ∨ · · · ∨ S n τq , and ∨ α S n α = lim −→ σ < ··· <σ r S n σ ∨ · · · ∨ S n σr . On the other hand, if σ ⊂ τ , collapsing the remaining spheres to the basepoint definesa retraction p τ,σ : S n τ ∨ · · · ∨ S n τq −→ S n σ ∨ · · · ∨ S n σr . Definition.
A connected space, Y , is rationally wedge-like if for some S as above Y ≃ −→ lim ←− σ < ··· <σ r ( S n σ ∨ · · · ∨ S n σr ) Q , where the maps ρ τ,σ defining the inverse system satisfy ρ τ,σ = ( p τ,σ ) Q . Remark.
If each sphere has dimension ≥
2, and if there are only finitely many in eachdimension, then Y is Sullivan rational. Proposition 20. (i) A minimal Sullivan algebra ∧ V with homotopy Lie algebra, L , isthe Sullivan model of a wedge of spheres if and only if L is profree and H ( ∧ V ) isthe dual of a graded vector space, H ∗ .(ii) A connected space Y is rationally wedge-like if and only if for some minimal Sullivanalgebra, ∧ V , Y ≃ < ∧ V > and the homotopy Lie algebra of the minimal Sullivanalgebra, ∧ V , is profree. Lemma 15.
The following conditions on a minimal Sullivan algebra, ( ∧ V, d ) , with V = 0 ,are equivalent:(i) There is a quasi-isomorphism ( ∧ V, d ) ≃ → Q ⊕ S with zero differential in S and S · S =0 .(ii) The generating space V can be chosen so that d : V → ∧ V and Q ⊕ ( V ∩ ker d ) ∼ = −→ H ( ∧ V, d ) . (iii) Q ⊕ V ∩ ker d ∼ = → H ( ∧ V, d ) .(iv) The homotopy Lie algebra, L , is profree.If they hold then ( ∧ V, d ) ∼ = ( ∧ V, d ) and, if V is chosen to satisfy (iii), then V ∩ ker d = V ∩ ker d . proof. (i) ⇔ (ii). First, let S = S ≥ be any graded vector space. Then Q ⊕ S , with zerodifferential and S · S = 0 is a cdga, and its minimal model has the form( ∧ V, d ) ≃ −→ ( Q ⊕ S, . On the other hand, a successive adjoining of variables w to S constructs a quadraticSullivan algebra ( ∧ W, d ) such that H [1] ( ∧ W, d ) = S and H [2] ( ∧ W ) = 0. (Here as inPart II, H [ k ] ( ∧ W, d ) denotes the homology classes represented by cycles in ∧ k W .) NowLemma 9 asserts that H [ k ] ( ∧ W ) = 0, k ≥
2. Thus division by ∧ ≥ W and by a directsummand of S in W defines a quasi-isomorphism( ∧ W, d ) ≃ → Q ⊕ S. Since minimal models are unique it follows that there is an isomorphism( ∧ W, d ) ∼ = ( ∧ V, d ) . Now choose V so this isomorphism takes W ∼ = → V .On the other hand, if (ii) holds, then with the given choice of V , division by ∧ ≥ V and by a direct summand of V ∩ ker d in V defines a quasi-isomorphism( ∧ V, d ) ≃ → Q ⊕ ( V ∩ ker d ) . This gives (i).(i) ⇔ (iii). If (i) holds, the observations above yield an isomorphism ( ∧ W, d ) ∼ =( ∧ V, d ). This induces an isomorphism ( ∧ W, d ) ∼ = → ( ∧ V, d ), which for appropriate choiceof V preserves wedge degree. It follows that with this choice ( ∧ V, d ) is quadratic and V ∩ ker d = V ∩ ker d . This proves (iii), and the same argument shows that (iii) ⇒ (ii).Finally, the assertion (iiii) ⇔ (iv) is Theorem 1. (cid:3) X , is formal if its minimal Sullivan model is also aSullivan model for ( H ( X ) , Corollary 1. If X is a connected space and H ≥ ( X ) = 0 then the homotopy Lie algebra L X is profree if and only if X is formal and H ≥ ( X ) · H ≥ ( X ) = 0. Corollary 2.
