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Jun 10, 2004 - Since we work with graded objects, we will pay a special attention to signs. ...... where εσ is the graded signature and Sh(j) denotes the set of (j, ...
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J. Eur. Math. Soc. 9, 123–156

c European Mathematical Society 2007

Yves F´elix · Jean-Claude Thomas · Micheline Vigu´e-Poirrier

Rational string topology Received June 10, 2004 and in revised form July 28, 2006 Abstract. We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a simply connected closed manifold M. We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the cochain algebra C ∗ (M) with coefficients in C ∗ (M). Some explicit computations of the loop product and the string bracket are given. Keywords. String homology, rational homotopy, Hochschild cohomology, free loop space, loop space homology

1. Introduction Let C∗ (X) (respectively H∗ (X)) be the singular chains (respectively the singular homology) of a space X with coefficients in Q. For simplicity we identify H∗ (X × X) with H∗ (X) ⊗ H∗ (X) and the singular cohomology H ∗ (X) with the graded dual of the homology, i.e. H ∗ (X) := H (C ∗ (X)) = (H∗ (X))∨ . Let M be a simply connected closed oriented m-manifold and let LM (respectively M I and M) be the space of free loops (respectively paths and based loops). M. Chas and D. Sullivan [6] have constructed a product, called the loop product H∗ (LM) ⊗ H∗ (LM) → H∗−m (LM),

x ⊗ y 7 → x • y,

so that H∗ (LM) := H∗+m (LM) is a commutative graded algebra, and a morphism, called the I -homomorphism I : H∗ (LM) → H∗ (M), The authors are partly supported by INTAS program 03 51 3251. Y. F´elix: D´epartement de math´ematique, Universit´e Catholique de Louvain, 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium; e-mail: [email protected] J.-C. Thomas: D´epartement de math´ematique, Facult´e des Sciences, 2, Boulevard Lavoisier, 49045 Angers, France; e-mail: [email protected] M. Vigu´e-Poirrier: D´epartement de math´ematique, Institut Galil´ee, Universit´e de Paris-Nord, 93430 Villetaneuse, France; e-mail: [email protected] Mathematics Subject Classification (2000): 55P35, 54N45, 55N33, 17A65, 81T30, 17B55

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which relates the loop product structure on H∗ (LM) with the Pontryagin algebra H∗ (M). The loop product induces the string bracket on the S 1 -equivariant homology S 1 (LM), of LM, H∗ = H∗+m H∗ ⊗ H∗ → H∗−2 ,

a ⊗ b 7 → [a, b],

such that (H, [−, −]) is a graded Lie algebra of degree 2. Few things are known about this bracket. For surfaces of genus larger than zero, Chas and Sullivan recover formulae proved in the context of symplectic geometry. The purpose of this paper is to provide explicit computational tools for the duals of these two operations in cohomology and for the I -homomorphism. First we will describe the dual of the loop product (Theorem A) and the dual of the string bracket (Theorem B) in terms of Sullivan models, [21]. It follows from Theorem A that the loop product structure on H∗ (LM) is invariant under orientation preserving maps which are quasi-isomorphisms. In Theorem C, we adapt the technics of Theorem A to a special type of Sullivan models, the cochains of a differential graded Lie algebra with coefficients in a differential graded Lie module. The main point in this paper is Theorem D which allows us to describe the dual of the loop product in terms of chains (instead of cochains) of a differential graded Lie algebra. This translation is performed by means of the “cap-homomorphism”. This homomorphism realizes, in some sense, the Poincar´e duality at the level of free loop spaces. The other interest for considering the caphomomorphism is that it allows us, in Theorem E, to identify the loop product in H∗ (LM) with the Gerstenhaber product on the Hochschild cohomology HH ∗ (C ∗ (M); C ∗ (M)). Here C ∗ (M) denotes the cochain algebra of singular cochains on M. Theorem E. Let M be a simply connected closed oriented manifold. There exists a natural isomorphism of graded algebras ∼ =

J : H∗ (LM) → HH ∗ (C ∗ (M); C ∗ (M)). Such an identification has been proved by Merkulov [19] for the field of real numbers using iterated path integrals. Finally, we show Theorem F. Let J be the map in Theorem E. There exists a canonical isomorphism of graded algebras, J¯, making the diagram H∗ (LM)

/ HH ∗ (C ∗ (M); C ∗ (M))

J ∼ =

HH ∗ (C ∗ (M),ε)

I

 H∗ (M)



 / HH ∗ (C ∗ (M))

commutative, where ε : C ∗ (M) → Q denotes the usual augmentation. Theorems E and F are complementary results to our previous paper [14]. In particular, Theorems 2 and 3 in [14] apply verbatim.

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For the convenience of the reader we repeat the relevant material of rational homotopy theory, without proof, thus making our exposition self-contained. Moreover, we roughly indicate here the basic ideas of the paper. Let L be a graded differential Lie model of M. We denote by U La its enveloping algebra considered as an L-module for the adjoint representation, and U L∨ a its graded dual. Then the cochain algebra C ∗ (L; U L∨ ) is a Sullivan model for the free loop space a LM and the map C ∗ (L;µ∨ )

∗ ∨ ∨ C ∗ (L; U L∨ a ) −−−−−→ C (L; U La ⊗ U La )

is a model for the composition of paths LM ×M LM → LM where µ denotes the multiplication in U L. V On the other hand, if (A ⊗ W, d) is a (Sullivan) model of LM with A a finitedimensional model for M satisfying Poincar´e duality, there exists a linear map (A ⊗

V

W, D) ⊗A (A ⊗

V

W, D) → (A ⊗

V

W, D)⊗2

which induces the cohomology Gysin map (see 4.5) i ! : H ∗ (LM ×M LM) → H ∗+m (LM × LM) of the finite-codimensional embedding i : LM ×M LM ,→ LM × LM. Since the dual of the loop product is the composition H ∗ (cM ) ◦ i ! , the above constructions enable explicit calculations of the loop product. The paper is organized as follows. 2. 3. 4. 5. 6.

Preliminaries on differential homological algebra. The dual of the loop product. Theorem A. Dual of the loop product in terms of Sullivan models. Theorem B. Dual of the string bracket in terms of Sullivan models. Theorem C. Dual of the loop product in terms of cochains on a differential graded Lie algebra. 7. Theorem D. Dual of the loop product in terms of chains on a differential graded Lie algebra. 8. Theorem E. Loop product and Gerstenhaber product. 9. Theorem F. I -homomorphism and Hochschild homology.

2. Preliminaries on differential homological algebra All the graded vector spaces, algebras, coalgebras and Lie algebras V are defined over Q and are supposed to be of finite type, i.e. dim Vn < ∞ for all n.

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2.1. Graded vector spaces If V = {Vi }i∈Z is a (lower) graded Q-vector space (when we need upper graded vector space we put Vi = V −i as usual) then V ∨ denotes the graded dual vector space and sV denotes the suspension of V : V ∨ = Hom(V , Q),

(sV )n = Vn−1 ,

(sV )n = V n+1 .

Since we work with graded objects, we will pay a special attention to signs. Recall that if P = {Pi } and N = {Ni } are differential graded vector spaces with ∂P : Pi → Pi−1 and ∂N : Ni → Ni−1 then • P ⊗ N is a differential graded vector space: M (P ⊗ N)r = Pp ⊗ Nq , ∂P ⊗N = ∂P ⊗ idN + idP ⊗ ∂N , p+q=r

• Hom(P , N) is a differential graded vector space: Y Homn (P , N) = Hom(Pl , Nk ), DHom (P ,N ) f = ∂N ◦ f − (−1)|f | f ◦ ∂P . k−l=n

The same formulae hold if P = {P i } and N = {N i } are differential graded vector spaces with dP : P i → P i+1 and dN : N i → N i+1 . Later on we will omit subscripts on the differentials. 2.2. Differential graded algebras, coalgebras and Lie algebras For precise definitions we refer to [12, §3 c,d and §21]. Recall that T V denotes the tensor algebra on V , while T c (V ) is the free supplemented coalgebra generated by V . If C is a differential graded coalgebra with coproduct 8 and A is a differential graded algebra with product µ, then the cup product, f ∪ g = µ ◦ (f ⊗ g) ◦ 8, gives the differential graded vector space Hom(C, A) a structure of differential graded algebra. 2.3. Sullivan rational homotopy theory We refer the reader to [12, §12] for notation, terminology and results concerning Sullivan models. However, we recall here that if V = {V i }i≥0 is a graded Q-vector space we V denote by V the free graded commutative algebra generated by V . Any path-connected space X admits a Sullivan model V ' ρX : MX := ( V , d) → AP L (X) where AP L denotes the contravariant functor of piecewise linear differential forms, and ρX is a quasi-isomorphism [12, §10 and §12]. If X and Y are two path-connected spaces then any continuous map f : X → Y admits a Sullivan representative Mf . Hereafter we will make the following identifications: H ∗ (X) = H (AP L (X)) = H (MX ),

H ∗ (f ) = H (AP L (f )) = H (Mf ).

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2.4. Semifree modules Let A be a differential graded algebra. A differential graded A-module P is called semiS free if P is equipped with a filtration P = n≥0 P (n) satisfying P (0) = 0, P (n) ⊂ P (n + 1) and such that P (n)/P (n − 1) is free on a basis of cycles [12, §6]. For any A-module N , there exists a semifree module P and a quasi-isomorphism ϕ : P → N. The module P is called a semifree resolution of N. 3. The dual of the loop product 3.1. A convenient definition of the dual of the loop product While M. Chas and D. Sullivan [6] have defined the loop product by using “transversal geometric chains” it is convenient for our purpose to define directly the dual of the loop product in the following way. First we replace the space LM (respectively M I and M) by a Hilbert manifold ([1] or [7, Proposition 2.3.1]). Secondly, we consider the commutative diagram LM ×2 o

i

LM ×M LM

Comp

p0

p0 ×p0

 M ×M o

1

 M

/ LM (1)

p0

 M

where • • • •

Comp denotes composition of free loops, p0 is the evaluation at 0 and is thus a locally trivial fibre bundle [3], the left hand square is a pullback diagram of locally trivial fibrations, the inclusion i is the smooth embedding of the Hilbert manifold of composable loops into the product of the two Hilbert manifolds LM × LM.

