A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY RALPH L. COHEN AND JOHN D.S. JONES Abstract.Let M be a closed, oriented manifold of dimension d. Let LM be * *the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the hom* *ology H*(LM) of degree -d. They then investigated other structure that this product i* *nduces, including a Lie algebra structure on H*(LM), and an induced product on the S1 equiva* *riant homology, HS1*(LM). These algebraic structures, as well as others, came under the * *general heading of the "String topologyö f M. In this paper we will describe a realizat* *ion of the Chas - Sullivan loop product in terms of a ring spectrum structure on the Tho* *m spectrum of a certain virtual bundle over the loop space. We show that this ring spe* *ctrum structure extends to an operad action of the the äc ctus operad", originally defin* *ed by Voronov, which is equivalent to operad of framed disks in R2. We then describe a* * cosimplicial model of this spectrum and, by applying the singular cochain functor to * *this cosimplicial spectrum we show that this ring structure can be interpreted as the cup * *product in the Hochschild cohomology, HH*(C*(M); C*(M)). Introduction Let Md be a closed, oriented d - dimensional manifold, and let LM = C1 (S1, M* *) be the space of smooth loops in M. In [2] Chas and Sullivan described an intersection * *product on the homology, H*(LM), having total degree -d, O : Hq(LM) Hr(LM) ! Hq+r-d(LM). In this paper we show that this product is realized by a geometric structure, n* *ot on the loop space itself, but on the Thom spectrum of a certain bundle over LM. We de* *scribe this structure both homotopy theoretically and simplicially, and in so doing de* *scribe the relationship of the Chas - Sullivan product to the cup product in Hochshild coh* *omology. We now make these statements more precise. Consider the standard parameterization of the circle by the unit interval, ex* *p : [0, 1] ! S1 defined by exp(t) = e2iit. With respect to this parameterization we can regard* * a loop fl 2 LM as a map fl : [0, 1] ! M with fl(0) = fl(1). Consider the evaluation map ___________ Date: July 25, 2001. The first author was partially supported by a grant from the NSF . 1 2 R.L. COHEN AND J.D.S JONES ev : LM! M fl! fl(1). Now let ' : M ! RN+d be a fixed embedding of M into codimension N Euclidean s* *pace. Let N ! M be the N - dimensional normal bundle. Let T h( N ) be the Thom spac* *e of this bundle. Recall the famous result of Atiyah [1] that T h( N ) is Spanier -* * Whitehead dual to M+ . Said more precisely, let M-TM be the spectrum given by desuspendi* *ng this Thom space, M-TM = -(N+d)T h( N ). Then if M+ denotes M with a disjoint basepoint, there are maps of spectra S0 ! M+ ^ M-TM and M+ ^ M-TM ! S0 that establish M-TM as the S - dual of M+ . Said another way, these maps indu* *ce an equivalence with the function spectrum M-TM ' Map(M+ , S0). In particular in h* *omology we have isomorphisms Hq(M+ ) ~=H-q(M-TM ) H-q(M-TM ) ~=Hq(M+ ) for all q 2 Z. These duality isomorphism are induced by the compositions H-q(M-TM ) - -fi-!~H-q+d(M) - -j-! Hq(M) = ~= where ø is the Thom isomorphism, and æ is Poincare duality. Notice by duality, the diagonal map : M ! M x M induces a map of spectra * : M-TM ^ M-TM ! M-TM that makes M-TM into a ring spectrum with unit. The unit S0 ! M-TM is the map* * dual to the projection M+ ! S0. Now let T h(ev*( N )) be the Thom space of the pull back bundle ev*( N ) ! LM* *. Define the spectrum LM-TM = -(N+d)T h(ev*( N )) The goal of this paper is to define and study a product structure on the spectr* *um LM-TM which among other properties makes the evaluation map ev : LM-TM ! M-TM a map* * of A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 3 ring spectra. Here, by abuse of notation, ev is referring the map of Thom spect* *ra induced by the evaluation map ev : LM ! M. We will prove the following theorem. Theorem 1. The spectrum LM-TM is a homotopy commutative ring spectrum with u* *nit, whose multiplication ~ : LM-TM ^ LM-TM -! LM-TM satisfies the following properties. 1. The evaluation map ev : LM-TM ! M-TM is a map of ring spectra. 2. There is a map of ring spectra æ : LM-TM ! 1 ( M+ ) where the target is* * the suspension spectrum of the based loop space with a disjoint basepoint. Its* * ring structure is induced by the usual product on the based loop space. In homology the * *map æ* is given by the composition æ* : Hq(LM-TM )- fi--!~Hq+dLM --'-! Hq( M) = where like above, ø is the Thom isomorphism, and the map ' takes a (q + d)* * - cycle in LM and intersects it with the based loop space viewed as a codimension d -* * submanifold. 3. The ring structure is compatible with the Chas - Sullivan homology product* * in the sense that the following diagram commutes: Hq(LM-TM ) Hr(LM-TM ) ---! Hq+r(LM-TM ^ LM-TM ) --j*-!Hq+r(LM-TM ) ? ? fi?y~= ~=?yfi Hq+d(LM) Hr+d(LM) - --! Hq+r+d(LM) O Remark. In [2] Chas and Sullivan define a regrading of the homology of the loop* * space Hq = Hq+d(LM) with respect to which the product O is of total degree zero. We observe that th* *e Thom isomorphism defines an isomorphism H* ~=H*(LM-TM ) which respects gradings, and by the above theorem is an isomorphism of rings, w* *here the ring structure on the right hand side comes from the ring spectrum structure of* * LM-TM . Next we show that the ring structure ~ : LM-TM ^ LM-TM ! LM-TM extends to * *an operad structure of the äc ctus operad" C defined originally by Voronov. This * *operad is 4 R.L. COHEN AND J.D.S JONES homotopy