Loop coproducts in string topology and triviality of higher genus TQFT operations

LOOP COPR ODUCTS IN STRING TO POLOGY AND TRIVIALITY OF HIGHER GENUS TQ FT OPERA TIONS Hir ot aka T amanoi Univ ersit y of California, San ta Cru z A b s t rac t . Cohen and Godin constructed p ositive boundary top ological quantum field theory (TQFT) structure on the ho mology of free loop spaces of orien ted closed smo oth manifolds by a sso ciating a certain o p erations called string o p erations to orien table surfaces with parametrized b oundaries. W e show that all TQFT string operations asso ciated to surfaces of genus at least one v anish iden tically . This is a simple consequence of prop erties of the lo op copro duct whic h will b e discussed in detail. One in teresting prop erty is tha t the lo op copro du ct is nontrivial only on the degree d homology group of the connected comp onent of LM co n sisting of con tractible lo ops, where d = dim M , with v alues in the degree 0 homology group of constant lo o ps. Thus the loop coproduct b ehav es in a dramatically s impler wa y than the lo op pro duct. Contents 1. I ntro duction and trivialit y of higher genus TQFT string op erations . . . . . . . 1 2. T he lo op copro duct and its F rob enius compatibilit y . . . . . . . . . . . . . . . . . . . . . . 4 3. P r op erties of the lo op copro duct and their consequences . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 § 1. In tro duction and trivialit y of higher gen us T QFT string op erations Let M b e a connected closed orien table smo oth manifold of dimension d , and let LM = Map( S 1 , M ) b e its free loop space of con tinuous maps fr om the circle S 1 to M . Chas and S ulliv an [CS] s h o wed that its homology H ∗ ( LM ) = H ∗ + d ( LM ) come s equipp ed with an asso ciativ e graded comm u tativ e pro duct of d egree − d , and a compatible Lie brac k et of degree 1. These t w o pro ducts together with an op erator ∆ of degree 1 with ∆ 2 = 0, coming from the n atural S 1 action on LM , giv e H ∗ ( LM ) the stru cture of a Batalin-Vilk ovisky algebra. The asso ciativ e pro d uct calle d the lo op pr o duct w as generalized to so called strin g op erations b y Cohen and Godin [CG]. Let Σ b e an orien table connected surface of genus g with p incoming and q outgoing parametrized b oundary circles, wh ere we require that q ≥ 1. T o such a surface Σ, they asso ciated an op erator µ Σ of the f orm µ Σ : H ∗  ( LM ) p  − → H ∗ + χ (Σ) d  ( LM ) q  , in su c h a w a y that µ Σ dep end s only on the top ological t yp e of the su rface Σ and µ Σ is compatible with sewing of sur f aces along p arametrized b oundaries. These op erations giv e rise to topological 1991 Mathematics Subje ct Cla s sific ation . 55P35. Key words and phr as es. loop copro duct, lo op product, string op erations, string to p ology. 1 2 HIROT AKA T AMANOI quan tum field theory (TQFT) without a counit. When Σ is a pair of pant s w ith either 2 incoming or 2 outgoing circles, we get a pro duct and and a coprod u ct: µ : H ∗ ( LM × LM ) − → H ∗− d ( LM ) , Ψ : H ∗ ( LM ) − → H ∗− d ( LM × LM ) , where the pr o duct µ coincides with the lo op pro d u ct of Chas and Sulliv an. See form ula (2-8) for a homotop y th eoretic definition of the lo op pro d uct. Since an y surface Σ can b e decomp osed in to pairs of pant s and capping discs, we can compute th e string op eration µ Σ b y comp osing lo op pro du cts and lo op copro ducts according to p an ts decomp ositions of Σ. In this p ap er, w e stud y prop er ties of coprod uct in detail, and as a consequ ence we sho w that for higher gen us surfaces Σ, the strin g op erations µ Σ are alw a y s trivial. Theorem A. L et Σ b e an oriente d c onne cte d c omp a ct surfac e of genus g with p inc oming and q ≥ 1 outgoing p ar ametrize d b oundary cir cles. If g ≥ 1 or q ≥ 3 , then the asso ciate d string op er atio n µ Σ vanishes. Th us the on ly nontrivial TQ FT str ing op erations corresp ond to gen us 0 surfaces w ith at most 2 outgoing circles. T o elemen ts a 1 , a 2 , . . . , a p ∈ H ∗ ( LM ), su c h op erations associate either their lo op p r o duct a 1 a 2 · · · a p or its loop copro duct Ψ( a 1 a 2 · · · a p ). T h us once w e und erstand the lo op copro duct Ψ, we kno w the b eha vior of all string op erations asso ciated to orientable su rfaces with parametrized b ound aries. F or a ∈ H ∗ ( LM ), let | a | d enote its homological d egree. Let c 0 b e the constan t lo op at the base p oin t x 0 in M , and let [ c 0 ] its h omology class in H 0 ( LM ). The connected comp onents of LM are parametrized by conjugacy classes of π 1 ( M ). Let ( LM ) [1] b e the comp onen t corresp ond ing to the conjugacy class of 1 ∈ π 1 ( M ). This is the sp ace of con tractible lo ops in M . In addition to the F r ob enius compatibilit y (Theorem 2.2), p r op erties of the lo op coprodu ct are describ ed in Theorem B, whose p art (2) sho ws dramatic simp licit y of the lo op copr o duct compared with the lo op pr o duct. Theorem B is th e main result of this pap er. Theorem A is only one of the consequences of Theorem B. W e will discuss some of the other consequences in Th eorem C. Theorem B. (1) L et p ≥ 0 , and let a 1 , a 2 , . . . , a p ∈ H ∗ ( LM ) b e arbitr ary p elements. Then the image of t he lo op c opr o duct Ψ lie s in the subset H ∗ ( LM ) ⊗ H ∗ ( LM ) ⊂ H ∗ ( LM × LM ) of cr oss pr o ducts, and for any 0 ≤ ℓ ≤ p it is given by Ψ( a 1 · a 2 · · · a p ) = χ ( M )[ c 0 ] a 1 · a 2 · · · · · a ℓ ⊗ [ c 0 ] a ℓ +1 · · · · · a p ∈ H ∗ ( LM ) ⊗ H ∗ ( LM ) , wher e χ ( M ) is the Eu ler char acteristic of M . (2) The lo op c opr o duct Ψ is nontrivial only on H d  ( LM ) [1]  , the de gr e e d homolo gy gr oup of the c omp onent o f c ontr actible lo ops in M . On H d  ( LM ) [1]  , th e lo o p