The embeddings 1 and i both have codimension m. Thus, using the Thom–Pontryagin theory (see 4.5 for a precise definition), we obtain the Gysin maps 1! : H k (M) → H k+m (M ×2 ),

i ! : H k (LM ×M LM) → H k+m (LM ×2 ).

Thus diagram (1) yields the following diagram: ×2 ) o H k+m (LM O

i!

o H k (LM × O M LM)

H k (Comp)

H k (LM) O

H ∗ (p0 )

H ∗ (p0 )⊗2

H k+m (M ×2 ) o

1!

H k (M)

H ∗ (p0 )

(2)

H k (M)

Following [22], [8] or [9], the dual of the loop product is defined by composition of maps in the upper line: i ! ◦ H ∗ (Comp) : H ∗ (LM) → H ∗+m (LM ×2 )

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while the map in the lower line is the dual of the intersection product. The commutativity of this diagram expresses the fact that H ∗ (p0 ) is a homomorphism between two cocommutative graded coalgebras.

3.2. Why is it possible to express this definition in terms of Sullivan models? First we remark that diagram (1) is the pullback diagram of the diagram i0

(M I )×2 o

M I ×M M I

Comp0

/ MI

q

(p0 ,p1 )×2

 M ×4 o

 M ×3

12,3

(10 )

(p0 ,p1 )

 / M ×2

π 1,3

along the diagram o M ×4 O

12,3

M ×3 O

π 1,3

/ M ×2 O

10

1×2

M ×2 o

1

1

M

M

where 10 = (1 ⊗ id) ◦ 1, π 1,3 (x, y, z) = (x, z), q(α, β) = (α(0), α(1) = β(0), β(1)) and 12,3 (x, y, z) = (x, y, y, z). The other maps are the obvious ones. The second observation that once again the Thom–Pontryagin theory yields the commutative diagram i0!

I ×2 o H k+m ((M O ) )

I o H k (M I × O MM ) H ∗ (q)

H ∗ (p0 ,p1 )⊗2

H k+m (M ×4 ) o

H ∗ (Comp0 )

idM ⊗1! ⊗idM

I H k (M O ) H ∗ (p0 ,p1 )

H k (M ×3 ) o

H ∗ (π 1,3 )

(20 )

H k (M ×2 )

with H ∗ (π 1,3 )(a ⊗ b) = a ⊗ 1 ⊗ b. It is then natural to define the path product as the composition of the maps in the upper line of the diagram (20 ). Since the loop product appears as a “pullback” of the path product it is worthwhile to determine the path product. This is the purpose of the last observation: There is a commutative diagram (M IO )×2 o

i0

σ0

σ ×σ

M ×2 o

I M I ×M O M

1

M

Comp0

/ MI O σ

M

(3)

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where σ (x) is the constant path at x ∈ M and σ 0 (x) = (σ (x), σ (x)). From the properties of the Thom–Pontryagin construction, diagram (3) converts into a commutative diagram I ×2 o H k+m ((M O ) )

i0!

I o H k (M I × O MM )

Comp0

H ∗ (σ )

H ∗ (σ 0 )

H ∗ (σ )⊗2

H k+m (M ×2 ) o

1!

I H k (M O )

(30 )

H k (M)

H k (M)

Since 1! is the multiplication by the Euler class of the diagonal embedding (see 4.3) and since σ is a homotopy equivalence we obtain a description of the path product in terms of Sullivan models. 4. Theorem A. The dual of the loop product in terms of Sullivan models 4.1. Relative Sullivan model for loop fibrations It is convenient to consider a cofibrant Sullivan representative V of a map f : X → Y called a relative Sullivan model [12, §14]: λf : MY ,→ (MY ⊗ V , d) = MX . Our primary example is the following diagram: σ M M I Eo ' { EE { { EE { E {{ (p0 ,p1 ) EE" }{{ 1 M ×2

(4)

which describes the fibration associated to the diagonal map, and where σ (x) is the constant path at x ∈ M. This diagram converts into

MM I = (MM ×2

MM ×2 = M⊗2 LLML j λ(p0 ,p1 ) jjjj LM j LL1L=µ jjjj LLL j j j ujj % V Mσ ⊗ sV , d) o MM '

(M(4))

In this diagram: V a) MM = ( V , d) is V a Sullivan model of M. b) µ is the product on V V . V c) The differential in ( V )⊗2 ⊗ sV , d) = MM I is defined as in [12, §15, Example 1]: for v ∈ V , sv ∈ sV ,  ¯ = dv ⊗ 1 ⊗ 1, ¯ d(v ⊗ 1 ⊗ 1)     d(1 ⊗ v ⊗ 1) ¯ = 1 ⊗ dv ⊗ 1, ¯ ∞ X (sd)i   ¯− ¯ d(1 ⊗ 1 ⊗ sv) = (v ⊗ 1 − 1 ⊗ v) ⊗ 1 (v ⊗ 1 ⊗ 1).   i! i=1

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V V Here 1 and 1¯ denote respectively V Vthe unit V of V and of sV , and s is the unique degree −1 derivation of V ⊗ V ⊗ sV defined by  ¯ = 1 ⊗ 1 ⊗ sv = s(1 ⊗ v ⊗ 1), ¯ s(v ⊗ 1 ⊗ 1) s(1 ⊗ 1 ⊗ sv) = 0. V V V d) Mσ = µ ⊗ ε¯ : (( V )⊗2 ⊗ sV , d) = MM I V → MM = ( V , d) is a quasiisomorphism with ε¯ the canonical augmentation of sV . The use of relative Sullivan models is interesting because it converts a pullback diagram of fibrations into a pushout diagram in the category of differential graded commutative algebras [12, Proposition 15.8]: Each pullback diagram f0

f ∗E p0

/E (∗)

p

 B0

f

 /B

where p is any fibration, converts into the pushout diagram in the category of differential graded commutative algebras, Mf

MB λp

/ MB 0 (M(∗))

λp0



ME = (MB ⊗

V

Mf ⊗id

W, d)

/ (MB 0

V  ⊗ W, d 0 ) = Mf ∗ E

The differential on Mf ∗ E is explicitly defined by the canonical isomorphism of graded algebras MB 0 ⊗MB ME = MB 0 ⊗MB (MB ⊗

V

∼ =

W ) → MB 0 ⊗

V

W = Mf ∗ E .

As a first example, the pullback diagram LM



j

/ MI

p0

 M

(5)

(p0 ,p1 )

 / M ×M

1

converts into the pushout diagram M1 =µ

M⊗2 m λ(p0 ,p1 )



MM I = (M⊗2 m ⊗

V

sV , d)

Mj

/ MM 

λp0

/ MLM = (MM ⊗ V sV , d)

(M(5))

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where 0 is a Sullivan relative model for p0 [12, §12, c]. The differential on MLM = V λpV ( V ⊗ sV , d) is defined by ds = −sd where s denotes the degree −1 derivation which extends the linear isomorphism V → sV and satisfies s ◦ s = 0. In particular Mj = µ ⊗ id. In the same way, the pullback diagram LM ×2 o

i

LM ×M LM (6)

p0

p0 ×p0

 M ×M o

1

 M

converts into the pushout diagram µ

M⊗2 M

/ MM

λp0 ×p0

M⊗2 LM

(M(6))

λp0

 V ⊗2 ∼ = (MM ⊗ ( sV )⊗2 , d)

µ⊗id⊗id

/ (MM

V ⊗ ( sV )⊗2 , d)

V The differential on MLM×M LM = (MM ⊗ ( sV )⊗2 , d) is defined by  ¯ = dv ⊗ 1¯ ⊗ 1, ¯  d(v ⊗ 1¯ ⊗ 1) ¯ = −s(dv ⊗ 1¯ ⊗ 1), ¯ v ∈ V, v ∈ V . d(1 ⊗ sv ⊗ 1)  0 ¯ ¯ ¯ d(1 ⊗ 1 ⊗ sv) = −s (dv ⊗ 1 ⊗ 1), V V Here s and s 0 are the unique degree −1 derivations of V ⊗ ( sV )⊗2 such that s ◦ s = 0 = s 0 ◦ s 0 and for v ∈ V , sv ∈ sV ,  ¯ = 1 ⊗ sv ⊗ 1, ¯ s(v ⊗ 1¯ ⊗ 1)    0 ¯ ¯ ¯ s (v ⊗ 1 ⊗ 1) = 1 ⊗ 1 ⊗ sv, ¯ = 0 = s 0 (1 ⊗ sv ⊗ 1), ¯  s(1 ⊗ sv ⊗ 1)   0 ¯ ¯ s(1 ⊗ 1 ⊗ sv) = 0 = s (1 ⊗ 1 ⊗ sv). p0

These two examples provide us with relative Sullivan models for LM → M and p0 LM ×M LM → M as well as for Mi = µ ⊗ id ⊗ id : MLM → MLM×M LM = MLM ⊗MM MLM . In a similar way, the pullback diagram which appears as the left part in diagram (10 )— call it (7)—converts into the pushout diagram M⊗4 M λ(p

0 ,p1 )

M⊗2 MI

id⊗µ⊗id

×2

 V ⊗4 ∼ (M ⊗ ( sV )⊗2 , d)⊗2 = M

/ M⊗3 M λq

 id⊗µ⊗id⊗id / (M⊗3 ⊗ (V sV )⊗2 , d) M

(M(7))

