View of On Additive K-Theory with the Loday - Quillen *-Product

ON ADDITIVE K-THEORY WITH THE LODAY - QUILLEN *-PRODUCT

KATSUHIKO KURIBAYASHI and TOSHIHIRO YAMAGUCHI

Abstract

The *-product defined by Loday and Quillen [17] on the additive K-theory (the cyclic homology with shifted degrees) K^‡…A† for a commutative ring A is naturally extended to a product (*-product) on the additive K-theory K^‡…†for a differential graded algebra …;d†over a commutative ring. We prove that Connes' B-maps from the additive K-theoryK^‡…†to the negative cyclic homology HC^ÿ…†and to the Hochschild homology HH…†are morphisms of algebras under the *-product onK^‡…†. Applications to topology of Connes' B-maps are also described.

xxx0. Introduction

LetAbe an algebra over a commutative ring. Let HC^ÿ_n…A†and HHn…A†de- note the negative cyclic homology and the Hochschild homology of A, re- spectively. In the algebraic K-theory, C. Hood and J. D. S. Jones [11] have constructed the Chern characterch_n:K_n…A† !HC^ÿ_n…A†which is a lift of the Dennis trace map Dtr:Kn…A† ! HHn…A† by modifying basic construction due to Connes [5] and Karoubi [12]. When the algebraAis commutative, the usual pairing of K…A† and the product on HC^ÿ…A† defined by Hood and Jones in [11] make the character ch into a morphism of algebras. In con- sequence, we can have the following commutative diagram in the category of graded algebras:

…0:1†

Here h is the map induced from the natural projection to the Hochschild complex from the cyclic bar complex. The Chern character ch:K0…A† !HC^ÿ₀…A† ˆ HC^per₀ …A† is connected with the ordinary Chern character K…X† !H_{de Rham}^even …X;C† when A is the ring consisting of smooth

MATH. SCAND. 87 (2000), 5^21

Received January 12, 1998.

functions from a compact manifold X to the complex number C (see, for example, [19, 6.2.9. Example]). Therefore, one may expect that the Chern character chn:Kn…A† !HC^ÿ_n…A† or the Dennis trace map Dtr:Kn…A† ! HHn…A†becomes a map with value in the de Rham (singular) cohomology of some manifold (space) by replacing the algebraAwith an appropriate object concerning with the space.

Hochschild and (negative) cyclic homologies can be extended to functors defined on the category of commutative differential graded algebras (DGAs) over a commutative ring (see [8], [11], [4]). In particular, if we choose the de Rham complex……X†;d†of a simply connected manifoldXas the DGA, the Hochschild and the negative cyclic homology of…X†can be regarded as the real cohomology and the realT-equivariant cohomology of the space of free loops onX respectively (see [8]), whereTdenotes the circle group. However, in algebraic K-theory, we can not expect such an extension. What is ``K- theory'' which addmits an extension to a functor on the category of DGAs and in which there is a commutative diagram corresponding to (0.1)? We can consider the additive K-theoryK<sup>‡</sup>…A†(see [6]) as ``K-theory'', which is iso- morphic to the positive cyclic homology group HC_ÿ1…A†. Let be the iso- morphism form K^‡…A† to HCÿ1…A† defined by Loday and Quillen in [17]

and independently Tsygan in [21]. Tillmann's commutative diagram [20, Theorem 1] connects the dual of the Dennis trace map with the Connes B- map by the dual of the isomorphism : K^‡…A† ! HC_ÿ1…A† when A is a Banach algebra. Therefore it is natural to choose the Connes B-map BHH:HCÿ1…A† !HH…A†as a map in the additive K-theory corresponding to the Dennis trace map in algebraic K-theory. The Connes' B-map BHH:K^‡…A†  HCÿ1…A† !HH…A† has a natural lift B, which is also called Conne's B-map, to the negative cyclic homology HC^ÿ…A†. Moreover functors HC, HC^ÿ, HH and the connecting maps can be extend on the category of DGAs by using the cyclic bar complex in [7] and [8]. In the consequence, we can obtain the following commutative diagram corre- sponding to (0.1) in the category of graded modules:

…0:2†

whereis a DGA. We propose a natural question that whether the diagram (0.2) is commutative in the category of graded algebras, as well as the dia- gram (0.1), under an appropriate product on K^‡…†. To answer this ques- tion, we extend the *-product defined by Loday and Quillen [17] to a product on the additive K-theory (the cyclic homology with shifted degrees) of a 6 katsuhiko kuribayashi and toshihiro yamaguchi

DGA, which is an explicit version of that of Hood and Jones [11, Theorem 2.6]. Since the product is defined at chain level, we can see that

Theorem 0.1. The diagram(0.2)is commutative in the category of graded algebras when the product of K^‡…†is given by the *-product.

LetM be a simply connected manifold andLM the space of all smooth maps from circle groupTtoM. By using the Connes' B-map B_HH, we con- sider the vanishing problem of string class of a loop group bundle LSpin…n† !LQ!LM. In the consequence, a generalization of the main theorem in [14] is obtained when the given manifoldM is formal (see Theo- rem 2.1).

We also show that the algebra structure of HC^ÿ…†can be described with the *-product onK^‡…†via Connes' B-mapB:K^‡…† !HC^ÿ…†when the DGA …;d†over a field kof characteristic zero is formal. This fact allows us to deduce the following theorem.

