Cap Products in String Topology

CAP PR ODUCTS IN STRING TOPOLOGY HIROT AKA T A MANOI Abstract. Chas and Sulliv an show ed that the homology of the free lo op space LM of an orien ted closed smooth manifold M admits t he structure of a Batalin- Vilko visky (BV) algebra equipped with an associative product (loop pro duct) and a Lie brac ket (loop brack et). W e show that the cap pro duct i s com- patible with the abov e t w o pro ducts in the loop homology . Namely , the cap product with cohomology classes coming from M via the circle action acts as deriv ations on the lo op pro duct as well as on the loop brack et. W e show tha t Po isson identities and Jacobi ident ities hold for the cap pro duct action, tu rn- ing H ∗ ( M ) ⊕ H ∗ ( LM ) in to a BV algebra. Finally , we descri be cap products in terms of the BV algebra structure in the lo op homology . Contents 1. Int ro duction 1 2. Cap pro ducts and intersections of lo ops 4 3. The in tersection pro duct and the lo op pro duct 6 4. Cap pro ducts and extended BV algebra structure 11 5. Cap pro ducts in terms of BV algebra structure 18 References 19 1. Introduction Let M be a closed o riented smo oth d -manifold. Let D : H ∗ ( M ) ∼ = − → H d −∗ ( M ) be the Poincar´ e dualit y map. F o llowing a pr actice in str ing top olo gy , we shift the homology gr ading down ward b y d and let H −∗ ( M ) = H d −∗ ( M ). The Poincar´ e duality no w takes the form D : H −∗ ( M ) ∼ = − → H ∗ ( M ). F or a homology element a , let | a | denote its H ∗ -grading of a . The in tersection pro duct · in homolo gy is defined as the P oincar´ e dual of the cup pro duct. Namely , for a, b ∈ H ∗ ( M ), D ( a · b ) = D ( a ) ∪ D ( b ). If α ∈ H ∗ ( M ) is dual to a , then α ∩ b = a · b , and its Poincar´ e dual is α ∪ D ( b ). Thus, through Poincar´ e duality , the in tersection pro duct, the cap pro duct, and the cup pr o duct are all the same. In par ticular, the cap pro duct a nd the int ersectio n pro duct commute: (1.1) α ∩ ( b · c ) = ( α ∩ b ) · c = ( − 1) | α || b | b · ( α ∩ c ) . In fact, the dir ect sum H ∗ ( M ) ⊕ H ∗ ( M ) can be made into a gr aded commutative asso ciative algebra with unit, given by 1 ∈ H 0 ( M ), using the cap and the cup pro duct. 2000 Mathema tics Subje c t Classific ation. 55P35. Key wor ds and phr ases. Batalin-Vilko visky algebra; cap products; intersection pro duct; lo op brac ket s; lo op pro ducts; string topology . 1 2 HIR OT AKA T AMANOI F or an infinite dimensional ma nifold N , there is no longer Poincar´ e duality , and geometric intersections of finite dimens io nal cycles ar e all trivia l. Howev er, cap pro ducts can still be non trivial and the ho mo logy H ∗ ( N ) is a mo dule o ver the cohomolog y r ing H ∗ ( N ). When the infinite dimensional manifold N is a free loop space LM of contin uous maps from the circle S 1 = R / Z to M , the homolo gy H ∗ ( LM ) = H ∗ + d ( LM ) ha s a great deal more structur e . As befor e, | a | denotes the H ∗ -grading of a homolo gy ele- men t a of LM . Chas and Sulliv an [1] showed that H ∗ ( LM ) has a degree pres erving asso ciative graded comm utative pro duct · calle d the loo p product, a Lie brack et { , } of deg ree 1 called the loop brac ket compatible with the lo op pro duct, and the BV opera to r ∆ of degree 1 coming from the homolo gy S 1 action. These structures turn H ∗ ( LM ) into a Ba talin-Vilko visky (BV) algebra. The purp os e of this pa p er is to clarify the interplay betw een the cap pr o duct with cohomolo gy elements and the BV structure in H ∗ ( LM ). Let p : L M → M be the base p o int map p ( γ ) = γ (0) for γ ∈ LM . F or a cohomolog y cla ss α ∈ H ∗ ( M ) in the base manifold, its pull-back p ∗ ( α ) ∈ H ∗ ( LM ) is also denoted b y α . Let ∆ : S 1 × L M − → LM b e the S 1 -action ma p. This map induces a degr ee 1 map ∆ in homolo gy given by ∆ a = ∆ ∗ ([ S 1 ] × a ) for a ∈ H ∗ ( LM ). F or a cohomolog y cla ss β ∈ H ∗ ( LM ), the for mula ∆ ∗ ( β ) = 1 × β + { S 1 } × ∆ β defines a degree − 1 map ∆ in co homology , where { S 1 } is the fundamental cohomolog y class. A lthough we use the same notatio n ∆ in three different but clos ely related situations, what is meant b y ∆ should b e cle a r in the context. Theorem A. L et b, c ∈ H ∗ ( LM ) . The c ap pr o duct with α ∈ H ∗ ( M ) gr ade d c om- mutes with the lo op pr o duct. Namely (1.2) α ∩ ( b · c ) = ( α ∩ b ) · c = ( − 1) | α || b | b · ( α ∩ c ) . F or α ∈ H ∗ ( M ) , the c ap pr o duct with ∆ α ∈ H ∗ ( LM ) acts as a derivation on the lo op pr o duct and the lo op br acket : (∆ α ) ∩ ( b · c ) = (∆ α ∩ b ) · c + ( − 1) ( | α |− 1) | b | b · (∆ α ∩ c ) , (1.3) (∆ α ) ∩ { b , c } = { ∆ α ∩ b, c } + ( − 1) | α |− 1)( | b | +1) { b, ∆ α ∩ c } . (1.4) The op er ator ∆ ac ts as a derivatio n on the c ap pr o duct. Namely, for α ∈ H ∗ ( M ) and b ∈ H ∗ ( LM ) . (1.5) ∆( α ∩ b ) = ∆ α ∩ b + ( − 1) | α | α ∩ ∆ b. W e recall tha t in the BV alg ebra H ∗ ( LM ), the following iden tities ar e v a lid for a, b, c ∈ H ∗ ( LM ) [1]: ∆( a · b ) = (∆ a ) · b + ( − 1) | a | a · ∆ b + ( − 1) | a | { a, b } (BV identit y ) { a, b · c } = { a, b } · c + ( − 1) | b | ( | a | +1) b · { a, c } (Poisson identit y) a · b = ( − 1) | a || b | b · a, { a, b } = − ( − 1) ( | a | +1)( | b | +1) { b, a } (Commut ativity) { a, { b, c }} = {{ a, b } , c } + ( − 1) ( | a | +1)( | b | +1) { b, { a, c }} (Jacobi iden tity) Here, deg a · b = | a | + | b | , deg ∆ a = | a | + 1, a nd deg { a, b } = | a | + | b | + 1. W e c a n extend the lo o p pro duct and the lo op bracket in H ∗ ( LM ) to include H ∗ ( M ) in the following way . F or α ∈ H ∗ ( M ) and b ∈ H ∗ ( LM ), w e define the lo o p pro duct and the lo o p brack et of α and b b y (1.6) α · b = α ∩ b, { α, b } = ( − 1) | α | (∆ α ) ∩ b. CAP P RODUCTS IN STRING TOPOLOGY 3 F urthermor e, the BV structur e in H ∗ ( LM ) can be ex tended to the direct sum A ∗ = H ∗ ( M ) ⊕ H ∗ ( LM ) b y defining the BV opera tor ∆ on A ∗ to be trivial on H ∗ ( M ) and to be the usua l homolog ical S 1 action ∆ on H ∗ ( LM ). Here in A ∗ , elements in H k ( M ) are regar ded as ha ving homological degree − k . Theorem B. The dir e ct sum H ∗ ( M ) ⊕ H ∗ ( LM ) has a structu re of a BV algebr a. In p articular, for α ∈ H ∗ ( M ) and b, c ∈ H ∗ ( LM ) , the fol lowing form of Poisson identity and t he Jac obi identity hold : { α · b, c } = α · { b, c } + ( − 1) | b | ( | c | +1) { α, c } · b = α · { b, c } + ( − 1) | α || b | b · { α, c } , (1.7) { α, { b, c }} = {{ α, b } , c } + ( − 1) ( | α | +1)( | b | +1) { b, { α, c }} . (1.8) All the other p ossible forms of Poisson and Jacobi identities ar e also v alid, and the ab ov e tw o iden tities ar e the most nontrivial ones. The s e identit ies are prov ed by using standa rd pro