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V ⊗2 where the differential on MM I ×M M I = (M⊗3 M ⊗ ( sV ) , d) is defined for v ∈ V and v ∈ V by:  ¯ = dv ⊗ 1 ⊗ 1 ⊗ 1¯ ⊗ 1, ¯ d(v ⊗ 1 ⊗ 1 ⊗ 1¯ ⊗ 1)     ¯ = 1 ⊗ dv ⊗ 1 ⊗ 1¯ ⊗ 1, ¯  d(1 ⊗ v ⊗ 1 ⊗ 1¯ ⊗ 1)      ¯ ¯ ¯ ¯   d(1 ⊗ 1 ⊗ v ⊗ 1 ⊗ 1) = 1 ⊗ 1 ⊗ 1 ⊗ dv ⊗ 1 ⊗ 1, ∞ X (sd)i ¯ = (v ⊗ 1 − 1 ⊗ v) ⊗ 1 ⊗ 1¯ ⊗ 1¯ − ¯ d(1 ⊗ 1 ⊗ 1 ⊗ sv ⊗ 1) (v ⊗ 1 ⊗ 1 ⊗ 1¯ ⊗ 1),   i!   i=1    ∞  X (s 0 d)i   ¯ ¯ ¯ ¯  d(1 ⊗ 1 ⊗ 1 ⊗ 1 ⊗ sv) = 1 ⊗ (v ⊗ 1 − 1 ⊗ v) ⊗ 1 ⊗ 1 − (1 ⊗ v ⊗ 1 ⊗ 1¯ ⊗ 1).  i! i=1 V V Here s and s 0 are the unique degree −1 derivations of ( V )⊗3 ⊗ ( sV )⊗2 such that s ◦ s = 0 = s 0 ◦ s 0 and  ¯ ¯ ¯ ¯ ¯   s(v ⊗ 1 ⊗ 1 ⊗ 1 ⊗ 1) = 1 ⊗ 1 ⊗ 1 ⊗ sv ⊗ 1 = s(1 ⊗ v ⊗ 1 ⊗ 1 ⊗ 1), ¯ = 0 = s 0 (v ⊗ 1 ⊗ 1 ⊗ 1¯ ⊗ 1), ¯ s(1 ⊗ 1 ⊗ v ⊗ 1¯ ⊗ 1)   0 ¯ = 1 ⊗ 1 ⊗ 1 ⊗ 1¯ ⊗ sv = s(1 ⊗ 1 ⊗ v ⊗ 1¯ ⊗ 1). ¯ s (1 ⊗ v ⊗ 1 ⊗ 1¯ ⊗ 1) In particular Mi 0 = id ⊗ µ ⊗ id ⊗ id.

4.2. Sullivan representatives of compositions of free paths and free loops Consider the next diagram which relates the right parts of diagrams (1) and (10 ):

LM

LM ×M LM rrr r r r rrComp r y rr ⊂



σ0

p0

 p0 M r r rrrr ' rrrrrr r r r rr  rrrrrrr 1 M

'

/ MI A

/ MI × MI 4 s M ss s ss 0 s s sy s Comp q

(p0 ,p1 ) (id×1)◦1 σ

 / M ×2



(8)

/ M ×3 r r π 1,3rrr rr ry rr

Lemma 1. There exists a unique homomorphism MComp0 of differential graded algebras such that, in the next diagram, the upper square commutes while the lower square com-

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mutes up to homotopy: MO M

MO M

Mσ 0

Mσ =µ.¯ε

M M I ⊗M M M M I o O

MComp0

MM I O

(9)

λ(p0 ,p1 )

λq

o M⊗3 M

Mπ 1,3

M⊗2 M

Moreover, Mσ 0 = µ.¯ε ⊗ µ.¯ε is a surjective quasi-isomorphism. Proof. Let us introduce the following notation: Vˆ = V , a) VV= sV , V b) ( V ⊗ Vˆ , d) is the acyclic Sullivan algebra whose differential d is defined by d(v) ¯ V = vˆ and ˆ V = 0, Vd(v) V c) π : ( V ⊗ V ⊗ Vˆ , d) → ( V , d) is the quasi-isomorphism defined by π(v) = v and π(v) ¯ = π(v) ˆ =V 0, V V ¯ s(v) ¯ = s(v) ˆ = 0. d) s is the derivation on ( V , d)⊗( V ⊗ Vˆ , d) defined by s(v) = v, Following [12, §15 Example 1], there is an isomorphism ϕ defined by  ¯ = v,  ϕ(v ⊗ 1 ⊗ 1)   ∞  X V V V (sd)n ¯ = v + vˆ + (v), ϕ : MM I → ( V , d) ⊗ ( V ⊗ Vˆ , d), ϕ(1 ⊗ v ⊗ 1)  n!  n=1   ϕ(1 ⊗ 1 ⊗ sv) = v. ¯ This isomorphism gives the relative Sullivan model of (p0 , p1 ) considered in diagram (4). The composite MM I ⊗MM MM I

ϕ⊗MM ϕ V

/(

V V V V V V , d)⊗( V ⊗ Vˆ , d)⊗V V ( V , d)⊗( V ⊗ Vˆ , d)

V V ( V , d)⊗( V ⊗

 V

id⊗V V π

V Vˆ , d)⊗V V ( V , d) π⊗V V id

V ( V , d) o

id·π·π

 V V ( V , d)⊗V V ( V , d)

is precisely Mσ 0 . From the lifting lemma [12, Lemma 12.4], we deduce that there exists a unique homomorphism MComp0 such that MComp0 ◦ Mσ 0 = Mσ . It follows from [12, Proposition 14.6] that the lower square of (9) is commutative up to homotopy. t u

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Therefore, using diagram (M(5)) of 4.1, we obtain the commutative diagram MM I

MComp0

V ⊗2 / MM I ⊗MM MM I ∼ = (M⊗3 M ⊗ ( sV ) , d) = MM I ×M M I

Mj =µ⊗id



MLM



MComp

Mj ⊗MM Mj

µ◦(µ⊗id)⊗id

/ MLM ⊗MM MLM ∼ = (MM

V  ⊗ ( sV )⊗2 , d) = MLM×M LM

Thus we have proved: Proposition 1. The homomorphism MComp is a Sullivan representative of Comp : LM ×M LM → LM. 4.3. A representative for i ! : H ∗ (LM ×M LM) → H ∗+m (LM)⊗2 First recall [2, VI, Theorem 12.4] that the Euler class of the diagonal embedding 1 : M → M × M (also called the diagonal class) is the cohomology class X e1 = (−1)|βl | βˆl ⊗ βl ∈ H m (M × M) = (H ∗ (M) ⊗ H ∗ (M))m l

where {βl } denotes a homogeneous linear basis of H ∗ (M) and {βˆl } its Poincar´e dual basis j (hβi ∪ βˆj , [M]i = δi 1). Observe here that H ∗ (M) is an H ∗ (M)⊗2 -module via the multiplication H ∗ (1). Thus ! 1 is mutiplication by e1 and an H ∗ (M)⊗2 -linear map. The crucial point in our construction of a representative of i ! is the following result: Proposition 2. There exists an M⊗2 M -linear map f : MM I = (M⊗2 M ⊗

V

sV , d) → M⊗2 M

of degree m such that f (d(x)) = (−1)m d ◦ f (x). Moreover, f is unique up to an M⊗2 M linear homotopy and 1! = H (f ) ◦ H (σ )−1 . V Proof. The homomorphism defined in 4.1, Mσ : MM I = (M⊗2 sV , d) → MM , M ⊗ ⊗2 is an MM -semifree resolution (see 2.4). Thus (see [11, Appendix]) we have the Moore spectral sequence p,q

p+q (MM , M⊗2 M ) M⊗2 M

ExtH ⊗H (H, H ⊗ H ) ⇒ Ext

:= H ∗ (HomM⊗2 (MM I , MM )), M

V with H = H ( V , d). The spectral sequence can also be constructed by replacing the minimal models by Halperin–Stasheff filtered models ([16]) and by using the induced filtration on the Hom complex. Since H ⊗ H is a Poincar´e duality algebra of formal

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dimension 2m, by [11, Theorem 3.1] we have  {0} if p + q 6 = 2m, p,q ∼ ExtH ⊗H (Q, H ⊗ H ) = Q if p + q = 2m. p,q

Induction on the dimension of E proves that ExtH ⊗H (E, H ⊗ H ) = 0 if 0 ≤ p + q < 2m − d for any finite-dimensional H ⊗ H -module E concentrated in degrees ≤ d. In particular, p,q ExtH ⊗H (H, H ⊗ H ) = 0 if 0 ≤ p + q < m. (C1 ) On the other hand, since M is simply connected, we have H m−1 = 0, and from the long exact sequence associated to the short exact sequence 0 → H m → H → H /H m → 0 we deduce that  {0} if p + q = m + 1, p,q ∼ ExtH ⊗H (H, H ⊗ H ) = (C2 ) Q for (p, q) such that p + q = m. Conditions (C1 ) and (C2 ) and the convergence of the Moore spectral sequence imply that Extm

M⊗2 M

∼ (MM , M⊗2 M ) = Q.

Multiplication by e1 defines a generator of Extm H ⊗H (H, H ⊗ H ) which survives in ∼ Q. Any cocycle in Hom = M⊗2 (MM I , MM ) can be viewed as a

Extm ⊗2 (MM , M⊗2 M ) MM

M

map f satisfying the conclusion of Proposition 2. If f and f 0 are two such cocycles then f − f 0 = Dh and h is the required homotopy between f and f 0 . t u The proof of the next result is postponed to Subsection 4.5. Proposition 3. The map M!i , defined as the composition of the following natural maps: ' / MM I ⊗ ⊗2 M⊗2 MM I ⊗MM MLM×M LM ML MM GG GG GG f ⊗id GG GG  GG GG M⊗2 ⊗ ⊗2 M⊗2 GG M LM MM M!i GG GG GG GG ' # 

M⊗2 LM

is such that i ! = H (M!i ) ◦ H (Mσ ⊗ id)−1 : H k (LM ×M LM) → H ∗+m (LM ×2 ).

