MODELS AND STRING HOMOLOGY A. L. GARCIA PULIDO AND J. D. S. JONES

Abstract. We give a method for doing systematic computations of the full structure of the string topology of a large family of manifolds. By building upon [Jon87], [CJ02] this problem is translated into the purely algebraic problem of calculating Hochschild homology and cohomology complete with its natural extra structure. Models provide us with a good technique for calculating these Hochschild theories with their extra structure and so much of our work is devoted to models for the Hochschild theories. Then our method for calculating string homology is to use this algebraic theory to give a very efficient way to compute the differentials in the spectral sequence of [CJY04].

1. Introduction Let M be a closed, oriented manifold of dimension d and let LM be the free loop space of M . In [CS99], Chas and Sullivan prove that the shifted homology groups H∗ (LM ) = H∗+d (LM ) have a very interesting product, and bracket (known respectively as the string product and string bracket) and an additional operator called the Batalin-Vilkovisky operator. This product and bracket make H∗ (LM ) into a Gerstenhaber algebra and together with the Batalin-Vilkovisky operator give H∗ (LM ) the structure of a Batalin-Vilkovisky algebra (see the definitions below). This structure on H∗ (LM ) is known as the string homology of M . In this paper, we give a method for doing systematic computations of the full structure of the string topology for a large family of manifolds. Using the results of [Jon87] and [CJ02] the topological problem of computing string topology is equivalent to the algebraic problem of calculating Hochschild homology and cohomology. Our approach is to use models (see below) to compute Hochschild homology and cohomology complete with the extra structure corresponding to the Batalin-Vilkovisky structure of string homology. Thus, much of this paper is devoted to calculating Hochschild homology and cohomology via models. Moreover, from these models we are able to obtain all of the extra Hochschild structure in a very transparent and effective way. We indicate how to adapt these techniques for the case of non zero characteristic. Finally, to complete the calculations of string homology we use these models, together with the structure obtained from them, as a very efficient method for calculating the differentials in the spectral sequence of [CJY04], when working in a field of characteristic zero. The first author was fully supported by Consejo Nacional de Ciencia y Tecnolog´ıa, M´exico. We wish to thank James Griffin for the helpful conversation regarding Keller’s work used in Section 5, Theorem 5.2. 1

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A. L. GARCIA PULIDO AND J. D. S. JONES

There is a lot of work on loop spaces, models, Hochschild homology and cohomology, and even cyclic homology, some of which overlaps with our work, for example [Tat57], [Smi81], [CJY04], [FTVP07]. Our aim is to give a direct and transparent way to compute the Hochschild homology and cohomology with all of its structure, adding to it what is needed to apply it to calculating string homology. We now present the central ideas of our work in more detail. A Gerstenhaber algebra V (see [Ger63]) is a graded vector space with an associative and graded commutative product • and a bracket [, ] that satisfy the following properties. (1) The bracket is antisymmetric. That is, [a, b] = −(−1)(|a|+1)(|b|+1) [b, a]. (2) The bracket has degree 1. That is, |[a, b]| = |a| + |b| + 1. (3) The bracket satisfies the Poisson identity. That is, it is a graded derivation of the product [a, b • c] = [a, b] • c + (−1)|b|(|a|+1) b • [a, c]. (4) The bracket satisfies the Jacobi identity. That is, it is a graded derivation of the bracket. [a, [b, c]] = [[a, b], c] + (−1)(|a|+1)(|b|+1) [b, [a, c]]. where a, b, c ∈ V . Note that conditions (1), (2) and (4) imply that [, ] is a graded Lie bracket of degree +1. Sometimes we find further structure in a Gerstenhaber algebra. A BatalinVilkovisky algebra (see [Get94, Chapter 1]) is a Gerstenhaber algebra with an additional operator ∆ : Vn → Vn+1 , the Batalin-Vilkovisky operator, such that (1) ∆ ◦ ∆ = 0 (2) [a, b] = (−1)|a| ∆(a • b) − (−1)|a| (∆a) • b − a • (∆b) for every a, b ∈ V . Notice that, in a Batalin-Vilkovisky algebra, the Lie bracket measures how much the Batalin-Vilkovisky operator fails to be a derivation. In [CS99] Chas and Sullivan investigated the structure of H∗ (LM ) and proved the following important theorem. Theorem 1.1 ([CS99], Theorem 5.4). The shifted homology groups H∗ (LM ) = H∗+d (LM ) form a Batalin-Vilkovisky algebra. We call this structure the string homology of M and we refer to its product and bracket as the string product and string bracket, respectively. We now describe two key results which constitute the basis of our work. The first is the relationship between Hochschild homology and the cohomology of the loop space of a simply connected topological space X. Theorem 1.2 ([Jon87]). Let X be a simply connected topological space. There exists an isomorphism of graded algebras ρ∗ : HH∗ (S ∗ (X), S ∗ (X)) → H ∗ (LX).

MODELS AND STRING HOMOLOGY

3

Moreover, via ρ, the B-operator defined on HH∗ (S ∗ (X), S ∗ (X)) is the dual of the Batalin-Vilkovisky operator on H∗ (LX). The second is a dual statement to the above. Theorem 1.3 ([CJ02]). Let M be a simply connected, oriented, closed smooth manifold. There exists an isomorphism of graded algebras, f : H∗ (LM ) → HH ∗ (S ∗ (M ), S ∗ (M )). Further, we have the following theorem Theorem 1.4 (Tradler, [Tra08]). There is a Batalin-Vilkovisky algebra structure in HH ∗ (S ∗ (M ), S ∗ (M )). Finally, in [FT08] Felix and Thomas proved the following. Theorem 1.5 (Felix, Thomas). Assume the hypothesis of Theorem 1.3 and in addition suppose that the coefficient field is of characteristic zero. Then the map f is an isomorphism of Batalin-Vilkovisky algebras. Now we introduce the definition of a model for an algebra A. Definition 1.6. Let A be a graded commutative algebra. A model P for A is a differential graded commutative algebra, together with a map of differential graded algebras  : P → A, where A is considered as a differential graded algebra with zero differential, such that • P is free as a graded commutative algebra • the induced homomorphism H ∗ (P ) → A is an isomorphism. Theorems 1.2 and 1.3 motivate us to study HH∗ (A, A) and HH ∗ (A, A) for a graded algebra A. In particular, given a description of A it would be desirable to use it to obtain a description of HH∗ (A, A) and HH ∗ (A, A). Indeed, amongst other results, we use a model for A to obtain models for HH∗ (A, A) and HH ∗ (A, A) with their additional algebraic structure. The first results of this kind are due to Smith in [Smi81], based on work of Tate (see [Tat57]). However, the B-operator is not discussed and it is assumed that K = Q and that A is a graded complete intersection algebra. The starting point for our work was to try to adapt these methods to calculate string topology with all its structure. Our main results combined with Theorems 1.2 and 1.3, make the homology and cohomology of the free loop space much easier to manipulate when working over a field of zero characteristic. We illustrate this by giving results regarding the free loop space with proofs that are reduced into a very simple algebraic setting when using our theorems. We also provide examples where we compute the full Batalin-Vilkovisky structure of the free loop space of several topological spaces using our theorems. These examples include the Grassmannian of 2-planes in C4 , which should give the reader the insight and techniques to obtain the string topology of any Grassmann manifold. The outline of the paper is as follows. In Sections 2 and 3 we turn our attention to the algebraic preliminaries. We discuss some basic facts in these theories - the product and the Connes or B-operator in Hochschild

