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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).

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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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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