CARTAN HOMOTOPY FORMULAS AND THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY Ezra Getzler Department of Mathematics, MIT, Cambridge, Mass. 02139 USA It is well-known that the periodic cyclic homology HP (A) of an algebra A is homotopy invariant (see Connes [3], Goodwillie [8] and Block [1]). Let A be an algebra over a field k and let A ν be a formal deformation of A, that is, an associative product m ν Hom(A 2, A)[ν 1,..., ν n ] such that m ν=0 is the product on A. We will define a connection on the periodic cyclic bar complex of A ν for which the differential is covariant constant, thus inducing a connection on the periodic homology HP (A ν ), thought of as a module over k[ν 1,..., ν n ]. This connection generalizes the classical Gauss-Manin connection, and indeed we will prove that it has curvature chain homotopic to zero. The Gauss-Manin connection is obtained by a generalization of Rinehart s result: if D is a derivation on A, then the operator ul(d) on the cyclic bar complex C(A)[u] of A is chain homotopic to zero (see Rinehart [10], and also Goodwillie [8]). Inspired by the work of Nistor [9], we prove a more general result on the action of the cochains C (A, A) on the cyclic bar complex C(A), where A is an A -algebra. (Recall that A -algebras are a generalization of differential graded algebras. It is shown in [6] that A -algebras form a natural setting for the study of cyclic homology.) The author would like to thank J. Block, V. Nistor, D. Quillen and B. Tsygan for helpful conversations on the subject of this article; in particular, the term Gauss-Manin connection is used at Quillen s suggestion. Note that some similar formulas, in the setting of Hochschild homology, have been obtained by Gelfand, Daletskiĭ and Tsygan [4]. 1. Hochschild cochains and A -algebras In this paper, all vector spaces will be over a field k. If V and W are graded vector spaces, we denote by V W the graded tensor product: thus, if A End(V ) and B End(W ), then (A B)(v w) = ( 1) B v (Av) (Bw). If V is a graded vector space, we denote by V (k) the tensor power V k. Let A be a graded vector space, and let sa be its suspension The bar coalgebra of A is the direct sum (sa) i = A i 1. B(A) = (sa) (n) ; n=0 This work was partially funded by the IHES and the NSF. 1 Typeset by AMS-TEX
2 EZRA GETZLER we denote the element (sa 1 )... (sa n ) B(A) by [a 1... a n ]. The coproduct is given by the formula n [a 1... a n ] = [a 1... a i ] [a i+1... a n ], and the counit ε sends [ ] to 1, and [a 1... a n ] to 0 if n 1. i=0 Definition 1.1. If A and B are graded vector spaces, the space of Hochschild cochains on A with values in B is C (A, B) = Hom(B(A), sb). If D is a homogeneous function of A, we denote its degree of homogeneity by d(d). Given D C (B, C) and D i C (A, B), 1 i k, define an element D{D 1,..., D k } C (A, C) with D{D 1,..., D k } = D + k D i, given for homogeneous D i by the formula D{D 1,..., D k }[a 1... a n ] = ( 1) k ω j i D i where η i = a 1 + + a i i and (j 1,...,j k ) J D[a 1... a j1 D 1 [a j1 +1... a j1 +d(d 1 )] a j1 +d(d 1 )+1...... a jk D k [a jk +1... a jk +d(d k )] a jk +d(d k )+1... a n ], J = {(j 1,..., j k ) 0 j 1, j i + d i j i+1 for 1 i k 1, j k n d k }. In the case k = 1, this operation was introduced by Gerstenhaber [5], and is denoted Lemma 1.2. If D 0, D 1, D 2 C (A, A), then D 0 {D 1 } = D 0 D 1. (D 0 D 1 ) D 2 D 0 (D 1 D 2 ) = D 0 {D 1, D 2 } + ( 1) D 1 D 2 D 0 {D 2, D 1 }. It follows from this lemma that the bracket [D 0, D 1 ] = D 0 D 1 ( 1) D 0 D 1 D 1 D 0 gives C (A, A) the structure of a graded Lie algebra. Recall that a coderivation on a coalgebra C is a linear map δ : C C such that (δ 1 + 1 δ) a = (δa) for a C. The space of coderivations Coder(C) is a graded Lie algebra, with bracket the graded commutator. Proposition 1.3. There is an isomorphism of graded Lie algebras given for homogeneous D by the formula δ : C (A, A) Coder(B(A)), n d δ(d)[a 1... a n ] = ( 1) ω i D [a 1... a i D[a i+1... a i+d(d) ] a i+d(d)+1... a n ]. i=0 The following definition is due to Stasheff [11] (see also [6]).
THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY 3 Definition 1.4. An A -algebra structure on a graded vector space A is a codifferential δ of degree 1 on B(A) such that δ[ ] = 0 A codifferential δ on B(A) corresponds to a Hochschild cochain m C (A, A), which may be viewed as a sequence of multilinear maps m k : A (k) A, k 1, of degree k 2. The equation δ 2 = 0 translates to a sequence of identities which are summed up in the formula m m = 0, or more explicitly, the sequence of identities for k 1,, k j 1 ( 1) ω i m j (a 1,..., a i, m k j+1 (a i+1,..., a i+k j+1 ), a i+k j+2,..., a k ) = 0. j=1 i=0 Lemma 1.5. An A -algebra such that m k = 0 for k > 2 is the same as a differential graded algebra, with product a 1 a 2 = ( 1) a 1 m 2 (a 1, a 2 ) and differential m 1. An algebra is analogous to a connection: the cochain m C 2 (A, A) is homogeneous of degree 2, just as a connection is homogeneous of degree 1. In this language, an A -structure m is the analogue of a superconnection. The augmention A + of an A -algebra A is the A -algebra whose underlying space is A ke, where the element e acts as an identity for A + ; that is, m C (A, A) is extended to A + by setting m 2 (e, a) = ( 1) a m 2 (a, e) = a, m 2 (e, e) = e, m k (..., e,... ) = 0, for k 2. The following lemma follows from Lemma 1.2, and is analogous to Steenrod s formula a 1 0 a 2 ( 1) a 1 a 2 a 2 0 a 1 = δ(a 1 1 a 2 ) (δa 1 ) 1 a 2 ( 1) a 1 a 1 1 (δa 2 ). In particular, it implies that the cup product is graded commutative on the Hochschild cohomology H (A, A). Lemma 1.6. (δd 1 ) D 2 δ(d 1 D 2 ) + ( 1) D 1 D 1 (δd 2 ) = m{d 1, D 2 } + ( 1) D 1 D 2 m{d 2, D 1 } Let A be an A -algebra, with A -structure m C (A, A). Define a Hochschild cochain M C (C (A, A), C (A, A)) by 0, k = 0, M[D 1... D k ] = m D 1 ( 1) D1 D 1 m, k = 1, m{d 1,..., D k }, k > 1. Proposition 1.7. The cochain M is an A -structure on C (A, A). Proof. By the definition of M M, we see that, if k > 1, (M M)[D 1... D k ] = 0 i j k ( 1) i l=1 D l m{d 1,..., D i, m{d i+1,..., D j }, D j+1,..., D k } k ( 1) i l=1 Dl m{d 1,..., D i m, D i+1,..., D k } + ( 1) k D i m{d 1,..., D k } m.
4 EZRA GETZLER The last two terms add up to k ( 1) i l=1 Dl m{d 1,..., D i, m, D i+1,..., D k }. Combining this with the first term, we obtain (m m){d 1,..., D k }, which clearly vanishes. To complete the proof, we must check that (M M)[D 1 ] vanishes: (M M)[D 1 ] = [m, [m, D 1 ]] = [m m, D 1 ] = 0. When A is a differential graded algebra, the A -algebra C (A, A) is a differential graded algebra; our construction generalizes that of Gerstenhaber [5]. The operator D M[D] is a differential on C (A, A) which is usually denoted D δd, and its cohomology is the Hochschild cohomology H (A, A) of the A -algebra A. If A and B are A -algebras, a morphism between them is a map of differential graded coalgebras f : B(A) B(B), such that f[ ] = [ ]. Proposition 1.8. Let A and B be A -algebras with A -structures m C (A, A), and n C (B, B). A twisting cochain on A with values in B is a cochain ρ C (A, B) of degree zero such that ρ[ ] = 0 and n 1 (ρ) + n 2 (ρ, ρ) + n 3 (ρ, ρ, ρ) + = ρ m. There is a correspondence between A -morphisms and twisting cochains, given by the formula f(ρ)[a 1... a k ] = k l=1 0 j 