c opr o duct Ψ has values in the homolo gy classes of c onstant lo ops H 0  ( LM ) [1]  ⊗ H 0  ( LM ) [1]  ∼ = Z [ c 0 ] ⊗ [ c 0 ] . Theorem B is p ro v ed in T heorem 3.1. Note that if M has v anishing Eu ler c haracteristic, for example if M is o d d dimensional, then its lo op copro duct is identical ly 0. Befo re w e prov e the ab o v e result in § 3, in § 2 we w ill pro v e v arious ge neral results on the lo op copro du ct including F r ob enius compatibilit y (Theorem 2.2) with precise treatmen t of signs, and F rob en ius compatibilit y and co deriv ation compatibilit y with resp ect to cap pro d u cts (Theorem 2.4). Since the pro of o f Theorem A is more o r less straigh tforward, we giv e its proof b elo w. Th is v anishing prop ert y is the basis of trivialit y of stable higher str in g op erations [T2] in the con text of homologica l conformal field theory in w h ic h homology classes of mo duli spaces of Riemann s u rfaces giv e rise to strin g op erations [G]. LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 3 As a consequence of Theorem B, w e obtain the follo win g result on torsion elemen ts pro ved in § 3. Let ι : Ω M − → LM b e the inclusion map from the based loop space to the free loop sp ace. Reca ll that the transfer map ι ! : H ∗ + d ( LM ) − → H ∗ (Ω M ) obtained by intersecting cycles with Ω M is an algebra map with resp ect to the loop pro duct in H ∗ ( LM ) an d the Pon trjagin pro d uct in H ∗ (Ω M ). Theorem C. L et M b e an even dimensional manifold with χ ( M ) 6 = 0 . Consider the f ol lowing c omp ositio n map ι ∗ ◦ ι ! : H p + d ( LM ) ι ! − → H p (Ω M ) ι ∗ − → H p ( LM ) . If p 6 = 0 , then the image of ι ∗ ι ! c onsists of torsion elements of or der a divisor of χ ( M ) . Namely, χ ( M ) ι ∗ ι ! ( a ) = χ ( M )[ c 0 ] · a = 0 if | a | 6 = d for a ∈ H ∗ ( LM ) . Thus, r ational ly, the c omp os ition is a trivial map if p 6 = 0 . See Example 3.6 for explicit examples of this fact when M is S 2 n or C P n . Since Theorem A can b e quic kly pro ved fr om T heorem B, w e giv e its pro of here in th e remainder of this in tro d uction. Pr o of of The or em A fr om The or em B. Let S ( p, q ) b e a gen u s 0 s urface with p in coming and q outgoing parametrized b ound ary circles, and let T b e a torus with 1 incoming and 1 outgoing parametrized b ound ary circles. Then any su rface Σ of gen us g with p incoming b oundary circles and q outgoing b oundary circles can b e d ecomp osed as S ( p, 1)# T # · · · # T # S (1 , q ), where T app ears g times. Corr esp ond ingly , the asso ciated string op eration µ Σ can b e d ecomp osed as µ Σ = µ S (1 ,q ) ◦ µ T ◦ · · · ◦ µ T ◦ µ S ( p, 1) . Assume g ≥ 1. W e compute µ T using a decomp osition of T into t wo pairs of pan ts corresp onding to the loop copr o duct and the lo op pr o duct. F or an y a ∈ H ∗ ( LM ), µ T ( a ) = µ ◦ Ψ( a ) = µ  χ ( M )[ c 0 ] ⊗ ([ c 0 ] · a )  = ( − 1) d χ ( M )([ c 0 ] · [ c 0 ]) · a. Since [ c 0 ] · [ c 0 ] = 0 ∈ H − d ( LM ) by dimensional reason, w e ha v e µ T ( a ) = 0 for all a ∈ H ∗ ( LM ). In view of the ab ov e decomp osion of µ Σ , this pro v es the v anishin g of string op erations associated to surfaces of gen us g ≥ 1. Next we assume q ≥ 3. T hen µ S (1 ,q ) = ( µ S (1 ,q − 2) ⊗ 1 ⊗ 1) ◦ (Ψ ⊗ 1) ◦ Ψ. F or an y a ∈ H ∗ ( LM ), (Ψ ⊗ 1) ◦ Ψ( a ) = (Ψ ⊗ 1)( χ ( M )[ c 0 ] ⊗ [ c 0 ] · a ) = χ ( M )Ψ([ c 0 ]) ⊗ [ c 0 ] · a = 0 , since Ψ([ c 0 ]) = 0 ∈ H − d ( LM × LM ) b y dimensional reason. Hence µ S (1 ,q ) = 0 for q ≥ 3. Again, in view of the ab ov e decomp osition of µ Σ , this pro ve s q ≥ 3 case of Theorem A.  In § 2, w e discuss general p rop erties of the lo op copro duct in detail and pro ve F rob enius com- patibilit y (Theorem 2.2), a symmetry pr op erty (Proposition 2. 3), and co deriv ation prop ert y of certain cap pro ducts (Theorem 2.4). In § 3. we pr o ve Theorem B and relate d results in Th eorem 3.1, and deduce their c onsequences including Theorem C pro ve d in Corollary 3.3 and Corollary 3.4. W e also discuss torsion prop erties of certin lo op br ac ket elemen ts in Corollary 3.5, and other miscellaneous p rop erties of image elemen ts of the loop copro duct in P rop ositions 3.7 and 3.8. All homology group s in this pap er ha ve inte ger co efficients. 4 HIROT AKA T AMANOI § 2. The lo op coproduct a nd its F rob enius compatibilit y As b efore, let LM b e the free lo op space of con tin uous maps from the c ircle S 1 = R / Z to a connected oriented closed smo oth d -manifold M . C ohen and Jones [CJ] ga v e a homotop y theoretic description of th e lo op pro d u ct. The lo op copro duct can b e d escrib ed in a similar w a y , an d w e study its prop erties in this section. A description of the lo op copro duct usin g transv ersal chains is giv en in [S]. Let p, p ′ : LM − → M b e ev aluation maps giv en b y p ( γ ) = γ (0) and p ′ ( γ ) = γ ( 1 2 ) for γ ∈ LM . W e consider the follo wing diagram where S M = ( p, p ′ ) − 1  φ ( M )  consists of lo ops γ su c h that γ (0) = γ ( 1 2 ), and q is the restriction of ( p, p ′ ) to this subsp ace. Let ι : S M − → LM b e th e inclusion map and let j : S M − → LM × LM b e giv en by j ( γ ) = ( γ [0 , 1 2 ] , γ [ 1 2 , 1] ). The map φ : M − → M × M is the d iagonal map. LM ι ← − − − − S M j − − − − → LM × LM ( p,p ′ )   y q   y M × M φ ← − − − − M Then the copr o duct map Ψ is defined b y the follo wing comp osition of maps: Ψ = j ∗ ◦ ι ! : H ∗ + d ( LM ) ι ! − → H ∗ ( LM ) j ∗ − → H ∗ ( LM × LM ) , where ι ! is the transf er map, also called a push-forw ard m ap, defined in the f ollo wing w ay . Let π : ν − → φ ( M ) b e the normal bundle to φ ( M ) in M × M and w e orien t ν so that w e ha ve an orien ted isomorphism ν ⊕ T φ ( M ) ∼ = T ( M × M ) | φ ( M ) . Let N b e a closed tubular neighborh o o d of φ ( M ) such that D ( ν ) ∼ = N , where D ( ν ) is the closed disc bu ndle. Let c : M × M − → N/∂ N b e the Thom collapse map. W e ha v e th e follo wing comm u tativ e diagram: H d ( M × M , M × M − φ ( M )) − − − − → H ∗ ( M × M ) ∼ =   y excision c ∗ x   H d ( N , N − φ ( M )) ∼ = − − − − → H d ( N , ∂ N ) ∼ = ˜ H d ( N/∂ N ) . Let u ′ ∈ ˜ H ( N/∂ N ) b e the Thom class of the n ormal b u ndle ν . Let u ′′ ∈ H d  M × M , M × M − φ ( M )  and u ∈ H d ( M × M ) b e corresp onding Thom classes. Th e class u is charact erized b y the p rop erty u ∩ [ M × M ] = φ ∗ ([ M ]). Since u comes from u ′′ , it is represen ted by a co cycle f whic h v anish on simplices in M × M whic h d o not intersect with φ ( M ). Let ˜ N = ( p, p ′ ) − 1 ( N ) b e a tubu lar neighborh o o d of S M in LM , and let ˜ c : LM − → ˜ N /∂ ˜ N b e the Thom collapse map. L et ˜ u ′ ∈ ˜ H d ( ˜ N /∂ ˜ N ) an d ˜ u ∈ H d ( LM ) b e pull-bac ks of corresp onding classes. W e h a ve ˜ u = ˜ c ∗ ( ˜ u ′ ). Let ˜ π : ˜ N − → S M ⊂ LM × LM b e a pro jection map corresp onding to π , and is give n as follo ws. Sup p ose γ ∈ ˜ N is suc h that ( p, p ′ )( γ ) = ( x 1 , x 2 ) ∈ N . Let η ( t ) = ( η 1 ( t ) , η 2 ( t )) b e a p ath in N fr om ( x 1 , x 2 ) to π ( x 1 , x 2 ) = ( y , y ) ∈ φ ( M ) corresp onding to the straigh t r a y in the bund le ν . Then ˜ π ( γ ) = ( η − 1 1 · γ [0 , 1 2 ] · η 2 ) · ( η − 1 2 · γ [ 1 2 , 1] · η 1 ) ∈ S M . F rom this descrip tion, it is ob vious that ˜ π is a deformation retraction. The trans fer map ι ! is defined by the follo wing comp osition of maps: ι ! : ˜ H ∗ + d ( LM ) ˜ c ∗ − → ˜ H ∗ + d ( ˜ N /∂ ˜ N ) ˜ u ′ ∩ ( · ) − − − − → H ∗ ( ˜ N ) ˜ π ∗ − → ∼ = H ∗ ( S M ) . Let s : M − → LM b e the constan t lo op map giv en by s ( x ) = c x , where c x is the constant lo op at x ∈ M . Since p ◦ s = 1 M , w e h a ve s ∗ ◦ p ∗ = 1. The transfer map ι ! has the follo wing prop erties. LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 5 Prop osition 2.1. (1) The c ohomolo gy clas s ˜ u ∈ H d ( LM ) is given by ˜ u = p ∗ ( e M ) , wher e e M ∈ H d ( M ) is the Euler class of M . (2) F or any element a ∈ H ∗ ( LM ) , (2-1) ι ∗ ι ! ( a ) = p ∗ ( e M ) ∩ a. In p ar ticular, ι ∗ ι ! ( s ∗ ([ M ])) = χ ( M )[ c 0 ] , wher e c 0 is the c onsta nt lo op at the b ase p oint x 0 in M , and χ ( M ) is the Euler char acteristic of M . (3) F or any α ∈ H ∗ ( LM ) and b ∈ H ∗ ( LM ) , (2-2) ι ! ( α ∩ b ) = ( − 1) d | α | ι ∗ ( α ) ∩ ι ! ( b ) . Pr o of. (1) S ince the map ( p, p ′ ) : LM − → M × M can b e factored as LM φ − → LM × LM p × p ′ − − − → M × M and p and p ′ are homotopic, we ha ve ˜ u = ( p, p ′ ) ∗ ( u ) = φ ∗ ◦ ( p × p ′ ) ∗ ( u ) = φ ∗ ◦ ( p × p ) ∗ ( u ) = p ∗ ◦ φ ∗ ( u ). Since φ ∗ ( u ) is, by definition, ( − 1) d times the Euler class e M of M and the Eu ler class is of order 2 when d is o dd, we ha v e ( − 1) d e M = e M . So we ha ve ˜ u = p ∗ ( e M ). (2) Al though w e ca n use a certa in comm utativ e d iagram for a pro of (see b elo w ), w e fi rst do a c h ain argumen t here in the spirit of [CS] and [S]. By barycentric sub divisions on the cycle ξ r epresen ting a ∈ H ∗ ( LM ), we may assume that ev ery simplex of ξ in tersecting with S M is con tained in In t ( ˜ N ). Since cohomology classes u ∈ H d ( M × M ) and u ′ ∈ ˜ H d ( N/∂ N ) come from the class u ′′ in H d  M × M , M × M − φ ( M )  , they can b e repr esen ted b y co cycles f and f ′ so that f v anishes on simp lices in M × M not intersecting φ ( M ), and f ′ v anishes on simplices in N /∂ N not in tersecting φ ( M ). So the cocycle ˜ f ′ = ( p, p ′ ) # ( f ′ ) representi ng ˜ u ′ v anishes on simplices in N not intersect ing with S M . Similarly , the cocycle ˜ f = ( p, p ′ ) # ( f ) = ˜ c # ( ˜ f ′ ) repr esen ting ˜ u = ˜ c ∗ ( ˜ u ′ ) v anishes on simplices in LM not inte rsecting with S M , and has the same v alues as ˜ f ′ on simplices in ˜ N intersect ing with S M . Since the cycle ξ is fine enough, the cycle s ˜ f ∩ ξ and ˜ f ′ ∩ ˜ c # ( ξ ) represent ing ˜ u ∩ a and ˜ u ′ ∩ ˜ c ∗ ( a ) are in fact iden tical. Since ι ! ( a ) = ˜ π ∗  ˜ u ′ ∩ ˜ c ∗ ( a )  is represen ted b y a cycle ˜ π # ( ˜ f ∩ ξ ), and ˜ π is a deformation r etraction, the t wo cycles ˜ π # ( ˜ f ∩ ξ ) and ˜ f ∩ ξ are homologous inside of Int ˜ N . Thus, ι ∗ ι ! ( a ) = [ ˜ π # ( ˜ f ∩ ξ )] and ˜ u ∩ a = [ ˜ f ∩ ξ ] represent the same homology class. Hence ι ∗ ι ! ( a ) = ˜ u ∩ a = p ∗ ( e M ) ∩ a , by (1). W e also giv e a homological pro of, using the follo wing comm utativ e d iagram. H ∗ ( LM ) ˜ c ∗ − − − − → H ∗ ( ˜ N , ∂ ˜ N ) ˜ u ′ ∩ ( · ) − − − − − → H ∗− d ( ˜ N ) ˜ π ∗ − − − − → ∼ = H ∗− d ( S M )    ( ι N ) ∗   y ∼ = ( ι N ) ∗   y ι ∗   y H ∗ ( LM ) j ∗ − − − − → H ∗ ( LM , LM − S M ) ˜ u ′′ ∩ ( · ) − − − − − → H ∗− d ( LM ) H ∗− d ( LM ) where ι N : ˜ N − → LM is an in clusion map. Here, the class ˜ u ′′ is giv en by ˜ u ′′ = ( p, p ′ ) ∗ ( u ′′ ), and it satisfies ˜ u ′ = ι ∗ N ( ˜ u ′′ ). Thus, for a ∈ H ∗ ( LM ), the comm utativ e diagram sho ws ι ∗ ι ! ( a ) = ˜ u ′′ ∩ j ∗ ( a ) = j ∗ ( ˜ u ′′ ) ∩ a = ˜ u ∩ a . The ab o ve chain argument giv es geometric m eaning to the comm u tativ e diagram ab o ve. When a = s ∗ ([ M ]), we ha ve ι ∗ ι !  s ∗ ([ M ])  = p ∗ ( e M ) ∩ s ∗ ([ M ]) = s ∗  s ∗ p ∗ ( e M ) ∩ [ M ]  . Since p ◦ s = 1, th is is equal to s ∗  χ ( M )[ x 0 ]  = χ ( M )[ c 0 ]. (3) W e compute. By definition of ι ! , w e h a ve ι ! ( α ∩ b ) = ˜ π ∗  ˜ u ∩ ˜ c ∗ ( α ∩ b )  = ˜ π ∗  ˜ u ′ ∩ ( ι ∗ N ( α ) ∩ ˜ c ∗ ( b )  = ( − 1) | α | d ˜ π ∗  ι ∗ N ( α ) ∩  ˜ u ′ ∩ ˜ c ∗ ( b )  . 6 HIROT AKA T AMANOI Since ι ∗ N ( α ) = ˜ π ∗ ι ∗ ( α ), the last form ula b ecomes ι ∗ ( α ) ∩ ˜ π ∗  ˜ u ′ ∩ ˜ c ∗ ( b )  = ι ∗ ( α ) ∩ ι ! ( b ), times the sign. Th is completes the pr o of.  