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4.4. Statement of Theorem A and Examples From diagram (2) and Propositions 1 and 3 we deduce our first result: Theorem A. The dual of the loop product H ∗ (Comp)

i!

H ∗ (LM) −−−−−→ H ∗ (LM ×M LM) → H ∗+m (LM ×2 ) is induced in homology by the maps MLM MComp

 Mσ ⊗id MLM×M LM = MM ⊗MM MLM×M LM o ' MM I ⊗MM MLM×M LM ' MLM×M LM M!i

 MLM ⊗ MLM Example: Formal V spaces. Let M be a formal space, i.e. a space M whose minimal model MM = ( V , d) is quasi-isomorphic to (H = H ∗ (M), 0). Examples of formal spaces are given by connected compact K¨ahler manifolds ([10]) and quotients of compact connected Lie groups by closed subgroups of the same rank. When M is a formal space the dual of the loop product is induced in homology by the maps (H ⊗

V

sV , D)

H (MComp )

(H ⊗ (

µ.¯ε⊗id V V V sV )⊗2 , d) o (H ⊗2 ⊗ sV ) ⊗H (H ⊗ ( sV )⊗2 ) M!i

V (H ⊗2 ⊗ ( sV )⊗2 , d) Let us describe case M = CP n . The minimal model of CP n is V explicitly the particular n+1 given by ( (x, y), d), d(y) = V x , |x| = 2, |y| = 2n + 1. Thus the relative Sullivan model of the free loop space is (V (x, x, ¯ y, y), ¯ d),Vd(x) ¯ = 0, d(y) ¯ = −(n + 1)x n x. ¯ Since n+1 we have a quasi-isomorphism ( (x, y), d) → ( (x)/(x ), d) =: H , the space CP n is formal. A linear basis of V V≥1 V n+1 n+1 n H ∗ (LM) = H ( (x, x, ¯ y)/(y ¯ ), d) ∼ (x, x)/(x ¯ , x x) ¯ ⊗ y) ¯ =Q·1⊕( is formed by the elements 1, x p y¯ [q] , and x r x¯ y¯ [s] , p = 1, . . . , n, q ≥ 0, s ≥ 0, r = 0, . . . , n − 1, with y¯ [s] = y¯ s /s!.

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A Sullivan representative of the composition of free loops Comp : LM ×M LM → LM is given by MComp (x) ¯ = x¯ + x¯ 0 ,

MComp (y) ¯ = y¯ + y¯ 0 −

n(n + 1) n−1 0 x x¯ x¯ . 2

The dual of the loop product is induced by the map V V V θ : (H ∗ (M) ⊗ (x, ¯ y), ¯ d) → (H ∗ (M) ⊗ (x, ¯ y), ¯ d) ⊗ (H ∗ (M) ⊗ (x, ¯ y), ¯ d),  n X s X  [s]  θ (α ⊗ y ¯ ) = αx p y¯ [j ] ⊗ x n−p y¯ [s−j ]     p=0 j =0    n X s−1  n(n + 1) X αx n−1+p x¯ y¯ [j ] ⊗ x n−p x¯ y¯ [s−j ] , − 2   p=0 j =0   n X s  X    [s] p [j ] n−p [s−j ]  θ (α ⊗ x ¯ ⊗ y ¯ ) = (1 ⊗ x ¯ + x ¯ ⊗ 1) · αx y ¯ ⊗ x y ,   p=0 j =0

with α ∈ H ∗ (M). The dual basis 1, ap,q , br,s ,

p = 1, . . . , n, q ≥ 0, s ≥ 0, r = 0, . . . , n − 1,

with |ap,q | = 2p + 2qn, |br,s | = 2r + 1 + 2sn, is a linear basis of H∗ (LM). Thus the loop product H∗ (LM × LM) → H∗−m (LM), x ⊗ y 7 → x • y, is described by the formulae ap,q • ar,s = ap+r−n,q+s ,

ap,q • br,s = bp+r−n,q+s ,

(an−1,0 )n = 1,

1 • an,1 = 0 .

This shows that V H∗ (L(CP n ); Q) ∼ = (a, b, t)/(a n+1 , a n b, a n t), with |a| = −2, |b| = −1 and |t| = 2n, a = an−1,0 , b = bn−1,0 , t = an,1 (cf. [9]).

4.5. Proof of Proposition 3 Let M and N be (smooth Banach and without boundary) connected manifolds and f : M → N be a (smooth) closed embedding [18, II, §2]. Then we have the exact sequence of fiber bundles Tf 0 → T M → T N|M → νf → 0 where T M and T N are the tangent bundles and νf is the normal fiber bundle of f . By definition of an immersion, this exact sequence splits [18, II, Proposition 2.3]. Hereafter we will identify νf with a factor bundle of T N|M . When the fiber of νf is of finite dimension k, the embedding has codimension k. Consider the associated disk and sphere bundles, ν D , ν S , and the Thom class of the oriented normal fiber bundle pair (ν D , ν S ). The exponential map D ⊂ T N → N restricted to νf is a local isomorphism on the zero

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section of the bundle T N → N. Since f is a closed embedding and since N admits a partition of unity, by [18, IV, Theorem 5.1], there exists an open neighborhood Z of the zero section of νf , an open neighborhood U of f (M) in N, and an isomorphism θ : Z → U which identifies the zero section of νf with f (M). Since f has finite codimension k we θ

identify Z with νfD , and the isomorphism νfD → U =: tube f restricts to an isomorphism νfS ∼ = θ (νfS ) =: ∂ tube f . The above discussion is summarized in the commutative diagram j (N,N−f (M))

H ∗ (N, N − f (M)) ∼ = Excision

 H ∗ (tube f, ∂ tube f )

H ∗ (νfD , νfS )

/ H ∗ (N )

 / H ∗ (tube f )

j (tube f,∂ tube f )

j

(νfS ,νfS )

/ H ∗ (ν D ) O f

O

π D ◦f|f−1(M)

H ∗ (π D ◦f )(−)∪τf ∼ =

H ∗−k (M)

−∪e

πS

/ H ∗ (M)

The H ∗ (N)-linear map f ! defined as the composition of the natural homomorphisms H ∗+k (νfD , νfS ) = H ∗ (tube f, ∂tubef ) O

(Excision)−1 ∼ =

H ∗ (π D ◦f )(−)∪τf

H ∗ (M)

/ H ∗ (N, N − f (M)) j (N,N−f (M))

 / H ∗ (N )

f!

is called the cohomology Gysin map and f ! (1) = ef ∈ H k (N) is called the Euler class of the embedding f ([20]). End of proof of Proposition 3. First consider the commutative diagram / LM ×2 E eLL LLL {= q8 q { q { i qq ' { LLLσ ×id q { LLL {{ qqq q { ' q { ' / LM ×M LM p0×2 F p0

  p0 ×p1 / M ×2 I M e L LL {= qq8 LL σ ' {{{ 1 qqq LL q { q q { ' LLL  {{ L  qqqq p0 / M MI where the front face and the back face of the cube are pullback diagrams.

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Secondly, observe that ∗ H ∗ (E) = H (MM I ⊗M⊗2 M⊗2 LM ) and H (F ) = H (MM I ⊗MM MLM×MM LM ). M

From Proposition 2 we have the associated commutative diagram H (f ⊗id)

/ H (LM)⊗2 H ∗ (E) O PPP O r m6 m r PPHP(σ ×id) r i !mmmm r P r PPP rr∼ ∼ mmm PP( = yrrr = mmm ∗ (p ) H 0 o H ∗ (F H ∗ (LM × O ) O M LM) H (f )

/ H ∗ (M)⊗2 H ∗ (M I )P P n6 s n P s PPPH (σ ) ∼ 1! nnnn = sss P n PPP ss ∼ nnn PP' = ysss nnn H ∗ (p0 ) H ∗ (M) H ∗ (M I ) o H ∗ (p0 )

This ends the proof of Proposition 3.

t u

5. Theorem B. String bracket in terms of Sullivan models 5.1. Statement and proof of Theorem B The string homology (respectively string cohomology) is the desuspended equivariant homology of the free loop space (respectively the graded dual of the desuspended equivariant homology) 1

S H∗ = H∗+m (LM) = H∗+m (LM ×S 1 ES 1 )

(respectively H∗ = (H∗ )∨ ).

Let ξ ∈ H 2 (LM ×S 1 ES 1 ) be the characteristic class of the circle bundle S 1 → p

LM × ES 1 → LM ×S 1 ES 1 and consider the associated Gysin sequence H (p)

ξ ∩−

M

· · · → Hn −→ Hn −→ Hn−2 −→ Hn−1 → · · · .

(10)

The string bracket on H∗ is the bilinear map (see [6]) [−, −] : H∗ ⊗ H∗ → H∗−2 ,

a ⊗ b 7 → [a, b] = (−1)|a| H∗ (p)(M(a) • M(b)),

where • denotes the loop product on H∗ (LM). λp0 V p0 Let MM ,→ MLM = (MM ⊗ sV , d) be a Sullivan model of LM → M (see diagram (5)). Then a Sullivan representative for the inclusion LM → LM ×S 1 ES 1 is given by the MLM -linear map (see [4]) V π : (MLM ⊗ u, D) → MLM , 1 ⊗ u 7 → 0,

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V where (MLM ⊗ u, D) is a Sullivan model for the equivariant free loop space LM ×S 1 ES 1 whose differential D is given by D(u) = 0, |u| = 2, D(v) = d(v) + us(v), D(sv) = −s(dv). This proves that the dual of the Gysin sequence (10) is the homology long exact sequence associated to the short exact sequence of cochain complexes V V µu π 0 → (MLM ⊗ u, D) → (MLM ⊗ u, D) → MLM → 0. The connecting map s˜ is induced by the derivation s considered in the definition of MLM (see 4.1) and µu denotes multiplication by the cocycle u. Then Theorem B follows from Theorem A and the definition of the string bracket: Theorem B. The dual of the string bracket B ∨ : H∗ → (H∗ ⊗ H∗ )∗+2 is induced in homology by the homomorphisms of complexes: MLM o

π

(MLM ⊗

V

u, D)

MComp

 µ.¯ε⊗id MLM×M LM = MM ⊗MM MLM×M LM o ' MM I ⊗MM MLM×M LM ' MLM×M LM M!i

(MLM ⊗

V

u, D)⊗(MLM ⊗

V

u, D) o

s˜ ⊗˜s

 MLM ⊗MLM

5.2. Examples V V 1) Assume that H ∗ (M) is equal to u with |u| odd, or to u/(un+1 ) with |u| even. From [4], we have Vthe following facts: If H ∗ (M) = V u with |u| = 2p + 1 then H2i = 0 for all i. If H ∗ (M) = u/(un+1 ) with |u| = 2p, then dimQ H2i = 1 for all i. Furthermore, the space M is formal and it is shown in [23] that the map ξ ∩ − is an isomorphism. This proves, in the two cases, the nullity of the maps E = H∗ (p) : H2i → H2i

and M : H2i −→ H2i+1 .