1 j l 1 k [ρ[a 1... a j1 ]... ρ[a ji 1 +1... a ji ]... ρ[a jl 1 +1... a k ]]. If B is a differential graded algebra, the formula for a twisting cochain becomes δρ + ρ ρ = 0, where k (f 1 f 2 )(a 1,..., a k ) = ( 1) η i f 2 f 1 (a 1,..., a i )f 2 (a i+1,..., a k ), (δf)(a 1,..., a k ) = df(a 1,..., a k ) ( 1) f (f m)(a 1,..., a k ). The Hochschild chain complex C(A) = n=0 C n(a) of a graded vector space A is the graded vector space such that { A, n = 0, C(A) = A + (sa) (n), n > 0. The element a 0... a n of C n (A) will be denoted (a 0,..., a n ); it has degree a 0 + + a n + n. For such an element, let η j = a 0 + + a j j. In the rest of this section, we will construct a twisting cochain of the A -algebra C (A, A) with values in End(C(A)). Given D 1,..., D k C (A, A), define an operator b{d 1,..., D k } on C(A) by the formula b{d 1,..., D k }(a 0,..., a n ) = l=k+1 (j 0,...,j k ) J(l) ε(j 0,..., j k ) (m l (a j0 +1,..., D 1 [... ],..., D k [... ],..., a n, a 0,... ),..., a j0 ) n l ( 1) ηj 1 (a 0,..., m l (a j+1,..., a j+l ),..., a n ), k = 0, + l=0 j=0 0, k > 0,
THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY 5 where ε(j 0,..., j k ) = ( 1) η n(η n η j0 )+ k D i (η ji η j0 ), we write D i [... ] as an abbreviation for D i [a ji +1... a ji +d(d i )], and J(l) = { (j 0,..., j k ) n (l 1) k D i j 0 j 1, } j i + d(d i ) j i+1 for 1 i k 1, j k n d(d k ). For k = 0, b{} is the Hochschild boundary b on C(A). For k = 1, we obtain an operator b{d}, which in the special case where A is a differential graded algebra may be written b{d}(a 0,..., a n ) = ( 1) (η n 1)(η n η n d(d) )+ D +1 (D[a n d(d)+1... a n ]a 0,..., a n d(d) ) This operator, with D C 1 (A, A), is considered by Rinehart, where it is denoted by e(d). Theorem 1.9. b is a twisting cochain of C (A, A) with values in End(C(A)). Proof. Apply the cochains δb and b b to {D 1,..., D k }. If k = 0, the term δb{} vanishes, and we must show that b{}b{} = 0. This is the standard fact that the Hochschild boundary b = b{} is a differential on C(A), and follows from the formula m m = 0. Thus, take k 1. We use a rather abbreviated notation in the proof, but the reader will have little difficulty in reconstituting the full formulas, if so desired. Observe that (δb){d 1,..., D k } = P + Q, where P = 0 i<j k Q = 1 i k ( 1) i l=1 D l b{d 1,..., m{d i+1,..., D j },..., D k }, ( 1) i 1 l=1 D l b{d 1,..., D i m,..., D k }. Using the formula m m = 0, we see that b{}b{d 1,..., D k } = R, where R = ε(j 0,..., j k ) ( 1) k D i +(η n η j0 )+η j (j 0,...,j k ) j (m(..., D 1 [... ],..., D k [... ],..., a 0,... ),..., m(a j+1,... ),... ). Similarly, we check that ( 1) i l=1 Dl b{d 1,..., D i }b{d i+1,..., D k } = S, where S = ε(j 0,..., j k ) ( 1) k D i +(η j η j0 ) (j 0,...,j k ) j (m(..., D 1 [... ],..., m(a j+1,..., D i+1 [... ],..., D k [... ],..., a 0,... ),... ),... ). Finally, we check that ( 1) k D i b{d 1,..., D k }b{} + Q + R = T + U + V, where T = k ( 1) i l=1 Dl b{d 1,..., D i, m, D i+1,..., D k }, i=0 U = ε(j 0,..., j k ) ( 1) k D i +η j η j0 (m(..., D 1 [... ],..., D k [... ],..., m(a j+1,..., a 0,... ),... ),... ), V = ε(j 0,..., j k ) ( 1) k D i +(η n η j0 )+(η j 1) (m(..., D 1 [... ],..., D k [... ],..., a 0,..., m(a j+1,... ),... ),... ).