Next we recall a homotop y theoretic descrip tion of the lo op pro du ct from [CJ]. W e consider the follo wing diagram, where LM × M LM denotes the set ( p × p ) − 1  φ ( M )  consisting of pairs ( γ , η ) of lo ops suc h that γ (0) = η (0), and ι ( γ , η ) denotes the usual lo op multiplicatio n γ · η . LM × LM j ← − − − − LM × M LM ι − − − − → LM p × p   y q   y M × M φ ← − − − − M Then for a, b ∈ H ∗ ( LM ), the lo op pro du ct a · b is defi ned b y a · b = ( − 1) d ( | a |− d ) ι ∗ j ! ( a × b ) . Here, as b efore, the transfer map j ! is defined using the Thom class u ′ ∈ ˜ H d ( N/∂ N ) and its pull bac k to the tubular neigh b orh o o d ˜ N = ( p × p ) − 1 ( N ). T he sign ( − 1) d ( | α |− d ) is natural since on the righ t hand s id e, the map j ! , whic h r epresen ts the con ten t of th e lo op pro duct, is in fron t of a , whereas on the le ft hand side, the dot represent ing the loop pro d uct is b et wee n a and b . Switc h ing the ord er of j ! and a yields the sign ( − 1) d | α | . The s ign ( − 1 ) d comes from our c hoice of the orienta tion of the normal bundle ν so that [ M ] ∈ H d ( LM ) acts as the unit. Note that the | a | − d is the degree of a in the loop algebra H ∗ ( LM ) = H ∗ + d ( LM ). F or further discuss ion, we need transfer maps defined in the follo wing general con text. Let ι : K − → M b e a smo oth em b edding of orien ted closed smo oth m anifolds and let ν be its normal bund le orien ted by ν ⊕ ι ∗ ( T K ) ∼ = T M | ι ( K ) . Let u ′ b e the T h om class of ν and let u ∈ H d − k ( M ) b e the corresp ondin g Thom class for the emb edding ι , where d and k are dimensions of M and K . With the ab o v e choice of the orient ation on ν , we h a ve u ∩ [ M ] = ι ∗ ([ K ]), whic h charact erizes the class u . Had w e orien ted ν by ι ∗ ( T K ) ⊕ ν ∼ = T M | ι ( K ) , then we wo uld ha ve obtained u ∩ [ M ] = ( − 1) k ( d − k ) ι ∗ ([ K ]). Let p : E − → M b e a Hurewicz fibr ation, and let E K b e its p ull-bac k o v er K via the em b edding ι . Let ι : E K − → E b e the inclusion of fibr ations. Pro ceeding as b efore, we can define a transfer map. ι ! : H ∗ + d ( E ) − → H ∗ + k ( E K ) , suc h that ι ∗ ι ! ( a ) = p ∗ ( u ) ∩ a for any a ∈ H ∗ ( E ) . W e remark that with the ab o ve c hoice of the orien tation on the normal bundle ν , the transfer map b et wee n base manifolds satisfies ι ! ([ M ]) = [ K ]. Also, it can b e verified that for a comp osition of s mo oth emb eddings K g − → L f − → M and th e asso ciated in duced inclusions of fibrations E K g − → E L f − → E , w e hav e ( f ◦ g ) ! = g ! ◦ f ! . The lo op pro du ct enjo ys the F roben ius compatibilit y with resp ect to the lo op copro du ct, in the follo wing sense. This is discussed in [S ] fr om the p oin t of view of c hains. Here, we giv e a homotop y theoretic pr o of with p recise determination of signs. F or a ∈ H ∗ ( LM ) and c ∈ H ∗ ( LM × LM ), let a · c b e defined b y ( ι × 1) ∗ ◦ ( j × 1) ! ( a × c ) = ( − 1) d ( | a |− d ) a · c using the follo win g diagram ( LM × LM ) × LM j × 1 − − − − → ( LM × M LM ) × LM ι × 1 − − − − → LM × LM p 1 × p 2   y p 1   y M × M φ ← − − − − M LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 7 where p 1 × p 2 denotes pr o jections from the first and second factor. If c is of the form of a cross pro du ct b × c , then a · ( b × c ) = ( a · b ) × c . Similarly , an elemen t c · a is defined b y (1 × ι ) ∗ (1 × j ) ! ( c × a ) = ( − 1) d ( | c |− d ) c · a us ing a similar diagram. Theorem 2.2. The lo op pr o d uct and the lo op c opr o duct satisfy F r ob enius c omp atibility, namely, for a, b ∈ H ∗ ( LM ) , (2-3) Ψ( a · b ) = ( − 1) d ( | a |− d ) a · Ψ( b ) = Ψ( a ) · b. Pr o of. F or conv enience, we introdu ce a space L r M of cont inuous lo ops from a circle of length r > 0 to M . W e let L ′ M = L 1 3 M and L ′′ M = L 2 3 M . W e identify S M ⊂ L 2 r M with L r M × M L r M . W e ha ve the follo win g comm utativ e diagram of inclusions: L ′ M × L ′′ M 1 × ι ← − − − − L ′ M × L ′ M × M L ′ M 1 × j − − − − → L ′ M × L ′ M × L ′ M j x   j 1 =( j × M 1) x   j × 1 x   L ′ M × M L ′′ M ι 1 =(1 × M ι ) ← − − − − − − − L ′ M × M L ′ M × M L ′ M j 2 =(1 × M j ) − − − − − − − → L ′ M × M L ′ M × L ′ M ι   y ι 2 =( ι × M 1)   y ι × 1   y LM ι ← − − − − L ′′ M × M L ′ M j − − − − → L ′′ M × L ′ M . The b ase man if olds of fibrations in the ab o ve diagram form the follo win g d iagram wh ic h we use to compute Thom classes of em b edd ings, which in turn are u sed to constru ct transf er maps. M × M × M 1 × φ ← − − − − M × M 1 × φ − − − − → M × M × M φ × 1 x   φ x   φ × 1 x   M × M φ ← − − − − M φ − − − − → M × M φ × 1   y φ   y φ × 1   y M × M × M φ 13 ← − − − − M × M φ 13 − − − − → M × M × M where φ 13 ( x, y ) = ( x, y , x ), or φ 13 = (1 × T )( φ × 1) and T : M × M − → M × M is the switc h ing map. Here, for example, the fibration p : L ′′ M × M L ′ M − → M × M is giv en b y p ( γ , η ) = ( γ (0) = η (0) , γ ( 1 3 )), and the fib ration p : L ′ M × M L ′′ M − → M × M is giv en b y p ( γ , η ) = ( γ (0) = η (0) , η ( 1 3 )). T o pro v e Ψ( a · b ) = ( − 1) d ( | a |− d ) a · Ψ( b ), w e examine the f ollo wing induced homology diagram with transfers in whic h we replaced L ′ M and L ′′ M b y their homeomorphic cop y LM . H ∗ ( LM × LM ) (1 × ι ) ! − − − − − − − → =( − 1) d 1 × ι ! H ∗− d ( LM × LM × M LM ) (1 × j ) ∗ − − − − → H ∗− d ( LM × LM × LM ) ˜ j ! = j !   y ( j 1 ) !   