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homology and the Gerstenhaber algebra structure of Hochschild cohomology. We also discuss the Hochschild homology and cohomology of a free graded commutative algebra. Building on the free case, in Section 4 (respectively, Section 5), we describe a model for the Hochschild homology (respectively, cohomology) of a graded commutative algebra and the corresponding B-operator (respectively Gerstenhaber algebra structure), see Theorem 4.2 (respectively, Theorem 5.2). In Sections 4 and 5 we use our theorems to prove further results concerning the nature of the free loop space, see Theorem 4.7 and Corollary 5.4. The first of these statements was proven in a weaker form by Smith in [Smi81, Theorem 4.1]. Here we give a different and more general proof to exemplify the use of our techniques. In Sections 6 and 7, we use our theory to highlight how these models simplify working with string topology in zero characteristic. We give explicit use of our theory by computing the cohomology of the free loop space and the full Batalin-Vilkovisky structure of the spheres, the projective spaces and the Grassmann manifold of two planes in C4 . We conclude this introduction with a few general conventions. Throughout this paper we will use cohomological grading conventions which means that boundary operators will increase degree by 1. For example, the differential in a differential graded algebra will always increase degree by 1. This means that the Hochschild boundary operator b will increase degree by 1 whereas the Connes operator B will decrease degree by 1. Additionally, we will assume all the algebras A considered here to be finitely generated. If V is a graded vector space, we denote by Σk V the graded vector space defined by (Σk V )n = Vn−k and by σ k : V → Σk V the linear isomorphism which raises degree by k. 2. Hochschild Homology For a detailed account of Hochschild homology, the reader is referred to [CE56, Chapters IX and XI], [Wei94, Section 9.9.1], and [Lod92, Sections 5.3 & 5.4]. Let (A, ∂) be a differential graded algebra and define the double complex Cn,p (A, A) = (A⊗n+1 )p with horizontal differential n X n ∂(a0 , . . . , an ) = (−1) (−1)|a0 |+···+|ai−1 | (a0 , . . . , ai−1 , ∂ai , ai+1 , . . . , an ), i=0

induced by ∂, and vertical differential n X b(a0 , . . . , an ) = (−1)i (a0 , . . . , ai ai+1 , . . . , an ) i=0

+ (−1)n+|an |(|a0 |+|a1 |+...+|an−1 |) (an a0 , a1 , . . . , an−1 ). The operator b is often called the (algebraic) Hochschild boundary map. Definition 2.1. The Hochschild chain complex C∗ (A, A) of the differential graded algebra (A, ∂) is the, total complex of the above double complex.

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5

Taking account of the bidegrees of ∂ and b, M Cn,p (A, A), Cr (A, A) = p−n=r

and the differential (which raises total degree by 1) is b + d. The Hochschild homology of (A, ∂), denoted by HH∗ (A, A), is the homology of the Hochschild complex. We will normally restrict attention to positively graded simply connected differential graded algebras (A, ∂) of finite type over the ground field K. Here simply connected means that H 0 (A, ∂) = K and H 1 (A, ∂) = 0. In this case we will use the normalised Hochschild complex and make no distinction in notation. For a detailed discussion of this see [CE56, Chapter IX, Section 6]. In addition, we have Connes’ B-operator B : HHn (A, A) → HHn−1 (A, A), see [Lod92, Section 2.1.7]. This operator provides us with important additional structure. In particular when A is graded commutative there are products HHp (A, A) ⊗ HHq (A, A) → HHp+q (A, A) which makes HH∗ (A, A) into a graded commutative algebra, [Lod92, Section 4.2.6]. In this case, the B operator is a derivation of the product in Hochschild homology [Lod92, Corollary 4.3.4]. Thus, when A is a differential graded commutative algebra this gives us extra structure in HH∗ (A, A) which by Theorem 1.2 is related to the Batalin - Vilkoviski operator in string homology. Example 2.2. Let K be a field of characteristic zero and A = K[V ] be the free graded commutative algebra generated by the graded vector space V with zero differential. Notice that C0,∗ (A, A) = K[V ] and C1,∗ (A, A) = K[V ] ⊗ K[V ]. The inclusion of V in K[V ] and the map v 7→ 1 ⊗ v give maps V → C∗ (A, A) and Σ−1 V → C∗ (A, A) (taking account of the grading in the total complex). This gives a map of chain complexes V ⊕ Σ−1 V → C∗ (A, A) where the boundary map in V ⊕Σ−1 V is zero. Using the graded commutative product in HH∗ (A, A) this gives a map of algebras K[V ] ⊗ K[Σ−1 V ] = K[V ⊕ Σ−1 V ] → HH∗ (A, A). A classical result of Hochschild, Kostant and Rosenberg (see [Lod92, Proposition 5.4.6] and [HKR62, Theorem 5.2]) tells us that this map is an isomorphism of graded algebras K[V ] ⊗ K[Σ−1 V ] → HH∗ (K[V ], K[V ]). It also follows that under this isomorphism the Connes’ B operator corresponds to the derivation of K[V ] ⊗ K[Σ−1 V ] defined on the generators by B(v) = σ −1 v, B(σ −1 w) = 0,

v∈V

w ∈ V.

Using a homogeneous basis x1 , . . . , xn of V we see that the elements of HH∗ (K[V ], K[V ]) of the form p(x1 , . . . , xn )Bxi1 , . . . Bxim where p ∈ K[V ] form a basis for HH∗ (K[V ], K[V ]).

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Remark 2.3. When the field K is of non zero characteristic, there exists an analogous isomorphism HH∗ (K[V ], K[V ]) ∼ = K[V ] ⊗ Γ[Σ−1 V ]. Here Γ[Σ−1 V ] denotes the tensor product of the exterior algebra in the odd generators of Σ−1 V and the divided polynomial algebra, or divided power algebra, on the even generators of Σ−1 V . For a definition of the divided polynomial algebra, see [Ber73, Definition 1.5, Chapter I]. In the generators, the B-operator is given by the same formulas as above. 3. Hochschild Cohomology Let (A, ∂) be a differential graded algebra over K and let C n,p (A, A) = Homp (A⊗n , A) be the set of K-linear maps f : A⊗n → A that raise degree by p. Observe that C n,p (A, A) is a double complex with the differentials defined by the following formulas. For f ∈ Hom(A⊗n+1 , A) the horizontal differential δ is defined by n X δ(f )(a1 ⊗ · · · ⊗ an ) = (−1) (−1)|a1 |+···+|ai−1 | f (a1 ⊗ · · · ⊗ ∂(ai ) ⊗ · · · ⊗ an ) n

i=1

and the vertical differential is defined by β(f )(a1 ⊗ . . . ⊗ an+1 ) = a1 f (a2 ⊗ . . . ⊗ an+1 ) X + (−1)i f (a1 ⊗ . . . ⊗ ai ai+1 ⊗ . . . ⊗ an+1 ) 0
+ (−1)n+1+|an+1 |(|a1 |+...+|an |) f (a1 ⊗ . . . ⊗ an )an+1 . Definition 3.1. The Hochschild cochain complex C ∗ (A, A) of the differential graded algebra (A, ∂) is the total complex of the double complex C n,p (A, A). Taking account of the bidegrees of δ and β, M C r (A, A) = C n,p (A, A), p+n=r

and the differential (which raises degree by 1) is β + δ. The Hochschild cohomology of (A, ∂), denoted by HH ∗ (A, A), is the homology of C ∗ (A, A). Hochschild cohomology also has a product, the cup product. At the level of cochains, this product is given by (f ∪ g)(a1 ⊗ . . . ⊗ ap ⊗ b1 ⊗ . . . ⊗ bq ) = f (a1 ⊗ . . . ⊗ ap )g(b1 ⊗ . . . ⊗ bq ), where f ∈ C p,∗ (A, A) and g ∈ C q,∗ (A, A). This cochain level product descends to cohomology to give a graded commutative product in HH ∗ (A, A). Also HH ∗ (A, A) has the structure of a Lie algebra, with the bracket defined as follows. Given f ∈ C m (A, A) and g ∈ C n (A, A) we define f ◦ g(a1 , . . . , am+n−1 ) = m X i=1

(−1)(i−1)(n−1) f (a1 , . . . , ai−1 , g(ai , . . . , ai+n−1 ), ai+n , . . . , am+n−1 )

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and corresponding bracket operation [f, g] = f ◦ g − (−1)(m−1)(n−1) g ◦ f. Once more this bracket operation descends to cohomology and in [Ger63], Gerstenhaber proved that this bracket together with the cup product make HH ∗ (A, A) into a Gerstenhaber algebra. In view of Theorem 1.3 this Gerstenhaber algebra is related to string homology. Example 3.2. Let A = K[V ] be the free graded commutative algebra over the field K. Then C 0,∗ (A, A) = K[V ] and C 1,∗ (A, A) = Hom(K[V ], K[V ]) and in this case we get a map of chain complexes V ⊕ ΣV ∗ → C ∗ (A, A) where V ⊕ ΣV ∗ has zero differential. Once more this gives us an isomorphism of graded algebras K[V ⊕ ΣV ∗ ] = K[V ] ⊗ K[ΣV ∗ ] → HH ∗ (A, A) ([HKR62, Theorem 5.2]). Furthermore, using the elementary properties of the Gerstenhaber bracket in low degrees we see that [v, w] = 0,

v, w ∈ V

[σφ, v] = φ(v), [σφ, σψ] = 0,

v ∈V φ∈V∗

φ, ψ ∈ V ∗ .