Theorem0.2.Let X be a formal simply connected manifold. Then H_T…LX;R†  fH…LX;R†=Im…BHHI†g^‡1R‰uŠ

as an algebra, where I :H…LX;R† ˆHHÿ……X†† !K_ÿ^‡……X††is the map in Connes' exact sequence (1,1) mentioned in x1 for the de Rham complex …X†with negative degrees andR‰uŠis the polynomial algebra over u with de- gree2. The multiplication of the algebra on the right hand side is given as fol- lows; wuⁱˆ0and ww⁰ˆwBIw⁰, whereis the cup product on H…LX;R†.

In particular,

(i) if H…X;R† R‰xŠ=…x^s‡1†and s>1, then

H_T…LX;R†  k0;1jsRf…j;k†g R‰uŠ

as an algebra, where deg…j;k† ˆj degx‡k……s‡1†degxÿ2† ÿ1, …j;k† …j⁰;k⁰† ˆ0and…j;k† uˆ0for any j;k;j⁰;k⁰, and

(ii) if H…X;R† …y†, then

H_T…LX;R†  _k0Rf…k†g R‰uŠ

as an algebra, wheredeg…k† ˆ …k‡1†…degyÿ1†,…k† …j† ˆ…k‡j‡1†

and…k† uˆ0for any j;k.

As for the algebra structure of H_T…LX;R†, the above results cover [13, Theorem 2.4].

This paper is set out as follows. In Section 1, we define the additive K- Theory K^‡…† of a DGA …;d† over a commutative ring and a product (*-product) onK^‡…†. Some properties of the *-product will also be studied.

on additive k-theory with the loday - quillen *-product 7

In Section 2, we will describe the applications of Connes' B-mapsBandB_HH which are mentioned above.

xxx1. The *-product onK^‡

Let …;d† be a commutative differential graded algebra (DGA) over a commutative ringk, ˆL

i0i, with unit 1 in 0, endowed with a dif- ferential d of degree ÿ1 satisfying d…1† ˆ0. We assume that differential graded algebras are non-positively graded algebras with the above proper- ties unless otherwise stated. We recall the cyclic bar complex defined in [7]

and [8]. The complex …C…†‰u^ÿ1Š;b‡uB† is defined as follows:

C…† ˆX¹

kˆ0

^k;

b…!₀; ::; !_k† ˆ ÿX^k

iˆ0

…ÿ1†^"iÿ1…!₀; ::; !_iÿ1;d!_i; !_i‡1; ::; !_k†

ÿX^kÿ1

iˆ0

…ÿ1†^"i…!₀; ::; !_iÿ1; !_i!_i‡1; !_i‡2; ::; !_k†

‡…ÿ1†^{deg!kÿ1†"kÿ1}…!_k!₀; ::; !_kÿ1†; b…u^ÿ1† ˆ0 and

B…!₀; ::; !_k†ˆX^k

iˆ0

…ÿ1†…"_iÿ1‡1†…"_kÿ"_iÿ1†…1; !_i; ::; !_k; !₀; ::; !_iÿ1†; B…u^ÿ1† ˆ0;

where ˆ=k, deg…!₀; ::; !_k† ˆdeg!₀‡ ‡deg!_k‡k, for …!₀; ::; !_k†in C…†,"_iˆdeg!₀‡ ‡deg!_iÿiand deguˆ ÿ2. Note that the formulas bB‡Bbˆ0 and b² ˆB² ˆ0 hold, see [7]. The negative cyclic homology HC^ÿ…†, the periodic cyclic homology HC^per …† and the Hochschild homology HH…† of a DGA …;d† are defined as the homology of the complexes…C…†‰‰uŠŠ;b‡uB†,…C…†‰‰u;u^ÿ1Š;b‡uB†and…C…†;b†, respec- tively. Since a DGA in our case has negative degrees, the power series alge- bra C…†‰‰uŠŠ agrees with the polynomial algebra C…†‰uŠ, similarly, C…†‰‰u;u^ÿ1Š ˆC…†‰u;u^ÿ1Š.

We define thenth additive K-theoryK_n^‡…;d†of a DGA …;d†to be the …nÿ1†-th cyclic homology HCnÿ1…;d†which is the …nÿ1†-th homology of the cyclic bar complex…C…†‰u^ÿ1Š;b‡uB†:

K^‡…† ˆHC_ÿ1…† ˆH_ÿ1…C…†‰u^ÿ1Š;b‡uB†:

8 katsuhiko kuribayashi and toshihiro yamaguchi

Unless we note the differentiald of a DGA in particular, K_n^‡…;d† will be denoted by K_n^‡…†. We define a product (*-product) on the complex …C…†‰u^ÿ1Š;b‡uB†as follows:

Xⁿ

iˆ0

xiu^ÿiX^m

jˆ0

yju^ÿjˆXⁿ

iˆ0

xiBy0u^ÿi; whereis the shuffle product onC…†.

Proposition 1.1. (i) The *-product induces a degree‡1map of complexes C…†‰u^ÿ1Š C…†‰u^ÿ1Š !C…†‰u^ÿ1Šwhich is associative.

(ii) The *-product on the cyclic bar complex defines an associative graded commutative algebra structure on K^‡…†.

In [7], to give anA1-algebra structure to the gradedk-moduleC…†‰‰uŠŠ, E. Getzler and J. D. S. Jones have defined a sequence of operators B_k:C…†^k!C…†of degree k and have clarified relation of B_k,B_kÿ1 and the shuffle products on C…†. In particular, in order to prove Proposition 1.1, we need the following formula representing the relation of the operator B2, Connes' B-operatorB:C…† !C…†and the shuffle products.