equivalent to the operad of framed little disks in R2. We will recall * *the definition of the cactus operad C in section 2, where we will prove the following theorem. Theorem 2. There are maps of spectra ik : (Ck)+ ^ k (LM-TM )(k)-! LM-TM giving LM-TM the structure of a C - operad ring spectrum, compatible with the * *ring struc- ture ~ : LM-TM ^ LM-TM ! LM-TM . Our next result has to do with the simplicial structure of LM-TM , and the r* *esulting simplicial description of the product. Let S1*be the simplicial set decomposition of the circle which has one zero s* *implex and one nondegenerate one simplex. In this decomposition there are n + 1 n - simpli* *ces, all of which are degenerate for n > 1. We write this as S1n= {n + 1}. Now given any sp* *ace X, there is a resulting cosimplicial model for the free loop space, LX = Map(S1, X* *), X*. The n - simplices of X* are given by maps Xn = Map(S1n, X) = Map({n + 1}, X) = Xn+1. Of course the coface and codegeneracy maps of (LX)* are dual to the face and de* *generacy maps of S1*. Our next result states that there is a similar cosimplicial model for LM-TM . Theorem 3. For M a closed, oriented manifold, the spectrum LM-TM has the str* *ucture of a cosimplicial spectrum which we write as TX*. The k simplices of TX* are gi* *ven by TXk = (Xk)+ ^ M-TM . This cosimplicial structure has the following properties. 1. The ring structure of LM-TM is realized on the (co)simplicial level by pa* *irings i j ~k : (Xk)+ ^ M-TM ^ (Xr)+ ^ M-TM ! (Xk+r)+ ^ M-TM defined by ~k(x1, . .,.xk; u) ^ (y1, . .,.yr; v) = (x1, . .,.xk, y1, . .y.r; *(* *u ^ v)) where * is the ring structure defined on M-TM described earlier (dual to* * the diagonal map : M ! M x M). A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 5 2. Applying the singular chain functor C*(-) to the cosimplicial space TX*, w* *e get a natural chain homotopy equivalence between the chains of LM-TM and the Ho* *chshild cochain complex f* : C*(LM-TM ) ~=C*(TX*) ~=CH*(C*(M); C*(M)). Here the notation CH*(A; A) refers to the Hochshild cochain complex of the* * form A ! Hom(A; A) ! . .!.Hom(A n; A) ! Hom(A n+1; A) ! . . . Furthermore, the pairing on the chains C*(LM-TM ) induced by the ring spe* *ctrum structure corresponds via the chain homotopy equivalence f* to the cup pro* *duct pairing in CH*(C*(M); C*(M)). This gives ring isomorphisms in homology, H* ~=H*(LM-TM ) --f*-!~HH*(C*(M); C*(M)). = Remark. The fact that the Chas - Sullivan product is realized as the cup produc* *t in Hochshild cohomology was also observed by T. Tradler, and will appear in his CU* *NY Ph.D thesis. The paper will be organized as follows. In section 1 we will show how to realiz* *e the Chas - Sullivan product using the Pontrjagin - Thom constuction for embedded, finite c* *odimension manifolds. We will use this to prove theorem 1. In section 2 we define the cact* *us operad C and prove theorem 2. In section 3 we will recall the cosimplicial study of th* *e loop space done by the second author in [4], apply the Thom spectrum construction to it an* *d use it to prove theorem 3. The authors are grateful to I. Madsen, J. Morava, G. Segal, D. Sullivan, and * *U. Tillmann for helpful conversations regarding this material. 6 R.L. COHEN AND J.D.S JONES 1. The ring structure on LM-TM : The proof of theorem 1 In this section we will describe the ring spectrum structure of the Thom spec* *trum LM-TM defined in the introduction, discuss some its properties, and prove theo* *rem 1. We begin by restating it. Theorem 4. The spectrum LM-TM is a homotopy commutative ring spectrum with u* *nit, whose multiplication ~ : LM-TM ^ LM-TM -! LM-TM satisfies the following properties. 1. The evaluation map ev : LM-TM ! M-TM is a map of ring spectra. 2. There is a map of ring spectra æ : LM-TM ! 1 ( M+ ) where the target is* * the suspension spectrum of the based loop space with a disjoint basepoint. Its* * ring structure is induced by the usual product on the based loop space. In homology the * *map æ* is given by the composition æ* : Hq(LM-TM )- u*--!~Hq+dLM --'-! Hq( M) = where u* is the Thom isomorphism, and the map ' takes a (q + d) - cycle in* * LM and intersects it with the based loop space viewed as a codimension d - subman* *ifold. 3. The ring structure is compatible with the Chas - Sullivan homology product* * in the sense that the following diagram commutes: Hq(LM-TM ) Hr(LM-TM ) ---! Hq+r(LM-TM ^ LM-TM ) --~*-!Hq+r(LM-TM ) ? ? u*?y~= ~=?yu* Hq+d(LM) Hr+d(LM) - --! Hq+r+d(LM) O Proof.The multiplicative structure ~ : LM-TM ^ LM-TM ! LM-TM will be defined using the Pontrjagin - Thom construction. We therefore begin by recalling some * *properties of this construction. Let e : P k,! Nn+k be an embedding of closed, oriented manifolds. Let e be a tubular neighborhoo* *d of of e(P k), which we identify with the total space of the normal bundle of the embe* *dding. Let ø : P k! e [ 1 A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 7 be the Pontrjagin - Thom collapse map to the one point compactification, define* *d by 8 < x ifx 2 e ø(x) = : 1 ifx =2 e. If we identify the compactification with the Thom space of the normal bundle,* * e[ 1 ~= P e, then in homology, after applying the Thom isomorphism u* : Hq(P e)-~=--!* *Hq-n(P k), we get the üp sh-forward", or ü mkehr" map, e!: Hq(Nk) - fi*--!Hq(P e)-u*--!~Hq-n(P k). = Recall that in the case of the diagonal embedding of a d - dimensional closed* * oriented manifold, : Md ! Md x Md that the normal bundle is isomorphic to the tangent bundle, ~=T M so that the Pontrjagin - Thom map is a map ø : M x M ! MTM . Furthermore, the p* *ush - forward map in homology, !: H*(Md x Md) - fi*--!H*(MTM ) --u*-!