p erties o f the cap product and the BV ident ity ab ov e in H ∗ ( LM ) relating the BV op era tor ∆ and the lo op brack et { , } , but without using Poisson ident ities nor Jaco bi iden tities in the BV algebra H ∗ ( LM ). The ab ove identit ies may seem r ather sur pr ising, but they b ecome tr anspare nt once we pr ov e the follo wing result. Theorem C. F or α ∈ H ∗ ( M ) , let a = α ∩ [ M ] ∈ H ∗ ( M ) b e its Poinc ar´ e dual. Then for b ∈ H ∗ ( LM ) , (1.9) α ∩ b = a · b, ( − 1) | α | ∆ α ∩ b = { a, b } . Mor e gener al ly, for c ohomolo gy elements α 0 , α 1 , . . . α r ∈ H ∗ ( M ) , let a 0 , a 1 , . . . a r ∈ H ∗ ( M ) b e their Poinc ar´ e duals. Then for b ∈ H ∗ ( LM ) , we have (1.10) ( α 0 ∪ ∆ α 1 ∪ · · · ∪ ∆ α r ) ∩ b = ( − 1) | a 1 | + ··· + | a r | a 0 · { a 1 , { a 2 , . . . { a r , b } · · · }} . Since the cohomology H ∗ ( M ) and the homology H ∗ ( M ) are iso morphic through Poincar´ e dualit y and H ∗ ( M ) is a subring of H ∗ ( LM ), the first formula in (1.9 ) is not surprising. How ever, the main differ ence b etw ee n H ∗ ( M ) and H ∗ ( M ) in our context is that the homolog y S 1 action ∆ is trivial on H ∗ ( M ) ⊂ H ∗ ( LM ), although cohomolog y S 1 action ∆ is nontrivial on H ∗ ( M ) and is related to lo op br ack et as in (1.9). Theorem A and Theorem C describes the cap pro duct action of the co homolog y H ∗ ( LM ) o n the BV algebra H ∗ ( LM ) for most elemen ts in H ∗ ( LM ). F or example, for α ∈ H ∗ ( M ), the cap pro duct with ∆ α is a de r iv atio n on the lo op algebra H ∗ ( LM ) given by a lo o p brack et, and consequently the cap pro duct with a cup pro duct ∆ α 1 ∪ · · · ∪ ∆ α r acts on the lo op algebra as a comp os ition of deriv ations , which is equa l to a comp osition of lo o p brack ets, according to (1.10). If H ∗ ( LM ) is generated b y elements α and ∆ α for α ∈ H ∗ ( M ) (for example, this is the case when H ∗ ( M ) is an exterio r alg ebra, see Re ma rk 5 .3), then Theo rem C gives a co mplete description of the cap pro duct with a rbitrary element s in H ∗ ( LM ) in terms of the BV a lgebra s tr ucture in H ∗ ( LM ). How ever, H ∗ ( LM ) is gener a l bigger than the subalgebra generated by H ∗ ( M ) and ∆ H ∗ ( M ). Since H ∗ ( LM ) is a BV algebra, in view of Theor em C, the v alidity of Theorem B may seem obvious. Ho wev er, in the pro of of Theorem B, we only used stan- dard prop erties of the cap pro duct and the BV iden tity . In fact, Theorem B gives 4 HIR OT AKA T AMANOI an alter nate elementary a nd purely homotopy theoretic proo f of Poisso n and Ja- cobi identities in H ∗ ( LM ), when at least one o f the elements a, b, c are in H ∗ ( M ). Similarly , Theo r em C gives a purely ho mo topy theore tic interpretation of the lo op pro duct and the lo o p brack et if one of the elements a re in H ∗ ( M ). Our interest in cap pro ducts in s tring top olog y comes from an intuitiv e geo metric picture that cohomolo g y cla s ses in LM a re dual to finite co dimensio n submanifolds of LM co nsisting of certain lo op configurations. W e can consider configurations of lo o ps intersecting in pa rticular ways (for e xample, tw o lo ops having their base po ints in commo n), or we can co nsider a family of lo ops intersecting tr a nsversally with s ubmanifolds of M at certa in p o ints of lo ops . In a given family of lo ops, taking the cap pro duct with a co homology class selects a subfamily of a cer tain lo op c onfiguratio n, which are r eady for cer tain lo op interactions. In this context, roughly s p ea king, c omp osition of tw o interactions o f lo ops cor resp ond to the cup pro duct of cor r esp onding cohomology classes. The o rganiza tion of this pap er is as follows. In section 2, w e describ e a geometric problem of describing certain family of in tersection configuratio n of loops in terms of ca p products. This gives a geometric motiv a tion for the remainder of the pap e r . In s ection 3, we review the loo p product in H ∗ ( LM ) in detail from the po int of view of the intersection product in H ∗ ( M ). Here we pay ca reful atten tion to sig ns. In particular, we give a homotopy theoretic pro o f of grade d commutativit y in the BV algebra H ∗ ( LM ), which turned out to be not so trivial. In s ection 4, we pr ov e compatibility re la tions b etw een the cap pro duct and the BV alg e br a structure, a nd prov e Theorems A and B. In the last section, we prov e Theo rem C. W e thank the refer ee for numerous sugg estions which lead to clarification and improv emen t of exp osition. 2. Cap products and intersections of loops Let A 1 , A 2 , . . . A r and B 1 , B 2 , . . . B s be oriented closed submanifolds of M d . Let F ⊂ LM b e a compac t family of lo ops. W e cons ide r the following question. Question : Fix r po ints 0 ≤ t ∗ 1 , t ∗ 2 , . . . , t ∗ r ≤ 1 in S 1 = R / Z . Describ e the homology cla ss of the subset I of the compact family F consisting of lo ops γ in F such that γ intersects submanifolds A 1 , . . . A r at time t ∗ 1 , . . . t ∗ r and intersects B 1 , . . . B s at some unsp ecified time. This subset I ⊂ F can be describ ed as follows. W e consider the fo llowing diagram of an ev aluation map and a pro jection map: (2.1) s z }| { ( S 1 × · · · × S 1 ) × LM e − − − − → r z }| { M × · · · × M × s z }| { M × · · · × M π 2   y LM given by e  ( t 1 , . . . t s ) , γ  =  γ ( t ∗ 1 ) , . . . γ ( t ∗ r ) , γ ( t 1 ) , . . . γ ( t s )  . Then the pull-back set e − 1 ( Q i A i × Q j B j ) is a clos ed subset of S 1 × · · · × S 1 × L M . Let ˜ I = e − 1 ( Y i A i × Y j B j ) ∩ ( S 1 × · · · × S 1 × F ) . CAP P RODUCTS IN STRING TOPOLOGY 5 The set I in ques tio n is giv en by I = π 2 ( ˜ I ). W e w ant to under stand this set I homolo g ically , including multiplicit y . Although e − 1 ( Q i A i × Q j B j ) is infinite dimensional, it has finite c o dimension in ( S 1 ) r × LM . So we w ork coho mologica lly . Let α i , β j ∈ H ∗ ( M ) b e cohomolog y c lasses dual to [ A i ] , [ B j ] for 1 ≤ i ≤ r and 1 ≤ j ≤ s . Then the subset e − 1 ( Q i A i × Q j B j ) is dual to the co homology clas s e ∗ ( Q i α i × Q j β j ) ∈ H ∗ (( S 1 ) s × LM ). Supp os e the family F is pa rametrized by a closed orie nted manifold K by an onto map λ : K − → F and let b = λ ∗ ([ K ]) ∈ H ∗ ( LM ) b e the ho mology cla ss of F in LM . Then the homolog y class of ˜ I in ( S 1 ) s × L M is given b y (2.2) [ ˜ I ] = e ∗ ( Y i α i × Y j β j ) ∩ ([ S 1 × · · · × S 1 ] × b ) . Note that the homology class ( π 2 ) ∗ ([ ˜ I ]) repre s ents the homolog y clas s of I with m ultiplicity . Prop ositi on 2.1. With the ab ove notation, ( π 2 ) ∗ ([ ˜ I ]) is given by the f ol lowing formula in terms of the c ap pr o duct or in terms of t he BV st ructur e : (2.3) ( π 2 ) ∗ ([ ˜ I ]) = ( − 