Given these formulas the Jacobi and Poisson identities completely determine the bracket in HH ∗ (K[V ], K[V ]). 4. Models and Hochschild Homology Given a model P = (K[V ], ∂, ) for a graded commutative algebra A we will now construct a model L(P ) for HH∗ (A, A). As in Example 2.2, when K has characteristic zero, the underlying graded commutative algebra for this model is given by K[V ⊕ Σ−1 V ]. The following lemma will be very important for the construction of the differential of the model L(P ). Lemma 4.1. Let P = (K[V ], ∂) be a differential graded algebra. There exist unique derivations δ, β : K[V ⊕ Σ−1 V ] → K[V ⊕ Σ−1 V ] of degree +1 and −1 respectively, such that (1) (2) (3) (4)

δ 2 = β 2 = 0, δβ + βδ = 0, δv = ∂v for every v ∈ V , βv = σ −1 v for every v ∈ V .

We now prove the following theorem. Theorem 4.2. Let A be a graded commutative algebra and P = (K[V ], ∂, ) be a model for A. Then there is a map of differential graded algebras L() : (K[V ⊕ Σ−1 V ], δ) → (HH∗ (A, A), 0) such that • the map L() makes L(P ) = ((K[V ⊕ Σ−1 V ], δ), L()) into a model for HH∗ (A, A) and

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A. L. GARCIA PULIDO AND J. D. S. JONES

• the following diagram commutes (4.1)

Hn (K[V ⊕ Σ−1 V ], δ) β



Hn−1 (K[V ⊕ Σ−1 V ], δ)

/ HHn (A, A) 

B

/ HHn−1 (A, A)

where β is defined as in Lemma 4.1. Remark 4.3. In [Smi81], Smith proves the first result of this kind, based on work of Tate (see [Tat57]). However, the B-operator is not discussed and it is assumed that A is a graded complete intersection algebra. Remark 4.4. When the characteristic of K is non zero, an analogous theorem holds by replacing K[V ⊕ Σ−1 V ] with K[V ] ⊗ Γ[Σ−1 V ]. The corresponding construction is the same as that given below. In this case, the underlying graded commutative algebra K[V ] ⊗ Γ[Σ−1 V ] is inspired by Remark 2.3. Note that this agrees with the case when K has zero characteristic, as in this case, K[V ⊕ Σ−1 V ] = K[V ] ⊗ Γ[Σ−1 V ]. Proof. Throughout this proof we will be using the normalised Hochschild complex without making any distinction in notation. First define the map ψ : K[V ⊕ Σ−1 V ] → C∗ (K[V ], K[V ]) that arises from the extension as a map of algebras of the linear maps V ,→ C0 (K[V ], K[V ]) = K[V ] and

Σ−1 V ,→ C1 (K[V ], K[V ]) σ −1 v 7→ 1 ⊗ v = Bv where B represents the normalised Connes operator at the chain level. In this case it is easy to check that b ◦ ψ = 0. Indeed we have that b is zero in C1 (K[V ], K[V ]) because K[V ] is commutative and that b is zero in C0 (K[V ], K[V ]) by definition. These facts together with the fact that b is a derivation of the product in C∗ (K[V ], K[V ]) show that b ◦ ψ = 0. Thus the map ψ : (K[V ⊕ Σ−1 V ], 0) → (C∗ (K[V ], K[V ]), b) is a map of differential graded algebras. We now claim that ψ gives a map of chain complexes ψ : (K[V ⊕ Σ−1 V ], δ) → (C∗ (K[V ], K[V ]), b + ∂) where, abusing notation, we denote by ∂ the differential in C∗ (K[V ], K[V ]) that satisfies ∂b + b∂ = 0, ∂B + B∂ = 0 and which is induced by the differential ∂ in K[V ] (see Subsection 2). The formula ψ ◦ δ = (b + ∂) ◦ ψ can be verified directly on the generators v ∈ V and σ −1 v ∈ Σ−1 V of K[V ⊕ Σ−1 V ] and from this follows that it is satisfied in the whole of K[V ⊕Σ−1 V ]. By Example 2.2, ψ is a quasi-isomorphism when restricted to the columns,

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9

so the induced map of the total complexes is a quasi-isomorphism, by a standard lemma of double complexes. Similarly, the map of chain complexes ˜: (C∗ (K[V ], K[V ]), b + ∂) → (C∗ (A, A), b), induced by the map in the model  : (K[V ], ∂) → (A, 0), is a quasi-isomorphism restricted to the columns and therefore, by a standard argument of double complexes, the map of the total complexes is a quasi-isomorphism. Finally, let η : (K[V ⊕ Σ−1 V ], δ) → (C∗ (A, A), b) be the composition η = ˜ ◦ ψ. It is clear that the induced homomorphism η∗ : H∗ (K[V ⊕ Σ−1 V ], δ) → HH∗ (A, A) is an isomorphism of graded algebras. We now proceed to prove the second part of the theorem. Although β is a derivation at the chain level, B is not. Nevertheless, B is a derivation up to homotopy (see [Lod92, Corollary 4.3.4]). Using this fact we prove the following lemma: Lemma 4.5. The following diagram commutes up to chain homotopy K[V ⊕ Σ−1 V ]s,t β



K[V ⊕ Σ−1 V ]s−1,t

/ Cs,t (K[V ], K[V ]) 

B

/ Cs−1,t (K[V ], K[V ])

Proof. By definition of ψ we have ψ(p(σ −1 v1 )α1 · · · (σ −1 vn )αn ) = p(Bv1 )α1 · · · (Bvn )αn where p ∈ K[V ] and vi ∈ V for i = 1, . . . , n. In [Lod92, Corollary 4.3.5] Loday proves that, at the chain level, B(xBy) = BxBy. From this it follows that B(ψ(p(σ −1 v1 )α1 · · · (σ −1 vn )αn )) = (B(p))(Bv1 )α1 · · · (Bvn )αn . On the other hand, from the definition of β one gets ψ ◦ β(p(σ −1 v1 )α1 · · · (σ −1 vn )αn ) = ψ(β(p)(σ −1 v1 )α1 · · · (σ −1 vn )αn ) = ψ(β(p))(Bv1 )α1 · · · (Bvn )αn . It is sufficient to prove that there is a chain homotopy that makes the following diagram commute K[V ] β



K[V ]

/ K[V ⊕ Σ−1 V ] ψ

/ K[V ⊕ Σ−1 V ]

ψ

/ C∗ (K[V ], K[V ]) 

B

/ C∗ (K[V ], K[V ])

Since B is a derivation up to homotopy, there exists a chain homotopy h(2) : (C∗ (K[V ], K[V ]), b + ∂)⊗2 → (C∗ (K[V ], K[V ]), b + ∂) between the maps u ⊗ v 7→ B(u · v) and u ⊗ v 7→ (Bu)v + (−1)|u| uBv,

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where · denotes the inner product in C∗ (K[V ], K[V ]). This induces a chain homotopy h(n) : (C∗ (K[V ], K[V ]), b + ∂)⊗n → (C∗ (K[V ], K[V ]), b + ∂) between the maps u1 ⊗ · · · ⊗ un 7→ B(u1 · . . . · un ) and n X u1 ⊗ · · · ⊗ un 7→ (−1)|u1 |+...+|ui−1 | u1 . . . ui−1 (Bui )ui+1 . . . un . i=1

Finally define h : K[V ⊕ Σ−1 V ] → C∗ (K[V ], K[V ]) in terms of a homogeneous basis x1 , . . . , xm of V as 1 m )β(xi1 )γi1 · · · β(xis )γis ⊗· · ·⊗x⊗α h(xα1 1 · · · xαmm )β(xi1 )γi1 · · · β(xis )γis = h(n) (x⊗α m 1

where n = α1 + . . . + αm . This is precisely a chain homotopy between ψβ and Bψ.   Remark 4.6. Using the isomorphism from Theorem 1.2, we will later prove that Theorem 4.2 implies, given a model for a simply connected topological space M , we can obtain a model for LM . For the remainder of this section we will work in a field of characteristic zero. This is because, in the non zero characteristic case, it is not true that there is a graded commutative model for the cochains S ∗ (M ). Working with a model for LM makes H ∗ (LM ) an object which is much easier to handle. It helps to reduce topological problems into a very easy algebraic setting. An example of this is the following theorem, which was first proved by Smith for the case when H ∗ (X) is a graded complete intersection algebra in [Smi81, Theorem 4.1], but we give a different proof using Theorem 4.2. i∗