6 EZRA GETZLER It only remains to observe that P + S + T + U + V = l=k+1 (j 0,...,j k ) J(l) which vanishes, because m m = 0. ε(j 0,..., j k ) ((m m) l (a j0 +1,..., D 1 [... ],..., D k [... ],..., a 0,... ),..., a j0 ), 2. The Cartan homotopy formula Let t be the operator on C(A) defined by the formulas t(a 0,..., a n ) = ( 1) η n( a n 1) (a n, a 0,..., a n 1 ), t(e, a 1,..., a n ) = 0. If D is a Hochschild k-cochain on A, define the operator D : C n (A) C n k+1 (A) by D(a 0,..., a n ) = (D[a 0... a k 1 ], a k,..., a n ). Given D 1,..., D k C (A, A), define an operator B{D 1,..., D k } on C(A) by the formula B{D 1,..., D k } = σ t j 1 j0 D 1 t j 2 j1 D 2... t j k j k 1 D k t jk 1, (j 0,j 1,...,j k ) J where σ(a 0,..., a n ) = (e, a 0,..., a n ), and J = {(j 0,..., j k ) 0 j 0, j i + d(d i ) j i+1 for 1 i k, j k n d(d k )}. More explicitly, we may write B{D 1,..., D k }(a 0,..., a n ) = (j 0,...,j k ) J ( 1) η n(η j0 1)+ k D i (η ji η j0 ) (e, a j0 +1,..., D 1 [... ],..., D k [... ],..., a 0,..., a j0 ), where we write D i [... ] as an abbreviation for D i [a ji +1... a ji +d(d i )]. For k = 0, the operator B{} is Connes s differential B. For k = 1, we obtain an operator B{D}. This operator, with D C 1 (A, A), is considered by Rinehart, where it is denoted by E(D). Let C(A)[u] be the space of power series in a variable u of degree 2. Consider b and B to be cochains on C (A, A) with values in the algebra End(C(A))[u], extending it linearly over k [u]. Definition 2.1. (1) Let ι C (C (A, A), End(C(A))[u]) equal b ub. (2) Let L C + (C (A, A), End(C(A))[u]) be the curvature of ι, defined by the formula δι + ι ι = ul. Since L is the curvature of ι, it satisfies the Bianchi identity δl + ι L L ι = 0. For example, this shows that [b ub, L{D}] = L{δD}. From the definition of the curvature L{D}, we see that (2.1) [b ub, ι{d}] = ul{d} ι{δd}. This formula is the non-commutative analogue of the Cartan homotopy formula in differential geometry: if X is a vector field on a smooth manifold, [d, ι(x)] = L(X). The main results of this paper is the calculation of L.
THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY 7 Theorem 2.2. For k = 0, L{} = 0 (that is, (b ub) 2 = 0). For k 1, L{D 1,..., D k } is given by the formula L{D 1,..., D k }(a 0,..., a n ) = ( 1) k D i (η ji 1) (a 0,..., D 1 [... ],..., D k [... ],..., a n ) where + (j 1,...,j k ) J k (j 1,...,j k ) J(i) ( 1) η n(η n η j1 + i l=2 D l )+ k l=2 D l (η jl η j1 + i l=2 D l ) (D 1 [a j1 +1... D i+1 [... ]... D k [... ]... a 0... ],... J = {0 j 1, j l + d(d l ) j l+1, j k + d(d k ) n},..., D 2 [... ],..., D i [... ],..., a j1 ), J(i) = {0 j 2, j l + d(d l ) j l+1, j k + d(d k ) n, j i + d(d i ) j 1 j i+1 }. Proof. We first calculate that b{}b{d 1,..., D k } + k ( 1) i l=1 Dl B{D 1,..., D i }b{d i+1,..., D k } = P + Q + R + S, where P = Q = 0 i j k ( 1) i l=1 D l B{D 1,..., m{d i+1,..., D j },..., D k }, k ( 1) i l=1 Dl B{D 1,..., D i m,..., D k }, R = S = (j 1,...,j k ) J {(j 0,...,j k ) J j 0 =j 1 } ( 1) k D i (η ji 1) (a 0,..., D 1 [... ],..., D k [... ], a n ), ( 1) η n(η n η j0 )+ k l=2 D l (η jl η j0 ) (D 1 [... ],..., D k [... ],..., a 0,... ). Next, we check that (δb){d 1,..., D k } + P + Q = 0, and that ( 1) D 1 b{d 1 }B{D 2,..., D k } + R + S = L{D 1,..., D k }. The proof is completed by observing that b{d 1,..., D i }B{D i+1,..., D k } = 0 for i > 1, and B{D 1,..., D i }B{D i+1,..., D k } = 0. We will need an explicit formula for L{D}: L{D}(a 0,..., a n ) = n d(d) j=0 ( 1) D (η j 1) (a 0,..., D[a j+1... a j+d(d) ],..., a n ) + n j=n d(d) ( 1) η n(η n η j ) (D[a j+1... a 0... a j+d(d) n 1 ],..., a j ).