y ( j × 1) ! = j ! × 1   y H ∗− d ( LM × M LM ) ( ι 1 ) ! − − − − → H ∗− 2 d ( LM × M LM × M LM ) ( j 2 ) ∗ − − − − → H ∗− 2 d ( LM × M LM × LM ) ι ∗   y ( ι 2 ) ∗   y ( ι × 1) ∗   y H ∗− d ( LM ) ˜ ι ! =( − 1) d ι ! − − − − − − − → H ∗− 2 d ( LM × M LM ) j ∗ − − − − → H ∗− 2 d ( LM × LM ) 8 HIROT AKA T AMANOI In the ab ov e, the tr an s fer maps ˜ j ! , ˜ ι ! indicate that Th om classes used to defin e these transfer maps ma y b e d ifferent in signs f rom Thom classes used to defin e transfers ι ! and j ! . The top left square and the b ottom righ t square comm ute b ecause of the fun ctorial prop erties of transfer maps and in duced maps. W e examine the c omm utativit y of the b ottom left squ are. Since the corresp onding squ are of fibrations comm u tes, the homology square with induced maps and transfer maps comm utes up to a sign. T o determine this s ign, f or a ∈ H ∗ ( LM × M LM ), w e compare ι ∗ ( ι 2 ) ∗ ( ι 1 ) ! ( a ) and ι ∗ ˜ ι ! ι ∗ ( a ) in H ∗ ( LM ). Let u ∈ H d ( M × M ) b e the T hom class for the em b edding φ : M − → M × M . Then the Thom class for the em b edding φ 13 : M × M − → M × M × M is giv en by ( − 1) d u 13 , w here u 13 = (1 × T ) ∗ ( u × 1) = P i ( u ′ i × 1 × u ′′ i ) if u = P i u ′ i × u ′′ i . Hence ( ι ∗ ˜ ι ! ) ι ∗ ( a ) = ( − 1) d p ∗ ( u 13 ) ∩ ι ∗ ( a ), where the map p : LM − → M × M × M is a fibr ation giv en b y p ( γ ) = ( γ (0) , γ ( 1 3 ) , γ ( 2 3 )). On the other h and, using the comm utativit y of the indu ced homology square, we hav e ι ∗ ( ι 2 ) ∗ ( ι 1 ) ! ( a ) = ι ∗ ( ι 1 ) ∗ ( ι 1 ) ! ( a ) = ι ∗ ( p ∗ ( u ) ∩ a ), s in ce th e Thom cla ss for th e em b edding ι 1 is p ∗ ( u ). Since u = ( φ × 1) ∗ ( u 13 ), w e ha ve p ∗ ( u ) = p ∗ (( φ × 1) ∗ ( u 13 )) = ι ∗ ( p ∗ ( u 13 )). Hence ι ∗ ( p ∗ ( u ) ∩ a ) = p ∗ ( u 13 ) ∩ ι ∗ ( a ). Collecting our computations, w e ha ve that ι ∗ ( ι 2 ) ∗ ( ι 1 ) ! ( a ) = p ∗ ( u 13 ) ∩ ι ∗ ( a ). Comparing with the formula ab ov e for ι ∗ ˜ ι ! ι ∗ ( a ), we see that the sign d ifference b et w een ( ι 2 ) ∗ ( ι 1 ) ! ( a ) and ˜ ι ! ι ∗ ( a ) is giv en b y ( − 1) d . Hence th e square commute s up to ( − 1) d . Similar argument sho ws that the top right square in the homology diagram actually comm u tes. Next we examine trans f er maps in the diagram. F or the top horizont al left transfer (1 × ι ) ! , since the T hom class of the em b eddin g 1 × φ : M × M − → M × M × M is ( − 1) d (1 × u ), (1 × ι ) ∗ (1 × ι ) ! ( a × b ) = ( − 1) d p ∗ (1 × u ) ∩ ( a × b ) = ( − 1) d + d | a | a × ( p ∗ ( u ) ∩ b ) = ( − 1) d + d | a | a × ι ∗ ι ! ( b ) = ( − 1) d (1 × ι ) ∗ (1 × ( ι ) ! )( a × b ) . for a, b ∈ H ∗ ( LM ), Thus, (1 × ι ) ! = ( − 1) d 1 × ( ι ) ! , as indicated in the diagram. Similarly , we can v erif y that for the v ertical top r igh t transfer map, w e hav e ( j × 1) ! = j ! × 1. F or the v ertical top left transfer ˜ j ! asso ciated to the Thom class for the embedd ing φ × 1 : M × M − → M × M × M coincides with the tr an s fer j ! asso ciated to the Thom class for the em b eddin g φ : M − → M × M . The b ottom left horizon tal transfer map ˜ ι ! asso ciated to the Thom class ( − 1) d u 13 for the em b edding φ 13 : M × M − → M × M × M coincides with ( − 1) d ι ! , where ι ! is the transfer asso ciated t o the Thom class u of the embedd ing φ : M − → M × M . Hence for a, b ∈ H ∗ ( LM ), tracing the diagram fr om the top left corner to the b ottom righ t corner via b ottom left corner, w e get j ∗ ( ˜ ι ) ! ι ∗ ( ˜ j ) ! ( a × b ) = j ∗ ( − 1) d ι !  ( − 1) d ( | a |− d ) a · b  = ( − 1) d + d ( | a |− d ) Ψ( a · b ) . F ollo wing the d iagram via the top right corner, we get ( ι × 1) ∗ ( j × 1) ! (1 × j ) ∗ (1 × ι ) ! ( a × b ) = ( ι ∗ × 1)( j ! × 1)(1 × j ∗ )( − 1) d (1 × ι ! )( a × b ) = ( − 1) d + | a | d ( ι ∗ j ! × 1)( a × Ψ( b )) = ( − 1) d + | a | d + d ( | a |− d ) a · Ψ( b ) . Since the en tire d iagram comm utes up to ( − 1) d , we fi nally get Ψ ( a · b ) = ( − 1) d ( | a |− d ) a · Ψ( b ). T o pro v e the other iden tity Ψ( a · b ) = Ψ( a ) · b , w e consider the induced homology d iagram with LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 9 transfers fl owing from the b ottom right corner to the top left corner giv en b elo w. H ∗− 2 d ( LM × LM ) (1 × ι ) ∗ ← − − − − H ∗− 2 d ( LM × LM × M LM ) (1 × j ) ! ← − − − − − − − − − =( − 1) d (1 × j ! ) H ∗− d ( LM × LM × LM ) ˜ j ∗ x   ( j 1 ) ∗ x   ( j × 1) ∗ x   H ∗− 2 d ( LM × M LM ) ( ι 1 ) ∗ ← − − − − H ∗− 2 d ( LM × M LM × M LM ) ( j 2 )! ← − − − − H ∗− d ( LM × M LM × LM ) ˜ ι ! = ι ! x   ( ι 2 ) ! x   ( ι × 1) ! = ι ! × 1 x   H ∗− d ( LM ) ι ∗ ← − − − − H ∗− d ( LM × M LM ) ˜ j ! =( − 1) d j ! ← − − − − − − − H ∗ ( LM × LM ) where the transfer maps along the p erimeter has b een iden tified as sh o w n. Using similar metho ds, all the squares comm ute except the top right one whic h comm utes up to ( − 1) d . With this informa- tion, follo win g the diagram via top right corner giv es ( − 1) d + d | a | Ψ( a ) · b , and follo w ing the diagram via the b ottom left corner give s ( − 1) d + d ( | a |− d ) Ψ( a · b ). Since the en tire diagram comm utes up to ( − 1) d , we obtain the identit y Ψ( a · b ) = Ψ( a ) · b . This completes the pro of.  Note that in the same diagram of fib rations, if we consider an induced homology diagram w ith transfers flo wing from the top right corner to th e b ottom left corner, or a diagram flo wing from the b ottom left corner to the top r igh t corner, we obtain homotop y theoretic pro ofs of asso ciativit y of the lo op pro duct [CJ] and the coasso ciativit y of the lo op copro du ct. Next we sho w that Ψ is symmetric. Let T : LM × LM − → LM × LM b e the sw itc hin g map. Prop osition 2.3. The lo op c opr o duct is symmetric in the sense that T ∗  Ψ( a )  = Ψ ( a ) for any a ∈ H ∗ ( LM ) . Pr o of. W e consider the follo wing comm utativ e diagram: LM ι ← − − − − LM × M LM j − − − − → LM × LM R 1 2   y T   y T   y LM ι ← − − − − LM × M LM j − − − − → LM × LM Here, as b efore, w e identify S M with LM × M LM , and R 1 2 is the r otation of lo ops b y 1 2 , that is R 1 2 ( γ )( t ) = γ ( t + 1 2 ). The left square comm utes b ecause R 1 2 ◦ ι ( γ , η ) = R 1 2 ( γ · η ) = η · γ = ι ◦ T ( γ , η ). The Thom class for the embed d ing ι is giv en by ˜ u = p ∗ ( e M ). Since R 1 2 ≃ 1, w e ha ve R ∗ 1 2 ( ˜ u ) = ˜ u . Th us the T h om classes for t w o ι ’s are compatible and we ha v e T ∗ ◦ ι ! = ι ! ◦ R 1 2 ∗ = ι ! . Th us the ab o ve comm u tativ e diagram imp lies T ∗  Ψ( a )  = T ∗ ◦ j ∗ ◦ ι ! ( a ) = j ∗ ◦ T ∗ ◦ ι ! ( a ) = j ∗ ◦ ι ! ( a ) = Ψ( a ).  