~H*-d(Md). = is simply the intersection product. Now the Pontrjagin - Thom construction also applies when one has a vector bun* *dle over the ambient manifold of an embedding. That is, if one has an embedding e : P k,* *! Nn+k as above, and if one has a vector bundle (or virtual bundle) i ! N, then one ob* *tains a Pontrjagin - Thom map ø : i [ 1 ! ('*(i)) [ 1 where ('*(i)) is the tubular neighborhood of the induced embedding of total sp* *aces '*(i) ,! i. Now i [ 1 is the Thom space N``, and ('*(i)) [ 1 is the Thom space P '*(``)* *. eSo the Pontrjagin map is a map *(``) ø : N``! P ' . e Moreover this construction works when i is a virtual bundle over N as well. I* *n this case when i = -E, where E ! N is a k - dimensional vector bundle over N, then the Th* *om spectrum N``= N-E is defined as follows. Suppose the bundle E is embedded in a* * k + M 8 R.L. COHEN AND J.D.S JONES dimensional trivial bundle, E ,! N x Rk+M . Let E? be the M - dimensional orth* *ogonal complement bundle to this embedding E? ! N. Then ? N-E = -(N+k)NE . Notice that the Thom isomorhism is of the form u* : Hq(N) ~=Hq-k(N-E ). In particular, applying the Pontrjagin - Thom construction to the diagonal em* *bedding : M ,! M x M, using the virtual bundle -T M x -T M over M x M, we get a map of Thom spectra, *(-TMx-TM) ø : (M x M)-TMx-TM ! MTM or, ø : M-TM ^ M-TM ! M-TM . In homology, this map still realizes the intersection pairing on H*(M), after a* *pplying the Thom isomorphism. The map ø defines a ring spectrum structure on M-TM that is * *well known to be the Spanier - Whitehead dual of the diagonal map : M ! M x M. To construct the ring spectrum pairing ~ : LM-TM ^ LM-TM ! LM-TM , we basi* *cally üp ll back" the structure ø over the loop space. To make this precise, let ev x ev : LM x LM ! M x M be the product of the eva* *luation maps, and define LM xM LM to be the fiber product, or pull back: ~ LM xM LM ---! LM x LM ? ? (1.1) ev?y ?yevxev M ---! M x M. Notice that LM xM LM is a codimension d submanifold of the infinite dimensional* * manifold LM x LM, and can be thought of as LM xM LM = {(ff, fi) 2 LM x LM such thatff(0) = fi(0)}. Notice that there is also a natural map from LM xM LM to the loop space LM defi* *ned by first applying ff and then fi. That is, (1.2) fl : LM xM LM ! LM (ff, fi)! ff * fi A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 9 where 8 < ff(2t) if0 t 1_ ff * fi(t) = 2 : fi(2t - 1) if1_ 2 t 1. Notice that when restricted to the product of the based loop spaces, M x M * *LM xM LM, then fl is just the H - space product on the based loop space, M x M ! M. Notice that by its definition (1.1) the embedding ~ : LM xM LM ,! LM x LM has* * a tubular neighborhood ( ~) defined to be the inverse image of the tubular neigh* *borhood of the diagonal : M ,! M x M: ( ~) = ev-1( ( )). Therefore this neighborhood is homeomorphic to the total space of the d - dimen* *sional vector bundle given by pulling back the normal bundle of the embedding , which* * is the tangent bundle of M: ( ~) ~=ev*( ) = ev*(T M). Thus there is a Pontrjagin - Thom construction *(TM) ø : LM x LM ! LM xM LMev . As described earlier, we ease the notation by refering to this Thom spectrum as* * (LM xM LM)TM . By the naturality of the Pontrjagin - Thom construction, we have a comm* *utative diagram of spectra, LM x LM --fi-! (LM xM LM)TM ? ? (1.3) ev?y ?yev M x M ---! MTM fi Since, as observed before, ø* : H*(M x M) ! H*(MTM ) ~=H*-d(M) is the interse* *ction product, then in homology, H*(LM x LM) --fi*-!H*((LM xM LM)TM ) --u*-!H*-d(LM xM LM) can be viewed (as is done in Chas - Sullivan [2]) as taking a cycle in LM x LM,* * and "intersecting" it with the codimension d submanifold LM xM LM. 10 R.L. COHEN AND J.D.S JONES Now observe that the map fl : LM xM LM ! LM defined above (1.2) preserves the evaluation map. That is, the following diagram commutes: LM xM LM --fl-! LM ? ? (1.4) ev?y ?yev M ---! M = Thus fl induces a map of bundles fl : ev*(T M) ! ev*(T M), and therefore a map * *of Thom spectra, fl : (LM xM LM)TM ! LMTM . Now consider the composition (1.5) ~~: LM x LM --fi-!(LM xM LM)TM --fl-!LMTM In homology, the homomorphism (1.6) H*(LM x LM) - ~~*--!H*(LMTM ) - u*--!~H*-d(LM) = takes a cycle in LM xLM, intersects in with the codimension d - submanifold LM * *xM LM, and maps it via fl to LM. This is the definition of the Chas - Sullivan product* * H*(LM). Now as we did before with the diagonal embedding, we can perform the Pontrjag* *in - Thom construction when we pull back the virtual bundle -T M x -T M over LM x LM. That is, we get a map of Thom spectra *(-TMx-TM) ev*(TM) ev*( *(-TMx-TM)) ø : (LM x LM)(evxev) -! (LM xM LM) . But since ev*( *(-T M x -T M)) = ev*(-2T M), we have ø : LM-TM ^ LM-TM -! (LM xM LM)TM -2TM = (LM xM LM)-TM . Now by the commutativity of (1.4), fl induces a map of Thom spectra, fl : (LM xM LM)-TM ! LM-TM and so we can define the ring structure on the Thom spectrum LM-TM to be the c* *ompo- sition (1.7) ~ : LM-TM ^ LM-TM --fi-! (LM xM LM)-TM - -fl-!LM-TM . A few properties of this map ~ are now immediately verifiable. First, ~ is associative. This follows from the naturality of the Pontrjagin -* * Thom con- struction, and the fact that the map fl is associative. (Strictly speaking, for* *mula (1.2) is A1 - associative as is the usual formula for the product on the based loop spa* *ce, M. A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 11 However the standard trick