If a connected co-H-space, X , satisfies H ≥ ( X ) = 0 (in particular, if X isa wedge of spheres), then L X is profree. proof. Let U be a contractible open neighbourhood of a base point in X . Then X ∨ U and U ∨ X form an open cover for X ∨ X Since X is a co-H-space, the diagonal map ∆ : X → X × X factors up to homotopythrough X ∨ X , X ∆ ( ( PPPPPPPPPPPPP f / / X ∨ X (cid:15) (cid:15) X × X. The open sets f − ( X ∨ U ) and f − ( U ∨ X ) make then a covering of X by contractibleopen sets, and cat X = 1. It follows [8, Theorem 9.2] that the minimal Sullivan model, ∧ V, of X satisfies cat ( ∧ V ) = 1 . By Proposition 12, this implies the result. (cid:3) proof of Proposition 20 . (i) Suppose first that ∧ V is the minimal Sullivan model of awedge of spheres, X . Then Corollary 2 asserts that the homotopy Lie algebra, L , of ∧ V is profree. Moreover, H ( ∧ V ) = H ( X ) = H ∗ ( X ; Q ) ∨ is the dual of a graded vector space.In the reverse direction, since L is profree, by Lemma 15, there is a quasi-isomorphism ∧ V ≃ → Q ⊕ S . Moreover, by hypothesis S = H ≥ ( ∧ V ) is the dual of a graded vector space E . Let X = ∨ α S n α be a wedge of spheres satisfying H ≥ ( X ; Q ) = E . By Corollary 2 forthe minimal Sullivan model, ∧ W , of X , there is a quasi-isomorphism ∧ W ≃ → Q ⊕ H ≥ ( X ) = Q ⊕ E ∨ = Q ⊕ S. It follows that ∧ W ∼ = ∧ V .(ii) Suppose first that the homotopy Lie algebra, L , of a minimal Sullivan algebra, ∧ V ,is profree. By Lemma 15 there is a quasi-isomorphism ∧ V ≃ −→ Q ⊕ S, in which the differential in S is zero and S · S = 0. Fix a linearly ordered basis, { z α } α ∈ S of S and denote deg z α = n α . Then for each finite subset σ ⊂ S write σ = { σ , . . . , σ r } with σ < · · · < σ r and set | σ | = r .Now by induction on | σ | we construct quasi-isomorphisms λ σ : ∧ V ( σ ) ≃ −→ Q ⊕ ( ⊕ i Q z σ i ) , and for γ ⊂ τ ⊂ S , wedge degree preserving Sullivan representatives λ τ,γ : ∧ V ( γ ) → ∧ V ( τ )50or the inclusions ⊕ j Q z γ j → ⊕ i Q z τ i . These will be constructed so that λ τ,γ ◦ λ γ,ω = λ τ,ω and λ τ ◦ λ τ,γ = λ γ . In fact, suppose these have been constructed for all τ, γ such that | τ | , | γ | ≤ r . If | σ | = r + 1 we set ∧ V ( σ ) = lim −→ τ ⊂6 = σ ∧ V ( τ ) and λ σ = lim −→ τ ⊂6 = σ λ τ . Then set λ σ,τ to be the corresponding inclusion ∧ V ( τ ) → ∧ V ( σ ) . Finally, for any σ < · · · < σ r , ∧ V ( σ ) is the minimal Sullivan model of S n σ ∨ · · · ∨ S n σr and λ τ,σ is a Sullivan representative for the retraction p τ,σ corresponding to the inclusion σ ⊂ τ . All together this identifies ∧ V = lim −→ σ ∧ V ( σ ) and < ∧ V > = lim ←− < ∧ V ( σ ) > = lim ←− < S n σ ∨ · · · ∨ S n σr > Q . It follows that < ∧ V > is rationally wedge-like.In the reverse direction, suppose X is rationally wedge like, so that X ≃ lim ←− σ < ··· <σ k ( S n σ ∨ · · · ∨ S n σk ) Q . The explicit construction above then identifies X = < ∧ V > , where ∧ V is the minimalSullivan model of ∨ σ S n σ . (cid:3) Corollary. If X = ∨ σ S n σ is a wedge of spheres, then X Q is rationally wedge-like. If allthe spheres are circles then X Q is aspherical. Remark.
Rationally wedge-like spaces provide examples of minimal Sullivan algebras ∧ Z for which h∧ Z i is not the Sullivan completion of a space. For example, suppose Z = Z has a countably infinite basis, so that π ∗ h∧ Z i = π h∧ Z i = ( Z ) ∨ . But if ∧ V were theminimal model of a space X then we would have V ∼ = H ( X ) = H ( X ) ∨ and so eitherdim V < ∞ or card ( V ) ≥ card R .