1) P j j | β j |− s  α 1 · · · α r (∆ β 1 ) · · · (∆ β s )  ∩ b = ( − 1) P j j | β j |− s [ A 1 ] · · · [ A s ] · { [ B 1 ] , {· · · { [ B s ] , b } · · · } ∈ H ∗ ( LM ) . Pr o of. The ev aluation map e in (2.1) is given b y the following composition. s z }| { S 1 × · · · × S 1 × L M 1 × φ − − − → ( S 1 × · · · × S 1 ) × r + s z }| { LM × · · · × LM T − → r z }| { LM × · · · × L M × s z }| { ( S 1 × L M ) × · · · × ( S 1 × L M ) 1 r × ∆ s − − − − → ( LM × · · ·× LM ) × ( LM × · · · × L M ) p r × p s − − − − → ( M × · · · × M ) × ( M × · · ·× M ) , where φ is a diago nal map, T moves S 1 factors. Since we apply ( π 2 ) ∗ later, we only ne e d terms in e ∗ ( Q A i × Q B j ) containing the factor { S 1 } × · · · × { S 1 } . Since ∆ ∗ p ∗ ( β j ) = 1 × p ∗ ( β j )+ { S 1 } × ∆ β j for 1 ≤ j ≤ s , following the ab ov e deco mpo sition of e , we ha ve e ∗ ( α 1 × · · · × α r × β 1 × · · · × β s ) = ε { S 1 } s ×  α 1 · · · α r (∆ β 1 ) · · · (∆ β s )  + other ter ms , where the sign ε is g iven by ε = ( − 1) P s ℓ =1 ( s − ℓ )( | β ℓ |− 1)+ s P r ℓ =1 | α ℓ | . Thus, taking the cap pro duct with [ S 1 ] s × b and applying ( π 2 ) ∗ , w e get π 2 ∗  e ∗ ( α 1 × · · · × α r × β 1 × · · · × β s ) ∩ ([ S 1 ] × · · · × [ S 1 ] × b )  = ( − 1) P s ℓ =1 ℓ | β ℓ |− s α 1 · · · α r (∆ β 1 ) · · · (∆ β s ) ∩ b. The second equality follo ws from the formula (1.10).  R emark 2.2 . In the dia gram (2.1), in terms o f cohomolog y transfer π 2 ! we ha ve (2.4) π 2 ! e ∗ ( α 1 × · · · × α r × β 1 × · · · × β s ) = ± α 1 · · · α r (∆ β 1 ) · · · (∆ β s ) , where π 2 ! ( α ) ∩ b = ( − 1) s | α | π 2 ∗  α ∩ π 2 ! ( b )  for a ny α ∈ H ∗ (( S 1 ) s × LM ) a nd b ∈ H ∗ ( LM ). Here π 2 ! ( b ) = [ S 1 ] s × b . 6 HIR OT AKA T AMANOI 3. The intersection product and the loop product Let M b e a closed oriented smo o th d -manifold. The lo op pro duct in H ∗ ( LM ) w as discov ered by Chas a nd Sulliv a n [1], in ter ms of transversal chains. Later , Cohen and Jo nes [2] ga ve a ho mo topy theoretic description o f the lo op pro duct. The lo op pro duct is a h ybrid of the intersection pro duct in H ∗ ( M ) and the Pon trjagin pro duct in the ho mology of the based loop spaces H ∗ (Ω M ). In this section, we review and pr ov e so me prop erties of the lo o p pro duct in prepara tion for the nex t section. Our treatmen t of the loop product follows [2]. How ever, w e will b e precise with signs and give a homo topy theoretic pro of of the g raded co mm utativity of the lo op pr o duct, which [2] did not include. F or the F rob enius compatibility formula with car eful disc us sion of signs, see [10]. F or homotopy theoretic deduction of the BV iden tity , see [11]. F or our purp ose, the free lo op space LM is the s pace of c ont inuous maps from S 1 = R / Z to M . Our discussion is homotopy theo r etic and does not require smo othness of lo ops, altho ugh w e do need smoo thness of M whic h is enough to allows us to hav e tubular neighbor ho o ds for certain s ubmanifolds in the space o f contin uo us lo o ps. Rec all that the s pace LM of contin uous loops can b e given a structure of a smo oth ma nifold. See the discus sion b efore Definition 3.2. Let p : LM − → M b e the base p oint map given b y p ( γ ) = γ (0). Let s : M − → LM be the constan t lo o p map given by s ( x ) = c x , where c x is the constant lo op at x ∈ M . Since p ∗ ◦ s ∗ = 1, H ∗ ( M ) is contained in H ∗ ( LM ) thro ugh s ∗ and we often regar d H ∗ ( M ) as a subset o f H ∗ ( LM ). W e start with a discus sion on the in ter section r ing H ∗ ( M ) and la ter we compare it with the lo o p ho mology a lgebra H ∗ ( LM ). An exp os itio n on in tersection pro ducts in homology of manifolds ca n be found o n Dold’s b o ok [4], Chapter VI I I, § 13. Our sign conv ention (which follows Milnor [6 ]) is slight ly different from Dold’s. W e giv e a fair ly detailed discussio n of the intersection ring H ∗ ( M ) b ec a use the discussion for the loop homology algebra go es almost in parallel, and because our choice of the sign for the lo op pro duct comes fro m and is compatible with the intersection pro duct in H ∗ ( M ). Compar e form ulas (3.6) and (3.11). Those who are familiar with intersection pro duct and lo op pro ducts can s k ip this section after chec king Definition 3.2. Let D : H ∗ ( M ) ∼ = − → H d −∗ ( M ) be the Poincar´ e duality map suc h that D ( a ) ∩ [ M ] = a for a ∈ H ∗ ( M ). W e discuss t wo wa ys to define intersection pro duct in H ∗ ( M ). The first method is the officia l one and we simply define the intersection pro duct as the Poincar ´ e dual of the cohomology cup product. Thus, D ( a · b ) = D ( a ) ∪ D ( b ) for a, b ∈ H ∗ ( M ). F o r example, we hav e a · b = ( − 1) | a || b | b · a . The seco nd method uses the tr ansfer map induced from the diagona l map φ : M − → M × M . Let ν be the normal bundle to φ ( M ) in M × M , and we orie nt ν by ν ⊕ φ ∗ ( T M ) ∼ = T ( M × M ) | φ ( M ) . Let u ′ ∈ H d ( φ ( M ) ν ) be the Thom class of ν . Let N b e a clos ed tubular neighborho o d of φ ( M ) in M × M so that D ( ν ) ∼ = N , where D ( ν ) is the ass o ciated closed disc bundle of ν . Let π : N − → M b e the pro jection map. Then the ab ov e Thom cla ss can b e thought of as u ′ ∈ ˜ H d ( N / ∂ N ), and we hav e the following commutativ e diagram, where c : M × M − → N / ∂ N is the Thom CAP P RODUCTS IN STRING TOPOLOGY 7 collapse map, and ι N and j are obvious maps. (3.1) H d ( N , N − φ ( M )) ∼ = − − − − → H d ( N , ∂ N ) ∋ u ′ ∼ = x   ι ∗ N   y c ∗ u ′′ ∈ H d  M × M , M × M − φ ( M )  j ∗ − − − − → H d ( M × M ) ∋ u Let u ′′ ∈ H d  M × M , M × M − φ ( M )  and u ∈ H d ( M × M ) be the cla s ses corres p o nding to the Thom clas s. W e hav e u = c ∗ ( u ′ ) = j ∗ ( u ′′ ). This class u is characterized b y the pr op erty u ∩ [ M × M ] = φ ∗ ([ M ]), and φ ∗ ( u ) = e M ∈ H d ( M ) is the Euler class of M . See for example sec tio n 1 1 of [6]. The tra ns fer map φ ! is defined as the following compos ition: (3.2) φ ! : H ∗ ( M × M ) c ∗ − → ˜ H ∗ ( N / ∂ N ) u ′ ∩ ( ) − − − − → ∼ = H ∗− d ( N ) π ∗ − → ∼ = H ∗− d ( M ) . F or a homology element a , let | a | ′ denote its reg ular homolog y degr ee o f a , s o that w e hav e a ∈ H | a | ( M ) and | a | ′ = | a | + d . Prop ositi on 3.1. Supp ose M is a c onne cte d oriente d close d d -manifold with a b ase p oint x 0 . The tr ansfer map φ ! : H ∗ ( M × M ) − → H ∗− d ( M ) s atisfies the fol lowing pr op ert ies. F or a, b ∈ H ∗ ( M ) , φ ∗ φ ! ( a × b ) = u ∩ ( a × b ) (3.3) φ ! φ ∗ ( a × b ) = χ ( M )[ x 0 ] (3.4) F or α ∈ H ∗ ( M × M ) and b, c ∈ H ∗ ( M ) , we have (3.5) φ !  