Infinite Product Decomposition of Orbifold Mapping Spaces

INFINITE PR ODUCT DECOMPO SITION OF ORBIF OLD MAPPING SP A CES HIROT AKA T A MANOI Abstract. Ph ysicists sho wed that the generating function of orbif old ellip- tic genera of symmetric orbifolds can be written as an infinite product. W e sho w that there exists a geometric factorization on space lev el b ehind this in- finite product formula, and we do this in a m uch mor e general framework of orbifold mapping spaces, where factors in the infinite pro duct corresp ond to finite connect ed co verings of domain spaces whose fundamen tal groups are not necessarily abelian. F r om this formula, a concept of geometric Hec ke op er- ators for functors emerges. This is a non-ab elian geometric generalization of usual Hec k e operators. W e show that these generalized Heck e op erators indeed satisfy the iden tity of usual Heck e operators for the case of 2- dimensional tori. Contents 1. Int ro duction and summar y of results 1 2. Infinite pro duct decomp o s ition of o rbifold mapping space s 5 3. Generating functions of finite or bifo ld inv ar iants 11 4. Geometric Heck e o p erator s for functors 13 References 18 1. Introduction and summar y o f resul ts The elliptic genus of a Spin manifold M refers to the signature of L M [8], [14]. The elliptic gen us of a co mplex manifo ld M refers to the S 1 -equiv ariant χ y -characteristic of its fre e loo p spa ce L M = Ma p( S 1 , M ) [6]. These are some of the versions of elliptic gener a o f M . Since L M is infinite dimens io nal, the ab ove statements must b e b e interpreted using a lo ca lization for mu la [17]. Let G be a finite group. F o r a G -manifold M , we can consider an orbifo ld v ersio n of the elliptic g enus. Howev er, the free lo op space L ( M / G ) on the orbit s pa ce is not well b ehav ed. F ollowing [7 ], we define the orbifold lo op spa ce L orb ( M /G ) by (1.1) L orb ( M /G ) def =  a g ∈ G L g M  /G = a ( g ) ∈ G ∗  L g M /C G ( g )  , where G ∗ is the set of conjugacy cla s ses in G , C G ( g ) is the c e nt ra lizer of g in G , and L g M is the space of g -twisted lo ops in M given by (1.2) L g M = { γ : R → M | γ ( t + 1) = g − 1 γ ( t ) fo r all t ∈ R } . 2000 Mathematics Subject Classific ation. 55 N20, 55N91. Key wor d s and phr ases. Hec k e op erators, orbifold elliptic gen us, orbifold Euler c haracteristic, orbifold mapping space, orbifold loop space, symmetric orbif ol d, w r eath pro duct. 1 2 HIR OT AKA T AMANOI The centralizer C ( g ) acts on L g M . Also note that if the or der o f g is finite and is equal to s , then e a ch t wisted lo op γ in L g M is in fact a closed lo op o f length s . Thu s, L g M also admits an actio n of a circle S 1 = R /s Z of length s . One could use mo r e sophisticated languages on orbifolds (see for exa mple, [11]), but for our pur p o se, the ab ov e definition suffices. Now the o rbifold elliptic genus of ( M , G ), denoted by e ll orb ( M /G ), is defined a s the S 1 -equiv ariant χ y -characteristic o f L orb ( M /G ): (1.3) ell orb ( M /G ) = χ S 1 y  L orb ( M /G )  = X ( g ) ∈ G ∗ χ y ( L g M ) C ( g ) , where χ y ( L g M ) is thought of a s R  C ( g )  -v alued S 1 -equiv ariant χ y -characteristic computed and made sense thro ugh a use of lo c a lization for mulae. Counting the dimension of co efficient v ector s paces, we hav e (1.4) ell orb ( M /G ) ∈ Z [ y , y − 1 ][[ q ]] , where the p ow ers of q are characters of S 1 . Dijkgraaf, Mo ore, V er linde and V er linde [3] e ssentially proved a remark able fo r- m ula for the generating function o f orbifold elliptic g enera of symmetr ic pr o ducts. This was subsequently extended to symmetric orbifold case b y Bo r isov-Libgob er [1]. Here, for an integer n ≥ 0, the n -th symmetric pro duct of a space X is defined as S P n ( X ) = X n / S n , where the n - th s ymmetric group S n acts on X n by p er mut - ing n factor s. The DMVV and BL fo r mula for the genera ting function of orbifold elliptic ge nera of