Let a ∈ H2i−1 and b ∈ H2j −1 , for some (i, j ) ∈ Z2 . Then M(a) • M(b) ∈ H2(i+j +1) , so we have [a, b] = −E(M(a) • M(b)) = 0. Thus, for such manifolds M the string bracket is trivial. 2) If M := S 2k+1 × S 2k+1 , k ≥ 1, then models for M, LM and LM ×S 1 ES 1 are given by V M: (V(x, y), 0), LM : ( (x, y, x, ¯ y), ¯ 0),  V Du = D(x) ¯ = D(y) ¯ = 0, 1 LM ×S 1 ES : ( (x, y, x, ¯ y, ¯ u), D), Dy = uy, ¯ Dx = ux. ¯

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A set of cocycles representing a basis of the vector space H˜ S∗1 (LM) := HS∗1 (LM)/Qu is formed by the elements ([4]) ea,b = x¯ a y¯ b ,

(a, b) ∈ N2 − (0, 0),

fa,b = (y x¯ − x y) ¯ x¯ a y¯ b ,

(a, b) ∈ N2 .

The vector space H˜ ∗ (LM) has the following basis:   0 a b 2 ea,b ,  ea,b = xy x¯ y¯ , (a, b) ∈ N , 0 a b fa,b , f = x x¯ y¯ , (a, b) ∈ N2 ,  a,b fb00 = y y¯ b , b ∈ N. From the above description of the Gysin sequence we deduce H ∗ (p)(ea,b ) = ea,b , 0 ) = f , M 0 (f 00 ) = e 0 = ea+1,b , M 0 (ea,b a,b 0,b+1 , M (ea,b ) b

0 ) H ∗ (p)(fa,b ) = fa,b , M 0 (fa,b 0 = 0, and M (fa,b ) = 0.

To fix signs, denote by xy the fundamental class of M. A straightforward computation shows that B ∨ (ur ) = 0, p X q    X p q ∨ p q (x¯ r y¯ s+1 ⊗ x¯ p−r+1 y¯ q−s − x¯ r+1 y¯ s ⊗ x¯ p−r y¯ q−s+1 ) B (x¯ y¯ ) = s r r=0 s=0 B ∨ (fp,q ) = (f0,0 ⊗ 1 + 1 ⊗ f0,0 )(x¯ ⊗ y¯ − y¯ ⊗ x) ¯ · B ∨ (x¯ p y¯ q ). To describe the string bracket in H∗ we choose the dual basis tr , ap,q and bp,q to ur , and fp,q . In that basis the string bracket satisfies    k+l m+t km − lt [bk,t , al,m ] = bk+l−1,t+m−1 , k t (k + l)(t + m)    k+l m+t lt − km ak+l−1,t+m−1 , [ak,t , al,m ] = (k + l)(t + m) k t [br,s , bm,n ] = 0.

x¯ p y¯ q

In particular the string Lie algebra H∗ is not nilpotent, since for instance [a1,1 , ar,s ] = (r − s)ar,s . 6. Theorem C. Dual of the loop product in terms of cochains on a differential graded Lie algebra 6.1. Chains and cochains on a differential graded Lie algebra Let (L, ∂) be a differential graded Lie algebra with ∂ : Li → Li−1 and (P , ∂) (respectively (Q, ∂)) be a left (respectively right) differential L-module. The two-sided chain complex C∗ (P ; L; Q) is defined as follows: V Ck (P ; L; Q) = (P ⊗ k sL ⊗ Q, ∂0 + ∂1 ).

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A generic element is written p ⊗ sx1 ∧ · · · ∧ sxk ⊗ q with degree |p| + |q| + The differential ∂ = ∂0 + ∂1 is defined by ∂0

Pk

i=1 |sxi |.

∂1

Ck (P ; L; Q) → Ck (P ; L; Q), Ck (P ; L; Q) → Ck−1 (P ; L; Q), ∂0 (p ⊗ c ⊗ q) = ∂p ⊗ sx1 ∧ · · · ∧ sxk ⊗ q −p⊗

k X (−1)|p|+i sx1 ∧ · · · ∧ s∂xi ∧ · · · ∧ sxk ⊗ q i=1 |p|+k

+ (−1) p ⊗ c ⊗ ∂q, ∂1 (p ⊗ sx1 ∧ · · · ∧ sxk ⊗ q) k X ci · · · ∧ sxk ⊗ q = (−1)|p|+|sxi |i p · xi ⊗ sx1 ∧ · · · sx i=1

+p⊗

X

ci · · · sx cj · · · ∧ sxk ⊗ ∂q (−1)|p|+ij s[xi , xj ] ∧ · · · sx

1≤i
P where i,j = i+1 + j for i < j and i = k
k X X

εσ (sxσ (1) ∧ · · · ∧ sxσ (j ) ) ⊗ (sxσ (j +1) ∧ · · · ∧ sxσ (k) ),

j =0 σ ∈Sh(j )

where εσ is the graded signature and Sh(j ) denotes the set of (j, k − j )-shuffles. Recall that if Q is a left L-module then it is also a right L-module for the action defined by a · x := −(−1)|a|·|x| x · a, a ∈ Q, x ∈ U L. For any left L-differential module Q (respectively for any right differential module P ) put C∗ (L; Q) = C∗ (Q; L; Q)

(respectively C∗ (P ; L) = C∗ (P ; L; Q)). V V Thus, as graded vector spaces, C∗ (L; Q) = sL ⊗ Q and C∗ (P ; L) = P ⊗ sL. The cochain complex of L with coefficients in a right L-module P is defined by C ∗ (L; P ) = HomU L (C∗ (L; U L), P ). V When Q = U L with the action induced by left multiplication then C∗ (L; U L) = ( sL ⊗ U L, ∂) is a left C∗ (L)-comodule and a right U L-module and both structures are compatible. Moreover, the inclusion Q ,→ C∗ (L; U L) is a quasi-isomorphism and C∗ (L; U L) is a semifree L-module [12, Proposition 22.3]. Thus, H (C∗ (L; P )) = TorL (Q, P ) and H (C ∗ (L; P )) = ExtL (Q, P ).

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6.2. Cochains with coefficients in the adjoint module U La and free loop space Let (L, ∂) be as in 6.1 and consider the adjoint modules U La and U L∨ a with actions defined by l · x = [l, x] on U La , (f · l)(x) = f ([l, x]) on U L∨ a,

l ∈ L, x ∈ U L, f ∈ U L∨ .

Moreover, since U L is a Hopf algebra, for any right (respectively left) differential Lmodules (P , ∂) (respectively (Q, ∂)) the left diagonal L-module P ⊗ Q is defined by X 0 x · (a ⊗ b) = (−1)|xi | |a| (xi · a) ⊗ (xi0 · b), x ∈ U L, a ∈ P , b ∈ Q, i

where x 7 → i xi ⊗ xi0 denotes the coproduct in U L. Now let (L, ∂) be such that ([12, Lemma 23.1 and (24.b)]) the differential graded algebra C ∗ (L) is quasi-isomorphic to a minimal model of M: V C ∗ (L) = ( V , d) ' MM . P

The differential graded Lie algebra is uniquely defined by this condition. We have V = s(L∨ ), H (L, ∂) = π∗ (M) and H∗ (M) = H (U L, ∂) as Hopf algebras [12, Theorem 21.15]. Moreover, by the Poincar´ 21.2], the natV e–Birkhoff–Witt theorem [12, Proposition P ural linear isomorphism γ : L → U L, x1 ∧ · · · ∧ xk 7 → (1/k!) σ εσ xσ (1) · · · xσ (k) , is an isomorphism of graded coalgebras. Therefore we have the following isomorphism of graded algebras: V U L∨ ∼ = sV . Lemma 4. Considering U La as a right L-module, the graded linear isomorphisms V ∨ ∼ ∗ ∨ ∼ ∗ ∼ ∗ C ∗ (L; U L∨ sV a ) = C (L; U L ) = C (L) ⊗ U L = C (L) ⊗ define a structure of graded algebra on C ∗ (L; U L∨ a ) which is compatible with the differential of C ∗ (L; U L∨ ). Moreover, the natural inclusion a V ∗ ∗ C (L) ,→ C ∗ (L; U L∨ sV , d), f 7→ f ⊗ 1U L∨ , a ) ' (C (L) ⊗ is a relative Sullivan model of p0 : LM → M. Proof. Consider the differential graded Lie algebra LS defined as follows: LSn = Ln ⊕ Ln ,

Ln = Ln+1 ,

dx = −dx,

(−1)|a| [a, b] = [a, b],

[a, b] = 0.