α ∩ ( b × c )  = ( − 1) d | α | φ ∗ ( α ) ∩ φ ! ( b × c ) . The interse ction pr o duct and the tr ansfer map c oincide up to a sign. (3.6) a · b = ( − 1) d ( | a | ′ − d ) φ ! ( a × b ) . Pr o of. F or the fir s t identit y , we co nsider the following commutativ e diag ram, where M 2 denotes M × M . H ∗ ( M 2 ) c ∗ − − − − → H ∗ ( N , ∂ N ) u ′ ∩ ( ) − − − − → ∼ = H ∗− d ( N ) π ∗ − − − − → ∼ = H ∗− d ( M )    ∼ =   y ι N ∗   y ι N ∗   y φ ∗ H ∗ ( M 2 ) j ∗ − − − − → H ∗  M 2 , M 2 − φ ( M )  u ′′ ∩ ( ) − − − − → ∼ = H ∗− d ( M 2 ) H ∗− d ( M 2 ) The co mmutativit y implies that for a, b ∈ H ∗ ( M ), w e have φ ∗ φ ! = u ′′ ∩ j ∗ ( a × b ) = j ∗ ( u ′′ ) ∩ ( a × b ) = u ∩ ( a × b ). T o chec k the seco nd formula, w e first co mpute φ ∗ φ ! φ ∗ ([ M ]). By the first formula, φ ∗ φ ! φ ∗ ([ M ]) = u ∩ φ ∗ ([ M ]) = φ ∗ ( φ ∗ ( u ) ∩ [ M ]). Since φ ∗ ( u ) is the Euler class e M , this is equal to φ ∗ ( e M ∩ [ M ]) = χ ( M )[( x 0 , x 0 )]. Since M is assumed to be connected, φ ∗ is an isomor phism in H 0 . Hence φ ! φ ∗ ([ M ]) = χ ( M )[ x 0 ] ∈ H 0 ( M ). F or the nex t formula, w e examine the following comm utative diagr am. H ∗ ( M 2 ) c ∗ − − − − → H ∗ ( N , ∂ N ) u ′ ∩ ( ) − − − − → ∼ = H ∗− d ( N ) ι ′ ∗ ← − − − − ∼ = H ∗− d ( M ) α ∩ ( )   y ι ∗ N ( α ) ∩ ( )   y ι ∗ N ( α ) ∩ ( )   y ι ∗ ( α )   y H ∗−| α | ( M 2 ) c ∗ − − − − → H ∗−| α | ( N , ∂ N ) u ′ ∩ ( ) − − − − → ∼ = H ∗− d −| α | ( N ) ι ′ ∗ ← − − − − ∼ = H ∗− d −| α | ( M ) 8 HIR OT AKA T AMANOI where ι ′ : M → N is an inclusion ma p and ι ′ ∗ = ( π ∗ ) − 1 . The middle squar e commutes up to ( − 1) | α | d . The commutativit y of this diagra m immediately implies that ι ∗ ( α ) ∩ φ ! ( a × b ) = ( − 1) | α | d φ !  α ∩ ( a × b )  . F or the la st identit y , w e apply φ ∗ on b oth sides and compa re. Since a · b = φ ∗ ( D ( a ) × D ( b )) ∩ [ M ], we ha ve φ ∗ ( a · b ) =  D ( a ) × D ( b )  ∩ φ ∗ ([ M ]) =  D ( a ) × D ( b )  ∩  u ∩ [ M ]  = ( − 1) d ( | a | ′ − d ) u ∩ ( a × b ) = ( − 1) d ( | a | ′ − d ) φ ∗ φ ! ( a × b ) . Since φ ∗ is injectiv e, we ha ve a · b = ( − 1) d ( | a | ′ − d ) φ ! ( a × b ).  These tw o intersection pro ducts, one defined using the Poincar´ e dualit y , and the other us ing Pontrjagin Tho m construction, differ only in signs . Howev er, the formulas for g raded commutativit y take different forms. a · b = ( − 1) ( d −| a | ′ )( d −| b | ′ ) b · a (3.7) φ ! ( a × b ) = ( − 1) | a | ′ | b | ′ + d φ ! ( b × a ) (3.8) The sign ( − 1 ) d in the second formula ab ove comes fro m the fact that the Thom class u ∈ H d ( M × M ) sa tisfies T ∗ ( u ) = ( − 1) d u , wher e T is the switching map of factors. Next w e turn to the lo op pro duct in H ∗ ( LM ). W e consider the following dia gram. (3.9) LM × LM j ← − − − − LM × M LM ι − − − − → LM p × p   y q   y M × M φ ← − − − − M where L M × M LM = ( p × p ) − 1 ( φ ( M )) consists of pairs of loo ps ( γ , η ) with the same base p o int, and ι ( γ , η ) = γ · η is the product of compos able loops . Let e N = ( p × p ) − 1 ( N ) a nd let ˜ c : LM × L M − → e N /∂ e N be the Tho m colla pse ma p. Let ˜ π : e N − → LM × M LM b e a pro jection ma p defined as follows. F o r ( γ , η ) ∈ e N , let their base po ints b e ( x, y ) ∈ N . Let π ( x, y ) = ( z , z ) ∈ φ ( M ). Since N ∼ = D ( ν ) has a bundle structure, let ℓ ( t ) = ( ℓ 1 ( t ) , ℓ 2 ( t )) b e the straig ht ray in the fiber ov er ( z , z ) fro m ( z , z ) to ( x, y ). Then let ˜ π  ( γ , η )  = ( ℓ 1 · γ · ℓ − 1 1 , ℓ 2 · η · ℓ − 1 2 ). By considering ℓ [ t, 1] , we see that ˜ π is a defor mation retraction. In fact, more is true. Sta cey ([7], P rop os itio n 5.3) sho wed that when L smo oth M is the space o f smo oth lo ops, e N has a n actual structur e of a tubular neig hborho o d of L M × M LM inside of LM × LM equipped with a diffeomorphism p ∗  D ( ν )  ∼ = e N . His pro of only uses the smo o thness o f M and exactly the sa me pro of applies to the space LM of c ontinuous lo ops and e N still has the structure of a tubular neighborho o d and we ag ain hav e a diffeomorphism p ∗  D ( ν )  ∼ = e N b etw een s paces of contin uo us lo ops. But we do not need this muc h here. Let ˜ u ′ = ( p × p ) ∗ ( u ′ ) ∈ ˜ H d ( e N /∂ e N ), and ˜ u = ( p × p ) ∗ ( u ) ∈ H d ( LM × LM ) b e pull-backs of Thom classe s. Define the transfer ma p j ! by the following co mpo sition of maps. (3.10) j ! : H ∗ ( LM × LM ) ˜ c ∗ − → ˜ H ∗ ( e N /∂ e N ) ˜ u ′ ∩ ( ) − − − − → ∼ = H ∗− d ( e N ) ˜ π ∗ − → ∼ = H ∗− d ( LM × M LM ) . CAP P RODUCTS IN STRING TOPOLOGY 9 The tubular neighbo rho o d s tr ucture of e N implies that the middle ma p is a gen uine Thom isomorphism. Definition 3.2. Let M be a closed oriented d -ma nifo ld. F or a, b ∈ H ∗ ( LM ), their lo op pro duct, denoted by a · b , is defined by (3.11) a · b = ( − 1) d ( | a | ′ − d ) ι ∗ j ! ( a × b ) = ( − 1) d | a | ι ∗ j ! ( a × b ) . The sign ( − 1) d ( | a | ′ − d ) app ears in [3 ] in the co mmut ative diag ram (1-7). W e include this sign explicitly in the definition o f the lo op pro duct for a t least three reasons . The most trivial reason is that on the left hand side, the dot representing the lo op pro duct is betw ee n a and b . On the rig ht hand side, j ! of degree − d representing the lo o p pro duct is in front of a . Switching a and j ! gives the sign ( − 1) d | a | ′ . The other par t of the s ign ( − 1) d comes from our choice o f orientation of ν and ensures that s ∗ ([ M ]) ∈ H 0 ( LM ), with the + sign, is the unit of the lo op pro duct. W e quickly v erify the correctnes s of the sign. Lemma 3.3. The element s ∗ ([ M ]) ∈ H 0 ( LM ) is the u n it of t he lo op pr o duct . Namely for any a ∈ H ∗ ( LM ) , (3.12) s ∗ ([ M ]) · a = a · s ∗ ([ M ]) = a. Pr o of. W e conside r the following diagra m. LM LM π 2 x      M × M 1 × p ← − − − − M × LM j ′ ← − − − − M × M LM LM    s × 1   y s × M 1   y    M × M p × p ← − − − − LM × L M j ← − − − − LM × M LM ι − − − − → LM In the induced homology diagram with transfers j ! and j ′ ! , the b ottom middle square commutes b ec a use tr ansfers are defined using Thom classes pulled back from the same Thom class u of the bas e ma nifo ld. Thus, s ∗ ([ M ]) · a = ι ∗ j ! ( s × 1) ∗ ([ M ] × a ) = j ′ ! ([ M ] × a ) . Here, since [ M ] has degree d , the sign in (3.11 ) is +1. Since π 2 ◦ j ′ = 1, the identit y on LM , j ′ ! ([ M ] × a ) = π 2 ∗ j ′ ∗ j ′ ! ([ M ] × a ) = π 2 ∗  (1 × p ) ∗ ( u ) ∩ ([ M ] × a )  . Due to the way ν is oriented, the Tho m class u is of the form u = { M } × 1 + · · · + ( − 1) d 1 × { M } . Hence π 2 ∗  (1 × p ) ∗ ( u ) ∩ ([ M ] × a )  = π 2 ∗ ([ x 0 ] × a + · · · ) = a . The other identit y a · s ∗ ([ M ]) = a can b e pr ov ed similarly . This completes the pro of.  The seco nd reason is that this c hoice o f sign for the lo o p pr o duct is the same sign app ear ing in the formula for the intersection pro duct defined in ter ms of the transfer map (3.6). This makes the lo op pro duct compa tible with the in tersection pro duct in the following sense. Prop ositi on 3. 