Lemma 1.2. ([7, Lemma 4.3]) There exists an operator B₂:C…†²! C…†of rank 2satisfying

…ÿ1†^jj‡1bB₂…; † ‡B…† ˆ …ÿ1†^jbj‡1B₂…b; † ‡ …B†

…1:2:1†

‡ …ÿ1†^jjfB‡ …ÿ1†^jj‡1B2…;b†g:

The definitions of B2 (see [7, page 280]) and B enable us to deduce that, for any elementszandz⁰inC…†,

B2…z;Bz⁰† ˆB2…Bz;z⁰† ˆ0:

…1:2:2†

Proof of Proposition 1.1. (i) From the formulas (1.2.1) and (1.2.2), by replacing the elementwithB, it follows thatB…B† ˆBB. For any elements xˆP

x_iu^ÿi, yˆP

y_ju^ÿj and zˆP

z_ku^ÿk in C…†‰u^ÿ1Š, we see that, on C…†‰u^ÿ1Š, x …yz† ˆx …P

yjBz0†u^ÿjˆ P

xiB…y0Bz0†u^ÿiˆ …xy† z. We will prove that *-product is a map of complexes. Since the differential bis a derivation under the shuffle product on C…†‰u^ÿ1Š, by the formulabB‡Bbˆ0, we have

on additive k-theory with the loday - quillen *-product 9

…b‡uB†…xy† ˆ …b‡uB†X

i0

…x_iBy₀†u^ÿi

ˆX

i0

…bx_i† By₀u^ÿi‡X

i0

…ÿ1†^jxijx_ibBy₀u^ÿi‡X

i0

B…x_iBy₀†u^ÿi‡1

ˆX

i0

…bxi† By0u^ÿi‡X

i0

…ÿ1†^jxij‡1xiBby0u^ÿi‡X

i0

Bxi‡1By0u^ÿi: On the other hand, by the formulaBBˆ0, we have

…b‡uB†xy‡ …ÿ1†^jxj‡1x …b‡uB†y

ˆX

i0

…bxi† By0u^ÿi‡X

i0

Bxi‡1By0u^ÿi

‡…ÿ1†^jxj‡1xX

j0

by_ju^ÿj‡X

j0

By_j‡1u^ÿj

Thus we can conclude that …b‡uB†…xy† ˆ …b‡uB†xy‡ …ÿ1†^jxj‡1x …b‡uB†y. Note that…ÿ1†^jxjˆ …ÿ1†^jxijfor any i.

(ii) To prove that the *-product defines a graded commutative algebra structure onK^‡…†, it suffices to prove that, for any elementsxˆP

xiu^ÿi and yˆP

y_ju^ÿj in Ker…b‡uB†, there exists an element !ˆP

k0!_kÿ1u^ÿk such that

xkBy0ÿ…ÿ1†…jxj‡1†…jyj‡1†ykBx0ˆb!kÿ1‡B!k

for anyk0. We will verify that

!_kˆ …ÿ1†^jyj‡1†jxj X

i‡jˆk

y_ix_jÿ X

i‡jˆk‡1

…ÿ1†^jyijB₂…y_i;x_j†

fork0 and

!ÿ1ˆ …ÿ1†…jxj‡1†…jyj‡1†B2…y0;x0†

are factors of the required element. Since equalities byiˆ ÿByi‡1 and bxj ˆ ÿBj‡1xj‡1hold, it follows that, ifk0,

10 katsuhiko kuribayashi and toshihiro yamaguchi

…ÿ1†^jyj‡1†jxj…b!_kÿ1‡B!_k†

ˆ X

i‡jˆkÿ1

fÿByi‡1xj‡ …ÿ1†^jyijyi …ÿBxj‡1†g ÿ X

i‡jˆk

…ÿ1†^jyijbB2…yi;xj†

‡ X

i‡jˆk

fBy_ix_j‡ …ÿ1†^jyijy_iBx_j‡ …ÿ1†^jyijbB₂…y_i;x_j†g

ˆ ÿ X

i‡jˆk;i1

By_ix_jÿ X

i‡jˆk;j1

…ÿ1†^jyijy_iBx_j

‡ X

i‡jˆk

By_ix_j‡ X

i‡jˆk

…ÿ1†^jyijy_iBx_j

ˆBy₀x_k‡ …ÿ1†^jykjy_kBx₀

ˆ …ÿ1†^jyj‡1†jxj…xkBy0ÿ …ÿ1†…jyj‡1†…jxj‡1†ykBx0†

from the formulas (1.2.1) and (1.2.2). We can check that equality b!_ÿ1‡B!₀ˆx₀By₀ÿ …ÿ1†…jxj‡1†…jyj‡1†y₀Bx₀ holds in a similar way.

We define Connes' B-maps BHH:K_n^‡…†ÿ!HHn…† and B:K_n^‡…† ÿ!

HC^ÿ_n…†by BHH…P

i0xiu^ÿi† ˆBx0 and B…P

i0xiu^ÿi† ˆBx0. Note that the mapsB_HHandBare connecting maps in Connes' exact sequences ([16, The- orem 2.2.1 and Proposition 5.1.5])

!HH_n‡1…† ÿ!^I K_n‡2^‡ …† ÿ!^S K_n^‡…† ÿ!^BHH HH_n…† ! …1:1†

and

!HC^ÿ_n‡1…† ÿ!^u HC^per_nÿ1…†ÿ!K_n^‡…† ÿ!^B HC^ÿ_n…† ! …1:2†

respectively.

Proof of Theorem0.1. The product structure m2 on C…†‰uŠdefined by m2…1; 2† ˆ12 ‡…ÿ1†^j1j‡1 uB2…1; 2†induces the algebra structure of HC^ÿ…†. From (1.2.2), we see that the product m₂ agrees with the shuffle product if₁ or₂ belongs to the image of the operator B:C…† !C…†.