Then the inclusion V V C ∗ (L) ,→ C ∗ (LS ) = ( (V ⊕ sV ), d) = (C ∗ (L) ⊗ sV , d) with d(sv) = −sd(v) is a relative Sullivan model of p0 : LM → M (see 4.1, diagram (5)). The rest of the proof follows from the following sequence of isomorphisms of differential graded algebras: C ∗ (LS ) := Hom(C∗ (L; U La ), Q) ∼ = Hom(C∗ (L; U L) ⊗U L U La , Q) ∼ HomU L (C∗ (L; U L), U L∨ ) =: C ∗ (L; U L∨ ). u t = a a

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6.3. Statement and proof of Theorem C The multiplication ν : U L ⊗ U L → U L which induces the Pontryagin product on H∗ (M) is not a morphism of graded algebras. Nonetheless, ν : U La ⊗ U La → U La is a morphism of U L-modules when U La ⊗ U La is the diagonal left L-module. Thus the ∨ ∨ dual of the product on U L, ν ∨ : U L∨ a → U La ⊗U La , is a homomorphism of differential ∨ denotes the right diagonal module. graded right U L-modules when U L∨ ⊗ U L a a Proposition 4. The morphism of complexes V ∗ ⊗2 ∨ ∗ ∨ C ∗ (L; ν ∨ ) : C ∗ (L, U L∨ a ) → C (L, U La ⊗ U La ) ' (C (L) ⊗ ( sV ) , d) is a homomorphism of commutative differential graded algebras. This is a Sullivan model Comp

for the composition of free loops LM ×M LM −−−→ LM. Proof. The contractible chain complexes C∗ (L; U L) and C∗ (U L; L), defined in 6.1, are differential graded coalgebras via the graded linear isomorphisms C∗ (L; U L) ∼ = C∗ (L) ⊗ U L and C∗ (U L; L) ∼ = U L⊗C∗ (L). Therefore, the coproduct µ∨ on C∗ L induces a quasiisomorphism of differential graded coalgebras 8 : C∗ (L) → C∗ (L; U L) ⊗U L C∗ (U L; L) ∼ = (C∗ (L) ⊗ U L ⊗ C∗ (L), ∂) defined by 8(x) =

P

i

xi ⊗ 1 ⊗ xi0 when µ∨ (x) =

P

i

xi ⊗ xi0 , and the dual map

8∨ : (C ∗ (L) ⊗ U L∨ ⊗ C ∗ (L), d) := (C∗ (L; U L) ⊗U L C∗ (U L; L))∨ → C ∗ (L) (11) is a quasi-isomorphism of differential graded algebras. This implies that the natural injection λ

C ∗ (L) ⊗ C ∗ (L) → (C ∗ (L) ⊗ U L∨ ⊗ C ∗ (L), d) ∼ = (C ∗ (L)⊗2 ⊗ x ⊗ y 7 → x ⊗ 1 ⊗ y,

V

sV , d),

is a relative Sullivan model for the product µ : C ∗ (L) ⊗ C ∗ (L) → C ∗ (L) and thus, by dia(p0 ,p1 )

gram (M(4)) of 4.1, it is a Sullivan relative model for the path fibration M I −−−→ M ×2 . This fact allows us to follow the construction performed in 4.2 with MM I = (C ∗ (L) ⊗ U L∨ ⊗ C ∗ (L), d) and Mσ 0 = 8∨ . First observe that the cochain model of the vertical face on the right in diagram (8), M I ×M M I

Comp0

/ MI

q



M ×3

π 1,3



(p0 ,p1 )

/ M ×2

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145

is the pushout diagram EO

id⊗∇⊗id

/ C(L) ⊗ U L∨ ⊗ C(L) O

id⊗ε∨ ⊗id⊗ε∨ ⊗id

C ∗ (L)⊗3

id⊗ε∨ id⊗τ ⊗id

/ C ∗ (L)⊗2

where ε : U L → Q is the augmentation, τ : Q ,→ C(L) is the natural inclusion and ∇ is the composition ν∨

id⊗τ ⊗id

U L∨ → (U L∨ )⊗2 ' U L∨ ⊗ Q ⊗ U L∨ −−−−→ U L∨ ⊗ C ∗ (L) ⊗ U L∨ . Observe that E = C ∗ (L) ⊗ U L∨ ⊗ C ∗ (L) ⊗ U L∨ ⊗ C ∗ (L) ' (C∗ (L; U L) ⊗U L C∗ (U L; L; U L) ⊗U L C∗ (U L; L))∨ . Secondly, we use the machinery developed in (4.2) to translate diagram (8) in terms of Sullivan models. We deduce the explicit model of the map Comp by considering the pushout id⊗ε ∨ ⊗id

∨ ⊗2 o 6E (C ∗ (L)⊗(U O L ) , d) mmm O n6 MComp0 =id⊗∇⊗idmmmm MComp nnn nn mm mmm nnn m n m n λ m n M⊂ ∨ , d) o (C ∗ (L)⊗U L∨ ⊗C ∗ (L), d) = MM I λ (C ∗ (L)⊗U L O O λ λ

MM

nnn nnn n n nnn nnn o

C ∗ (L) o

µ

µ◦(id⊗µ)

C ∗ (L)⊗2

C ∗ (L)⊗3 m6 m m i13 mm m mmm m m m

The map MComp : (C ∗ (L) ⊗ U L∨ , d) → (C ∗ (L) ⊗ (U L∨ )⊗2 , d) is precisely C(L; ν ∨ ). t u Remark. Denote by LS1 and LS2 two copies of LS , and LT = LS1 ⊕L LS2 = L ⊕ L¯ 1 ⊕ L¯ 2 . We denote by π : LT → LS the projection obtained by mapping identically each LSi ∨ to LS . We observe that C∗ (LT ) is isomorphic to C∗ (L; U L∨ a ⊗U La ) and that the following

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diagram commutes: C ∗ (L;ν ∨ )

C ∗ (L; U L∨ a)

/ C ∗ (L; U L∨ ⊗ U L∨ ) a

Hom(C∗ (L; U La ), Q)

Hom(C∗ (L;ν),Q)

/ Hom(C∗ (L; U La ⊗ U La ), Q)

C ∗ (π)

C ∗ (LS )

a

/ C ∗ (LT )

This shows that π : LT → LS is a “Lie representative” for the composition of free loops LM ×M LM → LM. Proposition 4 and Theorem A yield Theorem C. If C ∗ (L) is a Sullivan model of M then the dual of the loop product H ∗ (Comp)

i!

H ∗ (LM) −−−−→ H ∗ (LM ×M LM) → H ∗+m (LM ×2 ) is induced in homology by the maps C ∗ (L; U L∨ a)

C ∗ (L;ν ∨ )

∨ ∗ ∨ / C ∗ (L; U L∨ ∗ C ∗ (L; U L∨ a ⊗U La ) = C (L)⊗ a ⊗U La ) O C (L) 8∨ ⊗id

∗ ∨ o C ∗ (L; U L∨ a )⊗C (L; U La )

M!i

∨ (C ∗ (L)⊗U L∨ ⊗C ∗ (L), d)⊗C ∗ (L) C ∗ (L; U L∨ a ⊗U La )

V ∗ ∨ (C ∗ (L; U L∨ a )⊗C (L; U La )⊗ sV , d) with 8∨ defined in (11).

6.4. Example Recall that a coformal space M is a space that admitsVa Sullivan minimal model with a purely quadratic differential. In this case C ∗ (L) = ( V , d), L = π∗ (M) ⊗ Q and V ∗ H (M) = sV . Thus, V If M is a coformal manifold with minimal model ( V , d), then a model for the path composition LM ×M LM → LM is given by V V V V id ⊗ ν ∨ : ( V ⊗ V , d) → ( V ⊗ ( V )⊗2 , d), V V where ν is the Pontryagin product on H∗ (M) and ( V ⊗ V , d) is the model of the free loop space, defined in 4.1.

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As a particular case consider the 11-dimensional manifold M obtained by taking the pullback of the tangent sphere bundle to S 6 along the map f : S 3 × S 3 → S 6 that collapses the 3-skeleton into a point / τ S6 S

M  S3 × S3

f

 / S6

The minimal model of M is V ( (x, y, z), d), dx = dy = 0, dz = xy,

|x| = |y| = 3, |z| = 5.

Thus M is a coformal space. A model for the path composition M I ×M M I → M I is given by V ϕ : ( (x, y, z, x 0 , y 0 , z0 , x, ¯ y, ¯ z¯ ), d) V → ( (x, y, z, x 0 , y 0 , z0 , x 00 , y 00 , z00 , x, ¯ y, ¯ z¯ , x¯ 0 , y¯ 0 , z¯ 0 ), d), with d(x) ¯ = x − x 0 , d(y) ¯ = y − y 0 , d(¯z) = z − z0 − 12 x(y ¯ + y 0 ) + 12 (x + x 0 )y, ¯ 1 0 0 0 0 00 0 0 00 0 0 00 d(x¯ ) = x − x , d(y¯ ) = y − y , d(¯z ) = z − z − 2 x¯ (y + y 00 ) + 12 (x 0 + x 00 )y¯ 0 , ϕ(x) ¯ = x¯ + x¯ 0 , ϕ(y) ¯ = y¯ + y¯ 0 , ϕ(¯z) = z¯ + z¯ 0 + 21 x¯ y¯ 0 − 12 x¯ 0 y. ¯ The induced model for the path composition LM ×M LM → LM is then given by V V ϕ : ( (x, y, z, x, ¯ y, ¯ z¯ ), d) → ( (x, y, z, x, ¯ y, ¯ z¯ , x¯ 0 , y¯ 0 , z¯ 0 ), d), with d(x) ¯ = 0, d(y) ¯ = 0, d(¯z) = −xy ¯ + x y, ¯ d(x¯ 0 ) = 0, d(y¯ 0 ) = 0, d(¯z0 ) = −x¯ 0 y + x y¯ 0 , 0 0 ϕ(x) ¯ = x¯ + x¯ , ϕ(y) ¯ = y¯ + y¯ , ϕ(¯z) = z¯ + z¯ 0 + 21 x¯ y¯ 0 − 12 x¯ 0 y. ¯ 7. Dual of the loop product in terms of chains on a differential graded Lie algebra 7.1. The cap-homomorphism Let (L, ∂) be a differential graded Lie algebra, P N be a differential graded L-module and C∗ (L; N), C ∗ (L; N) be as in 6.1. Let c = i sxi1 ∧ · · · ∧ sxiq ∈ Cq (L) be a cycle of degree m. We define the cap product by c: capc : C q−r (L; N) = HomU L (Cq−r (L; U L), N ) → Cr (L; N ), where

f ∩ c = (−1)m

X X i

f 7→ f ∩ c,

(−1)|f |·|sxσ (i1 ) ∧···∧sxσ (ir ) | εσ

σ ∈6q

· sxσ (i1 ) ∧ · · · ∧ sxσ (ir ) ⊗ f (sxσ (ir+1 ) ∧ · · · ∧ sxσ (iq ) ). A standard computation proves: Lemma 5. The morphism capc is a homomorphism of complexes which is natural with respect to homomorphisms of differential graded L-modules.