12 Wedges and free products of enriched Lie algebras
The category of enriched Lie algebras has free products (to be constructed immediatelybelow). Here we consider the inclusion i : X ∨ Y → X × Y , and establish Proposition 21.
Suppose X and Y are connected spaces. Then(i) The homotopy Lie algebra L X ∨ Y of the wedge X ∨ Y is the free product L X ∨ Y = L X b ∐ L Y of the homotopy Lie algebras of X and Y . In particular, the correspondence X L X preserves coproducts. ii) If one of H ( X ) , H ( Y ) is a graded vector space of finite type, then the homotopyfibre, F , of the map i Q : ( X ∨ Y ) Q → ( X × Y ) Q is rationally wedge-like.(iii) If X Q and Y Q are aspherical then so are F and ( X ∨ Y ) Q . Remark.
This result is analogous to the fact that the usual fibre of the injection X ∨ Y → X × Y is the join of Ω X and Ω Y and thus a suspension. (But note that ( X ∨ Y ) Q maybe different from X Q ∨ Y Q .)The main step in the proof of Proposition 21 is the explicit description of the homotopyLie algebra of a fibre product ∧ W × Q ∧ Q of any two minimal Sullivan algebras. Afterdefining L X b ∐ L Y we establish this description in Proposition 22, and return to Proposition21 and its application to Sullivan completions.First, note that the classical construction of the free product, L ∐ L ′ , of two gradedLie algebras extends naturally to enriched Lie algebras, ( L, { I α } ) and ( L ′ , { I ′ β } ), in whichthe enriched structure is given by the surjections ξ α,β,n : L ∐ L ′ → L α ∐ L β / ( L α ∐ L β ) n . Its completion will be denoted by L b ∐ L ′ , and almost by definition, L b ∐ L ′ ∼ = −→ L b ∐ L ′ . Lemma 16.
With the notation above, any two morphisms f : ( L, { I α } ) → E and g :( L ′ , { I ′ β } ) → E into a complete enriched Lie algebra extend uniquely to a morphism L b ∐ L ′ → E. This characterizes L b ∐ L ′ up to natural isomorphism. proof. Let { J γ } denote the enriched structure for E . Then for each γ there are indices α ( γ ) , β ( γ ) such that f and g factor to yield morphisms f γ : L α ( γ ) → E γ and g γ : L ′ β ( γ ) → E γ . Moreover, since E n ( γ ) γ = 0, some n ( γ ), f γ ∐ g γ factors to yield a morphism h γ : ( L α ( γ ) ∐ L ′ β ( γ ) ) / ( L α ( γ ) ∐ L ′ β ( γ ) ) n −→ E γ . Since E is complete these extend to h := lim ←− γ h γ : L b ∐ L ′ → E. The uniqueness is immediate. (cid:3)
Now, consider minimal Sullivan algebras, ∧ W and ∧ Q . The natural surjection ∧ W ⊗∧ Q → ∧ W × Q ∧ Q is surjective in homology, and so the corresponding Λ-extension, ϕ : ∧ T := ∧ W ⊗ ∧ Q ⊗ ∧ R ≃ −→ ∧ W × Q ∧ Q,
52n which ϕ ( R ) = 0, is a minimal Sullivan model for ∧ W ⊕ Q ∧ U . The sequence, ∧ W ⊗ ∧ Q → ∧ W ⊗ ∧ Q ⊗ ∧ R → ∧ R then ( §
8) induces the short exact sequence0 ← L W × L Q ← L T ← L R ← Proposition 22.