Therefore the formula B…B† ˆBB implies that the map B:K^‡…† !HC^ÿ…†is a morphism of algebras.

In study of the cyclic homology theory, it is often useful to consider the reduced theory. To prove some theorems below, we will use the reduced ad- ditive K-theory K~^‡…† defined by K~^‡…† ˆCoker…:K^‡…k† !K^‡…††, where:k!is the unit. The reduced additive K-theoryK~^‡…†is a direct summand of K^‡…† because the exact sequence 0!C…k†‰u^ÿ1Š ! C…†‰u^ÿ1Š !C…†‰u ^ÿ1Š !0 of cyclic chain complexes is a split sequence.

on additive k-theory with the loday - quillen *-product 11

More precisely, K^‡…† is isomorphic to K~^‡…† k‰u^ÿ1Š as a graded HC…k† ˆk‰u^ÿ1Š-module. When one notices the direct summand k‰u^ÿ1Š of K^‡…†, by definition of *-product, it follows that k‰u^ÿ1Š is included in the annihilator ideal ofK^‡…†. Therefore we can also conclude that the algebra …K^‡…†;†does not have an unit.

Let us consider a relation of the *-product on K^‡…† to the suspension map S:K^‡…† !K_ÿ2^‡ …† in Connes' exact sequence (1.1). Since the sus- pension map S is defined by SX

i0

x_iu^ÿi

ˆ X

i0

x_i‡1u^ÿi , it follows that SxyˆS…xy† on C…†‰u^ÿ1Š. From this fact and commutativity of the

*-product, we have

Proposition1.3. For any elements!andin K^‡…†, S!ˆS…!† ˆ!S:

For the rest of this paper, unless otherwise mentioned, we will assume that any DGA …;d† is a commutative algebra over a field k of characteristic zero, connected and simply connected, that is, ˆ i0i, 0ˆk, H₁…† ˆ0 andd…1† ˆ0. A DGA…;d†is said to beformal if there exists a DGA-morphism from the minimal model m of to the DGA …H…;d†;0† which induces a isomorphism between their homologies (see [10]).

For any DGA …;0† with the trivial differential, M. Vigue¨-Poirrier has given a decomposition of the negative cyclic homology HC^ÿ…;0†: HC^ÿ…;0† ˆ _q0H…c^q‰uŠ; b‡uB†, and has shown that the S-action on HCg^ÿ…;0† is trivial, see [22, Proposition 5], where c^q_nˆ f…a₀; ::;a_p†j Pdegaiˆ ÿq;ÿq‡pˆ ÿng. This fact implies that the S-action on HCg^ÿ…;d† for any formal DGA …;d† is trivial ([22, The¨ore©me A]). The proof of [22, Proposition 5] is based on Goodwillie's result [9, Corollary III.4.4], which is led from the following proposition.

Proposition1.4. [9].Let…;d†be aDGAover a commutative ring and D a derivation onwith degreejDjsatisfying that D…ab† ˆ …Da†b‡ …ÿ1†^jDjjaja…Db†

and ‰D;dŠ ˆ0. Then there exist chain maps eD:C…† !C…† of degree jDj ÿ1, E_D:C…† !C…† of degree jDj ‡1 and an operator L_D:C…† !C…† of degree jDj such that ‰u^ÿ1e_D‡E_D;b‡uBŠ ˆL_D in C…†‰u;u^ÿ1Š, where‰a;bŠ ˆabÿ …ÿ1†^jajjbjba for any operators a and b.

We can obtain a lemma by using Proposition 1.4 and the idea of the de- composition of cyclic homology due to Vigue¨-Poirrier [22].

Lemma1.5. Let…;0†be a DGAwith the trivial differential. For any ele- 12 katsuhiko kuribayashi and toshihiro yamaguchi

ment!inK~^‡…;0† ˆgHC_ÿ1…;0†, there exists an element₀inC…† \kerb such that!ˆ ‰₀ŠinK~^‡…;0†.

Proof. According to Vigue¨-Poirrier [22], we define a derivationD on by D…a† ˆ …deg a†a. Consider a decomposition K^‡…;0† ˆ P

q0K^‡…;0†^q defined byK^‡…;0†^q ˆHÿ1…c^q‰u^ÿ1Š;b‡uB†. SinceK~^‡…†is isomorphic to P

q1K^‡…†^q, in order to prove Lemma 1.5, it suffices to show that there exists an element 0 with the property in Lemma 1.5 for any element! in K^‡…†^q (q1). Since the operation L_D on C…† is defined by L_D…a₀; ::;a_p† ˆP

0ip…a₀; ::;Da_i; ::;a_p†, it follows that the operator L_D on K^‡…†^q is given by LD…!† ˆ ÿq! in our case. On the other hand, for any element ! in K^‡…†^q which is represented by P

i0!iu^ÿi in C…†‰u^ÿ1Š, we have that ‰u^ÿ1eD‡ED;b‡uBŠ!ˆ eDB!0‡uEDB!0ÿ …b‡uB†…u^ÿ1eD‡ E_D†! in C…†‰u;u^ÿ1Š. By virtue of Proposition 1.4, we can conclude that e_DB!₀ÿ …b‡uB†…u^ÿ1e_D‡E_D†!ˆ ÿq! in C…†‰u^ÿ1Š. Thus, we see that ÿ¹_qeDB!0 is the required element0.

We will consider the algebra structure ofK^‡…†by using a minimal model of …;d†. Let ':…m;d_m†ÿ!…;d† be a minimal model of a DGA …;d†.

Then ' induces an isomorphism of algebras K…'†:K^‡…m†ÿ!K^‡…†.