4. Both of the fol lowing maps ar e algebr a maps pr eserving units b etwe en t he lo op algebr a H ∗ ( LM ) and t he interse ction ring H ∗ ( M ) . (3.13) p ∗ : H ∗ ( LM ) − → H ∗ ( M ) , s ∗ : H ∗ ( M ) − → H ∗ ( LM ) . 10 HIR OT AKA T AMANOI Pr o of. The pro of is mo r e or le s s straig ht forward, but we disc uss it br iefly . W e consider the following diagram. LM × LM j ← − − − − LM × M LM ι − − − − → LM p × p   y p   y p   y M × M φ ← − − − − M M s × s   y s   y s   y LM × LM j ← − − − − LM × M LM ι − − − − → LM Since the Thom classes for embeddings j a nd φ are compatible via ( p × p ) ∗ , the induced homo logy dia gram with tra ns fers j ! and φ ! is co mmut ative. Then by dia- gram chasing, we can easily c heck that p ∗ and s ∗ preserve pro ducts beca use of the same signs app earing in (3 .6) and (3.11).  The third reas on of the sign for the lo o p pro duct is that it g ives the corr e ct graded commut ativity , a s given in [1] proved in terms of chains. W e dis cuss a ho - motopy theor etic pro o f of gra ded comm utativity becaus e [2] did not include it, and bec ause the homotopy theoretic pro of itself is not so trivial with car eful treatment of tra nsfers and signs. Contrast the pr esent homotopy theoretic pro of with the simple geometric pro o f given in [1]. Prop ositi on 3.5. F or a, b ∈ H ∗ ( LM ) , the following gr ade d c ommutativity r elation holds : (3.14) a · b = ( − 1) ( | a | ′ − d )( | b | ′ − d ) b · a = ( − 1) | a || b | b · a. Pr o of. W e consider the following commutativ e diagram, wher e R 1 2 is the r otation of lo ops by 1 2 , that is, R 1 2 ( γ )( t ) = γ ( t + 1 2 ). LM × LM j ← − − − − LM × M LM ι − − − − → LM T   y T   y R 1 2   y LM × LM j ← − − − − LM × M LM ι − − − − → LM Since R 1 2 is homotopic to the identit y map, we hav e R 1 2 ∗ = 1. Hence a · b = ( − 1) d ( | a | ′ − d ) ι ∗ j ! ( a × b ) = ( − 1) d ( | a | ′ − d ) ι ∗ T ∗ j ! ( a × b ) . Next we show that the induced homo logy squar e with tr ansfer j ! , we have T ∗ j ! = ( − 1) d j ! T ∗ . Since the left square in the above diagram commutes on space lev el, w e hav e that T ∗ j ! and j ! T ∗ coincides up to a sig n. T o determine this sign, w e co mpo se j ∗ on the left of these maps and compare . Since the homo lo gy squa re with induced homology maps commute, j ∗ T ∗ j ! ( a × b ) = T ∗ j ∗ j ! ( a × b ) = T ∗  ˜ u ∩ ( a × b )  . On the other hand, j ∗ j ! T ∗ ( a × b ) = ˜ u ∩ T ∗ ( a × b ) = T ∗  T ∗ ( ˜ u ) ∩ ( a × b )  . W e compare T ∗ ( ˜ u ) and ˜ u . Since ˜ u = ( p × p ) ∗ ( u ), we have T ∗ ( ˜ u ) = ( p × p ) ∗ T ∗ ( u ). Since u is c haracter ized by the prop erty u ∩ [ M × M ] = φ ∗ ([ M ]) and T ◦ φ = φ , w e CAP P RODUCTS IN STRING TOPOLOGY 11 hav e φ ∗ ([ M ]) = T ∗ φ ∗ ([ M ]) = T ∗ ( u ) ∩ T ∗ ([ M × M ]) = T ∗ ( u ) ∩ ( − 1) d [ M × M ] . Thu s T ∗ ( u ) = ( − 1) d u . Hence T ∗ ( ˜ u ) = ( − 1) d ˜ u . In view of the above t w o identities, this implies that j ∗ T ∗ j ! = ( − 1) d j ∗ j ! T ∗ , or T ∗ j ! = ( − 1) d j ! T ∗ . Contin uing o ur computation, a · b = ( − 1) d | a | ′ ι ∗ j ! T ∗ ( a × b ) = ( − 1) | a | ′ | b | ′ + d | α | ι ∗ j ! ( b × a ) = ( − 1) ( | a | ′ − d )( | b | ′ − d ) b · a. This completes the homotopy theoretic pro of of commutativit y formula.  R emark 3 .6 . If we let µ = ι ∗ j ! : H ∗ ( LM ) ⊗ H ∗ ( LM ) − → H ∗ ( LM ), then us ing the metho d in [1 0], we can show that the asso ciativity of µ takes the form µ ◦ (1 ⊗ µ ) = ( − 1) d µ ◦ ( µ ⊗ 1). With our choice of the s ign for the lo o p pro duct in Definition 3.2, we can get rid of the ab ov e sign and we ha ve a usual ass o ciativity rela tio n ( a · b ) · c = a · ( b · c ) for the lo op pro duct without any signs for a, b, c ∈ H ∗ ( LM ). This is yet another re ason of o ur c hoice of the sign in the definition of the lo op pro duct. The transfer map j ! enjoys the following pro p erties similar to thos e sa tisfies b y φ ! as given in Prop ositio n 3.1. The pro of is similar , and we omit it. Prop ositi on 3. 7. F or a, b ∈ H ∗ ( LM ) and α ∈ H ∗ ( LM × LM ) , the fol lowing formulas ar e valid. j ∗ j ! ( a × b ) = ˜ u ∩ ( a × b ) (3.15) j !  α ∩ ( a × b )  = ( − 1) d | α | j ∗ ( α ) ∩ j ! ( b × c ) (3.16) The second formula says that j ! is a H ∗ ( LM × LM )-mo dule map. 4. Cap products and extended BV algebra str ucture W e exa mine compatibility of the cap pro duct with the v a rious structures in the BV-algebra H ∗ ( LM ) = H ∗ + d ( LM ). W e recall that a BV-algebra A ∗ is an asso ciative gr aded co mmut ative alg ebra equipp e d with a degree 1 Lie brack et { , } and a deg r ee 1 opera tor ∆ satisfying the following r elations for a, b, c ∈ A ∗ : ∆( a · b ) = (∆ a ) · b + ( − 1) | a | a · ∆ b + ( − 1) | a | { a, b } (BV identit y ) { a, b · c } = { a, b } · c + ( − 1) | b | ( | a | +1) b · { a, c } (Poisson identit y) a · b = ( − 1) | a || b | b · a, { a, b } = − ( − 1) ( | a | +1)( | b | +1) { b, a } (Commut ativity) { a, { b, c }} = {{ a, b } , c } + ( − 1) ( | a | +1)( | b | +1) { b, { a, c }} (Jacobi iden tity) Here, degrees o f elements are given b y ∆ a ∈ A | a | +1 , a · b ∈ A | a | + | b | , and { a, b } ∈ A | a | + | b | +1 . One wa y to view these r elations is to consider oper ators D a and M a acting on A ∗ for each a ∈ A ∗ given b y D a ( b ) = { a, b } and M a ( b ) = a · b . Let [ x, y ] = xy − ( − 1) | x || y | y x b e the gra ded c ommutator of op era tors. Then the Poisson ident ity a nd the Ja cobi iden tit y take the following for ms: (4.1) [ D a , M b ] = M { a,b } , [ D a , D b ] = D { a,b } , where degrees of op erato rs are | D a | = | a | + 1 and | M b | = | b | . One nice context to understand BV ident ity is in the context o f o dd symplectic geometry ([5], § 2), where BV op erator ∆ app ear s as a mixed second order o dd 12 HIR OT AKA T AMANOI differential o p e r ator, and BV iden tity can b e simply understoo d as L e ipnitz rule in differential ca lculus. This con text actually arises in loop homology . In [9], we explicitly computed the BV structure of H ∗ ( LM ) for the Lie group SU( n + 1) and complex Stiefel ma nifolds. Ther e, the BV op era tor ∆ is given by seco nd or der mixed o dd differential op erato r as ab ov e, and H ∗ ( LM ) is interpreted as the spa c e of po lynomial functions on the o dd symplectic vector space. The fact that the loo p algebra H ∗ ( LM ) is a BV-algebra w as pr ov ed in [1]. Note that the ab ov e BV re lations ar e satisfied with resp ect to H ∗ -grading , rather tha n the usual homology grading . The s ame is true for compatibility relations with cap pro ducts. First w e dis cuss cohomological S 1 