The lo op copro duct b ehav es well with r esp ect to cap p ro du cts with cohomology classes in H ∗ ( LM ) arising from α ∈ H ∗ ( M ). L et p : LM − → M b e the base p oin t map. F or the ev al uation map e = p ◦ ∆ : S 1 × LM − → M , let e ∗ ( α ) = 1 × p ∗ ( α ) + { S 1 } × ∆  p ∗ ( α )  , where { S 1 } is the fundamental cohomology class for S 1 . 10 HIROT AKA T AMANOI Theorem 2.4. L et α ∈ H ∗ ( M ) and b ∈ H ∗ ( LM ) . (1) The c ap pr o duct with p ∗ ( α ) satisfies F r ob enius c omp atibility with r esp e ct to the lo op c opr o d- uct : (2-4) Ψ  p ∗ ( α ) ∩ b  = ( − 1) d | α |  p ∗ ( α ) × 1  ∩ Ψ( b ) = ( − 1) d | α |  1 × p ∗ ( α )  ∩ Ψ( b ) . (2) The c ap pr o duct with ∆  p ∗ ( α )  b ehaves as a c o d erivation with r esp e ct to the lo op c opr o d uct : (2-5) Ψ  ∆  p ∗ ( α )  ∩ b  = ( − 1) d ( | α |− 1)  ∆  p ∗ ( α )  × 1 + 1 × ∆  p ∗ ( α )  ∩ Ψ ( b ) . Pr o of. F rom th e d efinition of th e lo op copro du ct and a p rop erty (2-2) of the transfer ι ! , w e ha ve Ψ  p ∗ ( α ) ∩ b  = j ∗ ◦ ι !  p ∗ ( α ) ∩ b  = ( − 1) d | α | j ∗  ι ∗ p ∗ ( α ) ∩ ι ! ( b )  . T o understand ι ∗ p ∗ ( α ), we consider the follo wing commutativ e diagram. LM ι ← − − − − LM × M LM j − − − − → LM × LM p × p ′   y q   y p × p   y M × M φ ← − − − − M φ − − − − → M × M π 1   y    π i   y M M M where p ′ ( γ ) = γ ( 1 2 ), and π i for i = 1 , 2 is the pro j ection on to the i th f actor. F rom the diagram, w e hav e ι ∗ p ∗ ( α ) = q ∗ ( α ) = j ∗ ( p × p ) ∗ π ∗ i ( α ), wh ic h is equal, for i = 1 , 2, to j ∗  p ∗ ( α ) × 1  and to j ∗  1 × p ∗ ( α )  . F or i = 1 case, ( − 1) d | α | Ψ  p ∗ ( α ) ∩ b  = j ∗  j ∗  p ∗ ( α ) × 1) ∩ ι ! ( b )  =  p ∗ ( α ) × 1  ∩ j ∗ ι ! ( b ) =  p ∗ ( α ) × 1  ∩ Ψ ( b ) . Similarly , for the case i = 2, we obtain ( − 1) d | α | Ψ  p ∗ ( α ) ∩ b  =  1 × p ∗ ( α )  ∩ Ψ ( b ). F or (2) , first we note that Ψ  ∆( p ∗ ( α )) ∩ b  = j ∗ ι !  ∆( p ∗ ( α )) ∩ b  = ( − 1) d ( | α |− 1) j ∗  ι ∗ ∆( p ∗ ( α )) ∩ ι ! ( b )  . W e need to understand ι ∗  ∆( p ∗ ( α ))  . F or this purp ose, w e in tro du ce some n otations. Let I 1 = [0 , 1 2 ] and I 2 = [ 1 2 , 1]. Let r : S 1 = I /∂ I − → I / { 0 , 1 2 , 1 } = S 1 1 ∨ S 1 2 , where S 1 i = I i /∂ I i for i = 1 , 2, b e an identificat ion map. Let ι i : S 1 i − → S 1 1 ∨ S 1 2 b e the inclusion map for i = 1 , 2. W e consider the follo wing comm u tativ e diagram. S 1 i × ( LM × M LM ) 1 × j − − − − → S 1 i × ( LM × LM ) ι i × 1   y 1 × π i   y S 1 × ( LM × M LM ) r × 1 − − − − → ( S 1 1 ∨ S 1 2 ) × ( LM × M LM ) S 1 i × L M ∼ = S 1 × L M 1 × ι   y e ′   y e   y S 1 × L M e − − − − → M M where e ′ ( t, γ , η ) is giv en b y γ (2 t ) for 0 ≤ t ≤ 1 2 , and η (2 t − 1) f or 1 2 ≤ t ≤ 1. Let e ′ ∗ ( α ) = 1 × ι ∗ p ∗ ( α ) + { S 1 1 } × ∆ 1 ( α ) + { S 1 2 } × ∆ 2 ( α ), where the first term is due to a fact that e ′ restricted LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 11 to { 0 }× ( LM × M LM ) is giv en by p ◦ ι . Sin ce e ∗ ( α ) = 1 × p ∗ ( α )+ { S 1 }× ∆ p ∗ ( α ) and r ∗ ( { S 1 i } ) = { S 1 } for i = 1 , 2, the comm utativit y of th e left b ottom squ are implies that ι ∗ ∆  p ∗ ( α )  = ∆ 1 ( α ) + ∆ 2 ( α ) . W e need to ident ify ∆ i ( α ) for i = 1 , 2. The comm u tativit y of the righ t squ are implies that, for i = 1, (1 × j ) ∗ (1 × π 1 ) ∗ e ∗ ( α ) = 1 × j ∗ ( p ∗ ( α ) × 1) + { S 1 } × j ∗  ∆( p ∗ ( α )) × 1  is equal to ( ι 1 × 1) ∗ e ′ ∗ ( α ) = 1 × ι ∗ p ∗ ( α ) + { S 1 } × ∆ 1 ( α ). Hence ∆ 1 ( α ) = j ∗  ∆( p ∗ ( α )) × 1  . Similarly , the i = 2 case implies that ∆ 2 ( α ) = j ∗  1 × ∆( p ∗ ( α ))  . Com bining the ab o ve calculations, we ha v e Ψ  ∆( p ∗ ( α )) ∩ b  = ( − 1) d ( | α |− 1)  ι ∗ ∆( p ∗ ( α )) ∩ ι ! ( b )  = ( − 1) d ( | α |− 1) j ∗  j ∗  ∆( p ∗ ( α )) × 1 + 1 × ∆( p ∗ ( α ))  ∩ ι ! ( b )  = ( − 1) d ( | α |− 1)  ∆( p ∗ ( α )) × 1 + 1 × ∆( p ∗ ( α ))  ∩ Ψ ( b ) . This pr o ves the co deriv ation prop erty .  § 3 Propert ies of the lo op copro duct and their consequences So far we h a ve pro ved v arious algebraic prop erties of the lo op copro duct. These prop erties tu r n out to b e str on g enough to force the lo op copro d uct to b e giv en b y a very simple formula, giv en in th e next theorem. Let s : M − → LM b e the constan t lo op map giv en b y s ( x ) = c x , wh ere c x is the constant lo op at x ∈ M . Recall that we assume that M is connected w ith base p oin t x 0 , and let c 0 b e the co nstant lo op at the base p oin t. The connected comp on ents of LM are in 1:1 coresp ondence to the set of free homotop y classes of lo ops [ S 1 .M ], whic h is in 1:1 corresp ondence with conju gacy classes of π 1 ( M ). Let LM = ( LM ) [1] ∪ [ [ α ] 6 =[1] ( LM ) [ α ] , b e the decomposition of LM into its comp onent s, w here [ α ]’s are conjugacy classes in π 1 ( M ). Theorem 3.1. L et M b e a c onne cte d oriente d close d smo o th d - manifold. (1) L et p ≥ 0 and let a 1 , a 2 , . . . , a p ∈ H ∗ ( LM ) . The lo op c opr o duct on the lo op pr o duct of these elements is given by the fol lowing f ormula, f or e ach 0 ≤ ℓ ≤ p . (3-1) Ψ( a 1 a 2 · · · a p ) = χ ( M )  [ c 0 ] · a 1 · a 2 · · · a ℓ  ⊗  [ c 0 ] · a ℓ +1 · · · a p  ∈ H ∗ ( LM ) ⊗ H ∗ ( LM ) . In p articular, for the unit 1 = s ∗ ([ M ]) ∈ H d ( LM ) = H 0 ( LM ) of the lo op homolo gy al gebr a, its c opr o duct is give n by (3-2) Ψ(1) = χ ( M )[ c 0 ] ⊗ [ c 0 ] ∈ H 0 ( LM ) ⊗ H 0 ( LM ) ∼ = H 0 ( LM × LM ) . When p = 1 , the formula for a ∈ H ∗ ( LM ) for ℓ = 0 , 1 b e c omes (3-3) Ψ( a ) = χ ( M )  [ c 0 ] · a  ⊗ [ c 0 ] = χ ( M )[ c 0 ] ⊗  [ c 0 ] · a  . (2) If | a | 6 = d , then Ψ( a ) = 0 . If | a | = d , then Ψ( a ) = n [ c 0 ] ⊗ [ c 0 ] for some n ∈ Z . Thus, ImΨ = Z [ c 0 ] ⊗ [ c 0 ] . (3) Supp ose a ∈ H d  ( LM ) [ α ]  b e a de gr e e d homol o gy class in [ α ] -c omp o nent of LM . If [ α ] 6 = [1] , then Ψ( a ) = 0 . 