Theorem 4.7. The map H ∗ (LM ) −→ H ∗ (ΩM ) is surjective if and only if H ∗ (M ) is free. The idea of the proof is to use the facts that, from a minimal model P = (K[V ], ∂) for M , we get • a minimal model for ΩM which is given by ΩP = (K[Σ−1 V ], 0) (see Theorem 4.8) and • the model for LM described in Theorem 4.2, given by LP = (K[V ⊕ Σ−1 V ), ∂). We then prove that there is a map of chain complexes p∗ : (K[V ⊕ Σ−1 V ], δ) → (K[Σ−1 V ], 0),

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induced by the projection onto the second factor p : K[V ⊕Σ−1 V ] → K[Σ−1 V ], and a commutative diagram H ∗ (LM )

i∗

O

/ H ∗ (ΩM ) O

∼ =

∼ = p∗

H ∗ (LP )

/ H ∗ (ΩP )

Hence Theorem 4.7 is equivalent to the statement p∗ is surjective if and only if ∂ = 0, which we then prove directly. In order to prove Theorem 4.7 we require the following results. Theorem 4.8. Let M be a simply connected topological space. Then there exists an isomorphism of algebras ∼ HH∗ (S ∗ M, K). (4.2) H ∗ (ΩM ) = In particular, if P = (K[V ], ∂) is a minimal model for M , then ΩP := (K[Σ−1 V ], 0) is a model for H ∗ (ΩM ). For a proof of the first statement see [Ada56]. For the part concerning models, please refer to [FHT01]. The next three lemmas will translate Theorem 4.7 from a topological setting into an algebraic setting. Lemma 4.9. Let  : S ∗ M → K be the augmentation map. The following diagram commutes i∗

H ∗ (LM )

/ H ∗ (ΩM ) O

O

∼ =

∼ =

HH∗ (S ∗ M, S ∗ M )

∗

/ HH∗ (S ∗ M, K)

where the left vertical isomorphism is given by Theorem 1.2 and the right, by Equation 4.2. Proof. By construction of the vertical maps, the diagram above is commutative.  Lemma 4.10. Let P = (K[V ], ∂) be a minimal model for M , γ : P → S ∗ M the quasi-isomorphism obtained from the model and  : S ∗ M → K and η : P → K augmentation maps. Then the following diagram commutes. HH∗ (S ∗ M, S ∗ M )

∗

O

/ HH∗ (S ∗ M, K) O

γ∗

γ∗ η∗

HH∗ (P, P )

/ HH∗ (P, K)

Proof. First notice that the following diagram commutes. S ∗ OM



γ

P

/K O γ

η

/K

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A. L. GARCIA PULIDO AND J. D. S. JONES

Therefore, the following diagram is commutative (S ∗ M )⊗n+1

∗

O

γ∗

/ (S ∗ M )⊗n ⊗ K O γ∗

(K[V ])⊗n+1

η∗

/ (K[V ])⊗n ⊗ K

Finally, since the diagram commutes at the chain level, then the diagram commutes in homology.  Lemma 4.11. Let P = (K[V ], ∂) be a minimal model for M and η : P → K the augmentation map. Further, let LP = (K[V ⊕ Σ−1 V ], δ) be the model for LM given in Theorem 4.2 and ΩP = (K[Σ−1 V ], 0) be the model for ΩM given in Theorem 4.8. Finally, let p : K[V ⊕ Σ−1 V ] −→ K[Σ−1 V ] be the projection onto the second factor. Then p is a map of chain complexes and the following diagram is commutative HH∗ (P, P )

η∗

/ HH∗ (P, K) O

p∗

/ H ∗ (ΩP )

O

H ∗ (LP )

where the left vertical isomorphism is given by Theorem 4.2 and the right, by Theorem 4.8. Proof. We first prove that p is a map of chain complexes. Let β be as in Theorem 4.2. Note that p(δv) = 0 because δv = ∂v ∈ K[V ]. Since P is a minimal model, ∂v is a decomposable element and, using the fact that β is a derivation, β(δv) = β(∂v) ∈ I(V ) and so p(δ(σ −1 v)) = p(β(δv)) = 0. In conclusion, p induces a map of chain complexes. We now focus on the second part of the lemma. We will first analyse the composite map C ∗ (K[V ⊕ Σ−1 V ], δ) −→ C∗ (K[V ], K[V ]) −→ C∗ (K[V ], K). Recall that the map in the left is defined by the inclusions V → K[V ] = C0 (K[V ], K[V ]) and Σ−1 V → C1 (K[V ], K[V ]) = K[V ] ⊗ K[V ] σ −1 w 7→ 1 ⊗ w and extended as a map of algebras. The map induced by η is defined by η∗ (r0 ⊗ r1 ⊗ · · · ⊗ rn ) = η(r0 ) ⊗ r1 ⊗ · · · ⊗ rn and so v 7→ 0 and 1 ⊗ w 7→ 1 ⊗ w. It follows that the composite takes the generators v ∈ V to 0 and σ −1 w ∈ Σ−1 V to 1 ⊗ w and extends into a map of algebras. In a similar fashion we analyse the second composite map C ∗ (K[V ⊕ Σ−1 V ], δ) −→ C ∗ (K[Σ−1 V ], 0) −→ C∗ (K[V ], K).

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13

Note that the map in the right is defined by the inclusion Σ−1 V → C1 (K[V ], K) = K ⊗ K[V ] σ −1 w 7→ 1 ⊗ w and extended as a map of algebras. Also, recall that the map in the left is the induced by the projection onto the second factor. Thus, the generators v ∈ V and σ −1 w ∈ Σ−1 V are send to 0 and 1 ⊗ w, respectively and the composite extends as a map of algebras. We have just proved that the diagram is commutative.  Using the above results, the following theorem is reduced to a simple algebraic exercise. i∗

Theorem 4.7. The map H ∗ (LM ) −→ H ∗ (ΩM ) is surjective if and only if H ∗ (M ) is free. Proof. Let P = (K[V ], ∂) be a minimal model for M and LP , ΩP and p : K[V ⊕ Σ−1 V ] −→ K[Σ−1 V ] be as in Lemma 4.11. Suppose ∂ 6≡ 0. By combining the diagrams of Lemmas 4.9, 4.10 and 4.11, we have the following commutative diagram. H ∗ (LM )

i∗

O

∼ =

H ∗ (LP )

/ H ∗ (ΩM ) O ∼ =

p∗

/ H ∗ (ΩP )

Thus i∗ is not surjective if and only if p∗ is not surjective. Since ∂ 6≡ 0 there exist v ∈ V of minimal degree such that ∂v 6= 0. Therefore, δ(σ −1 v) = β(∂v) 6= 0 and so σ −1 v is not a cycle in LP but it is a cycle in ΩP . Notice that σ −1 v is also an element in Σ−1 V with lowest degree such that δ(σ −1 v) 6= 0. As a consequence, by the Leibniz rule and the linearity of δ, we can deduce that for any homogeneous element q ∈ K[V ⊕ Σ−1 V ] with |q| < |σ −1 v| = |v| − 1, then δq = 0. Suppose that there exists a cycle α in LP such that p(α) = p(σ −1 v). Thus α = σ −1 v + w with w ∈ I(V ), the ideal generated by V . Since α is a cycle we have 0 = δ(α) = δ(σ −1 v) + δ(w) and so δ(w) = −δ(σ −1 v) is not a cycle in LP . Since w ∈ I(V ) is of the form X xi vi with xi ∈ LP and vi ∈ V , then X X δ(w) = xi δ(vi ) + δ(xi )vi = δ(xi )vi = 0 because |xi |, |vi | < |w| = |α| < |v| and v was of minimal degree. This contradiction proves that i∗ is surjective implies that H ∗ (M ) is free. The converse is an easy exercise. 