Therefore if a DGA …;d† is formal, then there exist isomorphisms K^‡…;d† K^‡…m;d_M† K^‡…H…†;0†. It follows immediately that the iso- morphisms are compatible with the S-action. Since Lemma 1.5 asserts that any element ofK~^‡…;0†can be represented by an element with column de- gree 0, from the definition of S-action, we can get

Proposition 1.6. If a DGA …;d† is formal, then the suspension map S: ~K^‡…†ÿ!K~_ÿ2^‡ …†is trivial.

Let …m;d_m† be a free commutative differential graded algebra …V;d† over k. We denote by …e…m†; ; † the double complex defined in [4, Ex- ample 2] by D. Burghelea and M. Vigue-Poirrier. Namely, e…m† ˆ …VV†,is the unique derivation of degree‡1 defined byvˆvandis the unique derivation of degreeÿ1 which satisfiesj_Vˆd and ‡ˆ0.

HereV is the vector space withV_n‡1ˆV_n. We here mention that the double complex induces the complex…e…m†‰u^ÿ1Š; ‡u†with a product defined by P!iu^ÿiP

ju^ÿjˆP

!i0u^ÿi. By [4, Theorem 2.4 (i)], we see that the map :C…m† !e…m†defined by…a0;a1; ::;ap† ˆ1=p!a0a1 ap is a chain map between the double complexes…C…m†;b;B†and…e…m†; ; †. [4, Theo- rem 2.4 (iii)] shows that the induced map K…† from K^‡…m† to Hÿ1…e…m†‰u^ÿ1Š; ‡u†is an isomorphism of graded vector spaces. More- over we have

on additive k-theory with the loday - quillen *-product 13

Proposition 1.7. The map K…†:K^‡…m† !H_ÿ1…e…m†‰u^ÿ1Š; ‡u† is an isomorphism of algebras.

The following lemma will be needed to prove thatK…†is a morphism of algebras.

Lemma1.8.Let…m;d_m†be a free commutativeDGA.

(i) The chain map:C…m† !e…m†is compatible with B:C…† !C…† and:e…m† !e…m†:ˆB.

(ii) Let W be a subspace of C…m† consisting of the elements whose first factor have even degree: Wˆ fP

i…ai0; ::;a_ik…i†† 2C…m†jdegai0 is eveng.

Then …!!⁰† ˆ!!⁰ for any element !⁰ in W and any element ! in C…m†, herein the left hand side and right hand side are the shuffle product on C…m†and the natural product one…m†respectively.

Proof. It is straightforward to check that identities ˆB and …!!⁰† ˆ!!⁰ hold.

Proof of Proposition1.7. From the definition of Connes' B-operator, it follows that ImBis a subspace ofWin Lemma 1.8. By virtue of Lemma 1.8, we see that…!B!⁰† ˆ!!⁰for any element!and!⁰inC…m†. Thus we can conclude thatK…†is a morphism of graded algebras.

By virtue of Proposition 1.7, we can determineK^‡…†explicitly as an al- gebra when the homology of…;d†is generated with one generator.

Theorem1.9.For any formalDGA…;d†,

K^‡…† HHgÿ1…†=Im…BHHI :HHgÿ2…†

!HHg_ÿ1…†† kf1;u^ÿ1;u^ÿ2; ::g

as an algebra, where degu^ÿkˆ2k‡1, !!⁰ˆ!B!⁰, !u^ÿkˆ0 for any elements!and!⁰inHHg…†=Im…B_HHI†and u^ÿiu^ÿjˆ0. In particular,

(i) whendegx is even,

K^‡…k‰xŠ=…x^s‡1††  k0;1jskf…j;k†g kf1;u^ÿ1;u^ÿ2; ::g;

where deg…j;k† ˆjdegx‡k……s‡1†degx‡2† ‡1, !!⁰ˆ0 for any ele- ments!and!⁰in K^‡…k‰xŠ=…x^s‡1††, and

(ii) whendegy is odd,

K^‡……y††  _k0kf…k†g kf1;u^ÿ1;u^ÿ2; ::g

where deg…k†ˆdegy‡k…degy‡1†‡1; …k† …j†ˆ…k‡j‡1†;1…k† ˆ0 and…k† u^ÿl ˆ0.

Proof. By Proposition 1.6, the suspension map S: ~K^‡…† !K~_ÿ2^‡ …† is 14 katsuhiko kuribayashi and toshihiro yamaguchi

trivial. From this fact and Connes' exact sequence (1.1) obtained by using instead of a DGA, it follows that the mapI:HHg_ÿ1…† !K~^‡…†is sur- jective and that the kernel of I is the image of BHHI : HHgÿ2…† !HHgÿ1…†. Thus we can conclude that K^‡…† K~^‡…† kf1;u^ÿ1;u^ÿ2; ::g HHg_ÿ1…†=Im…B_HHI† kf1;u^ÿ1;u^ÿ2; ::g as algebras.

From Proposition 1.7 and the explicit formulas of the Hochschild homology ofk‰xŠ=…x^s‡1†and…y†in [15], we can get (i) and (ii).

Remark. In Theorem 1.9, the elements…j;k†and…k†correspond to the elementsx^j!^kandyy^kin [15, Proposition 1.1(ii)], respectively.

As mentioned before Proposition 1.3, the algebraK^‡…†does not have an unit. SinceK~₀^‡…†is non zero in general, the algebraK~^‡…†may be have an unit. However, the results of Theorem 1.9 (i) and (ii) enable us to conjecture that the reduced additive K-theoryK~^‡ does not have an unit for any DGA either. The first assertion in the following proposition is an answer to the conjecture.

Proposition1.10. (i) Let…;d†be aDGA. Assume thatK~^‡…† 6ˆ0. Then the algebraK~^‡…†does not have an unit.