8 EZRA GETZLER The Hochschild boundary b = b{} on C(A) is equal to L(m), where m C (A, A) defines the A -structure on A. Given cochains D 1 and D 2 on A, define ρ{d 1, D 2 } by the formula ρ{d 1, D 2 }(a 0,..., a n ) = j 1 j 2 ( 1) ηn(ηn ηj1 )+ D2 (ηj2 ηj1 ) Note that ρ{m, D} = b{d}. Lemma 2.3. L{D 1, D 2 } + ( 1) D 1 D 2 L{D 2, D 1 } + L{D 1 D 2 } Proof. It is easily checked that where (D 1 [a j1 +1... D 2 [a j2 +1... ]... a 0... ],..., a j1 ). = L{D 1 }L{D 2 } + ρ{d 1, D 2 } + ( 1) D 1 D 2 ρ{d 2, D 1 } L{D 1 }L{D 2 } = L{D 1 D 2 } + P 1 + ( 1) D 1 D 2 P 2 + Q 1 + ( 1) D 1 D 2 Q 2, P 1 = ( 1) D 1 (η n η j1 )+ D 2 (η n η j2 ) (a 0,..., D 1 [... ],..., D 2 [... ],..., a n ), P 2 = ( 1) D 1 (η n η j1 )+ D 2 (η n η j2 ) (a 0,..., D 2 [... ],..., D 1 [... ],..., a n ), Q 1 = ( 1) η n(η n η j1 )+ D 2 (η j2 η j1 ) (D 1 [... a 0... ],..., D 2 [... ],... ), Q 2 = ( 1) η n(η n η j2 )+ D 2 (η j1 η j2 ) (D 2 [... a 0... ],..., D 1 [... ],... ). Here, we abbreviate D i [a ji +1... a ji +d(d i )] to D i [... ], and in the definitions of Q i, the sum is taken over j i > n d(d i ). Since L{D 1, D 2 } = P 1 + Q 1 + ρ{d 1, D 2 } and ( 1) D 1 D 2 L{D 2, D 1 } = P 2 + Q 2 + ρ{d 2, D 1 }, the proof of the lemma is completed. It follows from this lemma that [L(D 1 ), L(D 2 )] = L([D 1, D 2 ]) which gives another proof that b 2 = 0. If W is a graded module over the ring k[u], the cyclic homology HC (A; W ) with coefficients in W is the homology of the complex (C(A) W, b ub). Let us list some examples with different coefficients W. (1) W = k[u] gives the negative cyclic homology HC (A); (2) W = k[u, u 1 ] gives the periodic cyclic homology HP (A); (3) W = k[u, u 1 ]/uk[u] gives the positive cyclic homology HC (A); (4) W = k[u]/uk[u] gives the Hochschild homology HH (A). The following theorem is a consequence of the results of Sections 1 and 2. Theorem 2.4. (1) The graded Lie algebra H (A, A) acts on HC (A; W ) by the Lie derivative D L(D), for any coefficients W. (2) If D Z (A, A) is a cocycle, then ul(d) is chain homotopic to zero on the cyclic bar complex C(A)[u]. In particular, L(D) acts as zero on the periodic cyclic homology HP (A).
THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY 9 3. The Gauss-Manin connection Let A ν be an n-parameter formal deformation of the A -algebra A; in other words, the A - structure on A ν is defined by a cochain m ν C (A, A + )[ν 1,..., ν n ] such that m ν=0 is the product on A and m ν m ν = 0. The cohomology of the periodic cyclic bar complex C(A)[ν 1,..., ν n ]((u)) with differential b ν ub is the periodic cyclic homology HP (A), which is a module over k[ν 1,..., ν n ]((u)). Let A i = m ν ν i C (A, A)[ν 1,..., ν n ]. Proposition 3.1. The Gauss-Manin connection = d + u 1 n ι ν {A i } dν i commutes with b ν ub, and thus induces a connection on the module HP (A ν ). Proof. Taking a partial derivative of the formula m ν m ν = 0 with respect to ν i, we see that [m ν, A i ] = 0. Observe that (b ν ub) = L{A i }, ν i since b ν = L{m ν }. Thus, it follows from (2.1) that [, b ν ub] = = n ( (bν ub) ν i n (L{A i } L{A i }) dν i = 0. ) [ι ν {A i }, b ν ub] dν i As an example, suppose we have a one-parameter family of associative products a 1 ν a 2 on the ungraded vector space A. Then ι ν {A ν } is given by the formula ι ν {A ν }(a 0,..., a n ) = (A ν (a n 1, a n ) ν a 0, a 1,..., a n 2 ) ( 1) ni+(j i) (e, a i,..., A ν (a j, a j+1 ),..., a 0,..., a i 1 ). 1 i j n 1 In the remainder of this section, we give an expression for the curvature of the Gauss-Manin connection. We show that it has the form [b ν ub, P ] for a certain operator P, and hence that it induces a flat connection on the periodic cyclic homology. Let σ{d 1, D 2 } be the operator on C(A) be defined by the formula σ{d 1, D 2 } = ι{d 1, D 2 } + ( 1) D 1 D 2 ι{d 2, D 1 } ι{d 1 D 2 }. The following lemma will enable us to calculate the curvature of. Lemma 3.2. [b ub, σ{d 1, D 2 }] + σ{δd 1, D 2 } + ( 1) D 1 σ{d 1, δd 2 } + ( 1) D 1 [ι{d 1 }, ι{d 2 }] = u(l{d 1 }L{D 2 } + ρ{d 1, D 2 } + ( 1) D 1 D 2 ρ{d 2, D 1 })
10 EZRA GETZLER Proof. The definition of L in terms of ι shows that [b ub, ι{d 1, D 2 }] + ι{δd 1, D 2 } + ( 1) D 1 ι{d 1, δd 2 } + ( 1) D 1 ι{d 1 } ι{d 2 } + ι{m{d 1, D 2 }} = ul{d 1, D 2 }. If we let σ 0 {D 1, D 2 } = ι{d 1, D 2 } + ( 1) D 1 D 2 ι{d 2, D 1 }, we see that [b ub, σ 0 {D 1, D 2 }] + σ 0 {δd 1, D 2 } + ( 1) D 1 σ 0 {D 1, δd 2 } + ( 1) D 1 [ι{d 1 }, ι{d 2 }] + ι { m{d 1, D 2 } + ( 1) D 1 D 2 m{d 2, D 1 } } On the other hand, by Lemma 1.6, we have [b ub, ι{d 1 D 2 }] + ι{δd 1 D 2 } + ( 1) D 1 ι{d 1 δd 2 ) = u ( L{D 1, D 2 } + ( 1) D 1 D 2 L{D 2, D 1 } ). = ι { m{d 1, D 2 } + ( 1) D 1 D 2 m{d 2, D 1 } } + ul(d 1 D 2 ). Combining these two equations with Lemma 2.4 proves the lemma. It is now easy to prove the following theorem. Theorem 3.3. The curvature 2 of the Gauss-Manin connection is given by the formula 2 = u 2 1 i j n = u 2 1 i j n and hence is chain homotopic to zero. Proof. Observe that ([b ν ub, σ ν {A i, A j }] ul{a i }L{A j }) dν i dν j [b ν ub, σ ν {A i, A j } + uι{a i }L{A j }] dν i dν j, ι ν {A j } ν i The formula for the curvature of is seen as follows: [ ] + ι ν {A i }, + ι ν {A j } ν i ν j ( 2 m ) ν = ι ν j ν i { 2 } m ν = ι + ρ{a i, A j }. ν i ν j ( 2 m ) ν ι + ρ{a i, A j } ρ{a j, A i } + [ι ν (A i ), ι ν (A j )]. ν i ν j The first two terms cancel, and the remaining terms are shown by Lemma 3.2 to equal [b ν ub, σ ν {A i, A j }] ul{a i }L{A j }. 4. The Gauss-Manin connection and iterated integrals In this section, we will illustrate the Gauss-Manin connection of the last section in a simple example. Let A be a differential graded algebra. There are three commuting differentials on the