of replacing M with öM ore loops" changes the A1 st* *ructure to a strictly associative structure. The same technique applies to the map fl. * *Otherwise, the spectrum LM-TM will have the structure of an A1 ring spectrum.) Also, notice tha LM-TM has a unit, ' : S0 ! LM-TM , defined by the composit* *ion ' : S0--j-! M-TM --ff-!LM-TM where j is the unit of the ring spectrum structure of M-TM , and oe is the map* * of Thom spectra induced by the section of the evaluation map ev : LM ! M defined by vie* *wing points in M as constant loops. Thus ~ : LM-TM ^ LM-TM ! LM-TM defines an associative ring spectrum. Notice furthermore that in homology, after applying the Thom isomorphism, ~* * *induces the same homomorphism as ~~*, and so by (1.6) the following diagram commutes: Hq-2d(LM-TM ^ LM-TM ) --~*-! Hq-2d(LM-TM ) ? ? (1.8) u*?y~= ~=?yu* Hq(LM x LM) ---! Hq-d(LM) O where O : Hq(LM x LM) ! Hq-d(LM) is the Chas - Sullivan product. This proves pa* *rt (3) of theorem 1. Now by the naturality of the Pontrjagin - Thom construction, the following di* *agram of Thom spectra commutes (compare (1.3) LM-TM ^ LM-TM --~-! LM-TM ? ? (1.9) evxev?y ?yev M-TM ^ M-TM ---! M-TM . fi Thus the evaluation map ev : LM-TM ! M-TM is a map of ring spectra, which pro* *ves part 1 of theorem 1. We now verify part 2 of theorem 1. Let x0 2 M be a base point, and consider * *the following pullback diagram: M - -j-! LM ? ? (1.10) p?y ?yev x0 - --! M. i 12 R.L. COHEN AND J.D.S JONES Thus the embedding j : M ,! LM is an embedding of a codimension d submanifold,* * and has tubular neighborhood, (j) equal to the inverse image of the tubular neighb* *orhood of the inclusion of the basepoint i : x0 ,! M This tubular neighborhood is simply * *a disk Dd, and so (j) ~= M x Dd ~=ffld where ffld reflects the d dimensional trivial bundle over M. Thus the Thom - P* *ontrjagin construction makes sense, and is a map d d ø : LM ! Mffl= ( M+ ) where the last space is the d - fold suspension of M with a disjoint basepoint* *. In homology, the homomorphism ø* : Hq(LM) ! Hq( d( M+ ) = Hq-d( M) denotes the map that is obtained by intersecting a q -cycle in LM with the codi* *mension d submanifold M. By performing the Pontrjagin - Thom construction after pulling back the virtu* *al bundle -T M over LM, we get a map of Thom spectra d j*ev*(-TM) ø : LM-TM ! ( M)ffl . But by the commutativity of diagram (1.10), j*ev*(-T M) = p*i*(-T M), which is * *the trivial, virtual -d dimensional bundle which we denote ffl-d. So the Pontrjagin* * - Thom map is therefore a map of spectra (1.11) æ : LM-TM --fi-!( M)ffld j*ev*(-TM)= ( M)ffld ffl-d= 1 ( M+ ) where here, like before, 1 ( M+ ) denotes the suspension spectrum of the based* * loop space of M with a disjoint basepoint. To complete the proof of theorem 1 we need to p* *rove that æ : LM-TM ! 1 ( M+ ) is a map of ring spectra. Toward this end, consider the * *following diagram of pull back squares: ~ M x M --'-! LM xM LM ---! LM x LM ?? ? ? (1.12) y ?yev ?yevxev x0 ---! M ---! M x M. i This gives Pontrjagin - Thom maps (1.13) LM-TM ^ LM-TM --fi-!(LM xM LM)-TM --fi-! 1 ( M+ ) ^ 1 ( M+ ). A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 13 Notice that by the naturality of the Pontrjagin - Thom construction, the above * *composition is equal to æ ^ æ : LM-TM ^ LM-TM -! 1 ( M+ ) ^ 1 ( M+ ). Now notice that by the formula for the map fl : LM xM LM ! LM, the following diagram commutes: M x M - -'-! LM xM LM ? ? m?y ?yfl M - --! LM where m is the usual multiplication on the based loop space. Pulling back the * *virtual bundle -T M over LM, and applying the Pontrjagin - Thom construction, we then g* *et a commutative diagram of spectra, LM-TM ^ LM-TM --fi-!(LM xM LM)-TM - fi--! 1 ( M+ ) ^ 1 ( M+ ) ? ? (1.14) fl?y ?ym LM-TM - --! 1 ( M+ ) j Now as observed above (1.13), the top horizontal composition ø O ø is equal to * *æ ^ æ : LM-TM ^ LM-TM ! 1 ( M+ ) ^ 1 ( M+ ). Also, fl O ø is, by definition, the * *ring structure ~ : LM-TM ^ LM-TM ! LM-TM . Thus the following diagram of spectra commutes: j^j LM-TM ^ LM-TM - --! 1 ( M+ ) ^ 1 ( M+ ) ? ? ~?y ?ym LM-TM - --! 1 ( M+ ). j * * __ Thus æ is a map of ring spectra, which completes the proof of theorem 1. * * |__| 14 R.L. COHEN AND J.D.S JONES 2. The operad structure In this section we describe an operad structure on the spectrum LM-TM . Thi* *s is the äc ctus operad" C. This operad was introduced by Voronov. It has the homotopy o* *f the operad of framed little disks in R2. According to Getzler's result [3] this is * *precisely what is needed to induce the Batalin - Vilkovisky algebra structure in homology. The* *refore this structure can be viewed as a homotopy theoretic realization of the BV - algebra* * structure Chas and Sullivan show to exist on the chains of the loop space. We begin by recalling the definition of cactus operad C. A point in the spac* *e Ck is a P k collection of k oriented circles c1, . .,.ck, with radii ri so that i-1ri= 1.* * Each circle has a marked point xi 2 ci. Moreover the circles can intersect each other at a fin* *ite number of points (vertices) to create a äc ctus - type configuration". Strictly speaki* *ng this means that the dual graph of this configuration is a tree. That is, the äc ctus" (i.e* * the union of the circles) must be connected and have no "extra loops". (This is the tree con* *dition on the dual graph.) The edges coming into any vertex are also equipped with a cyclic o* *rdering. Notice that a cactus (i.e a point in Ck) comes equipped with a well defined m* *ap from the unit circle to the boundary of the cactus. That is, the map begins at the m* *arked point x1 2 c1, then traverses the circle c1 in the direction of its orientation in a * *length preserving manner. When a point of intersection with another circle is reached, the loop t* *hen traverses that circle, in the direction of its orientation. This path is continued until* * it eventually arrives back at the original basepoint x1 2 c1. Given a cactus c = 2 Ck we let ffic : S1 ! c1 [ . .