With the hypotheses and notation above,(i) The surjection L T → L W × L Q factors as L T ∼ = → L W b ∐ L Q → L W × L Q . (ii) L R is a profree Lie algebra. In particular < ∧ R > is rationally wedge-like.(iii) When ∧ W and ∧ Q are quadratic Sullivan algebras, then, for appropriate choice of R , ∧ T is the quadratic model of L W b ∐ L Q .(iv) If f : L → L ′ and g : E → E ′ are surjections of complete enriched Lie algebras, then f b ∐ g : L b ∐ E → L ′ b ∐ E ′ is also surjective. Corollary. If L and L ′ are complete enriched Lie algebras then the kernel of L b ∐ L ′ → L × L ′ is profree. Moreover, if L and L ′ are both profree then L b ∐ L ′ is also profree. proof of the Corollary. For the first assertion apply Proposition 22 to the quadratic models ∧ W and ∧ W ′ of L and L ′ . If now L and L ′ are profree then by Theorem 1 there are quasi-isomorphisms ∧ W ≃ −→ Q ⊕ S and ∧ W ′ ≃ −→ Q ⊕ S ′ with S · S = 0 = S ′ · S ′ , and vanishing differentials in S and S ′ . Thus ∧ W × Q ∧ W ′ ≃ Q ⊕ S ⊕ S ′ , and its homotopy Lie algebra, L b ∐ L ′ , is profree by Theorem 1. (cid:3) proof of Proposition 22. (i) Denote by [ ∧ M, ∧ N ] ∗ the set of based homotopy classes ofmorphisms. Then for any minimal Sullivan algebra, ∧ V , composition with ϕ induces abijection ([8, Proposition 1.10]),[ ∧ V, ∧ T ] ∗ → [ ∧ V, ∧ W × Q ∧ Q ] ∗ , of based homotopy classes of morphisms. The surjections ∧ W × Q ∧ Q → ∧ W, ∧ Q thenidentify [ ∧ V, ∧ W × Q ∧ Q ] ∗ = [ ∧ V, ∧ W ] ∗ × [ ∧ V, ∧ Q ] ∗ . Now apply Lemma 16 to convertthese bijections to a bijection, C ( L T , L V ) ∼ = −→ C ( L W , L V ) × C ( L Q , L V ) , C ( − , − ) denotes the set of morphisms in the category of enriched Liealgebras. This identifies L T as L W b ∐ L Q and establishes (i).(ii) Let ∧ W ⊗ ∧ U W and ∧ Q ⊗ ∧ U Q denote the respective acyclic closures. Then ∧ R is quasi-isomorphic to ∧ T ⊗ ∧ W ⊗∧ Q ( ∧ W ⊗ ∧ U W ⊗ ∧ Q ⊗ ∧ U Q ) ≃ A := ( ∧ W × Q ∧ Q ) ⊗ ∧ U W ⊗ ∧ U Q . Now divide A by the ideal generated by W to obtain the short exact sequence0 → ∧ ≥ W ⊗ ∧ U W ⊗ ∧ U Q → A → ∧ Q ⊗ ∧ U W ⊗ ∧ U Q → . Next, decompose the differential in ∧ W ⊗ ∧ U W in the form d = d + d ′ with d ( W ) ⊂∧ W , d ( U W ) ⊂ W ⊗ ∧ U W , d ′ ( W ) ⊂ ∧ ≥ W and d ′ ( U W ) ⊂ ∧ ≥ W ⊗ ∧ U W . Then d is a differential and ( ∧ W ⊗ ∧ U W , d ) is the acyclic closure of ( ∧ W, d ). Choose a directsummand, S , of d ( ∧ ≥ U W ) in W ⊗ ∧ U W . Then I = ( ∧ ≥ W ⊗ ∧ U W ) ⊕ S is acyclic forthe differential d and therefore also for the differential d . Thus J = I ⊗ ∧ U Q is an acyclicideal in A and A ∼ = → A/J .On the other hand, consider the short exact sequence0 → d ( ∧ U W ) ⊗ ∧ U Q → A/J → ∧ Q ⊗ ∧ U W ⊗ ∧ U Q → d ( ∧ U W ) ⊗ ∧ U Q is an ideal with trivial multiplication and trivial differential. Itfollows from the long homology sequence that Q ⊕ d ( ∧ U W ) ⊗ ∧ ≥ U Q ≃ −→ A/J.