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5. Models and Hochschild Cohomology For a graded algebra A with a model P = (K[V ], ∂), we now construct a model G(P ) for HH ∗ (A, A). As in Example 3.2, the underlying graded commutative algebra of G(P ) is given by K[V ⊕ (Σ−1 V )∗ ]. In order to get the differential for G(P ) we will need the following simple lemma. Lemma 5.1. Let ξ be a derivation of K[V ] and χ : K[V ⊕ Σ−1 V ] → K[V ⊕ Σ−1 V ] be the extension of ξ described in Lemma 4.1. Then there is a unique derivation Dξ : K[V ⊕ (Σ−1 V )∗ ] → K[V ⊕ (Σ−1 V )∗ ] such that (1) If v ∈ K[V ] ⊂ K[V ⊕ (Σ−1 V )∗ ], then Dξ (v) = ξ(v). (2) If v, w ∈ V , Dξ , then Dξ ((σ −1 v)∗ )(σ −1 w) = (σ −1 v)∗ ◦ χ(σ −1 w). In the right hand side of this last expression, (σ −1 v)∗ denotes the extension of the linear map (σ −1 v)∗ : Σ−1 V → K to K[V ⊕ (Σ−1 V )∗ ] as a K[V ]-linear map and a derivation. Moreover, Dξ2 = 0 and for any u, v ∈ K[V ⊕ (Σ−1 V )∗ ] and the bracket defined in Example 3.2, Dξ satisfies Dξ [u, v] = [Dξ (u), v] + (−1)(|u|+1)(|v|+1) [u, Dξ (v)]. That is, Dξ is a graded derivation of the bracket. Proof. The existence and uniqueness of Dξ follows immediately from the definition of Dξ on the generators of K[V ⊕ (Σ−1 V )∗ ] and its extension as a derivation of the product. Given the properties of [, ] and Dξ , it is sufficient to check that the identity Dξ [u, v] = [Dξ (u), v] + (−1)(|u|+1)(|v|+1) [u, Dξ (v)] holds on pairs of generators of K[V ⊕ (Σ−1 V )∗ ]. A direct calculation shows that this identity is satisfied on the generators.  The same argument shows that Dξ2 = 0. Theorem 5.2. Let A be a graded commutative algebra and P = (K[V ], ∂, ) a model for A. For the bracket defined as in Example 3.2 there is a unique operator δ of K[V ⊕ (Σ−1 V )∗ ] such that δ 2 = 0, δ is a derivation of the product, δ is a derivation of the bracket and δ(v) = ∂(v),

v ∈ V.

Moreover, there is a natural isomorphism of Gerstenhaber algebras G() : H ∗ (K[V ⊕ (Σ−1 V )∗ ], δ) → HH ∗ (A, A). Proof. Set δ = D∂ as in lemma 5.1. Let G(P ) be the differential Gerstenhaber algebra with underlying free commutative algebra K[V ⊕ (Σ−1 V )∗ ] equipped with the differential δ, and the elementary bracket described in Example 3.2. There is a map of algebras η : K[V ⊕ (Σ−1 V )∗ ] → C ∗ (K[V ], K[V ]), which is the extension, as a map of graded algebras, of the K-linear maps V ,→ C 0 (K[V ], K[V ]) = K[V ]

MODELS AND STRING HOMOLOGY

15

and (Σ−1 V )∗ ,→ C 1 (K[V ], K[V ]) = Hom(K[V ], K[V ]) where an element (σ −1 v)∗ gets mapped to the unique derivation of K[V ] which agrees with v ∗ in V . This map is the antisymmetrisation map seen in Section 3. We claim that β ◦ η = 0, where β is the Hochschild coboundary map. Indeed β ◦ η = 0 in C 0 (K[V ], K[V ]) because K[V ] is graded commutative and β ◦ η((σ −1 v)∗ ) = 0 since η((σ −1 v)∗ ) ∈ C 1 (K[V ], K[V ]) is a graded derivation and Ker(β : C 1 (K[V ], K[V ]) → C 2 (K[V ], K[V ])) = Der(K[V ]). Given that β is a derivation of the cup product, we conclude that β ◦ η = 0 (see [Ger63, Section 7]). Thus the map η is compatible with the differentials and hence it gives a map of differential graded algebras η : (K[V ⊕ (Σ−1 V )∗ ], 0) → (C ∗ (K[V ], K[V ]), β). Furthermore, η induces a map of chain complexes η : (K[V ⊕ (Σ−1 V )∗ ], δ) → (C ∗ (K[V ], K[V ]), β + ∂), where, abusing notation, ∂ is the differential in C ∗ (K[V ], K[V ]) induced by ∂. In fact, to see that η ◦ δ = (β + ∂) ◦ η, first verify it directly on the generators v ∈ V and (σ −1 v)∗ ∈ (Σ−1 V )∗ and then notice that since η is a map of algebras and δ, β, ∂ are derivations this identity holds in the whole of K[V ⊕ (Σ−1 V )∗ ]. By the Hochschild-Kostant-Rosenberg theorem, η is an isomorphism of differential graded algebras. Now, since  : (K[V ], ∂) → (A, 0) is a quasi-isomorphism, by a result of Rickard (see [Ric91]), there exists an isomorphism of graded algebras ˜: HH ∗ (K[V ], K[V ]) → HH ∗ (A, A). By a result of Keller, [Kel04], this is also an isomorphism of Gerstenhaber algebras. Define G() : H ∗ (K[V ⊕ (Σ−1 V )∗ ], δ) → HH ∗ (A, A) as the composition G() = ˜◦η. By definition, G() is an isomorphism of Gerstenhaber algebras.  Remark 5.3. As in the case of Hochschild homology, if we have a model for a simply connected, oriented, closed and smooth manifold M , combining Theorems 5.2 and 1.3, we obtain a model for H∗ (LM ). Notice that, together with the model for H ∗ (LM ) ∼ = HH∗ (S ∗ (M ), S ∗ (M )) and its B-operator, we get the full Batalin-Vilkovisky structure on H∗ (LM ). From here until the end of this section we will work in a coefficient field of characteristic zero, since we will be assuming the existence of a model for the cochains S ∗ (M ). Again, as in the case of H ∗ (LM ), the model from Theorem 5.2 gives a simpler description of H∗ (LM ), turning it into a far less complicated object to deal with. To illustrate this, we combine the previous theorem and a result of Cohen-Jones-Yan [CJY04] to obtain a very straightforward proof of the following corollary.

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Corollary 5.4. Let M be a simply connected manifold and P = (K[V ], ∂) a (minimal) model for H ∗ (M ). If N denotes Ker(δ : (Σ−1 V )∗ → K[V ⊕ (Σ−1 V )∗ ]), where δ is defined as in Theorem 5.2, then K[N ] ⊂ H∗ (LM ). For the proof of this corollary, we will need the following theorem. We will also make extensive use of this theorem in future sections. Theorem 5.5 (Cohen-Jones-Yan). Let M be a closed, oriented, simply connected manifold. There is a second quadrant spectral sequence of algebras r , dr : p ≤ 0, q ≥ 0} such that {Ep,q r is an algebra and the differential dr : E r → E r (1) E∗,∗ ∗,∗ ∗−r,∗+r−1 is a derivation for each r ≥ 1. (2) The spectral sequence converges to the loop homology H∗ (LM ) as ∞ is the associated graded algebra to a natural algebras. That is, E∗,∗ filtration of the algebra H∗ (LM ). (3) For m, n ≥ 0, ∼ H m (M ; Hn (ΩM )). E2 = −m,n

Here ΩM is the space of base point preserving loops in M . Further2 ∼ more the isomorphism E−∗,∗ = H ∗ (M ; H∗ (ΩM )) is an isomorphism of algebras, where the algebra structure on H ∗ (M ; H∗ (ΩM )) is given by the cup product on the cohomology of M with coefficients in the Pontryagin ring H∗ (ΩM ). In fact we will make use of a dualised version of this spectral sequence. More precisely, we will consider the spectral sequence of Theorem 5.5 with the following adjustments: • We regrade this spectral sequence into a fourth quadrant spectral sequence and so we make a slight change of notation; the spectral sequence becomes {Erp,q , dr : p ≥ 0, q ≤ 0}

with differentials

dr : Er∗,∗ −→ Er∗+r,∗−r+1 .