(ii) IfdegQH…;d† n, then there exist n elements x₁, x₂, .. , x_nin K^‡…† such that x1x2 xn6ˆ0, where QH…;d† denotes the space of in- decomposable elements in the graded algebra H…;d†.

Proof. From the usual argument on a minimal model of , we can as- sume thatis free.

(i) Suppose that there exists an elementeinK~^‡…†such thatexˆxfor any x in K~^‡…†. Let us consider the Hodge decomposition of Hochschild homology ([3], [4, Theorem 3.1]): HHg…† ˆ _i0HHg^i† …†. Since B_HH: ~K^‡…† !HHg…†is a morphism of algebras by Theorem 0.1, it fol- lows that BHH…e† BHH…x† ˆBHH…x†. We see that BHH…e† belongs to HH^0† …† becausedegBHH…e† ˆ0. The definition of the Hodge decomposi- tion and Lemma 1.8 (i) enables us to deduce that ImB_HH is included in _i1HHg^i† …†. Thus we have B_HH…e† ˆ0. On the other hand, we see that S^Neˆ0 for some sufficient large integerN. IfB_HH…x† ˆ0 for allx2K~^‡…†, then the map S: ~K_‡2^‡ …† !K~^‡…† is epimorphism. Therefore, for any x2K~^‡…†, there is an element x⁰2K~^‡…† such that S^Nx⁰ˆx. It follows from Proposition 1.3 that xˆexˆeS^Nx⁰ˆS^Nex⁰ˆ0 for any x, which a contradiction. Thus B_HH…x† 6ˆ0 for some x2K~^‡…†. However, B_HH…x† ˆB_HH…e† B_HH…x† ˆ0. The result now follows.

(ii) We can choosen elements of corresponding to xi in K^‡…† which are part of generators of. We represent the elements with the same nota- tionx1; :::;xn, respectively. Under the isomorphismH…†in [4, Theorem 2.4 on additive k-theory with the loday - quillen *-product 15

(ii)], B_HH…x₁ x_n† ˆ B_HHx₁ B_HHx_nˆx₁ x_n in H…Tote…†; †.

Since Imconsists of elements whose factors have an element in , it fol- lows thatx1 xn6ˆ0 in HH…†  H…Tote…†; †. By virtue of Proposi- tion 1.7, we can see thatx1 xn 6ˆ0 in K^‡…†.

From Proposition 1.10 (ii), Theorem 1.9 (i) and (ii), we can conclude that K^‡…†has trivial algebra structure if and only if the homology of …;d†is generated by one element with even degree.

xxx2. Applications of Connes' B-mapsB_HHandB

LetMbe a simply connected manifold andLM the space ofC¹-free loops onM. When an SO…n†-bundleP!M overMhas a spin structureQ!M, the string class…Q†, which belongs toH³…LM;Z†, is defined as an obstruc- tion to lift the structure groupLSpin…n† of LQ!LM to LSpin…n†, for de-d tails see [18]. HereLSpin…n†d is the universal central extension ofLSpin…n†by the circle. One of important properties for the string class …Q†is the fact that the class …Q†is the image of ¹₂ p₁ by the map R

S¹ev:H…M;Z† ! H…LMS¹;Z† !H^ÿ1…LM;Z†, wherep₁is the first Pontrjagin class of the bundleP!M, ev:LMS¹!M is the evaluation map andR

S¹ is the in- tegration along S¹. Let Gbe a linear Lie group and:Q!M a G-bundle over M. Let Ch^p‡1…† be the Chern character of the bundle . The higher string classes C~^p…L† …p1† (see [2]) in H^2p‡1…LM;C† defined for the LG- bundle L:LQ!LM has a similar property to the ordinary string class …Q†. Indeed, the pth string class C~^p…L† is the image of ÿ…2 

pÿ1

†^p‡1p!Ch^p‡1…† by the map R

S¹ev. As mentioned in the in- troduction, in the study of the problem of whether the map R

S¹ev is in- jective, the Connes' B-map B_HH:K^‡……M†† !HH……M†† H…LM;R†

plays an important role. We will have the following theorem which is a generalization of [14, Theorem 2]. We may call a simply connected manifold formal if its de Rham complex is formal (see [10]).

Theorem2.1. Let M be a simply connected manifold and formal.

(i) For any SO…n†-bundle P!M with a spin structure Q!M, if H³…M;Z†

is torsion free, then the string class…Q†vanishes if and only if¹₂p1vanishes.

(ii) Let G be a linear Lie algebra and :Q!M a G-bundle. The string classC~^p…L†vanishes if and only if the Chern characterCh^p‡1…†of the bundle vanishes.

By virtue of [14, Proposition 2.1], we can regard the map R

S¹ev: H…M;R† ! H^ÿ1…LX;R† as the map :H…M;R† !HH_ÿ……M†;d† de- fined by …x† ˆ …1;x† under the identification by the iterated integral map :HHÿ……M†† !H…LM;R† ([8]), where ÿi…M† is the ith de Rham 16 katsuhiko kuribayashi and toshihiro yamaguchi

complexⁱ_{de Rham}…M†and the differentiald :_ÿi…M† !_ÿiÿ1…M†is the ex- terior differential on the de Rham complex _{de Rham}…M†. Thus, to prove Theorem 2.1, it suffices to show that the map is injective whenM is for- mal. Note that, for any DGA …;d†, we can define the map :H…† !HH…†by…x† ˆ …1;x†. The definition of the mapallows us to deduce that factors through Connes' B-map B_HH as follows:

ˆB_HHIi, where i:H…† !HH…† and I:HH…† !K^‡…† are the homomorphisms induced by the natural inclusions !C…† and C…† !C…†‰uÿ1Šrespectively. For any DGA…;d†, we have

Lemma2.2. The map Hÿ…† ÿ!ⁱ HHÿ…† ÿ!^I K_ÿ‡1^‡ …†is injective.