[.ck be this loop that traverses the boundary of the cactus. Notice that Ck has a free action of the symmetric group k defined by permuti* *ng the ordering of the circles. The operad action , : Ck x (Cj1x . .x.Cjk)! Cj A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 15 where j = j1 + . .+.jk is defined as follows. Let c x (c1, . .,.ck) 2 Ck x (Cj1* *x . .x.Cjk). Scale the cactus c1 down so that its total radius is the radius r1 of the first* * component c1 of c 2 Ck. Similarly, scale each of the cacti ci down so that its total radius * *is the radius ri of the ithcomponent ciof c. By using the loops ffici(scaled down appropriately)* * we identify the component circles ciwith the boundary of the cactis ci. This has the effect* * of replacing the k component circles c1, . .,.ck making up the cactus c, by the k cacti, c1,* * . .,.ck. This produces a cactus with j1 + . .+.jk = j component circles. Our goal in this section is to prove theorem 2 of the introduction. We now re* *state that theorem. Theorem 5. There are maps of spectra ik : (Ck)+ ^ k (LM-TM )(k)-! LM-TM giving LM-TM the structure of a C - operad ring spectrum, compatible with the * *ring struc- ture ~ : LM-TM ^ LM-TM ! LM-TM . Proof.Given a cactus c = 2 Ck, define LcM to be the mapping space LcM = Map(c, M). This space consists of maps from the union c1 [ . .[.ck ! M. The map from the c* *ircle ffic : S1 ! c1 [ . .[.ck defines a map from LcM to the loop space, (2.1) flc : LcM! LM f! f O ffic Now LC M can also be viewed as the pullback of an evaluation mapping of the p* *roduct (LM)k defined as follows. For each component of the cactus ci, let mi be the nu* *mber of points on the circle cithat intersect other components of the cactus. Let mc = * *m1+. .+.mk. We define an evaluation map evc : (LM)k -! (M)mc as follows. On the circle ci, let y1, . .,.ymi be the points that intersect oth* *er circles in the cactus c. Assume that these points are ordered according to the orientation of * *the circle ci beginning at the marked point xi 2 ci. Let si : S1 ! ci be the identification o* *f the unit circle with ci obtained by scaling down the unit circle so as to have radius ri* *, and rotating 16 R.L. COHEN AND J.D.S JONES it so the basepoint 1 2 S1 is mapped to the marked point xi 2 ci. Let u1, . .,.* *umi be the points on the unit circle corresponding to y1, . .y.mi2 ci under the map si. De* *fine (2.2) evci: LM ! (M)mi oe! (oe(u1), . .,.oe(umi)) Now define (2.3) evc = evc1x . .x.evcmi: LM ! (M)m1 x . .x.(M)mi = (M)mc Now let w1, . .,.wnc 2 c1 [ . .[.ck denote all the points in the cactus that * *lie in more than one component. For each such point wi, let ~i be the number of components * *of the cactus on which wi lies. We think of ~i as the üm ltiplicityö f the intersecti* *on point wi. Notice that we have the relation ncX (2.4) ~i= mc. i=1 The "tree" condition on the dual of the cactus also imposes the following relat* *ion: (2.5) mc- nc = k - 1 Now consider the diagonal mapping c : (M)nc -! (M)mc defined by the composition ~1x...x ~nc = c : (M)nc ---------! (M)~1x . .x.(M)~nc - --! (M)mc where ~i: M ! (M)~i is the ~i -fold diagonal. Observe that the following is a * *cartesian pull - back square: ~ c LcM ---! (LM)k ? ? (2.6) evint?y ?yevc (M)nc ---! (M)mc c where evint: LcM ! (M)nc evaluates a map f : c ! M at the nc intersection point* *s, w1, . .,.wnc 2 c1 [ . .[.ck. The normal bundle ( c) of the diagonal embedding ~1x...x ~nc = c : (M)nc ---------! (M)~1x . .x.(M)~nc - --! (M)mc A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 17 is equal to (~1 - 1)T M x . .x.(~nc- 1)T M -! (M)nc, where (q)T M denotes the q - fold direct sum of T M with itself as a bundle ove* *r M. Notice P n that since ic=1~i = mc and mc - nc = k - 1 (2.5), then if : M ,! (M)nc is t* *he full diagonal map then the pull back of this normal bundle (2.7) *( ( c)) ~=(k - 1)T M -! M. Now by pulling back the tubular neighborhood (and the resulting normal bundle* * ) of c over LcM, we have a resulting Pontrjagin - Thom to the Thom space, * ( ( )) ø : (LM)k -! (LcM)evint c Notice that by (2.7), we have a bundle isomorphism ev*int( ( c)) ~=ev*((k - 1)T M) where ev : LcM ! M is evaluation at x1 2 c = c1 [ . .[.ck. In particular the m* *ap flc : LcM ! LM (2.1) is covered by a map of bundles, ev*int( ( c))-flc--!(k - 1)T M ?? ? (2.8) y ?y LcM ---! LM. flc Now, like what we did in section 1, let (-T M)k ! (LM)k denote the pull back * *of the k -fold exterior product of the virtual bundle -T M ! T M via the k -fold produ* *ct of the evaluation map at the basepoint, (ev)k : (LM)k ! (M)k. Then performing the Pont* *rjagin - Thom construction on this bundle, we get a map of Thom spectra, * ( ( )) ~*((-TM)k) (2.9) ø : (LM-TM )(k)-! (LcM)evint c c Now by (2.8) we have a map of virtual bundles ev*int( ( c)) ~ *c((-T M)k)flc---!(k - 1)T M (-k)T-M=--!-T M ?? ? y ?y LcM - --! LM. flc and therefore a map of Thom spectra * ( ( )) ~*((-TM)k) -TM flc : (LcM)evint c c ! LM 18 R.L. COHEN AND J.D.S JONES We then define the map of spectra i(c) : (LM-TM )(k)! LM-TM to be the composi* *tion * ( ( )) ~*((-TM)k) -TM (2.10) i(c) = flcO ø : (LM-TM )(k)-! (LcM)evint c c ! LM . This then defines the basic structure map (2.11) ik : (Ck)+ ^ (LM-TM )(k)-! LM-TM (c ; u1 ^ . .