• We also observe that, in this case, for m, n ≥ 0 E m,−n ∼ = H m (M ; Hn (ΩM )) 2

and, from above, we have an isomorphism of algebras E2∗,−∗ ∼ = H ∗ (M, H∗ (ΩM )). • We will refer to this spectral sequence as the “Cohen-Jones-Yan spectral sequence”. Proof of Corollary 5.4. Notice that, for any r = 1, 2, . . ., the elements of (Σ−1 V )∗ in the Er -term of the Cohen-Jones-Yan spectral sequence appear in first column of that term. Therefore, if x ∈ (Σ−1 V )∗ , then x 6∈ Im(δr ). So, if x ∈ N , then x survives to E∞ . An analogous argument shows that, if x, y ∈ (Σ−1 V )∗ , then xy is not a boundary in any of the Er -terms of the Cohen-Jones-Yan spectral sequence

MODELS AND STRING HOMOLOGY

17

and, using the Leibniz rule, xy ∈ N whenever x, y ∈ N . This concludes the proof.  6. Examples: Cohomology of the Loop Space and the B-operator In this section we use Theorem 4.2 to describe the cohomology of the loop space of the sphere, projective spaces and Grassmann manifolds and to compute the respective B-operators. For the reasons mentioned above, we will work with a field of characteristic zero. We will use the same method for each manifold M . • First we obtain the model for the cochains on the free loop space of M by using Theorem 4.2. • Then we display the cohomology Leray-Serre spectral sequence for the fibration ΩM ,→ LM → M . (See [McC01, Chapter 5] for the construction of the cohomology Leray-Serre spectral sequence of a fibration.) Here the differential that we obtained from the model suggests differentials for this spectral sequence in the following fashion. We start with the E2 term of this spectral sequence and we calculate the differential δ from the model of each of the elements x of bidegree (p, q). For such x we then set d2 to be the leading terms of δ(x), that is, the elements of the right bidegree ((p + 2, q − 1)) that appear as summands in δ(x). If there are no such elements of the right bidegree, then d2 (x) = 0. Once knowing the second differential we can calculate the elements of the third page E3∗,∗ . Similarly, in Pn−1 di . We order to calculate dn we take the leading terms of δ − i=2 note that, in our examples, this will give all the differentials since there is a very limited number of elements of each bidegree in a particular term of this spectral sequence. In most of the examples, there is only one non trivial differential. • Once more we use the model to compute β, the derivation of degree −1 from Theorem 4.2, at the chain level. Using all of its properties this operator β is easy to compute and in homology these computations are equivalent to computing B. We will make no distinction in notation between β and B. • From this, we get a description of H ∗ (LM ) with its cup product and the corresponding B-operator. 6.1. H ∗ (LS 2n+1 ) and the B-operator. In this case, the cohomology of the sphere is a free algebra on one generator x with degree |x| = 2n + 1, that is, the exterior algebra A = K[x]. As in Example 2.2, we obtain H ∗ (LS 2n+1 ) = K[x, u] where |u| = 2n. In this case the B-operator is given by Bx = u and, since B 2 = 0, Bu = 0. These formulas completely determine the B-operator on H ∗ (LS 2n+1 ) as B is a derivation of the cup product. 6.2. H ∗ (LS 2n ) and the B-operator. Let A = H ∗ (S 2n ) =

K[x] (x2 )

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A. L. GARCIA PULIDO AND J. D. S. JONES

with |x| = 2n. It is easy to see that a model for A is given by P = (K[x, y], ∂), where |y| = 4n − 1 and the differential ∂ is given by ∂x = 0 and ∂y = x2 on the generators. By Theorem 4.2 a model for H ∗ (LS 2n ) is given by LP = (K[x, y, u, v], δ) where |x| = 2n, |y| = 4n − 1, |u| = 2n − 1, |v| = 4n − 2 and, by Theorem 4.1 (2), δ is defined on the generators by δx = 0 δy = x2 δu = 0 δv = −2xu. Figure 1 shows the cohomology Leray-Serre spectral sequence associated to the path fibration ΩS 2n ,→ LS 2n → S 2n . There we compute the differentials using the model for H ∗ (LS 2n ). From Lemma 4.5 we see that, at the chain level, B is given by Bx = u, By = v, Bu = B 2 x = 0 and Bv = B 2 y = 0 since B 2 = 0. Let zi = xv i , i = 0, 1, 2, . . .. Using the fact that B is a derivation of the product we have that B(zi ) = uv i . From Figure 1, notice that the algebra H ∗ (LS 2n ) is the graded commutative algebra K[zi , B(zi ) : i ≥ 0] with degrees |zi | = 2n + i(4n − 2), |B(zi )| = 2n − 1 + i(4n − 2) and with the following relations: zi z j = 0

for i, j = 0, 1, 2, . . . ,

zi B(zj ) = 0

for i, j = 0, 1, 2, . . . ,

B(zi )B(zj ) = 0

for i 6= j.

That is, all the products in H ∗ (LS 2n ) are trivial. 6.3. H ∗ (LCP n ) and the B-operator. Set A=

K[x] = H ∗ (CP n ) (xn+1 )

with |x| = 2. Let P = (K[x, y], ∂) be the differential graded algebra with ∂ defined on the generators by ∂x = 0 and ∂y = xn+1 and |y| = 2n + 1. Notice that P is a model for A. Using Theorem 4.2 the model for H ∗ (LCP n ) is given by LP = (K[x, y, u, v], δ), where δ is defined on the generators by δx = 0 δy = xn+1 δu = 0 δv = −(n + 1)xn u

MODELS AND STRING HOMOLOGY

19

10n-5

8n-4

6n-3

4n-2

2n-1

0

0

...

2n

Figure 1. The cohomology Leray-Serre spectral sequence for H ∗ (LS 2n ) with degrees |u| = 1, |v| = 2n and the degrees of x and y as above. Using the model for H ∗ (LCP n ) we compute the differentials of the cohomology LeraySerre spectral sequence associated to the fibration ΩCP n ,→ LCP n → CP n , which are pictured in Figure 2. Using the model for H ∗ (LCP n ) we see that, at the chain level, the Boperator is given by Bx = u, By = v, Bu = B 2 x = 0 and Bv = B 2 y = 0.

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A. L. GARCIA PULIDO AND J. D. S. JONES

8n+1 8n

6n+1 6n

4n+1 4n

2n+1 2n

1 0 0

1

2

3

4

...

2n

Figure 2. The cohomology Leray-Serre spectral sequence for H ∗ (LCP n )

Thus, if zi = xv i , then Bzi = uv i for i = 0, 1, . . . as B is a derivation of the product. From Figure 2 we see that H ∗ (LCP n ) is the graded commutative algebra K[zi , B(zi ) : i ≥ 0]

MODELS AND STRING HOMOLOGY

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where the generators satisfy the following relations zi1 zi2 · · · zin+1 Bzi Bzj zi zj zi1 zi2 · · · zin Bzj

=0 =0 = z0 zi+j = 0,

for any i, j, i1 , . . . , in , in+1 ≥ 0. 6.4. H ∗ (L(G2 (C4 )) and the B-operator. Let A = H ∗ (G2 (C4 )) =

(x3

K[x, y] − 2xy, x2 y − y 2 )

where |x| = 2 and |y| = 4. This expression for H ∗ (G2 (C4 )) derives from the following. In the book [BT82, Proposition 23.2], Bott and Tu give a description of the cohomology ring of the Grassmann manifolds in terms of the Chern classes of the tautological bundle and its orthogonal complement. According to this description, the cohomology of G2 (C4 ) is given by H ∗ (G2 (C4 )) =

K[c1 , c2 , d1 , d2 ] , (1 + c1 + c2 )(1 + d1 + d2 ) = 1

where |c1 | = |d1 | = 2 and |c2 | = |d2 | = 4. By writting d1 and d2 in terms of c1 and c2 we get the expression above. One can verify that a model for A is given by P = (K[x, y, f, g], ∂), where ∂ is given on the generators by ∂x = 0, ∂y = 0, ∂f = x3 − 2xy, ∂g = x2 y − y 2 , with degrees |f | = 5, |g| = 7. The model for H ∗ (L(G2 (C4 ))) described in Theorem 4.2 is the commutative graded algebra K[x, y, f, g, u, v, a, b] with the degrees of x, y, f, g as above and |u| = 1, |v| = 3, |a| = 4 and |b| = 6. The differential of this model is defined on the generators by δx = δy = δu = δv = 0, δf = x3 − 2xy, δg = x2 y − y 2 , δa = −3x2 u + 2yu + 2xv, δb = −2xu − x2 v + 2yv. Using this model we can calculate the differentials of the cohomology Leray-Serre spectral sequence for the path fibration Ω(G2 (C4 )) ,→ L(G2 (C4 )) →

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A. L. GARCIA PULIDO AND J. D. S. JONES