THE GAUSS-MANIN CONNECTION IN CYCLIC HOMOLOGY 11 cyclic bar complex C(A), which we denote d(a 0,..., a k ) = k ( 1) ηi 1+1 (a 0,..., da i,..., a k ), i=0 k 1 b(a 0,..., a k ) = ( 1) η i (a 0,..., a i a i+1,..., a k ) B(a 0,..., a k ) = i=0 + ( 1) η k( a k +1) (a k a 0, a 1,..., a k 1 ), k ( 1) η k(η i 1 +1) (e, a i,..., a k, a 0,..., a i 1 ); i=0 as usual, η i = a 0 + + a i i. The total differential on C(A) is d + b B. If A i = Ω i (M) is the differential graded algebra of differential forms on a smooth manifold M, there is a map of complexes σ C(Ω(M)) Ω(LM) d+b B d ι(t ) C(Ω(M)) σ Ω(LM) called the iterated integral (see Chen [2] and Getzler-Jones-Petrack [7]). If k is the k-simplex 0 t 0 t k 1, and a(t) is the pull-back of the differential form a Ω(M) by the evaluation map γ γ(t), then σ is defined on C(Ω(M)) by the formula σ(a 0,..., a k ) = ( 1) k k a 0 (0) ι(t )a 1 (t 1 )... ι(t )a k (t k ) dt. The cyclic bar complex algebra (C(C (M)), b B) of the algebra of smooth functions C (M) on M maps to the complex of differential forms (Ω(M), d) by the map (f 0,..., f k ) α ( 1)k f 0 df 1... df k. k! Now consider the operator ι{d} = b{d} B{d} : C(Ω(M)) C(Ω(M)) of Section 2; the operators b{d} and B{d} are given by the formulas b{d}(a 0,..., a k ) = ( 1) (η k 1)( a k +1) (da k a 0, a 1,..., a k 1 ) B{d}(a 0,..., a k ) = ( 1) η k(η i 1 1)+(η j 1 η i 1 ) 1 i j k (e, a i,..., da j,..., a 0,..., a i 1 ). We will be interested in the operator e ι{d}, which may be rewritten as a Volterra series e ι{d} = k=0 using the formula B{d} 2 = 0. k e t 1b{d} B{d}e (t 2 t 1 )b{d}... e (t k t k 1)b{d} B{d}e (1 t k)b{d} dt,
12 EZRA GETZLER Proposition 4.1. Let i : Ω(LM) Ω(M) denote the restriction of differential forms under the inclusion M LM. We have a commuting diagram of complexes ( C(C (M)), b B ) α ( ) Ω(M), d e ι{d} i ( ) σ C(Ω(M)), d + b B ( Ω(LM), d ι(t ) ) Proof. In Section 3, we proved that (d + b B) e ι{d} = e ι{d} (b B). Thus, it only remains to show that if f i C (M), then i σ e ι{d} (f 0,..., f k ) = ( 1)k f 0 df 1... df k. k! The key observation is that i σ(a 0,..., a k ) = 0 if k > 0. Thus, only the term proportional to b{d} k contributes, and we see that i σ e ι{d} (f 0,..., f k ) = 1 k! i σ b{d} k (f 0,..., f k ) = 1 k! i σ(df 1... df k f 0 ) = ( 1)k f 0 df 1... df k k! References 1. J.L. Block, Cyclic homology of filtered algebras, K-Theory 1 (1987), 515 518. 2. K.T. Chen, Iterated integrals of differential forms and loop space homology, Ann. Math. 97 (1973), 217 246. 3. A. Connes, Non-commutative differential geometry, Publ. Math. IHES 62 (1985), 41 144. 4. I.M. Gel fand, Yu.L. Daletskiĭ, and B.L. Tsygan, On a variant of noncommutative geometry, Soviet Math. Dokl. 40 (1990), 422-426. 5. M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math. 78 (1963), 59 103. 6. E. Getzler and J.D.S. Jones, A -algebras and the cyclic bar complex, Ill. Jour. Math. 34 (1989), 256 283. 7. E. Getzler, J.D.S. Jones and S. Petrack, Differential forms on loop spaces and the cyclic bar complex, Topology 30 (1991), 339 371. 8. T. Goodwillie, Cyclic homology, derivations and the free loopspace, Topology 24 (1985), 187 215. 9. V. Nistor, An bivariant Chern character for p-summable quasihomomorphisms, K-theory 5 (1991), 193 211. 10. G. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195 222. 11. J.D. Stasheff, Homotopy associativity of H-spaces, II, Trans. Amer. Math. Soc. 108 (1963), 293 312.