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7.2. Statement of Theorem D ∨ ∨ Since the dual of the multiplication ν ∨ : U L∨ a → U La ⊗ U La is a morphism of U Lmodules, Lemma 5 implies:

Proposition 5. The following diagram commutes: C∗ (L; U L∨ a) O

C∗ (L;ν ∨ )

/ C∗ (L; U L∨ ⊗ U L∨ ) aO a

capc

C ∗ (L; U L∨ a)

capc

C ∗ (L;ν ∨ )

/ C ∗ (L; U L∨ ⊗ U L∨ ) a a

The next result, whose proof is postponed to Subsection 7.3, furnishes a representative of the Gysin map i ! : H ∗ (LM ⊗M LM) → H ∗+m (LM ×2 ) at the level of chains. Let (L, ∂) be such that C ∗ (L) ' MM . The algebra H ∗ (C ∗ (L)) = H ∗ (M) is a Poincar´e duality algebra with a fundamental class [M] ∈ Hm (M). We denote by u a cycle in C∗ (L) representing [M]. Then an easy spectral sequence argument shows that capu : C ∗ (L; N ) → C∗ (L; N ) is a quasi-isomorphism. Proposition 6. With the notation of 6.2, the following diagram commutes in homology: C∗ (δ;id)

∨ C∗ (L; U L∨ aO ⊗ U La )

/ C∗ (L; U L∨ ) ⊗ C∗ (L; U L∨ ) a a O

capu ∨ C ∗ (L; U L∨ aO ⊗ U La )

capu ⊗capu

(80 )∨ ⊗id ∼ = ∗ ∨ (C ∗ (L; U L∨ a ) ⊗ C (L; U La ) ⊗

V

V , d)

M!i

/ C ∗ (L; U L∨ ) ⊗ C ∗ (L; U L∨ ) a a

Here δ : L → L ⊕ L, x 7 → (x, x). It follows directly from Propositions 5 and 6 that the coproduct C∗ (L;ν ∨ )

C∗ (δ;id)

∨ ∨ ∨ ∨ C∗ (L; U L∨ a ) −−−−→ C∗ (L; U La ⊗ U La ) −−−−→ C∗ (L; U La ) ⊗ C∗ (L; U La )

induces on H∗ (LM) a coproduct H∗ (LM) → H∗ (LM) ⊗ H∗ (LM) which is identified via the linear map H (capu ) with the dual of the loop product H ∗ (LM) → (H ∗ (LM) ⊗ H ∗ (LM))∗+m . Now, observe that this coproduct on C∗ (L; U L∨ a ) coincides with the coproduct obtained by the tensor product of the coalgebra C ∗ (L) with the coalgebra (U L∨ , ν ∨ ). Thus we have proved the following result:

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Theorem D. The homology of the differential graded coalgebra C∗ (L; U L∨ a ) is isomorphic, via the isomorphism induces by capu , to the coalgebra H∗ (LM) := H ∗−m (LM) equipped with the dual of the loop product. It is worthwhile to observe here that there exists an isomorphism of differential graded algebras between the dual algebra Hom(C∗ (L; U L∨ a ), Q) and the differential graded algebra C ∗ (L; U La ). The structure of graded algebra on C ∗ (L; U La ) is defined by the tensor product of the two graded algebras: C ∗ (L) ⊗ (U L, ν).

7.3. Proof of Proposition 6 We will use the following general result about modules over differential graded algebras. Lemma 6. Let R be a differential graded algebra, S be a left semifree R-module, and let f, g : N → P be homomorphisms of right R-modules. If H (f ) = H (g) then the maps f ⊗ 1, g ⊗ 1 : N ⊗R (R ⊗ V ) → P ⊗R (R ⊗ V ) induce the same map in homology. P P Proof. Recall from 2.4 that S = n≥0 R ⊗ W (n) with dW (n) ⊂ R ⊗ k
H∗ (C∗O (L))

H∗ (δ)

cap[u]

H ∗ (L; Q)

/ H∗ (C∗ (L ⊕ L)) O cap[u] ⊗cap[u]

1!

/ H ∗ (L; Q) ⊗ H ∗ (L; Q)

Proof. By direct computation using definitions introduced in 4.3.

t u

End of proof of Proposition 6. We define R = C ∗ (L) ⊗ C ∗ (L). We remark that R ⊗ V ∗ ∨ = C ∗ (L; U L∨ a ) ⊗ C (L; U La ) is a semifree R-module. Note also that the diagram in Proposition 6 is the tensor product of a diagram of R-modules by R ⊗ V over R. By Lemma 7, this diagram commutes in homology when V = Q. It then commutes in homology by Lemma 6. This completes the proof. t u

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8. Loop product and Gerstenhaber product 8.1. Bar construction and chains of a differential graded Lie algebra Let (A, d) be a differential graded supplemented algebra, A = Q ⊕ A, and (P , d) (resp. (Q, d)) be a right (respectively left) differential graded A-module. The (normalized) twosided bar construction B(P ; A; Q) is defined as follows: Bk (P ; A; Q) = P ⊗ T k sA ⊗ Q whereP T k V = V ⊗k . A generic element is written m[a1 |a2 | · · · |ak ]n with degree |m| + |n| + ki=1 |sai |. The differential d = d0 + d1 is defined by d0

d1

Bk (P ; A; Q) → Bk (P ; A; Q), Bk (P ; A; Q) → Bk−1 (P ; A; Q), d0 (p[a1 |a2 | · · · |ak ]q) = d(p)[a1 |a2 | · · · |ak ]q −

k X (−1)|p|+i p[a1 |a2 | · · · |d(ai )| · · · |ak ]q i=1

+ (−1)|p|+k+1 p[a1 |a2 | · · · |ak ]d(q), d1 (p[a1 |a2 | · · · |ak ]q) = (−1)|p| pa1 [a2 | · · · |ak ]q +

k X (−1)|p|+i p[a1 |a2 | · · · |ai−1 ai | · · · |ak ]q i=2

− (−1)|p|+k p[a1 |a2 | · · · |ak−1 ]ak q P where i = j
C∗ (P ; L; Q) → B(P ; U L; Q), p ⊗ sx1 ∧ · · · ∧ sxk ⊗ q 7 → p ⊗

X

 εσ [xσ (1) | · · · |xσ (k) ] ⊗ q,

σ ∈6k

is a quasi-isomorphism. In particular: a) if P = N = U L is the canonical L-bimodule then 8 : C∗ (U L; L; U L) → B(U L; U L; U L) is also a quasi-isomorphism of differential graded U L-bimodules, b) if P = Q = Q is the trivial L-module then ψ : C∗ (L) → B(U L) is a quasi-isomorphism of differential graded coalgebras.

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8.2. The Hochschild complex and chains with coefficients in an adjoint module Let A = Q ⊕ A be a supplemented algebra, Aop be the opposite algebra and Ae = A ⊗ Aop be the enveloping algebra of A; thus Ae -modules are A-bimodules. The Hochschild cochain complex of A with coefficients in the A-bimodule P is the differential module CC ∗ (A; P ) = (Hom(T c (sA), P ), D0 + D1 ) ∼ = HomAe (B(A; A; A), P ) where the differential D0 + D1 of f ∈ Hom(T c (sA), P ) is defined by D0 (f )([a1 |a2 | · · · |ak ]) = dM (f ([a1 |a2 | · · · |ak ])) −

k X (−1)|f |+i f ([a1 | · · · |dA ai | · · · |ak ]), i=1

D1 (f )([a1 |a2 | · · · |ak ]) = − (−1)|sa1 | |f | a1 f ([a2 | · · · |ak ]) −

k X (−1)|f |+i f ([a1 | · · · |ai−1 ai | · · · |ak ]) i=2

+ (−1)|f |+k f ([a1 |a2 | · · · |ak−1 ])ak . The Hochschild cohomology of A with coefficients in M is HH ∗ (A; M) = H (CC ∗ (A; M)) = H ((Hom(T c (A), M), D0 + D1 )). The product defined on Hom(T c (sU L), U L) (see 2.2) commutes with the differential D0 + D1 . This is the Gerstenhaber product [15]. The next result specifies how the Gerstenhaber product on CC ∗ (U L; U L) and the differential graded coalgebra C ∗ (L; U L∨ a ), as considered in Theorem D, are related. The proof of this result relies upon the “inverse process” of Cartan–Eilenberg [5]. Proposition 8. Let L be a connected differential graded Lie algebra. Then there exists a canonical isomorphism of graded algebras HH ∗ (U L; U L) ∼ = Hom(H∗ (C∗ (L; U L∨ a )), Q). Proof. We form the commutative diagram Hom(U L)e (B(U L; U L; U L), U L)

Hom(9,U L)

∼ =

 Hom(B(U L), U L)

/ Hom(U L)e (C∗ (U L; L; U L), U L) ∼ =

Hom(ϕ,U L)

 / Hom(C∗ (L), U L)

where the vertical maps are the canonical isomorphisms of graded vector spaces. These isomorphisms define the graded algebra structure on Hom(U L)e (B(U L; U L; U L), U L) and Hom(U L)e (C∗ (U L; L; U L), U L).