action oper ator ∆ on H ∗ ( LM ). Let ∆ : S 1 × LM − → LM b e the S 1 action map given by ∆( t, γ ) = γ t , where γ t ( s ) = γ ( s + t ) for s, t ∈ S 1 = R / Z . The degree − 1 op era to r ∆ : H ∗ ( LM ) − → H ∗− 1 ( LM ) is defined by the follo wing form ula for α ∈ H ∗ ( LM ): (4.2) ∆ ∗ ( α ) = 1 × α + { S 1 } × ∆ α where { S 1 } is the fundamental co homology clas s of S 1 . The ho mological S 1 action ∆ is not a deriv ation with respect to the lo op pro duct and the deviation from b eing a der iv ation is given b y the loop brack et. Ho w ever, the cohomology S 1 -op erato r ∆ is a deriv atio n with resp ect to the cup pro duct. Prop ositi on 4.1. The c ohomolo gy S 1 -op er ator ∆ satisfies ∆ 2 = 0 , and it acts as a derivatio n on the c ohomolo gy ring H ∗ ( LM ) . That is, for α, β ∈ H ∗ ( LM ) , (4.3) ∆( α ∪ β ) = (∆ α ) ∪ β + ( − 1) | α | α ∪ ∆ β . Pr o of. The prop erty ∆ 2 = 0 is straig htf orward using the following dia gram S 1 × S 1 × L M 1 × ∆ − − − − → S 1 × L M µ × 1   y ∆   y S 1 × L M ∆ − − − − → LM Comparing b oth sides of (1 × ∆) ∗ ∆ ∗ ( α ) = ( µ × 1) ∗ ∆ ∗ ( α ), we obtain ∆ 2 ( α ) = 0. F or the der iv ation pro p erty , w e consider the following diagram. S 1 × L M φ × φ − − − − → ( S 1 × S 1 ) × ( LM × LM ) 1 × T × 1 − − − − − → ( S 1 × L M ) × ( S 1 × L M ) ∆   y ∆ × ∆   y LM φ − − − − → LM × L M LM × LM On the one hand, ∆ ∗ φ ∗ ( α × β ) = ∆ ∗ ( α ∪ β ) = 1 × ( α ∪ β ) + { S 1 } × ∆( α ∪ β ). On the other hand, ( φ × φ ) ∗ (1 × T × 1) ∗ (∆ × ∆) ∗ ( α × β ) = 1 × ( α ∪ β ) + ( − 1) | α | { S 1 } ×  α ∪ ∆ β + ∆ α ∪ β  . Comparing the ab ov e t wo identities, we obtain the deriv ation formula.  W e can regard the coho mo logy ring H ∗ ( LM ) together with cohomo logical S 1 action ∆ as a BV algebr a with trivial brack et pro duct. Now we show that the c ap pro duct is compatible with the lo o p pro duct in the BV-algebra H ∗ ( LM ). The follo wing theorem describes the behavior o f the cap pro duct with those elemen ts in the subalgebr a of H ∗ ( LM ) gener ated by H ∗ ( M ) and ∆  H ∗ ( M )  . CAP P RODUCTS IN STRING TOPOLOGY 13 Theorem 4.2. L et α ∈ H ∗ ( M ) and b, c ∈ H ∗ ( LM ) . The c ap pr o duct with p ∗ ( α ) b ehaves asso ciatively and gr ade d c ommutatively wi th r esp e ct t o t he lo op pr o duct. Namely (4.4) p ∗ ( α ) ∩ ( b · c ) = ( p ∗ ( α ) ∩ b ) · c = ( − 1) | α || b | b · ( p ∗ ( α ) ∩ c ) . The c ap pr o duct with ∆  p ∗ ( α )  is a derivatio n on the lo op pr o duct. Namely, (4.5) ∆  p ∗ ( α )  ∩ ( b · c ) =  ∆  p ∗ ( α )  ∩ b  · c + ( − 1) ( | α |− 1) | b | b ·  ∆  p ∗ ( α )  ∩ c  . Pr o of. F or the fir st for mula, we consider the following diagram, where π i is the pro jection onto the i th factor for i = 1 , 2. LM π i ← − − − − LM × L M j ← − − − − LM × M LM ι − − − − → LM p   y p × p   y q   y p   y M π i ← − − − − M × M φ ← − − − − M M Since p ∗ ( α ) ∩ ( b · c ) = ( − 1) d | b | ι ∗  ι ∗ p ∗ ( α ) ∩ j ! ( b × c )  , we need to understand ι ∗ p ∗ ( α ). F rom the ab ove commutativ e diagr am, we have ι ∗ p ∗ ( α ) = j ∗ π ∗ i p ∗ ( α ), which is equal to either j ∗ ( p ∗ ( α ) × 1) or j ∗ (1 × p ∗ ( α )). In the first case, contin uting our computation using (3.1 6), we have p ∗ ( α ) ∩ ( b · c ) = ( − 1) d | b | ι ∗  j ∗ ( p ∗ ( α ) × 1) ∩ j ! ( b × c )  = ( − 1) d | b | + d | α | ι ∗ j !  ( p ∗ ( α ) × 1) ∩ ( b × c )  = ( − 1) d | b | + d | α | ι ∗ j !  ( p ∗ ( α ) ∩ b ) × c  = ( p ∗ ( α ) ∩ b ) · c. Similarly , using ι ∗ p ∗ ( α ) = j ∗  1 × p ∗ ( α )  , we get the other identit y . This pr ov es (1 ). F or (2), we first note that the e le ment ∆  p ∗ ( α )  ∩ ( b · c ) is equal to ∆  p ∗ ( α )  ∩ ( − 1) d | b | ι ∗ j ! ( b × c ) = ( − 1) d | b | ι ∗  ι ∗  ∆( p ∗ ( α ))  ∩ j ! ( b × c )  . Thu s, we need to under s tand the element ι ∗  ∆( p ∗ ( α ))  . W e need some notatio ns. Let I = I 1 ∪ I 2 , where I 1 = [0 , 1 2 ] a nd I 2 = [ 1 2 , 1], a nd set S 1 i = I i /∂ I i for i = 1 , 2 . Let r : S 1 = I /∂ I − → I / { 0 , 1 2 , 1 } = S 1 1 ∨ S 1 2 be an identification map, and let ι i : S 1 i − → S 1 1 ∨ S 1 2 be the inclusion ma p into the i th wedge summand. W e ex a mine the following dia gram S 1 × ( L M × M LM ) r × 1 − − − − → ( S 1 1 ∨ S 1 2 ) × ( LM × M LM ) ← − − − − { 0 } × ( LM × M LM ) 1 × ι   y e ′   y ι   y S 1 × L M e − − − − → M p ← − − − − LM where e = p ◦ ∆ is the ev aluation map for S 1 × LM , and the other ev a luation map e ′ is given b y e ′ ( t, γ , η ) = ( γ (2 t ) 0 ≤ t ≤ 1 2 , η (2 t − 1 ) 1 2 ≤ t ≤ 1 . F or α ∈ H ∗ ( M ), we let e ′ ∗ ( α ) = 1 × ι ∗ p ∗ ( α ) + { s 1 1 } × ∆ 1 ( α ) + { S 1 2 } × ∆ 2 ( α ) 14 HIR OT AKA T AMANOI for some ∆ i ( α ) ∈ H ∗ ( LM × M LM ) for i = 1 , 2. The first term in the rig ht hand side is identified using the right squa r e of the ab ov e co mm utative dia g ram. Since r ∗ ( { S 1 i } ) = { S 1 } for i = 1 , 2, ( r × 1) ∗ e ′ ∗ ( α ) = 1 × ι ∗ p ∗ ( α ) + { S 1 } ×  ∆ 1 ( α ) + ∆ 2 ( α )  . The commut ativity of the left square implies that this m ust b e eq ual to (1 × ι ) ∗ ∆ ∗ p ∗ ( α ) = 1 × ι ∗ p ∗ ( α ) + { S 1 } × ι ∗ ∆  p ∗ ( α )  . Hence w e have ι ∗  ∆( p ∗ ( α ))  = ∆ 1 ( α ) + ∆ 2 ( α ) ∈ H ∗ ( LM × M LM ) . T o understand elements ∆ i ( α ), we co nsider the following co mmu tative dia gram, where ℓ 1 ( t ) = 2 t for 0 ≤ t ≤ 1 2 and ℓ 2 ( t ) = 2 t − 1 for 1 2 ≤ t ≤ 1. S 1 i × ( L M × M LM ) ℓ i × j − − − − → S 1 × ( L M × LM ) 1 × π i − − − − → S 1 × L M ι i × 1   y ∆   y ( S 1 1 ∨ S 1 2 ) × ( LM × M LM ) e ′ − − − − → M p ← − − − − LM On the one hand, ( ι 1 × 1) ∗ e ′ ∗ ( α ) = 1 × ι ∗ p ∗ ( α ) + { S 1 1 } × ∆ 1 ( α ). On the other hand, ( ℓ 1 × j ) ∗ (1 × π 1 ) ∗ ∆ ∗ p ∗ ( α ) = 1 × j ∗  p ∗ ( α ) × 1  + { S 1 1 } × j ∗  ∆( p ∗ ( α )) × 1  . By the comm utativity of the diagr am, we get ∆ 1 ( α ) = j ∗  ∆( p ∗ ( α )) × 1  . Similarly , i = 2 case implies ∆ 2 ( α ) = j ∗  1 × ∆( p ∗ ( α ))  . Hence we finally obtain ι ∗  ∆( p ∗ ( α ))  = j ∗  ∆( p ∗ ( α )) × 1 + 1 × ∆( p ∗ ( α ))  . With this identification of ι ∗  ∆( p ∗ ( α ))  as j ∗ of some other element, we ca n co n- tin ue our initial co mputation. ∆  p ∗ ( α )  ∩ ( b · c ) = ( − 1 ) d | b | ι ∗  j ∗  ∆( p ∗ ( α )) × 1 + 1 × ∆  p ∗ ( α )  ∩ j ! ( b × c )  = ( − 1) d | b | +( | α |− 1) d ι ∗ j !   ∆( p ∗ ( α )) × 1 + 1 × ∆  p ∗ ( α )  ∩ ( b × c )  = ( − 1) d ( | α | + | b |− 1) ι ∗ j !   ∆( p ∗ ( α )) ∩ b  × c + ( − 1) ( | b | + d )( | α |− 1) b ×  ∆( p ∗ ( α )) ∩ c   =  ∆( p ∗ ( α )) ∩ b  · c + ( − 1) ( | α |− 1) | b | b ·  ∆( p ∗ ( α )) ∩ c  . This completes the proo f of the der iv ation pro p er ty of the cap product with res p ect to the lo op pro duct.  