12 HIROT AKA T AMANOI (4) Supp o se a ∈ H d  ( LM ) [1]  , and su pp ose it is of the form a = k s ∗ ([ M ]) + ( de c omp osa bles ) in the lo op algebr a H ∗ ( LM ) for some k ∈ Z , then Ψ( a ) = k χ ( M )[ c 0 ] ⊗ [ c 0 ] . Pr o of. First, we pro ve the formula for Ψ(1). Since 1 = s ∗ ([ M ]) has degree d , and ι ! decreases degree b y d , w e h av e ι ! (1) ∈ H 0 ( LM × M LM ). Since M is connected, connected comp onents of L M are in 1:1 corresp ondence with conjugacy classes of π 1 ( M ). Let L 0 M b e the comp onen t consisting of con tractible lo ops so that c 0 ∈ L 0 M . No te that L 0 M × M L 0 M is also connected, and H 0 ( L 0 M × M L 0 M ) ∼ = Z is generated by [( c 0 , c 0 )]. So we ma y write ι ! (1) = m [( c 0 , c 0 )] for some m ∈ Z . Since ι ∗ : H 0 ( L 0 M × M L 0 M ) − → H 0 ( L 0 M ) is an isomorph ism w ith ι ∗ ([( c 0 , c 0 )]) = [ c 0 ], and sin ce (2-1) implies ι ∗ ι ! (1) = p ∗ ( e M ) ∩ s ∗ ([ M ]) = s ∗ ( e M ∩ [ M ]) = χ ( M )[ c 0 ], w e h a ve ι ! (1) = χ ( M )[( c 0 , c 0 )]. Hence Ψ(1) = j ∗ ι ! (1) = χ ( M )[ c 0 ] ⊗ [ c 0 ]. F or a 1 , a 2 , . . . , a p ∈ H ∗ ( LM ) and for 0 ≤ ℓ ≤ p , the F r ob enius compatibilit y (2-3) imp lies Ψ( a 1 · a 2 · · · a p ) = ( − 1) d ( | a 1 | + ··· + | a ℓ |− dℓ ) ( a 1 · · · a ℓ ) · Ψ(1) · a ℓ +1 · · · a p = ( − 1) d ( | a 1 | + ··· + | a ℓ |− dℓ ) χ ( M ) a 1 · · · a ℓ · [ c 0 ] ⊗ [ c 0 ] · a ℓ +1 · · · a p = χ ( M )  [ c 0 ] · a 1 · · · a ℓ  ⊗  [ c 0 ] · a ℓ +1 · · · a p  . Here, we used the graded comm utativit y in the lo op homolog y alg ebra giv en b y a · b = ( − 1) ( | a |− d )( | b |− d ) b · a, a, b ∈ H ∗ ( LM ) . When p = 1, we get the form ula for Ψ( a ) giv en in (3 -3). Note that the form ula is co mpatible with the symmetry form ula T ∗ Ψ( a ) = Ψ( a ) in Prop osition 2.3. Note also that our formula tells us that the image of Ψ is contai ned in the tensor pro d uct H ∗ ( LM ) ⊗ H ∗ ( LM ) ⊂ H ∗ ( LM × LM ), essen tially b ecause Ψ(1) is b y (3- 2). (2) F rom the form ula (3-3), the v alue Ψ( a ) must b e an inte gral multiple of [ c 0 ] ⊗ [ c 0 ] ∈ H 0 ( LM ) ⊗ H 0 ( LM ). Since Ψ low ers degree by d , if | a | 6 = d , w e m ust h a ve Ψ( a ) = 0. (3) Let a ∈ H d  ( LM ) [ α ]  . W e sho w th at if Ψ( a ) 6 = 0, then [ α ] = [1]. By (2), Ψ( a ) m ust be of the form n [ c 0 ] ⊗ [ c 0 ] for some n ∈ Z . Compaing w ith (3-3), if Ψ( a ) 6 = 0, then [ c 0 ] · a = k [ c 0 ] for some k 6 = 0, whic h is a homology class of finite union of con tractible lo ops. Th us a must b e represent ed by a cycle in the space of con tractible loops ( LM ) [1] . Hence we ha v e [ α ] = [1]. (4) By (2), if | a | 6 = d , w e m ust ha v e Ψ( a ) = 0, whic h is equiv alen t to (3-4) χ ( M )[ c 0 ] · a = 0 , a ∈ H ∗ ( LM ) with | a | 6 = d. No w sup p ose | a | = d and a is d ecomp osable of the form a = P i b ′ i · b ′′ i with | b ′ i | 6 = d for all i , then Ψ( a ) = P i χ ( M )[ c 0 ] · b ′ i ⊗ [ c 0 ] · b ′′ i = 0 by (3-1 ). Thus, if a is of the form a = k s ∗ ([ M ]) + (decomp osables), then Ψ( a ) = Ψ  k s ∗ ([ M ])  = k χ ( M )[ c 0 ] ⊗ [ c 0 ].  Implications of Th eorem 3.1 are rather striking. First, we start with straigh tforward corolla ries whose pro ofs are ob vious. Corollary 3.2. L et M b e a c onne cte d c lose d oriente d smo oth manifold. If its Euler char acteristic is zer o, then the lo o p c opr o duct vanishes identic al ly. In p articular, if M is o d d dimensional, then the lo op c opr o duct vanishes identic al ly. F or example, the loop copro d uct v anish es in H ∗ ( LS 2 n +1 ). Th e ab o ve Corollary 3 .2 w as also observ ed in [S]. Next, we examine torsion elemen ts in lo op homolog y . LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 13 Corollary 3.3. Assume that χ ( M ) 6 = 0 for a c onne cte d close d oriente d smo oth d -manifold M . F or any element a ∈ H ∗ ( LM ) with | a | 6 = d , the element [ c 0 ] · a is either 0 or a torsion element of or der a divisor of χ ( M ) . Pr o of. In the pro of of Theorem 3.1, w e noted that χ ( M )[ c 0 ] · a = 0 if | a | 6 = d in (3-4). Since χ ( M ) 6 = 0, the conclusion follo ws.  When | a | = d , the elemen t [ c 0 ] · a lies in H 0 ( LM ), so it is either 0 or torsion free. Let ι : Ω M − → LM b e the inclusion map from t he based lo op sp ace to the free lo op space. Recall that w e ha v e an algebra map ι ! : H ∗ + d ( LM ) − → H ∗ (Ω M ) from the lo op algebra to the P ontrjagin ring, w here d = dim M . Corollary 3.4. Supp ose χ ( M ) 6 = 0 for a close d oriente d smo ot h d -manifold M . Then for p 6 = 0 , the image of the c omp osition ι ∗ ◦ ι ! : H p + d ( LM ) − → H p (Ω M ) − → H p ( LM ) c onsists entir ely of torsion elements of or der a divisor of χ ( M ) . Pr o of. Since ι ∗ ◦ ι ! ( a ) = [ c 0 ] · a for a ∈ H ∗ ( LM ), the assertion follo ws from Corollary 3.3.  