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Therefore they can be viewed as isomorphisms of graded algebras (but not as isomorphisms of differential graded algebras). Since Hom(ψ, U L) is a homomorphism of graded algebras the map Hom(9, U L) is a quasi-isomorphism of differential graded algebras. Considering the commutative diagram α

Hom(U L)e (C∗ (U L; L; U L), U L)

/ Hom(C∗ (U L; L; U L) ⊗(U L)e U L∨ , Q)

∼ =



 Hom(C∗ (L), U L)

∼ =

/ Hom(C∗ (L) ⊗ U L∨ , Q)

one proves, in the same manner, that the canonical isomorphism α is an isomorphism of differential graded algebras. Finally, a straightforward computation shows that the canonical linear isomorphism β

∨ C∗ (L; U L∨ a ) → C∗ (U L; L; U L) ⊗(U L)e U L

is an isomorphism of chain complexes. From the commutative diagram Hom(C∗ (U L; L; U L) ⊗U Le U L∨ , Q) 

∼ =

β∨

/ Hom(C∗ (L; U L∨ ), Q) a

∼ =

Hom(C∗ (L) ⊗ U L∨ , Q)

 Hom(C∗ (L; U L∨ ), Q)

we deduce that β ∨ is an isomorphism of differential graded algebras. The composition β ∨ ◦ α ◦ Hom(9, U L) induces a canonical isomorphism of graded algebras HH ∗ (U L; U L) ∼ t u = Hom(H∗ (C∗ (L; U L∨ a )), Q). 8.3. Statement and proof of Theorem E Assume that C ∗ (L) is a Sullivan model of M. Using the natural isomorphism of graded algebras ([13]) HH ∗ (U L; U L) ∼ = HH ∗ (C ∗ (L); C ∗ (L)), ∼ =

∼ =

and the existence of quasi-isomorphisms of differential graded algebras, C ∗ (L) ← A → C ∗ (M), we deduce from Proposition 8 and Theorem D: Theorem E. Let M be a simply connected closed oriented manifold. There exists a natural isomorphism of graded algebras ∼ =

J : H∗ (LM) → HH ∗ (C ∗ (M); C ∗ (M)).

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9. I -homomorphism and Hochschild homology 9.1. A convenient definition of the I -homomorphism For each subspace Z ,→ M, we denote by LZ M the space of loops in LM with base point in Z. We thus have the commutative pullback diagram LZ M

i

  Z

/ LM  /M

Denote now by D a closed disk around the base point x0 . Since D is contractible, LD M is homotopy equivalent to D × M. The I -homomorphism I : H∗ (LM) → H∗−m (M) is the composition Excision H˜ ∗ (LM) → H∗ (LM, LM−x0 M) −−−→ H∗ (LD M, LS m−1 M) ∼ = Hm (D, S m−1 ) ⊗ H∗−m (M).

Therefore the I -homomorphism is precisely the Gysin map of the embedding M ,→ LM. V V Let ( V ⊗ V sV , d) be a relative Sullivan model for the free loop space (4.1, diagram (5)), and letV ω ∈ ( V )m be a cocycle V representing the fundamental class. Then the direct >m ⊕ S, where ( V )m = Q · ω ⊕ S, is an acyclic differential ideal in sum J = ( V ) V V V V ; the quotient map q : ( V , d) → (A, d) = ( V /J , d) is a quasi-isomorphism. We form the tensor product (A ⊗

V

V V sV , d) := (A, d) ⊗V V ( V ⊗ sV , d).

V V Lemma 7. The natural injection i : ( sV , 0) → (A ⊗ sV , d) defined by i(a) = m (−1) ω ⊗ a is a morphism of complexes of degree m inducing in cohomology the dual of the I -homomorphism, I ∨ : H ∗ (M) → H ∗+m (LM). Proof. V A model for the injection VLM−x0 M ,→ LM is given by the quotient map q : (A ⊗ sV , d) → (A/(ω) ⊗ sV , d). Therefore V a model for the cochain complex C ∗ (LM, LM−x0 M) is given by Ker q = (Q · ω ⊗ V , 0). This implies that a model for the composition ∼ =

H ∗ (M) → H ∗+m (LM, LM−x0 M) → H ∗+m (LM) is given by the composition

V

a7→(−1)m ω⊗a

sV −−−−−−→ Q · ω ⊗

V

V → H ∗ (A ⊗

V

sV , d).

t u

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9.2. Statement and proof of Theorem F Theorem F. Let J be as in Theorem E. There exists a canonical isomorphism of graded algebras, J¯, making the diagram H∗ (LM)

J ∼ =

/ HH ∗ (C ∗ (M); C ∗ (M)) HH ∗ (C ∗ (M),ε)

I

 H∗ (M)



 / HH ∗ (C ∗ (M))

commutative, where ε : C ∗ M → Q denotes the usual augmentation. ∗ ∨ Proof. Let (L, ∂) be such that C ∗ (L) = MM . The algebra C ∗ (L; U L∨ a ) = C (L) ⊗ U L is then a relative Sullivan model for the free loop space LM, and we can apply Lemma 6 (see 7.3) to this model to have a model for the restriction morphism. Now, since the ∨ projection q : C ∗ (L; U L∨ a ) → A ⊗ U La is a quasi-isomorphism, q admits a lifting 0 ∨ ∗ ∨ q : U L → C (L; U La ). Let u be a cycle in C(L) representing the orientation class [M] such that hω, [M]i = 1. We obtain a commutative diagram

U L∨

U L∨

q0

e

/ C ∗ (L; U L∨ ) a 

(−1)m −∩c

/ C∗ (L; U L∨ ) a

where e(a) = 1 ⊗ a. The dual diagram yields in homology a diagram of graded algebras whose vertical maps are isomorphisms: o H∗ (M) O

I



H∗ (U L) o

H∗+m (LM) O J

e0

(∗)

∗ ∗ H ∗ (Hom(C∗ (L; U L∨ a ), Q)) = H (C (L; U La ))

Note that the two C∗ (L)-bicomodules C∗ (L; U L) ⊗U L C∗ (U L; L) and C∗ (L) are quasiisomorphic. Therefore we have by duality a quasi-isomorphism of C ∗ (L)-bimodules C ∗ (L) ⊗ U L∨ ⊗ C ∗ (L) = (C∗ (L; U L) ⊗U L C∗ (U L; L))∨ → C ∗ (L). We deduce the isomorphism of differential graded vector spaces CC ∗ (U L; U L) = Hom(U L)e (C∗ (U L; L; U L), U L), CC ∗ (C ∗ (L); C ∗ (L)) = HomC ∗ (L)e ((C∗ (L; U L) ⊗U L C∗ (U L; L))∨ , C ∗ (L)). The isomorphisms Hom(U L)e (C∗ (U L; L; U L), U L) ∼ = Hom(C∗ (L), U L), HomC ∗ (L)e ((C∗ (L; U L) ⊗U L C∗ (U L; L))∨ , C ∗ (L)) ∼ = Hom(U L∨ , C ∗ (L))

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induce differentials on the right hand side terms. Now a simple computation (see for instance [13]) shows that the map D that associates to a map in Hom(C∗ (L), U L) the dual map in Hom(U L∨ , C ∗ (L)) is an isomorphism of complexes. This induces the following commutative diagram of complexes: UL o

e0

UL o

e0

Hom(U L)e (C∗ (U L; L; U L), U L)

Hom(C∗ (L), U L) ∼ = D

∼ =

 Hom(U L∨ , Q) o

Hom(U L∨ ,ε)

CC ∗ (C ∗ (L),ε)

HomC ∗ (L)e ((C∗ (L; U L)⊗U L C∗ (U L; L))∨ , Q) o

 Hom(U L∨ , C ∗ (L))

CC ∗ (C ∗ (L), C ∗ (L))

CC ∗ (C ∗ (L), Q) where ε : C ∗ (L) → Q denotes the canonical augmentation. The result follows now directly from the induced diagram in cohomology, combined with diagram (∗). t u References ˜ [1] Baker, A., Oze, C.: Complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups. In: Geometry and Topology (Aarhus, 1998), Comtemp. Math. 258, Amer. Math. Soc., Providence, RI, 1–19 (2000) Zbl 0979.57013 MR 1778093 [2] Bredon, G. E.: Topology and Geometry. Grad. Texts in Math. 139, Springer (1993) Zbl 0791.55001 MR 1224675 [3] Brylinski, J.-L.: Loop Spaces, Characterisic Classes and Geometric Quantization. Progr. Math. 107, Birkh¨auser Boston, Boston, MA (1993) Zbl 0823.55002 MR 1197353 [4] Burghelea, D., Vigu´e-Poirrier, M.: A model for cyclic homology and algebraic K-theory. J. Differential Geom. 22, 243–253 (1985) Zbl 0595.55009 MR 0834279 [5] Cartan, H., Eilenberg, S.: Homological Algebra. Princeton Univ. Press (1999) Zbl 0933.18001 MR 1731415 [6] Chas, M., Sullivan, D.: String topology. Preprint CUNY, Nov. 1999, math.GT/9911159, to appear [7] Chataur, D.: A bordism approach to string topology. Int. Math. Res. Not. 2005, no. 46, 2829– 2875 Zbl pre02246597 MR 2180465 [8] Cohen, R. L., Godin, V.: A polarized view of string topology. In: Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press, Cambridge, 127–154 (2004) Zbl 1095.55006 MR 2079373 [9] Cohen, R. L., Jones, J. D. S., Yan, J.: The loop homology algebra of spheres and projective spaces. In: Categorical Decomposition Techniques in Algebraic Topology, Progr. Math. 215, Birkh¨auser, 77–92 (2004) Zbl 1054.55006 MR 2039760

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