Thus (ii) follows from Lemma 16.(iii) Assign ∧ W and ∧ Q wedge degree as a second degree and assign U W and U Q second degree 0. Then ( ∧ W ⊗ ∧ U W , d ) and ( ∧ Q ⊗ ∧ U Q , d ) are the respective acyclicclosures of ( ∧ W, d ) and ( ∧ Q, d ), and d increases the second degree by 1. Now ϕ and T may be constructed so that R is equipped with a second gradation for which d increasesthe second degree by one and ϕ is bihomogeneous of degree zero.The argument in the proof of (ii) now yields a sequence of bihomogeneous quasi-isomorphisms connecting Q ⊕ (cid:0) d ( ∧ + U W ) ⊗ ∧ + U Q (cid:1) ≃ ∧ R. Thus H ≥ ( ∧ R ) is concentrated in second degree 1. Therefore R is concentrated in seconddegree 1 and so the second degree in R (and therefore in T ) is just the wedge degree. Thus ∧ T is quadratic and therefore is the quadratic model of L W b ∐ L Q .(iv) It is sufficient to consider the case g = id E : E = −→ E . In this case we have therow-exact commutative diagram0 / / K / / h (cid:15) (cid:15) L b ∐ E f b ∐ id E (cid:15) (cid:15) / / L × E / / f × id E (cid:15) (cid:15) / / K ′ / / L ′ b ∐ E / / L ′ × E / / , and so it is also sufficient to show that h is surjective.54ow let ∧ W , ∧ W ′ and ∧ Q be, respectively the quadratic models of L , L ′ and E , andlet ϕ : ∧ W ′ → ∧ W be the morphism corresponding to f . Then the restriction ϕ : W ′ → W dualizes to f , and so ϕ is injective. In particular, ϕ extends to an inclusion ϕ : ∧ W ′ ⊗ ∧ U W ′ → ∧ W ⊗ ∧ U W of the acyclic closures, restricting to an inclusion U W ′ → U W .On the other hand, the diagram above is the diagram of homotopy Lie algebras asso-ciated with a commutative diagram, ∧ W ⊗ ∧ Q / / ∧ W ⊗ ∧ Q ⊗ ∧ R / / ∧ R ∧ W ′ ⊗ ∧ Q ϕ ⊗ id O O / / ∧ W ′ ⊗ ∧ Q ⊗ ∧ R ′ O O / / ∧ R ′ , σ O O which identifies L σ with h .Moreover, the computations in the proof of (ii) identify H ( σ ) = ϕ : Q ⊕ d ( ∧ U W ′ ) → Q ⊕ d ( ∧ U W ) . It follows that H ( σ ) is injective. But since L R and L R ′ are profree, Lemma 5 identifies L R /L (2) R = H ≥ ( ∧ R ) ∨ and L R ′ /L (2) R ′ = H ≥ ( ∧ R ′ ) ∨ . It follows that L σ : L R → L R ′ induces a surjection L R /L (2) R → L R ′ /L (2) R ′ . Now Lemma 8(i) implies that L σ is surjective. (cid:3) proof of Proposition 21. Let ∧ W and ∧ Q be the minimal Sullivan models of X and Y .Then recall from Proposition 22 the quasi-isomorphism ∧ T := ∧ W ⊗ ∧ Q ⊗ ∧ R ≃ −→ ∧ W × Q ∧ Q. Since H ( X ∨ Y ) = H ( X ) × Q H ( Y ) it follows that A P L ( X ∨ Y ) ≃ → A P L ( X ) × Q A P L ( Y ).Therefore ∧ T is a minimal Sullivan model of X ∨ Y , and so (i) follows from Proposition22(i).(ii) If one of H ( X ), H ( Y ) has finite type then H ( X × Y ) = H ( X ) ⊗ H ( Y ). It followsthat the inclusion ∧ W ⊗ ∧ Q → ∧ T is a Sullivan representative for the inclusion i : X ∨ Y → X × Y . This identifies i Q as themap < ∧ T > → < ∧ W ⊗ ∧ Q > . But by [6, Proposition 17.9] the sequence < ∧ R > → < ∧ T > i Q −→ < ∧ W ⊗ ∧ Q > is a fibration. Thus (Proposition 22) the fibre, < ∧ R > of i Q is rationally wedge-like.(iii) When X Q and Y Q are aspherical then W = W and Q = Q ; Now it follows fromProposition 22(iii) that R = R and so F = < ∧ R > is aspherical. But since R = R , T = T and so < ∧ T > = ( X ∨ Y ) Q is also aspherical. (cid:3) Let E and E ′ be weighted enriched Lie algebras with enriched structures given by mor-phisms E → E α and E ′ → E ′ β . Denote by L and L ′ the completions of E and E ′ . Theweight decomposition in E and E ′ then extends in the standard way to E ∐ E ′ . Thecorresponding decomposition of the quadratic model for E ∐ E ′ is given as follows. Thedecomposition of the models ∧ V and ∧ V ′ induce one in ∧ V ⊕ Q ∧ V ′ which then in turndefines the decomposition in its minimal quadratic model. Proposition 23.