Next, we sho w that similar statemen ts hold for loop brac ket pro ducts of the form { [ c 0 ] , a } for a ∈ H ∗ ( LM ). Corollary 3.5. Supp ose χ ( M ) 6 = 0 for a close d c onne cte d oriente d smo oth d -manifold M , and let a ∈ H ∗ ( LM ) . (1) If | a | 6 = d, d − 1 , then the element { [ c 0 ] , a } is either 0 or a torsion element of or der a divisor of χ ( M ) . (2) Supp ose further M is simp ly c onne cte d. Then if | a | 6 = d − 1 , then the element { [ c 0 ] , a } i s either 0 or a tor sion element of or der a divisor of χ ( M ) . Pr o of. Since χ ( M ) 6 = 0, M is ev en dimensional. The BV-identit y m ultiplied by χ ( M ) giv es ∆  χ ( M )[ c 0 ] · a  = χ ( M )∆([ c 0 ]) · a + χ ( M )[ c 0 ] · ∆( a ) + χ ( M ) { [ c 0 ] , a } . If | a | 6 = d, d − 1, then b y Corollary 3.3, we h a ve χ ( M )[ c 0 ] · a = 0 and χ ( M )[ c 0 ] · ∆( a ) = 0. Sin ce S 1 action on M is trivial, w e ha v e ∆([ c 0 ]) = 0. Thus χ ( M ) { [ c 0 ] , a } = 0, and th e conclusion of (1) follo ws. F or (2), when | a | = d , the elemen t ∆( a ) has degree d + 1. By C orollary 3.3, χ ( M )[ c 0 ] · ∆( a ) = 0. If M is s imply connected, LM h as a single comp onent L 0 M and so [ c 0 ] · a ∈ H 0 ( LM ) ∼ = Z generated by [ c 0 ]. Since ∆([ c 0 ]) = 0, we ha ve ∆([ c 0 ] · a ) = 0. Hence χ ( M ) { [ c 0 ] , a } = 0, f rom which the conclusion follo ws.  When | a | = d − 1, since { [ c 0 ] , a } ∈ H 0 ( LM ), this elemen t is either 0 or t orsion f ree. T o see what h ap p ens when M is not simply connected, f or eac h conjugacy class [ g ] of π 1 ( M ) we c h o ose a lo op γ g in M b elonging to [ g ]. When | a | = d , the elemen t [ c 0 ] · a is a linear com bin ation of classes [ γ g ] ∈ H 0 ( LM ). S ince ∆([ γ g ]) ∈ H 1 ( L [ g ] M ) can b e nonzero, the simple connectivit y a ssump tion is needed in (2) of C orollary 3.5. 14 HIROT AKA T AMANOI Example 3.6 . W e can ve rify Corollary 3.3 in actual examples. In [CJ Y], the lo op h omology algebra for LS 2 n and L C P n are computed. T h eir computation s h o w s H ∗ ( LS 2 n ) ∼ = Λ( b ) ⊗ Z [ a, v ] / ( a 2 , ab, 2 av ) , b ∈ H − 1 , a ∈ H − 2 n , v ∈ H 4 n − 2 , H ∗ ( L C P n ) ∼ = Λ( w ) ⊗ Z [ c, u ] / ( c n +1 , ( n + 1) c n u, wc n ) , w ∈ H − 1 , c ∈ H − 2 , u ∈ H 2 n . F or H ∗ ( LS 2 n ), we h a ve [ c 0 ] = a and χ ( S 2 n ) = 2. By the ab o ve computation, we can easily see that χ ( S 2 n )[ c 0 ] · x = 2 a · x = 0 for all x ∈ H ∗ ( LS 2 n ) not in H 0 . F or H ∗ ( L C P n ), w e hav e [ c 0 ] = c n and χ ( C P n ) = n + 1. Again we can easily s ee that the id en tit y χ ( C P n )[ c 0 ] · y = ( n + 1) c n · y = 0 for all y not in H 0 . W e discu s s t wo final related results. T he first one concerns an analogue of the BV identit y for the lo op copro duct. The BV identit y can b e understo o d b y sa ying that the failure of the comm u tativit y of th e follo wing diagram is the loop b rac ket: H ∗ ( LM ) ⊗ H ∗ ( LM ) lo op pro duct − − − − − − − − → H ∗ ( LM ) ∆ ⊗ 1+1 ⊗ ∆   y ∆   y H ∗ ( LM ) ⊗ H ∗ ( LM ) lo op pro duct − − − − − − − − → H ∗ ( LM ) . W e ask a similar question for th e lo op copro duct. Do es the follo wing diagram comm ute? If not, what is the measure of the failure of the commutativit y? H ∗ ( LM ) Ψ − − − − → H ∗ ( LM × LM ) ∆   y ∆ × 1+1 × ∆   y H ∗ ( LM ) Ψ − − − − → H ∗ ( LM × LM ) Unfortunately , things turn out to b e rather trivial for the loop coprod uct. Prop osition 3.7. F or every a ∈ H ∗ ( LM ) , the identity (∆ × 1 + 1 × ∆)Ψ( a ) = 0 hol ds. Pr o of. F or a ∈ H ∗ ( LM ), b y (2) of Theorem 3.1 , Ψ( a ) ∈ Z [ c 0 ] ⊗ [ c 0 ] ⊂ H 0 ( LM × L M ). Since ∆([ c 0 ]) = 0, th e ab o v e iden tit y holds.  F or the s econd result, recal l that the loop pro d uct and the lo op copro duct satisfy F robeniu s compatibilit y (Theorem 2.2). W e ask a similar question. What is the compatibilit y relation for the lo op brack et and th e lo op copro d uct? The result tur ns out to b e trivial when one of the elements is from H ∗ ( M ). Prop osition 3.8. L et M b e as b efor e with χ ( M ) 6 = 0 . Supp ose a ∈ H ∗ ( M ) . Then for any b ∈ H ∗ ( LM ) , we have Ψ( { a, b } ) = 0 . Pr o of. Let α ∈ H ∗ ( M ) b e the cohomology class dual to a . Since ∆ α ∩ b = ( − 1) | α | { a, b } (see [T1]), using the co d eriv ation p rop erty of the cap pro duct with r esp ect to the loop coprod uct (2-5), Ψ( { a, b } ) = ( − 1) | α | +( | α |− 1) d (∆ α × 1 + 1 × ∆ α ) ∩ Ψ( b ) = ( − 1) | α | +( | α |− 1) d  χ ( M )  ∆ α ∩ ([ c 0 ] · b )  ⊗ [ c 0 ] + χ ( M )[ c 0 ] ⊗  ∆ α ∩ ([ c 0 ] · b )  . Since the loop br ack et b eha ves as a deriv ation in eac h v ariable, and { a, [ c 0 ] } = 0 for a ∈ H ∗ ( M ), w e hav e ∆ α ∩ ([ c 0 ] · b ) = ( − 1) | α | { a, [ c 0 ] · b } = ( − 1) | α | +( | α | +1) d [ c 0 ] · { a, b } . T h e ab ov e iden tit y then b ecomes Ψ( { a, b } ) = χ ( M )([ c 0 ] · { a, b } ) ⊗ [ c 0 ] + χ ( M )[ c 0 ] ⊗ ([ c 0 ] · { a, b } ) = Ψ ( { a, b } ) + Ψ( { a, b } ) , using (3-3). Hence Ψ( { a, b } ) = 0.  LOOP COPRODUCTS AND HIGHER GE NUS TQFT STRING OPE RA TIONS 15 Referen ces [CS] Moira Chas and Dennis Sulliv an, String top olo gy , preprin t, CUN Y , to app ea r i n An n . of Math. (1999), math.GT/9911 159. [CG] Ralph C ohen a nd V eronique Go din, A polarize d view of string top olo gy , T op o logy , geometry and quantum field theory , London Ma th. So c. Lecture Notes, vol. 308, Cambridge Univ. Press, Cambridge, 20 04, pp. 127– 154, math.A T/ 03030 0 3. [CJ] R a lph Cohen and J.D.S. Jo nes, A homotopy the or etic r e ali z ation of st ring top olo gy , Ma th. Ann. 324 (2002), no.4 7 73–79 8, math.GT/010 7187. [CJY] Ralph C ohen, J.D.S. Jones, and J . Y an, The lo op homolo gy algebr a of spher es an d pr ojective sp ac es , P ro gr. Math., v ol. 215, Birkh¨ auser, Basel, 2003, pp. 7 7 –92. [G] V eronique Go din, Higher string top olo gy op erations , arXiv:0711.4859 . [S] Dennis Sulliv an, Op en an d close d string field the ory interpr eted in cl a ssic al algebr aic top olo gy , T opo logy , geometry and quan tum field theory , London Math. Soc. Lecture Notes, v ol. 308, Cambridge Univ. Press, Cambri dge, 2004, pp. 34 4 –357, math .QA/0 30233 2. [T1] Hirotak a tamanoi, Cap pr o ducts in string top olo gy , arXiv:0706.0937 . [T2] Hirotak a T amanoi, Stable s t r ing oper ations ar e t r ivial , D e p ar t m en t of M a t he m a ti c s , U ni ve r s it y of Ca l ifo rn ia S a nta C ruz , S a n t a C ruz , CA 9 5 0 64 E-mail addr ess : t amanoi @math .ucsc.ed u