^.uk)-! i(c)(u1 ^ . .^.uk). By checking the definiton one sees that these map descend to the orbit of the* * k - action, (2.12) ik : (Ck)+ ^ k (LM-TM )(k)-! LM-TM . Furthermore, checking the definitions, and in particular using the naturality o* *f the Pontr- jagin - Thom constructions, one sees that the maps ik fit together compatibly t* *o define the C - operad structure of the spectrum LM-TM , as claimed. We end with two observations. First, let c0 2 C2 be the 2- component cactus c* *0 = , where c1 and c2 are both circles of radii 1=2, both oriented counter - clockwis* *e, intersecting at one point. The point of intersection is defined to be the marked point of e* *ach circle, x1 = x2 2 c1[c2. Then observe that the induced pairing, i(c0) is equal to the r* *ing spectrum multiplication ~ defined in section 1, i(c0) = ~ : LM-TM ^ LM-TM -! LM-TM . Thus the operad structure is compatible with the ring spectrum structure. Second, let oe(c0) 2 C2 denote the same 2 - component cactus as c0, permuted * *by the action of the nontrivial element of the symmetric group, oe 2 2. So c1 = . Then notice that the following diagram of spectra commutes: ``(ff(c0))-TM LM-TM ^ LM-TM -----! LM ? ? T?y ?y= LM-TM ^ LM-TM ---! LM-TM ~ where in this diagram T switches the two factors. Since the operad space C2 is connected, a path between c0 and oe(c0) yields t* *he homotopy commutativity of the ring spectrum structure ~ of LM-TM . * *__ |* *__| A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 19 3. A cosimplicial description of LM and LM-TM and a proof of Theorem 3 In this section we describe a cosimplicial model for the spectrum LM-TM . W* *e then describe the ring spectrum structure simplicially. This cosimplicial model will* * then give a natural way of relating the singular chains C*(LM-TM ) to the Hochshild cochai* *n complex CH*(C*(M), C*(M)), and in particular relate the simplicial model for the ring s* *tructure of LM-TM to the cup product structure in this cochain complex. This will allow us* * to prove theorem 3. We begin by reviewing the cosimplicial model of the loop space LX for any spa* *ce X, coming from a simplicial decomposition of the circle S1. We refer the reader t* *o [4] for details. Let S1*be the simplicial set decomposition of the circle which has one zero s* *implex and one nondegenerate one simplex. In this decomposition there are n + 1 n - simpli* *ces, all of which are degenerate for n > 1. We write this as S1n= {n + 1}. Now given any sp* *ace X, there is a resulting cosimplicial model for the free loop space, LX, which we c* *all X*. The n - simplices of X* are given by maps Xn = Map(S1n, X) = Map({n + 1}, X) = Xn+1. Of course the coface and codegeneracy maps of X* are dual to the face and degen* *eracy maps of S1*. They are given by the formulas (3.1) ffii(x0, . .,.xn-1)= (x0, . .,.xi-1, xi, xi, xi+1, . .,.xn-1), 0 i* * n - 1 ffin(x0, . .,.xn-1)= (x0, x1, . .,.xn-1, x0) oei(x0, . .,.xn+1)= (x0, . .,.xi, xi+2, . .,.xn+1), 0 i n Since the geometric realization of S1*is homeomorphic to the circle, S1 ~=|S1*|, the öt tal complexö r geometric corealization of X* is homeomorphic to the loo* *p space, LX ~=T ot(X*). This was studied in detail by the second author in [4], and in particular the f* *ollowing interpretation of this result was given. For each k, let k be the standard k -* * simplex: k = {(x1, . .,.xk) : 0 x1 x2 . . .xk 1}. 20 R.L. COHEN AND J.D.S JONES Consider the maps (3.2) fk : k x LX- ! Xk+1 (x1, . .,.xk) x!fl(fl(0), fl(x1), . .,.fl(xk)). Let ~fk: LX ! Map( k, Xk+1) be the adjoint of fk. Then the following was prov* *en in [4]. Q Theorem 6. Let X be any space, and let f : LX -! k 0Map( k, Xk+1) be the pr* *oduct of the maps ~fk. Then f is a homeomorphism onto its image. Furthermore, the ima* *ge con- sists of sequences of maps {OEk} which commute with the coface and codegeneracy* * operators. We call this space of sequences of maps Map *( *, X*+1) and this is the total s* *pace of the cosimplicial space T ot(X*). By applying singular cochains to the maps fk, one obtains maps f*k: C*(X) k+1 ! C*-k(LX). The following was also observed in [4]. Theorem 7. For any space X, the homomorphisms f*k: C*(X) k+1 ! C*-k(LX) fit together to define a chain map from the Hochshield complex of the cochains of X* * to the cochains of the free loop space, f* : CH*(C*(X)) ! C*(L(X)) which is a chain homotopy equivalence when X is simply connected. Hence it indu* *ces an isomorphism in homology ~= f* : HH*(C*(X)) - --! H*(L(X)). Remark. Let us clarify some notation. Given an algebra (or differential graded * *algebra) A, the the Hochshild complex of A, CH*(A) is a complex of the form . . .-b--!A n+2 --b-! A n+1 - -b-! . .-.b--!A A --b-! A. The homology of this algebra is denoted HH*(A). More generally if M is a bimodu* *le over A, we denote by CH*(A; M) the Hochshild complex of the form . .-.-b-! A n+1 M - -b-! A n M - -b-! . .-.b--!A M --b-! M. A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 21 The homology of this complex is denoted HH*(A; M). So in particular if M = A we* * see that HH*(A; A) = HH*(A). Dually, we denote by CH*(A; M) the Hochshild cochain complex of the form * b* b* b* b* M - b--! Hom(A; M) ---! . .-.--! Hom(A n; M) ---! Hom(A n+1; M) - --!