The completion of E ∐ E ′ is L b ∐ L ′ . proof. This follows directly from Proposition 22(iii). (cid:3) ppendix : Connected spaces and minimal Sulli-van models For the convenience of the reader we recall briefly some of the definitions and mainelements of Sullivan’s approach ([17]) to the rational homotopy theory of a connectedspace. A more detailed survey and proofs are provided respectively in [4] and [8]. (Hereby space we mean either a CW complex or a simplicial set; these categories are identified nythe inverse homotopy equivalences provided by the singular simplex and Milnor realizationfunctor ([12, § ∧ V, d ). These are the commutative gradeddifferential algebras (cdga’s), for which • ∧ V is the free graded commutative algebra on a graded vector space V = V ≥ , and • the differential is constrained by the condition V = ∪ n ≥ V n , where V = V ∩ ker d and V n +1 = V ∩ d − ( ∧ V n ).Here ∧ V = ⊕ p ≥ ∧ p V , where ∧ p V is the linear span of the monomials in V of length p . In particular, ( ∧ V, d ) is minimal if d : V → ∧ ≥ V . We frequently suppress thedifferential from the notation, and write ∧ V for ( ∧ V, d ). Then two cdga morphisms ϕ , ϕ : ∧ V → A are homotopic if for some morphism ϕ : ∧ V → A ⊗ ∧ ( t, dt ) with deg t = 0 theaugmentations t , ϕ to ϕ , ϕ ; ϕ and ϕ are based homotopic with respectto an augmentation ε A : A → Q if ϕ : V → ker ε A ⊗ ∧ ( t, dt ).We also recall [6, §
6] that a ∧ V -module, P , is a graded vector space equipped witha differential, d and a multiplication ∧ V ⊗ P → P compatible with the differential. The ∧ V -module is semifree if it has the form P = ∧ V ⊗ M in which the module structure issimply multiplication by ∧ V and, in addition P = ∪ k ≥ ∧ V ⊗ M k is the increasing unionof submodules in which the quotient differentials in ∧ V ⊗ M k +1 /M k are just d ⊗ id .Now, suppose ( A, ε A ) is an augmented cdga satisfying H ( A ) = Q . Then a Λ-extensionof ( A, ε A ) is a sequence of morphisms A λ / / A ⊗ ∧ Z ρ / / ∧ Z in which λ ( a ) = a ⊗ ρ = Q ⊗ A − , and the differential is constrained by the conditions: Z = ∪ n ≥ Z n with Z = Z ∩ d − ( A ⊗
1) and Z n +1 = Z ∩ d − ( A ⊗ ∧ Z n ). It is minimal ifthe quotient cdga, ( ∧ Z, d ) satisfies d : Z → ∧ ≥ Z .Every cdga, A , with H ( A ) = Q admits a quasi-isomorphism σ : ∧ V ≃ → A from aminimal Sullivan algebra; this is a minimal Sullivan model for A , and ∧ V is unique upto isomorphism. Similarly, if ϕ : A → B is a morphism of augmented cdga’s satisfying H = Q then ϕ extends to a quasi-isomorphism A ⊗ ∧ Z ≃ → B from a minimal Λ-extension. Again, A ⊗ ∧ Z is unique up to isomorphism.Now consider the case in which A is a minimal Sullivan algebra ∧ V , and H ( ϕ ) isinjective. In this case the quotient ( ∧ Z, d ) = ∧ V ⊗ ∧ Z ⊗ ∧ V Q will also be a minimal57ullivan algebra. Write ∧ V ⊗ Z = ∧ ( V ⊕ Z ) and note that the differential is the sum d = X n ≥ d n of derivations in which d k raises the wedge degree by k . In particular, ∧ V ⊗ ∧ Z is aminimal Sullivan algebra if and only if d = 0.In general, d is a linear map Z → V . Its dual is a linear map ∂ : L Z ← L V betweenthe respective homotopy Lie algebras L V = ( sV ) ∨ and L Z = ( sZ ) ∨ defined in [8, Chap 2]and recalled in §
8. Here we have the
Lemma A.1