G2 (C4 ). Standard definitions for spectral sequences tells us that dr : Erp,q → Erp+r,q−r+1 and that p,q p+r,q−r+1 p,q ∼ Ker dr : Er → Er Er+1 , = Im dr : Erp−r,q+r−1 → Erp,q

see [McC01, Definition 1.1, Chapter 1]. To obtain E2 , E4 and E5 we first notice that E2pq ∼ = H p (G2 (C4 )) ⊗ H q (Ω(G2 (C4 ))) and we calculate the second differential d2 by taking the leading terms of the differential δ given by the model. This calculation leads us to the third term of this spectral sequence E3 , in which, we notice that Im d3 = {0} since p,q E3p,q = {0} when p is odd. Then by definition E4p,q ∼ = E3 . In fact, this same argument applies to all the differentials d2k+1 of an odd term E2k+1 p,q p,q ∼ and so d2k+1 = 0 and E2k+2 = E2k+1 for k ≥ 1. We take d4 to be the leading terms of δ − d2 − d3 , which determines E5p,q . We see that d6 = 0 because the leading terms of ∂ − d2 − d3 − d4 − d5 are zero. Analogously d8 = 0. For k ≥ 9 we have that dk = 0 since Ekp,q = 0 when p ≥ 9. Thus, the spectral sequence collapses in the E5 -term. The E2 , E4 and E5 terms of this spectral sequence are depicted in Figures 3, 4, 5, respectively. At the chain level, the B-operator is given by Bx = u, By = v, Bu = B 2 x = 0, Bv = B 2 y = 0, Bf = a, Bg = b, Ba = B 2 f = 0, Bb = B 2 g = 0, as B 2 = 0. Using that B is a derivation of the product, if we write αn = bn x, βn = bn y, δm = bm u − mbm−1 av,   n 1 m 2 m n,m = b a ux − a uy , 2 λn,m = bn am y 2 ,

MODELS AND STRING HOMOLOGY

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20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

0

1

2

3

4

5

6

Figure 3. The E2 -term of the cohomology Leray-Serre spectral sequence for H ∗ (L(G2 (C4 ))) for m ≥ 1 and n ≥ 0, then B(αn ) = bn u, B(βn ) = bn v =: γn , B(δm ) = 0, B(n,m ) = −bn am uv =: θn,m , B(λn,m ) = 2bn am vy =: 2ρn,m ,

8 7

8

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A. L. GARCIA PULIDO AND J. D. S. JONES

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

0

1

2

3

4

5

6

8 7

Figure 4. The E4 -term of the cohomology Leray-Serre spectral sequence for H ∗ (L(G2 (C4 )))

for m ≥ 1 and n ≥ 0. From Figure 5 we see that H ∗ (L(G2 (C4 ))) is the graded commutative algebra

K[αn , u, βn , γn , δm , n,m , θn,m , λn,m , ρn,m : m ≥ 1, n ≥ 0]

8

MODELS AND STRING HOMOLOGY

25

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

0

0

1

2

3

4

5

6

Figure 5. The E5 = E∞ -term of the cohomology LeraySerre spectral sequence for H ∗ (L(G2 (C4 ))) with degrees |αn | = 6n + 2, |u| = 1, |βn | = 6n + 4, |γn | = 6n + 3, |δm | = 6m + 1, |n,m | = 6n + 4m + 5, |θn,m | = 6n + 4m + 4, |λn,m | = 6n + 4m + 8, |ρn,m | = 6n + 4m + 7

8 7

8

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A. L. GARCIA PULIDO AND J. D. S. JONES

and satisfying the relations αn αm βn βm αn β m α0 α0 αn α0 α0 βn

= α0 αn+m , = β0 βn+m , = α0 βn+m = αn+m β0 , = 2α0 βn , = β0 βn , n+m δn δm = δ1 δn+m−1 = uδn+m , n βl βm u = 0, αn δm = uαn+m , βn δm = uβn+m − mρn+m−1,1 , γn δm = uγn+m ,

and all other double products are trivial. 7. Examples: String Homology of the Spheres, Projective Spaces and Grassmann Manifolds In this section, using the model described in Theorem 5.2 we obtain the homology of the loop space of the spheres, projective spaces and of G2 (C4 ) with its corresponding string product. We also compute the BatalinVilkovisky operator using the calculations from Section 6. For the reasons mentioned above, we will work with a field of characteristic zero. We will use the same method for each manifold M . • Using the isomorphism of graded algebras from Theorem 1.3 and then the model described in Theorem 5.2, we write the model for the (shifted) homology of the free loop space of M . • Then we display the Cohen-Jones-Yan spectral sequence from Theorem 5.5. Here the differential that we obtained from the model suggests differentials for this spectral sequence in the following fashion. We start with the E 2 term of this spectral sequence and we calculate the differential δ from the model of each of the elements x of bidegree (p, q). For such x we then set d2 to be the leading terms of δ(x), that is, the elements of the right bidegree ((p + 2, q − 1)) that appear as summands in δ(x). If there are no such elements of the right bidegree, then d2 (x) = 0. Once knowing the second differential 3 . Similarly, in we can calculate the elements of the third page E∗,∗ Pn−1 i order to calculate dn we take the leading terms of δ − i=2 d . We note that, in our examples, this will give all the differentials since there is a very limited number of elements of each bidegree in a particular term of this spectral sequence. In most of the examples, there is only one non trivial differential. • From this, we obtain a description of H∗ (LM ) with the string product. • Now we use the calculations from Section 6 and Poincar´e duality on M to give an isomorphism (of vector spaces) between H∗ (LM ) and H ∗ (LM ).

MODELS AND STRING HOMOLOGY

27

• We use the isomorphism between H∗ (LM ) and H ∗ (LM ) and the calculations from 6 of the B-operator to obtain the Batalin-Vilkovisky operator B ∗ = ∆ via the formula s(B ∗ t) = Bs(t),

(7.1)

where s ∈ H ∗ (LM ) and t ∈ H∗ (LM ). 7.1. H∗ (LS 2n+1 ) and the String Product. This calculation is valid only in the case where n > 0, since Theorem 1.3 requires the manifold to be simply connected. In this case, the cohomology of the sphere is a free algebra on one generator x of degree |x| = 2n + 1, that is, A is the exterior algebra K[x]. Following an analogous idea to Subsection 6.1, since A is free we can use Example 3.2 to get H∗ (LS 2n+1 ) = K[x, ν] with |ν| = −2n. Recall from Section 6 that H ∗ (LS 2n+1 ) is the free graded commutative algebra generated by x and u. Using Poincar´e duality on S 2n+1 we can construct an isomorphism of vector spaces H∗ (LS 2n+1 ) → H 2n+1−∗ (LS 2n+1 ) which is given by xν k 7→ uk ν k 7→ uk x for k ≥ 0. Thus B ∗ (1) = xν and B ∗ (xν k ) = 0 = B ∗ (ν m ) for k ≥ 0 and m ≥ 1. As algebras with their respective products and disregarding the grading, H∗ (LS 2n+1 ) = K[x, ν] and H ∗ (LS 2n+1 ) = K[x, u] are isomorphic. 7.2. H∗ (LS 2n ) and the String Product. Let A = H ∗ (S 2n ) =

K[x] (x2 )

with |x| = 2n. Recall from Subsection 6.2 that a model for A is P = (K[x, y], ∂), where the differential ∂ is defined on the generators by ∂x = 0 and ∂y = x2 and |y| = 4n − 1. By Theorem 5.2, a model for H∗ (LS 2n ) is given by GP = (K[x, y, a, b], δ), where |x| = 2n, |y| = 4n − 1, |a| = 1 − 2n, |b| = 2 − 4n and δ is defined on the generators by δx = 0 δy = x2 δa = −2xb δb = 0.

28

A. L. GARCIA PULIDO AND J. D. S. JONES

Figure 6 represents the Cohen-Jones-Yan spectral sequence from Theorem 5.5 when M = S 2n . From this we see that K[x, b, ν] H∗ (LS 2n ) = 2 , (x , νx, 2bx) where ν = ax and the generators have degrees |x| = 2n, |b| = 2−4n, |ν| = 1. Now we will compute the Batalin-Vilkovisky operator ∆ = B ∗ using Formula 7.1. In subsection 6.2 we saw that H ∗ (LS 2n ) has generators zi and B(zi ), i ≥ 0, with the relations described there. Using Poincar´e duality on S 2n we have, for every k, an isomorphism of vector spaces Hk (LS 2n ) → H 2n−k (LS 2n ) given by x 7→ 1 bi ν 7→ B(zi ) bi 7→ zi . From this isomorphism and formula 7.1, we see that B ∗ is given by B ∗ (x) = 0 B ∗ (bi ν) = bi B ∗ (bi ) = 0. Finally we observe that there is a substantial difference between the rings H∗ (LS 2n ) and H ∗ (LS 2n ). In the latter, all the products are trivial whereas the former contains the polynomial ring K[b]. 7.3. H∗ (LCP n ) and the String Product. As in Subsection 6.3, a model for K[x, y] A = H ∗ (CP n ) = n+1 (x ) is given by P = (K[x, y], ∂), where |x| = 2, |y| = 4n + 1, ∂x = 0 and ∂y = xn+1 . Using Theorem 5.2 we see that a model for H∗ (LCP n ) is given by G(P ) = (K[x, y, a, b], δ), with |a| = −1, |b| = −2n and δ defined on the generators by δx = 0, δy = xn+1 , δa = −(n + 1)xn b, δb = 0. In Figure 7 we find the Cohen-Jones-Yan spectral sequence (see Theorem 5.5) for M = CP n . Let ν = ax. Observe from Figure 7 that H∗ (L(CP n )) =

(xn+1 ,

K[x, ν, b] , νxn , (n + 1)bxn )

where the generators have degree |x| = 2, |ν| = 1 and |b| = −2n.