* * . . . Its cohomology is denoted HH*(A; M). By dualizing theorem 6 we obtain the follo* *wing. Corollary 8. For any simply connected space X, there is a chain homotopy equiva* *lence from the singular chains of the loop space to the Hochshild cochain complex f* : C*(LX) ! CH*(C*(X); C*(X)) and so an isomorphism in homology, ~= f* : H*(LX) ---! HH*(C*(X); C*(X)). Notice that the cochain complex CH*(C*(X); C*(X)) does not in general have a * *natural product structure. This is because the coefficients, C*(X), is not in general a* * ring. Notice however that the Hochshild complex CH*(C*(X), C*(X)) does in fact have a cup pr* *oduct coming from the algebra structure of C*(X). Of course when X is a closed, orien* *ted manifold of dimension d, Poincare duality gives a chain homotopy equivalence, C*(X) ~=Cd* *-*(X), and so the cochain complex CH*(C*(X); C*(X)) inherits an algebra structure. The* *refore by the above corollary, H*(LX) inherits an algebra structure in this case. We w* *ill see that this indeed realizes the Chas - Sullivan product. We will show this by showing * *that when M is a closed, oriented d - manifold, the Thom spectrum LM-TM inherits a cosim* *plicial structure from X* for which the analogue of theorem 7 will yield a natural chai* *n homotopy equivalence C*(LM-TM ) ~=CH*(C*(M), C*(M)). To begin, notice that by the definitions 3.2, the following diagrams commute: k x LM --fk-!Mk+1 ? ? e?y ?yp1 M ---! M = where the left hand vertical map is the evaluation, e((t1, . .,.tk); fl) = fl(0* *), and the right hand vertical map is the projection onto the first coordinate. Pulling back the* * virtual bundle -T (M) defines a map of virtual bundles (fk)* : e*(-T M) -! p*1(-T M), 22 R.L. COHEN AND J.D.S JONES and therefore maps of Thom spectra, (which by abuse of notation we still call f* *k) (3.3) fk : ( k)+ ^ LM-TM -! M-TM ^ (Mk)+ . By taking adjoints, we get a map of spectra, Q f Q f : LM-TM - k-k-! kMap(( k)+ ; M-TM ^ (Mk)+ ) where on the right hand side the mapping spaces are maps of unital spectra. Th* *is map Q is just the induced map of Thom spectra of the map f : LX -! k 0Map( k, Xk+1) described in theorem 6. The following result is induced by theorem 6 by passing* * to Thom spectra. Let TX* be the cosimplicial spectrum defined to be the cosimplicial Thom spec* *trum of the cosimplicial virtual bundle -T M. That is, the virtual bundle over the k s* *implices Xk = Mk+1 is p*1(-T M). Said more explicitly, TX* is the cosimplicial spectrum * *whose k - simplices are the spectrum TXk = M-TM ^ (Mk)+ . To describe the coface and codegeneracy maps, consider the maps ~L : M-TM ! M+ ^ M-TM and ~R : M-TM ! M-TM ^ M+ of Thom spectra induced by the diagonal map : M ! M x M. ~L and ~R are the ma* *ps of Thom spectra induced by the maps of virtual bundles * : -T M ! p*L(-T M) and * : -T M ! p*R(-T M), where pL and pR are the projection maps M x M ! M onto t* *he left and right coordinates respectively. We then have the following formulas fo* *r the coface and codegeneracy maps: (3.4) ffi0(u; x1, . .,.xk-1)= (vR ; yR , x1, . .,.xk-1) ffii(u; x1, . .,.xk-1)= (u; x1, . .,.xi-1, xi, xi, xi+1, . .,.xk-1), * * 1 i k - 1 ffik(u; x1, . .,.xk-1)= (vL; x1, . .,.xk-1, yL), where ~R (u) = (vR , yR ), ~L(u) = (yL, vL) and oei(u; x1, . .,.xk+1) = (u; x1, . .,.xi, xi+2, . .,.xk+1), 0 i * * k The following result is simply the application of the Thom spectrum functor f* *or the virtual bundle -T M to theorem 6. A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 23 Theorem 9. Let M be any closed, d - dimensional manifold, and let Q f Q f : LM-TM - k-k-! kMap(( k)+ ; M-TM ^ (Mk)+ ) be the product of the maps of spectra fk, as defined above (3.3). Then f is a h* *omeomor- phism onto its image. Furthermore, the image consists of sequences of maps {OE* *k} which commute with the coface and codegeneracy operators. We call this space of sequ* *ences of maps Map *( *, M-TM ^ (Mk)+ ) and this is the total space of the cosimplicial * *spectrum T ot(TX*). Now for M orientable, recall Atiyah's S - duality between M+ and M-TM descri* *bed in the introduction [1]. This defines a chain homotopy equivalence between the* * cochains C*(M-TM ) (defined to be the appropriate desuspension of the cochains of the T* *hom space of the normal bundle of a fixed embedding M ,! RN ) and the chains of the manif* *old C*(M-TM ) ~=C-*(M+ ). The maps fk : ( k)+ ^ LM-TM -! M-TM ^ (Mk)+ then define maps of cochains, f*k: C-*(M) C*(M) k ~=C*(M-TM ^ (Mk)+ ) -! C*-k(LM-TM ). Taking the dual we get a map of chain complexes (fk)* : C*-k(LM-TM )-! Hom(C*(M) k C-*(M); Z) ~=Hom(C*(M) k; C*(M)) = CHk(C*(M); C*(M)) The following is then a consequence of corollary 8, by passing to Thom spectr* *a. Corollary 10. For any oriented, closed manifold M, the chain maps (fk)* fit tog* *ether to define a chain homotopy equivalence from the chains of Thom spectrum LM-TM to * *the Hochshild cochain complex f* : C*(LM-TM ) ! CH*(C*(M); C*(M)) and so an isomorphism in homology, ~= f* : H*(LM-TM )---! HH*(C*(M); C*(M)). 24 R.L. COHEN AND J.D.S JONES As mentioned in the introduction, the Hochshild cochain complex CH*(C*(M); C** *(M)) has a cup product structure. Namely, for any algebra A, if OE 2 CHk(A; A) = Hom(A k; A) and _ 2 CHr(A; A) = Hom(A r; A), then OE [ _ 2 CHk+r(A; A) = Hom(A k+r; A) is defined by OE [ _(a1 . . .ak ak+1 . . .ak+r) = OE(a1 . . .ak)_(ak+1 . . .ak* *+r). For A = C*(M) (where the algebra stucture is the cup product in C*(M)), by taki* *ng adjoints, we can think of this as a pairing (3.5i) j [ : C*(M) k C*(M) C*(M) r C*(M) -! C*(M) k+r C*(M) (ff1 . . .ffk `) (fi1 . . .fir-! æ)ff1 . . .ffk fi1 . . .