Proof. It suffices to prove that Lemma 2.2 holds when is free. In this case, we can identify K^‡…† with the homology of the complex …e…†‰u^ÿ1Š; ‡u†by Proposition 1.7. Since Im…‡u† \is contained in Imd which is a subspace of, it follows that ifIi…x†is zero inK^‡…†, then so isxin H…†.

Proof of Theorem 2.1. The reduced additive K-theory K~^‡…† includes Im…Ii:H_ÿ1…† !K^‡…††for<1. By Proposition 1.6, Connes' B-map B_HH: ~K^‡…† !HHg…†is injective. Therefore we can have Theorem 2.1 by virtue of Lemma 2.2.

In general case, we can show that Ii…Ker†…ˆIm…Ii† \KerB_HH† is contained in the space of annihilators ofK^‡…†.

Proposition2.3. For any DGA…;d†,K^‡…† fIm…Ii† \KerB_HHg ˆ0.

Proof. For any element! in Im…Ii† \KerB_HH, we can write !ˆ for some elementine…†. For any element!⁰in Ker…u‡†which is the subspace ofe…†‰u^ÿ1Š,

…u‡†…!⁰† ˆ …ÿ1†^deg!0!⁰ …u‡†

ˆ …ÿ1†^deg!0!⁰ …0‡!†

ˆ …ÿ1†^deg!0!⁰!

Note thatˆ0 in e…†‰u^ÿ1Š. Thus we see that!⁰!ˆ0 inK^‡…†.

We will describe some applications of Connes' B-map B:K^‡…† ! HC^ÿ…†.

Proposition2.4. The following diagram is commutative:

on additive k-theory with the loday - quillen *-product 17

K^‡…† ÿÿÿ!^B HC^ÿ…†

S??y ??y^S

K_ÿ2^‡ …† ÿÿÿ! HC^ÿ_ÿ2…†

B

Proof. For any element!ˆP

!_iu^ÿi in Ker…b‡uB†, by the definition of the S-action, we have that BS!ˆB!₁. On the other hand, SB!ˆB!₀u.

Sinceb!0‡B!1ˆ0, it follows thatBw0uÿB!1ˆ …b‡uB†!0. Thus we have SB!ˆBS!in HC^ÿ…†.

If the S-action on HCg^ÿ…† is trivial, then we can represent the algebra structure of the negative cyclic homology HC^ÿ…† with the *-product on K^‡…†.

Theorem 2.5. (i) The map B: ~K^‡…†ÿ!HCg^ÿ…† induced by Connes' B- map is an isomorphism of algebras.

(ii) The S-action on K~^‡…† is trivial if and only if so is the S-action on HCg^ÿ…†.

(iii) If the S-action on HCg^ÿ…†is trivial, thenHC^ÿ…† k‰uŠ K~^‡…†  k‰uŠ HHgÿ1…†=Im…BHHI† as algebras. By the assertions (i) and (ii), we see thatk‰uŠ HCg^ÿ…† k‰uŠ K~^‡…†as an algebra.

Proof. (i) The result [9, Theorem III.5.1] enables us to conclude that HC^per…† k‰u;u^ÿ1Š. From Connes' exact sequence (1.2) for, we can get (i). From (i) and Proposition 2.4, we have (ii). Since the S-action on HC^ÿ…† is trivial, it follows that HC^ÿ…† k‰uŠ HCg^ÿ…†as an algebra. From the proof of Theorem 1.9, we deduce the results of (iii).

We can now prove Theorem 0.2.

Proof of Theorem 0.2. If H…X;R† is isomorphic to the algebra R‰xŠ=…x^s‡1†or…y†, thenX is a formal. By virtue of Theorem 2.5 (iii), we see that H_T…LX;R†HC^ÿ_ÿ……X†† HC^ÿ_ÿ…H…X;R†† R‰uŠ K~_ÿ^‡…H…X;R††.

Therefore, Theorem 1.9 yields Theorem 0.2. In particular, we deduce (i) and (ii) by virtue of Theorem 1.9 (i) and (ii).

LetMbe a simply connected manifold (simplicial complex) and…M†its de Rham algebra of differential forms (simplicial differential forms) with coefficient in k ˆR;C (kˆR;C;Q). Then the isomorphism B: ~K^‡…† ! HCg^ÿ…† in Theorem 2.5 (i) agrees with the isomorphism b_M :HCg_ÿÿ1……M†† !H~_T…LM;k†in [3, Theorem B]. Therefore, if we regard K~^‡…†as a graded algebra with the *-product, the isomorphismbM becomes a morphism of algebras.

18 katsuhiko kuribayashi and toshihiro yamaguchi

Let…;d†and…⁰;d⁰†be DGAs over a fieldkof characteristic zero. If one wants to know about thek-module structure of the negative cyclic homology HC^ÿ…0†, the use of the Ku«nneth theorem [11, Theorem 3.1 (a)] for ne- gative cyclic homology theory may be effective, because the exact sequence

0!HC^ÿ…† _k‰uŠHC^ÿ…0† !HC^ÿ…0† ! Tor_k‰uŠ…HC^ÿ…†;HC^ÿ…0††_ÿ1!0

is split. However, it is not easy to determine the algebra structure of HC^ÿ…0†from the exact sequence even ifand⁰are formal. Theorem 2.5 (ii) enables us to represent the graded algebra structure of HC^ÿ…0† with the Hochschild homologies HH…†, HH…⁰†and the *-product when and⁰are formal. In term of spaces, we also assert that theT-equivariant cohomology of the space of loops on the product spaceMM⁰ can be re- presented with the cohomologies of the spaces of loops onM andM⁰, Con- nes' B-mapBHHand *-product.