MODELS AND STRING HOMOLOGY

0

1

...

29

2n

0

1-2n

2-4n

3-6n

4-8n

5-10n

Figure 6. The Cohen-Jones-Yan spectral sequence for H∗ (LS 2n )

To compute the Batalin-Vilkovisky operator ∆ = B ∗ on H∗ (LCP n ) we recall from Section 6.3 that H ∗ (LCP n ) has generators zi and B(zi ), i ≥ 0, with the corresponding relations described there. Then, by Poincar´e duality

30

A. L. GARCIA PULIDO AND J. D. S. JONES

0

1

2

3

4

...

2n

0 -1

-2n -2n-1

-4n -4n-1

-6n -6n-1

-8n -8n-1

Figure 7. The Cohen-Jones-Yan spectral sequence for H∗ (LCP n )

on CP n , we get an isomorphism of vector spaces

H∗ (LCP n ) → H 2n−∗ (LCP n )

MODELS AND STRING HOMOLOGY

31

given by xk 7→ (z0 )n−k , xl bm 7→ (z0 )n−l−1 (zm ), νxl bm 7→ (z0 )n−l−1 B(zm ), where 0 ≤ k ≤ n, 0 ≤ l ≤ n − 1, m ≥ 0. Using this isomorphism and Formula 7.1 to compute B ∗ , we get the following formulas B ∗ (xk ) = 0, B ∗ (xl bm ) = 0, B ∗ (νxl bm ) = (n − l − 1)xl bm , for any 0 ≤ k ≤ n, 0 ≤ l ≤ n − 1, m ≥ 0. Notice the great difference between the rings H∗ (L(CP n )) and H ∗ (L(CP n )). In the latter, all the products of length n + 1 are trivial whereas this is not the case for the former. Indeed, the polynomial algebra K[b] is contained in H∗ (L(CP n )). Remark 7.1. The rings H∗ (LS 2n+1 ), H∗ (LS 2n ) and H∗ (L(CP n )), computed in Sections 7.1, 7.2, 7.3 were first computed by Cohen, Jones and Yan in [CJY04]. We observe that, up to regrading, both descriptions agree. 7.4. H∗ (L(G2 (C4 ))) with the String Product. Let A = H ∗ (G2 (C4 )) =

(x3

K[x, y] , − 2xy, y 2 − x2 y)

where |x| = 2 and |y| = 4. As noted in 6.4, a model for A is given by P = (K[x, y, f, g], ∂), where ∂ is defined on the generators by ∂x = 0, ∂y = 0, ∂f = x3 − 2xy, ∂g = x2 y − y 2 , and with degrees |f | = 5 , |g| = 7. Using Theorem 5.2, we obtain a model G(P ) for H∗ (L(G2 (C4 ))) with underlying free commutative algebra G(P ) = K[x, y, f, g, µ, ν, α, β] with differential given by δx = δy = δα = δβ = 0, δf = x3 − 2xy, δg = x2 y − y 2 , δµ = −3αx2 + 2αy − 2βxy, δν = 2αx − βx2 + 2βy. Figures 8, 9 and 10 represent the E 2 , E 4 and E 5 = E ∞ -terms of the Cohen-Jones-Yan spectral sequence (see Theorem 5.5) when M = G2 (C4 ). Notice that the spectral sequence collapses in the E 5 -term.

32

A. L. GARCIA PULIDO AND J. D. S. JONES

0

1

2

3

4

5

6

7

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

-13

-14

-15

-16

-17

-18

-19

-20

Figure 8. The E 2 -term of the Cohen-Jones-Yan spectral sequence for H∗ (L(G2 (C4 )))

8

MODELS AND STRING HOMOLOGY

0

1

2

3

4

5

33

6

7

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

-13

-14

-15

-16

-17

-18

-19

-20

Figure 9. The E 4 -term of the Cohen-Jones-Yan spectral sequence for H∗ (L(G2 (C4 )))

8

34

A. L. GARCIA PULIDO AND J. D. S. JONES

0

1

2

3

4

5

6

7

0

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-11

-12

-13

-14

-15

-16

-17

-18

-19

-20

Figure 10. The E 5 = E ∞ -term of the Cohen-Jones-Yan spectral sequence for H∗ (L(G2 (C4 )))

8

MODELS AND STRING HOMOLOGY

35

Using these figures we obtain the algebra structure of H∗ (L(G2 (C4 ))) with respect to the string product. Let γ = µx,  = µy, ζ = νy 2 , η = µνy 2 and θ = ανx2 − 2ανy. Then H∗ (L(G2 (C4 ))) is equal to the differential graded commutative algebra K[x, y, α, β, γ, , ζ, η, θ], where the following products are trivial η 2 , αx, βy 2 , ζx, ηx, θx, αy, ζy, ηy, γ, γη, γθ, γζ, αγ, ζ, η, ζη, ζθ, ηθ, and subject to the relations x3 = 2xy, y 2 = x2 y, γx2 = 2γy = 2x, γx3 = 2y, θy = −βy = −αζ, θ = −αη. The degrees of the generators are |x| = 2, |y| = 4, |α| = −4, |β| = −6, |γ| = 1, || = 3, |ζ| = 5, |η| = 4, |θ| = −3. Finally we will compute the Batalin-Vilkovisky operator ∆ = B ∗ on H∗ (L(G2 (C4 )). In Subsection 6.4 we saw that H ∗ (L(G2 (C4 )) is the quotient of the graded commutative algebra generated by αn , βn , δm , n,m , λn,m , and  u = B(α0 ), γn = B(βn ), θn,m = −B(n,m ), ρn,m = B

1 λn,m 2



subject to the relations described in the same subsection. We have an isomorphism of vector spaces H∗ (L(G2 (C4 )) → H 8−∗ (L(G2 (C4 ))

36

A. L. GARCIA PULIDO AND J. D. S. JONES

given by y 2 7→ 1 β m x3 7→ αm β m x2 7→ αm α0 β m x 7→ αm (α0 )2 β m y 7→ βm αk β m 7→ λk,n β m 7→ βm β0 β m γ 7→ αm (α0 )2 B(α0 ) β m γx 7→ αm (α0 )B(α0 ) β m γx2 7→ αm B(α0 ) αk β m  7→ m,k β m  7→ βm B(α0 ) β k y 7→ δk µy 2 7→ B(α0 ) β m ζ 7→ γn αk β m η 7→ θm,k β m η 7→ B(α0 )γm β m αn θ 7→ ρm,n+1 for any n, m ≥ 0, k ≥ 1. A direct calculation shows that B ∗ is given by the following formulas B ∗ (y 2 ) 7→ 0 B ∗ (β m x3 ) 7→ 0 B ∗ (β m x2 ) 7→ 0 B ∗ (β m x) 7→ 0 B ∗ (β m y) 7→ 0 B ∗ (αk β m ) 7→ 0 B ∗ (β m ) 7→ 0 B ∗ (β m γ) 7→ 0 B ∗ (β m γx) 7→ 3β m x B ∗ (β m γx2 ) 7→ 2β m x2 B ∗ (αk β m ) 7→ 0 B ∗ (β m ) 7→ 0

MODELS AND STRING HOMOLOGY

37

B ∗ (β k y) 7→ β m x3 B ∗ (µy 2 ) 7→ x3 B ∗ (β m ζ) 7→ β m y B ∗ (αk β m η) 7→ −αk β m  B ∗ (β m η) 7→ β m  B ∗ (β m αn θ) 7→ 2αn+1 β m for any n, m ≥ 0, k ≥ 1. Notice that the rings H∗ (L(G2 (CP n ))) and H ∗ (L(G2 (CP n ))) differ greatly. Indeed, the latter contains a polynomial algebra in two generators, whereas in the former every element has order at most 5.

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