With the hypotheses and notation above, the image of ∂ is contained in thecentre of L Z .proof. We may identify ∧ Z as a subspace of Z ⊗ Z . Then a single calculation gives < ( d ⊗ id ) d z, s∂x ⊗ sy > = ± < d z, s∂x ⊗ sy > . Since ( d ⊗ id ) d z is the component in V ⊗ Z of d (1 ⊗ z ) it follows that < d z, s∂x ⊗ sy > = 0.On the other hand, the Lie bracket in L Z is given by < d z, sy ⊗ sy > = ( − deg y < z , s [ y , y ] > . It follows that for x ∈ L V , y ∈ L S we have [ ∂x, y ] = 0. (cid:3) For the connection between Sullivan algebras and topology we identify spaces withsimplicial sets via the singular simplex functor, and refer to the objects in either categoryas ’spaces’. The connection between Sullivan algebras and topology is provided by thesimplicial commutative cochain algebra, A P L , defined as follows:( A P L ) n = ∧ ( t , . . . , t n , dt , . . . , dt n ) P t i − , P dt i . The simplicial faces ∂ i and degeneracies s j are defind by their restrictions to the t k ∂ i t k = t k if k < i k = it k − if k > i and s j t k = t k if k < jt k + t k +1 if k = jt k +1 if k > j. The algebra of polynomial forms on X is then defined by A nP L ( X ) = Simpl( X, A nP L ) , and there is a natural isomorphism H ( A P L ( X )) ∼ = H ∗ ( X ; Q ).A Sullivan minimal model of X is then a minimal model for A P L ( X ), i.e., a minimalSullivan algebra equipped with a quasi-isomorphism ϕ : ∧ V ≃ → A P L ( X ) . The minimal model exists always and is unique up to isomorphism.58imilarly, if f : X → Y is a map of pointed connected spaces then A P L ( f ) extends toa quasi-isomorphism A P L ( Y ) ⊗ ∧ Z ≃ → A P L ( X )from a minimal Λ-extension; here the quotient ( ∧ Z, d ) is the analogue of the homotopyfibre, F , of f . In fact, there is a natural morphism ( ∧ Z, d ) → A P L ( F ) which, with certainhypotheses is the minimal Sullivan model ([8, Theorem 5.1]).Sullivan ([17],[8]) also introduces the realization functor , A < A > , from cdga’s tosimplicial sets adjoint to A P L ( − ): < A > p = Cdga( A, ( A P L ) p ) , Since the realization functor converts tensor products to products, it follows that < A ⊗ B > = < A > × < B > .For a connected space X this yields a natural bijection,Cdga( A, A
P L ( X ) ∼ = Simpl( X, < A > ) , denoted ϕ < ϕ > . This extends, for Sullivan algebras, to a natural bijection of homotopyclasses [ ∧ V, A
P L ( X )] ∼ = [ X, < ∧ V > ] . Applied to a minimal model ϕ : ∧ V ≃ → A P L ( X ), this yields a natural map < ϕ > : X → X Q := < ∧ V >, the
Sullivan rationalization or completion of X . When X is simply connected and H ( X ) isa graded vector space of finite type, then ([17]) < ϕ > is a rational homotopy equivalence,and < ϕ > induces isomorphisms of graded vector spaces π n ( X ) ⊗ Q → π n ( X Q ) . The hypothesis on the H ( X ) is important as we see in the next Proposition Proposition.
When X is simply connected and H ( X ) is not finite type, then < ϕ > isnot a rational homotopy equivalence. proof . Since some Betti numbers are infinite, there is a smallest integer p with dim π p ( X ) ⊗ Q = ∞ . Suppose first that X is ( p − E = π p ( X ) ⊗ Q . Then byHurewicz theorem, H p ( X ; Q ) ∼ = E ∨ . Let ( ∧ W, d ) → A P L ( X ) be the minimal model of X ,then W p ∼ = H p ( X ; Q ) ∼ = E p . It follows that π p < ∧ W > is the bidual of E . Now since E is infinite, E and its bidual are not isomorphic, and this proves the result in this case.In the general situation, consider the Postnikov fibration X ( p ) i → X p → Y where π q ( Y ) = 0 for q ≥ p , π q ( p ) is an isomorphism for q < p and π q ( i ) is an isomorphismfor i ≥ p . By [8, Theorem 5.1], we have a commutative diagram in which ( ∧ V ⊗ ∧ W, d ) isa relative minimal model and the vertical arrows are quasi-isomorphisms, A P L ( Y ) / / A P L ( X ) / / A P L ( X ( p ) ) ∧ V / / ≃ O O ∧ V ⊗ ∧ W / / ≃ O O ( ∧ W, d ) ≃ O O
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