Corollary 2.6. Let M and M⁰ be formal simply connected manifolds.

Then

H_T…L…MM⁰†;R† 

f…H…LM;R† H…LM⁰;R†=Im…BI11BI††g^‡1R‰uŠ

as an algebra, wheredeguˆ2. Here the multiplicationof the algebra on the right hand side is given as follows: !!⁰uˆ0, !!⁰⁰ˆ

!!⁰ …BI⁰ ‡…ÿ1†^jjBI⁰† for any !!⁰ and ⁰ in H…LM;R† H…LM⁰;R†=Im …BI11BI†, where is the cup pro- duct on H…LM;R† H…LM⁰;R†.

Proof. Let…m;d†and…m⁰;d†be minimal models of de Rham complexes ……M†;d† and ……M⁰†;d† respectively. We know that HHÿ…m†  H…LM;R†and HC^ÿ_ÿ…m† H_T…LM;R† as algebras ([8]). By virtue of [22, Proposition 5], the S-action on HC^ÿ_ÿ…m† is trivial. Therefore, it follows from Theorem 2.5 (ii) that, as algebras,

on additive k-theory with the loday - quillen *-product 19

H_T…L…MM⁰†;R† HC^ÿ_ÿ…mm⁰†

HHÿÿ1…mm†=Im…BHHI† R‰uŠ

H_ÿÿ1…e…mm⁰††=Im…B_HHI† R‰uŠ

 fH…e…m† H…e…m⁰††=

Im…I11I†g_ÿÿ1R‰uŠ

 fH…LM;R† H…LM⁰;R†=

Im…BI11BI†g^‡1R‰uŠ:

REFERENCES

1. W. Andrzejewski and A. Tralle, Cohomology of some graded differential algebras,Fund.

Math. 145 (1994), 181^204.

2. A. Asada,Four Lectures on Geometry of Loop Group and Non Abelian de Rham Theory, Chalmers University of Technology/The University of Go«teborg, 1990.

3. D. Burghelea, Z. Fiedorowicz and W. Gajda,Adams operations in Hochschild and cyclic homology of de Rham algebra and free loop spaces, K-theory 4 (1991) 269^287.

4. D. Burghelea and M. Vigue¨-Poirrier,Cyclic Homology of Commutative Algebras I, Lecture Notes in Math. 1318, 51^72.

5. A. Connes,Non commutative differential geometry Part I, II, Publ. Math. Inst. Hautes Etud.

Soc. 62 (1985), 41^93, 94^144.

6. B. L. Feigin and B. L. Tsygan,Additive K-theory, Lecture Notes in Math. 1289, 67^209.

7. E. Getzler and J. D. S. Jones,A1- algebras and cyclic bar complex, Illinois J. Math. 34 (1990), 256^283.

8. E. Getzler, J. D. S. Jones and S. Petrack,Differential form on loop spaces and the cyclic bar complex, Topology 30 (1991), 339^371.

9. T. G. Goodwillie,Cyclic homology, derivations and the free loop space, Topology 24 (1985), 187^215.

10. P. Griffiths and J.Morgan,Rational homotopy theory and differential forms, Prog. Math. 16 (1981), Birkha«user.

11. C. Hood and J. D. S. Jones,Some algebraic properties of cyclic homology groups, K-theory 1 (1987), 361^384.

12. M. Karoubi,Homologie cyclique et K-theorie alge¨brique, I et II, C. R. Acad. Sci. Paris 297 (1983), 447^450, 513^516.

13. K. Kuribayashi,On the real cohomology of spaces of free loops on manifolds, Fund. Math.

150 (1996), 173^188.

14. K. Kuribayashi,On the vanishing problem of string classes, J. Austral. Math. Soc. Ser. A 61 (1996), 258^266.

15. K. Kuribayashi and T. Yamaguchi,The cohomology algebra of certain free loop spaces, Fund. Math. 154 (1997), 57^73.

16. J. L. Loday,Cyclic homology, Grundlehren Math. Wiss. 301 (1992).

17. J. L. Loday and D. Quillen,Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv. 59 (1984), 565^591.

18. D. A. McLaughlin, Orientation and string structures on loop space, Pacific J. Math. 155 (1992), 143^156.

19. J. Rosenberg,Algebraic K-theory and its applications, Graduate Texts in Math. 147, 1994.

20 katsuhiko kuribayashi and toshihiro yamaguchi

20. U. Tillmann,Relation of the van Est spectral sequence to K-theory and cyclic homology, Illi- nois J. Math. 37 (1993), 589^608.

21. B. L. Tsygan, Homology of matrix algebras over rings and the Hochschild homology (in Russian), Uspekhi Mat. Nauk. 38 (1983), 217^218.

22. M. Vigue¨-Poirrier,Homologie cyclique des espaces formels, J. Pure Appl. Algebra 91 (1994), 347^354.

DEPARTMENT OF APPLIED MATHEMATICS OKAYAMA UNIVERSITY OF SCIENCE 1-1 RIDAI-CHO

OKAYAMA 700-0005 JAPAN

DEPARTMENT OF MATHEMATICS COLLEGE OF SCIENCE

RYUKYU UNIVERSITY NISHIHARA-CHO OKINAWA 903-0213 JAPAN

on additive k-theory with the loday - quillen *-product 21