* *fir ` [ æ. Now recall that by S - duality, there is a ring spectrum structure * : M-TM ^ M-TM -! M-TM dual to the diagonal map : M ! M x M. Passing to chains, * : C*(M-TM ) C*(M-TM ) ! C*(M-TM ), is, with respect to the duality identification C*(M-TM* * ) ~= C*(M), therefore the cup product on cochains * = [ : C*(M) C*(M) ! C*(M). Thus formula (3.5) for the cup product in Hochshild cochains is therefore reali* *zed by the map h i (3.6) ~~k,r: M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+-! M-TM ^ (Mk+r)+ (u; x1, . .,.xk) ^ (v; y1, .-.,.yr)! ( *(u, v); x1, . .,.xk,* * y1, . .,.yr) The maps ~~k,rdefine maps of the simplices ~~k,r: TXk ^ TXr ! TXk+r and it is straight forward to check that these maps preserve the coface and cod* *egeneracy operators, and so define a map of the geometric corealization (öt tal spectra") ~~: T ot(TX*) ^ T ot(TX*) -! T ot(TX*). This proves the following. A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 25 Theorem 11. Using the homeomorphism f : LM-TM ~=T ot(TX*) of theorem 9, LM-TM inherits the structure of a ring spectrum, ~~: LM-TM ^ LM-TM -! LM-TM which is compatible with the cup product in Hochshild cohomology. That is, with* * respect to the chain homotopy equivalence f* : C*(LM-TM ) ! CH*(C*(M); C*(M)) of corollary 10, the following diagram of chain complexes commutes: C*(LM-TM ) C*(LM-TM ) --~~-! C*(LM-TM ) ? ? f* f*?y~= ~=?yf* CH*(C*(M); C*(M)) CH*(C*(M); C*(M)) ---! CH*(C*(M); C*(M)). [ In view of theorems 9, 10, and 11, theorem 3 will therefore be proven once we* * prove the following. Theorem 12. Let ~ : LM-TM ^ LM-TM ! LM-TM be the ring spectrum structure defined in section 1. Then the structures ~ and ~~: T ot(TX*) ^ T ot(TX*) ! T o* *t(TX*) are compatible in the sense that the following diagram homotopy commutes: LM-TM ^ LM-TM - ~--! LM-TM ? ? f^f?y' '?yf T ot(TX*) ^ T ot(TX*)---! T ot(TX*). ~~ Proof.Let wk,r: k+r ! k x r be the Alexander - Whitney diagonal map. That i* *s, for (x1, . .,.xk+r) 2 k+r, then wk,r(x1, . .,.xk+r) = (x1, . .,.xk) x (xk+1, .* * .,.xk+r) 2 k x r. By the definition of the cosimplicial structure of TX*, to prove the t* *heorem it suffices to prove that the following diagrams of spectra commute: 26 R.L. COHEN AND J.D.S JONES (a) k+r+^ LM-TM ^ LM-TM - 1^~--! k+r+^ LM-TM ? wk,r^1?y ? ( k x r)+ ^ LM-TM ^ LM-TM ?yfk+r ? fk^fr?y M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ - --! M-TM ^ (Mk+r)+ ~~ We verify the commutativity of these diagrams in several steps. First observe t* *hat the maps fk x fr : k x r x LM x LM ! Mk+1 x Mr+1 restrict to k x r x LM xM LM to define a map fk,rwhose image is in M x Mk x Mr making the following diagram commute: k x r x LM xM LM --,!-! k x r x LM x LM ? ? fk,r?y ?yfkxfr M x Mk x Mr ---! Mk+1 x Mr+1 where the bottom horizontal map is the diagonal map: (m x (x1, . .,.xk) x (y1, . .,.yr))= (m, x1, . .,.xk) x (m, y1, . .,* *.yr). By the naturality of the Pontrjagin - Thom construction, we therefore have a co* *mmutative diagram of spectra (b) ( k x r)+ ^ LM-TM ^ LM-TM -1^fi--!( k x r)+ ^ (LM xM LM)-TM ? ? fk^fr?y ?yfk,r M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ ---! M-TM ^ (Mk)+ ^ (Mr)+ . fi Notice further that by the definition of the maps fn, fk,rand the loop composit* *ion fl : LM xM LM ! LM defined in the last section, the following diagram commutes: A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 27 k+r x (LM xM LM) - 1xfl--! k+r x LM ? wk,rx1?y ? k x r x (LM xM LM) ?yfk+r ? fk,r?y M x (Mk) x (Mr) - --! Mk+r+1. = Passing to Thom spectra this yields the following commutative diagram: (c) ( k+r)+ ^ (LM xM LM)-TM - 1^fl--!( k+r)+ ^ LM-TM ? wk,r^1?y ? ( k x r)+ ^ (LM xM LM)-TM ?yfk+r ? fk,r?y M-TM ^ (Mk)+ ^ (Mr)+ - --! M-TM ^ Mk+r+. = Now consider the following diagram of spectra: ( k+r)+ ^ LM-TM ^ LM-TM - 1^fi--!( k+r)+ ^ (LM xM LM)-TM - 1^fl--!(* * k+r)+ ^ LM-TM ? ? wk,r^1?y wk,r^1?y * * ? ( k x r)+ ^ LM-TM ^ LM-TM - --! ( k x r)+ ^ (LM xM LM)-TM * * ?yfk+r ? 1^fi ? fk^fr?y ?yfk,r M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ - --! M-TM ^ (Mk+r)+ - --! M-T* *M ^ (Mk+r)+ . fi = Now the top left square in this diagram clearly commutes. The bottom left diag* *ram is diagram (b) above, and so it commutes. The right hand rectangle is diagram (c)* * above, so it commutes. Therefore the outside of this diagram commutes. Now the top hor* *izontal composition is, by definition the map 1 ^ ~ : ( k+r)+ ^ LM-TM ^ LM-TM -! ( k+r)+ ^ LM-TM . The bottom horizontal map is seen to be ~~: M-TM ^ (Mk)+ ^ M-TM ^ (Mr)+ -! M-TM ^ (Mk+r)+ 28 R.L. COHEN AND J.D.S JONES by recalling that the ring multiplication * : M-TM ^ M-TM ! M-TM is the Pon* *trjagin - Thom map ø : M-TM ^ M-TM ! M-TM applied to the diagonal embedding : M ,! M x M. With these identifications, the outside of this diagram is then diagram (a) a* *bove. As observed earlier, the commutativity of diagram (a) proves this theorem, and thi* *s completes * * __ the proof of theorem 2. * * |__| References [1]M.F. Atiyah, Thom complexes , Proc. London Math. Soc. (3) , no. 11 (1961), * *291-310. [2]M. Chas and D. Sullivan, String Topology, preprint: math.GT/9911159, (1999). [3]E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological fi* *eld theories., Comm. Math. Phys. 159 no. 2,, (1994), 265-285. [4]J.D.S Jones, Cyclic homology and equivariant homology Inventionnes Math. 87* *, no. 2, (1987), 403-423. [5]T. Tradler, Ph.D thesis, CUNY, to appear. Dept. of Mathematics, Stanford University, Stanford, California 94305 E-mail address, Cohen: ralph@math.stanford.edu Department of Mathematics, University of Warwick, Coventry, CV4 7AL England E-mail address, Jones: jdsj@maths.warwick.ac.uk