Next we describ e the rela tion betw een the ca p pro duct and the BV op era tor in homology and cohomolo gy . Prop ositi on 4.3. F or α ∈ H ∗ ( LM ) and b ∈ H ∗ ( LM ) , the BV-op era tor ∆ satisfies (4.6) ∆( α ∩ b ) = (∆ α ) ∩ b + ( − 1 ) | α | α ∩ ∆ b. Pr o of. On the one hand, the S 1 -action map ∆ : S 1 × L M − → LM satisfies ∆ ∗  ∆ ∗ ( α ) ∩ ([ S 1 ] × b )  = α ∩ ∆ ∗ ([ S 1 ] × b ) = α ∩ ∆ b. On the other hand, s inc e ∆ ∗ ( α ) = 1 × α + { S 1 } × ∆ α , we ha ve ∆ ∗  ∆ ∗ ( α ) ∩ ([ S 1 ] × b )  = ∆ ∗  ( − 1) | α | [ S 1 ] × ( α ∩ b ) + ( − 1 ) | α |− 1 [ pt ] × (∆ α ∩ b )  = ( − 1) | α | ∆( α ∩ b ) + ( − 1) | α |− 1 ∆ α ∩ b. CAP P RODUCTS IN STRING TOPOLOGY 15 Comparing the ab ov e tw o formulas, w e obtain ∆( α ∩ b ) = ∆ α ∩ b + ( − 1) | α | α ∩ ∆ b .  Since homo logy BV oper ator ∆ on H ∗ ( LM ) acts trivially on H ∗ ( M ), the follow- ing corolla r y is immediate. Corollary 4.4. F or α ∈ H ∗ ( M ) , the c ap pr o duct of ∆ α with H ∗ ( M ) ⊂ H ∗ ( LM ) is trivial. Pr o of. F or b ∈ H ∗ ( M ), the op erato r ∆ acts trivially on b oth α ∩ b and b . Hence formula (4.6 ) implies (∆ α ) ∩ b = 0.  Next, w e discuss a b ehavior of the cap pro duct with respect to the lo op br ack et. Theorem 4.5 . The c ap pr o duct with ∆  p ∗ ( α )  is a derivatio n on the lo op br acket. Namely, for α ∈ H ∗ ( M ) and b, c ∈ H ∗ ( LM ) , (4.7) ∆  p ∗ ( α )  ∩ { b, c } = { ∆  p ∗ ( α )  ∩ b, c } + ( − 1) ( | α |− 1)( | b |− 1) { b, ∆  p ∗ ( α )  ∩ c } . Pr o of. Our pro of is computational using previous re sults. W e use the B V ident ity as the definition of the lo op brack et. Thus, { b, c } = ( − 1) | b | ∆( b · c ) − ( − 1) | b | (∆ b ) · c − b · ∆ c. W e compute the right hand side of (4.7). F or simplicity , we write ∆ α for ∆  p ∗ ( α )  . Each term in the right hand side of (4.7 ) gives { ∆ α ∩ b, c } = ( − 1) | b |−| α | +1 ∆  (∆ α ∩ b ) · c  − ( − 1) | b | (∆ α ∩ ∆ b ) · c − (∆ α ∩ b ) · ∆ c, { b, ∆ α ∩ c } = ( − 1) | b | ∆  b · (∆ α ∩ c )  − ( − 1) | b | ∆ b · (∆ α ∩ c ) − ( − 1 ) | α |− 1 b · (∆ α ∩ ∆ c ) , Here we used (4.6) for the seco nd term in the first identit y and in the third term in the second identit y . Combining these form ulas, we get { ∆ α ∩ b, c } + ( − 1) ( | α |− 1)( | b | +1) { b, ∆ α ∩ c } =  ( − 1) | b |−| α | +1 ∆  (∆ α ∩ b ) · c  + ( − 1) | b | +( | α |− 1)( | b | +1) ∆  b · (∆ α ∩ c )  −  ( − 1) | b | (∆ α ∩ ∆ b ) · c + ( − 1 ) ( | α |− 1)( | b | +1)+ | b | ∆ b · (∆ α ∩ c )  −  (∆ α ∩ b ) · ∆ c + ( − 1) ( | α |− 1)( | b | +1)+ | α |− 1 b · (∆ α ∩ ∆ c )  . Using the deriv a tion for mula for ∆ α ∩ ( ) with resp ect to the lo o p pro duct (4.5), three pairs of terms ab ove b eco me ( − 1) | b |−| α | +1 ∆  ∆ α ∩ ( b · c )  − ( − 1) | b | ∆ α ∩ (∆ b · c ) − ∆ α ∩ ( b · ∆ c ) = ∆ α ∩  ( − 1) | b | ∆( b · c ) − ( − 1) | b | ∆ b · c − b · ∆ c  = ∆ α ∩ { b, c } . This completes the pro of of the deriv a tio n formula for the loop bra ck et.  Recall that in the B V alg e br a H ∗ ( LM ), for ev ery a ∈ H ∗ ( LM ) the op er ation { a, · } of ta k ing the lo op brack et with a is a der iv ation with resp ect to b o th the lo op pro duct and the lo op bra ck et, in view of the Poisson identit y and the Jacobi ident ity . Since we hav e proved that the cap pr o duct with ∆ p ∗ ( α ) for α ∈ H ∗ ( M ) is a deriv atio n with resp ect to both the lo op product and the lo o p br ack et, we w onder if we ca n extend the BV struc tur e in H ∗ ( LM ) to a BV structure in H ∗ ( M ) ⊕ H ∗ ( LM ). Indeed this is possible by e x tending the loop pro duct and the lo op brack et to elements in H ∗ ( M ) as follows. 16 HIR OT AKA T AMANOI Definition 4 .6. F or α, β ∈ H ∗ ( M ) and b ∈ H ∗ ( LM ), w e define their lo op pro duct and lo op brack et b y (4.8) α · b = α ∩ b, { α, b } = ( − 1) | α | (∆ α ) ∩ b, α · β = α ∪ β , { α, β } = 0 . This defines a n asso ciative graded commutativ e lo op pro duct by (4.4 ), and a brack et pro duct on H ∗ ( M ) ⊕ H ∗ ( LM ). Note that this loop pro duct on H ∗ ( M ) ⊕ H ∗ ( LM ) reduces to the r ing structure on H ∗ ( M ) ⊕ H ∗ ( M ) mentioned in the in tro duction. With this definition, Poisson iden tities and Jaco bi identities a re still v alid in H ∗ ( M ) ⊕ H ∗ ( LM ). Theorem 4. 7. L et α, β ∈ H ∗ ( M ) , and let b, c ∈ H ∗ ( LM ) . (I) The fol lowing Poisson identities ar e valid in H ∗ ( M ) ⊕ H ∗ ( LM ): { α, β · c } = { α, β } · c + ( − 1) | β | ( | α |− 1) β · { α, c } (4.9) { αβ , c } = α · { β , c } + ( − 1) | α || β | β · { α, c } (4.10) { α, b · c } = { α, b } · c + ( − 1) ( | b |− d )( | α |− 1) b · { α, c } (4.11) { α · b, c } = α · { b, c } + ( − 1) | α | ( | b |− d ) b · { α, c } . (4.12) (II) The fol lowing J ac obi identities ar e valid in H ∗ ( M ) ⊕ H ∗ ( LM ): { α, { β , c }} = {{ α, β } , c } + ( − 1) ( | α |− 1)( | β |− 1) { β , { α, c }} (4.13) { α, { b, c }} = {{ α, b } , c } + ( − 1) ( | α |− 1)( | b |− d +1) { b, { α, c }} . (4.14) Pr o of. If we unra vel definitions, w e se e that (4.9) and (4.13) are really the same as the graded commutativit y of the cup pro duct o f the following fo r m (∆ α ) ∩ ( b ∩ c ) = ( − 1) | β | ( | α |− 1) β ∩ (∆ α ∩ c ) , (∆ α ) ∩ (∆ β ∩ c ) = ( − 1) ( | α |− 1)( | β |− 1) (∆ β ) ∩  (∆ α ) ∩ c  . the identit y (4.1 0) is equiv alent to the deriv atio n formula (4.3) of the cohomolog y S 1 action op erato r with resp ect to the cup pr o duct. ∆( α ∪ β ) = (∆ α ) ∪ β + ( − 1) | α | α ∪ (∆ β ) . The identit y (4.11) says that ∆ α ∩ ( ) is a deriv ation with r e sp ect to the lo op pro duct, and the identit y (4.14) says that ∆ α ∩ ( ) is a der iv ation with resp ect to the loo p brack et. W e ha ve alr eady v erified b oth of these cases. Thus, what remains to be chec k ed is formula (4.12), whic h says { α ∩ b, c } = α ∩ { b, c } + ( − 1) | α || b | + | α | b · (∆ α ∩ c ) . Using the BV identit y , the deriv ation for mula (4.6) of the BV op era tor with respect to the cap pro duct, and proper ties of α ∩ ( ) and ∆ α ∩ ( ), we c an pro ve this iden tit y CAP P RODUCTS IN STRING TOPOLOGY 17 as follows. ( − 1) | b |−| α | { α ∩ b, c } = ∆  ( α ∩ b ) · c  − ∆( α ∩ b ) · c − ( − 1) | b |−| α | ( α ∩ b ) · ∆ c = ∆  α ∩ ( b · c )  − (∆ α ∩ b + ( − 1) | α | α ∩ ∆ b ) · c − ( − 1) | b |−| α | α ∩ ( b · ∆ c ) = (∆ α ) ∩ ( b · c ) − (∆ α ∩ b ) · c + ( − 1) | α | α ∩ ∆( b · c ) − ( − 1) | α | α ∩ (∆ b · c ) − ( − 1) | b |−| α | α ∩ ( b · ∆ c ) = ( − 1) ( | α |− 1) | b | b · (∆ α ∩ c ) + ( − 1) | α | + | b | α ∩ { b, c } . Canceling some signs, we get the desired formula. This completes the pro of.  Other Poisson and Jacobi identities with cohomo logy elements in the second argument for mally follow from above identities b y making following definitions for α ∈ H ∗ ( M ) and b ∈ H ∗ ( LM ): b · α = ( − 1) | α || b | α · b, { b, α } = − ( − 1) ( | α | +1)( | b | +1) { α, b } . F or α ∈ H ∗ ( M ) we show ed that ∆ α ∩ ( ) is a deriv ation for both the lo op pro duct and the lo op brack e t, and α ∩ ( ) is g raded commutativ e and asso ciative with resp ect to the lo op pro duct. What is the behavior of α ∩ ( ) is w ith respec t to the lo o p brack et? F ormula (4.12) s ays that α ∩ ( · ) on lo op bra ck et is not a der iv ation o r graded commutativit y : it is a Poisson identit y! Poisson identities and Jacobi identities we have just proved in A ∗ = H ∗ ( M ) ⊕ H ∗ ( LM ) sho w that A ∗ is a Ge r stenhab er algebra . In fact, A ∗ can be forma lly tur ned int o a BV algebra b y defining a BV opera tor ∆ on A ∗ to be trivial on H ∗ ( M ) and to be the usual one o n H ∗ ( LM ) coming from the homolo gical S 1 action. Corollary 4 .8. The dir e ct sum A ∗ = H ∗ ( M ) ⊕ H ∗ ( LM ) has the stru ct ur e of a BV algebr a. Pr o of. Since H ∗ ( LM ) is a BV alg ebra a nd since we hav e already verified Poisson ident ities a nd Ja cobi identities in A ∗ , we only hav e to verify BV ide ntities in A ∗ . F or α, β ∈ H ∗ ( M ), an identit y ∆ ( α ∪ β ) = ( ∆ α ) ∪ β + ( − 1 ) | α | α ∪ ( ∆ β ) + ( − 1) | α | { α, β } is triv ially satis fie d since all terms are zero by definition of BV op erator ∆ and the lo op brack et on H ∗ ( M ) ⊂ A ∗ . Next, let α ∈ H ∗ ( M ) a nd b ∈ H ∗ ( LM ). Since the BV op erator ∆ o n A ∗ acts trivially on H ∗ ( M ), an identit y ∆ ( α ∩ b ) = ( ∆ α ) ∩ b + ( − 1) | α | α ∩ ( ∆ b ) + ( − 1) | α | { α, b } is really a r estatement of the deriv ative formula o f the homology S 1 action oper ator ∆ on cap pro duct: ∆( α ∩ b ) = ( − 1) | α | α ∩ (∆ b ) + (∆ α ) ∩ b in formula (4.6).  In connection with the ab ov e Co rollar y , w e can ask whether H ∗ ( LM ) ⊕ H ∗ ( LM ) has a structure of a BV alg ebra. Of co urse, H ∗ ( LM ) together with the coho mo - logical S 1 action oper ator ∆, whic h is a deriv ation, is a BV alge br a with trivial brack et pr o duct. Th us, as a direct sum of BV a lg ebras, H ∗ ( LM ) ⊕ H ∗ ( LM ) is a BV algebra, although pr o ducts b etw een H ∗ ( LM ) and H ∗ ( LM ) are trivial. More meaningful q uestion would be to ask whether the direct s um H ∗ ( LM ) ⊕ H ∗ ( LM ) has a BV algebr a structure extending the one on A ∗ describ ed in C o rollar y 4 .8. If we want to use the cap pro duct as an extension of the lo op pro duct, the answer is 18 HIR OT AKA T AMANOI no. This is be cause the ca p pro duct with a n a rbitrar y element α ∈ H ∗ ( LM ) do es not b ehav e asso ciatively with resp ect to the loo p pro duct in H ∗ ( LM ): if α is of the for m α = ∆ β for some β ∈ H ∗ ( M ), then α ∩ ( · ) acts as a deriv atio n on lo o p pro duct in H ∗ ( LM ) due to (4.5) and do es no t satisfy asso ciativity . R emark 4 .9 . In the co urse o f our inv estigation, we noticed the follo wing cur ious ident ity , w hich is in some sense symmetric in three v aria bles, fo r α ∈ H ∗ ( M ) and b, c ∈ H ∗ ( LM ). { α, b · c } + ( − 1) | b | α · { b , c } = { α, b } · c + ( − 1) | b | { α · b, c } = ( − 1) ( | α | +1) | b |  b · { α, c } + ( − 1) | α | { b, α · c }  . (4.15) This identit y is ea sily pr ov ed using Poisson ident ities. But we wonder the meaning of this symmetry . 5. Cap products in terms of BV al gebra structure In the previous sectio n, w e showed that the BV algebra structure in H ∗ ( LM ) can be extended to the BV algebra structure in H ∗ ( M ) ⊕ H ∗ ( LM ) b y proving Poisson ident ities a nd Ja cobi identities. This may b e a bit s urprising. But this turns out to be very natural through Poincar´ e duality in the following wa y . F or a ∈ H ∗ ( M ), we denote the element s ∗ ( a ) ∈ H ∗ ( LM ) by a , where s : M → LM is the inclus ion map. Theorem 5 .1. F or a ∈ H ∗ ( M ) , let α = D ( a ) ∈ H ∗ ( M ) b e its Poinc ar ´ e dual. Then for any b ∈ H ∗ ( LM ) , the fol lowing identities hold. (5.1) p ∗ ( α ) ∩ b = a · b, ( − 1) | α | ∆  p ∗ ( α )  ∩ b = { a, b } . Pr o of. Let 1 = s ∗ ([ M ]) ∈ H 0 ( LM ) b e the unit of the lo op pro duct. Since p ∗ ( α ) ∩ b = p ∗ ( α ) ∩ (1 · b ) =  p ∗ ( α ) ∩ 1  · b b y (4.4), and since p ∗ ( α ) ∩ 1 = p ∗ ( α ) ∩ s ∗ ([ M ]) = s ∗  s ∗ p ∗ ( α ) ∩ [ M ]  = s ∗ ( α ∩ [ M ]) = a, we have p ∗ ( α ) ∩ b = a · b . This prov es the first identit y . F or the s e c ond ident ity , in the BV identit y ( − 1) | a | { a, b } = ∆( a · b ) − (∆ a ) · b − ( − 1) | a | a · ∆ b, the first term in the rig ht hand side gives ∆( a · b ) = ∆( p ∗ ( α ) ∩ b ) = ∆  p ∗ ( α )  ∩ b + ( − 1) | α | p ∗ ( α ) ∩ ∆ b in view o f the first ide ntit y we just prov ed and the deriv a tion proper ty of the homologica l A 1 action op er ator on cap pro ducts. Here p ∗ ( α ) ∩ ∆ b = a · ∆ b . Since a ∈ H ∗ ( M ) is a homolog y class o f constant lo o ps, we have ∆ a = 0. T hus, ( − 1) | a | { a, b } = ∆  p ∗ ( α )  ∩ b + ( − 1) | α | a · ∆ b − ( − 1) | a | a · ∆ b = ∆  p ∗ ( α )  ∩ b, since | α | = −| a | . Th us, { a, b } = ( − 1) | α | ∆  p ∗ ( α )  ∩ b . This completes the pr o of.  In view of this theorem, since H ∗ ( LM ) is alr e ady a BV a lgebra, the Poisson ident ities and Jacobi identities we proved in section 4 ma y seem ob vious. Ho wev er, what we did in section 4 is that w e gav e a new and elementary homotopy the or etic pr o of o f Poisson identities and Jac o bi identities using only basic prop er ties of the cap pro duct and the BV iden tit y , when at least o ne of the elemen ts is fro m H ∗ ( M ). The ab ov e theorem shows that lo op pro ducts and lo op brack ets with elements in H ∗ ( M ) can be wr itten as ca p products with cohomology elements in LM . Th us, CAP P RODUCTS IN STRING TOPOLOGY 19 comp ositions of loo p pro ducts and loo p brack ets with elements in H ∗ ( M ) corre- sp onds to a cap pro duct with the pro duct of c orresp o nding cohomolog y classes in H ∗ ( LM ). Namely , Corollary 5. 2. L et a 0 , a 1 , . . . , a r ∈ H ∗ ( M ) , and let α 0 , α 1 , . . . α r ∈ H ∗ ( M ) b e their Poinc ar´ e duals. Then for b ∈ H ∗ ( LM ) , (5.2) a 0 · { a 1 , { a 2 , . . . { a r , b } · · · } } = ( − 1 ) | a 1 | + ··· + | a r |  α 0 (∆ α 1 )(∆ α 2 ) · · · (∆ α r )  ∩ b. In section 2, we cons idered a problem of in tersections of lo ops with submanifolds in certain configurations, a nd w e sa w that the homolo gy cla ss o f the intersections of int erest can b e given b y a cap product with cohomo logy cup pro ducts o f the above form (Prop ositio n 2.1). The ab ov e corollary computes this homolo gy class in terms of BV structure in H ∗ ( LM ) using the homolog y classes of these submanifolds. R emark 5.3 . In genera l, elements α, ∆ α fo r α ∈ H ∗ ( M ) do not g enerate the en- tire coho mology r ing H ∗ ( LM ). 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T amanoi, A homotop y t he or e tic pr o of of the BV identity in lo op homolo gy , Dep ar tment of Ma thematic s, University of California Sant a Cruz Sant a Cruz, CA 950 64 E-mail addr ess : tamanoi@ma th.ucsc.edu