symmetr ic or bifolds is given by (1.5) X n ≥ 0 p n ell orb  S P n ( M /G )  = Y n ≥ 1 m ≥ 0 k ∈ Z (1 − p n q m y k ) − c ( mn,k ) , where ell orb ( M /G ) = X m ≥ 0 k ∈ Z c ( m, k ) q m y k ∈ Z [ y, y − 1 ][[ q ]] . The amazing thing ab o ut this formula is that the right hand s ide of (1.5) is a genu s 2 Sie g el mo dular form, up to a simple multiplicativ e factor . The main motiv ation of this pa p er is to understand a geometr ic o r igin of this infinite pr o duct formula. In fact, w e will prov e such a n infinite pro duct formula on a geometric level, not merely on an alg e braic level, as in (1.5). W e can describ e this geometric form ula in a gener al context. Let ( M , G ) b e as befo re, and let Σ b e an ar bitrary co nnected manifo ld with Γ = π 1 (Σ). Instead of a lo op space, we cons ider a mapping space Map(Σ , M /G ). As b efore, this space is not w ell b ehaved and the correc t space to c o nsider is the orbifold mapping space defined by (1.6) Map orb (Σ , M /G ) def =  a θ ∈ Hom(Γ ,G ) Map θ ( e Σ , M ) . G = a ( θ ) ∈ Hom(Γ ,G ) /G  Map θ ( e Σ , M ) /C ( θ )  . Here e Σ is the universal cov er of Σ, and Map θ ( e Σ , M ) is the s pa ce of θ -eq uiv ariant maps α : e Σ → M such that α ( p · γ ) = θ ( γ ) − 1 · α ( p ) for all p ∈ e Σ and γ ∈ Γ. Note here that we r egar d the universal cov er e Σ a s a Γ- pr incipal bundle ov er Σ. F or a v ariable t and a space X , let S t ( X ) = ` k ≥ 0 t k S P k ( X ) be the total sym- metric pro duct of X . F or conv enience, we o ften write this us ing the s umma tio n INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP A CES 3 symbol as S t ( X ) = P k ≥ 0 t k S P k ( X ). In this pa p er , s ummation s ymbol applied to top ologica l spaces means top o logical disjoint union. Theorem A (Infinite Pro duct Decompos ition of O r bifold Mapping Spaces of Sym- metric Pr o ducts) . L et M b e a G -manifold and let Σ b e a c o nne cte d manifold. Then, (1.7) X n ≥ 0 p n Map orb  Σ , S P n ( M /G )  ∼ = Y [Σ ′ → Σ] conn. S p | Σ ′ / Σ |  Map orb (Σ ′ , M /G ) / D (Σ ′ / Σ)  . Her e t he infinite pr o duct is taken over al l t he isomorphi sm classes of finite she ete d c onne cte d c overing sp ac es Σ ′ of Σ , and D (Σ ′ / Σ) is t he gr oup of al l de ck tr ansforma- tions of the c overing sp ac e Σ ′ → Σ ( which is not ne c essaril y Galois ) . The num b er of she ets of this c overing is denote d by | Σ ′ / Σ | . W e will e x plain the details o f the action of D (Σ ′ / Σ) on Map orb (Σ ′ , M /G ) in § 2. When Σ = S 1 , the ab ov e formula reduces to (1.8) X n ≥ 0 p n L orb  S P n ( M /G )  ∼ = Y r ≥ 1 S p r  L ( r ) orb ( M /G ) / Z r  , where L ( r ) orb ( M /G ) is the space of o rbifold lo ops of length r . This is the geo metric version of the fo rmula (1.5 ). This for mula itself is relatively ea sy to prov e. See [16]. The a b ov e formula (1.8) for orbifold lo op space is an ”ab elia n” case since π 1 ( S 1 ) ∼ = Z . The form ula in Theorem A is, in a sense, a no n-ab elian genera lization of this orbifold lo o p spa ce ca se. The most interesting case seems to b e the one in which Σ is a 2-dimensional surface (rega rding it as a w or ld-sheet of a moving string ). Here, the genus of the sur face can be ar bitrary . In physics litera ture, elliptic genus itself is computed as a path in tegr al ov er mapping spa ces fro m torus [3]. Restricting the global decompos ition for mula (1.7) to the subspace of constant orbifold maps and considering their numerical in v ariants, we recov er our previous results in [12, 13]. See section 3 for a description of these re s ults. W e remark that we can a pply (ge ne r alized) homolo gy and cohomology functors to (1.7) to obtain infinite pro duct decomp o sition for mulas of these homolog y and cohomolo gy theories. Another surprising for mu la discov ered by physicists [3] is its connection to Hec ke op erator s. They show ed that the right hand side of for mula (1.5) ca n b e written in terms of Hecke o p er ators in a very nice wa y: (1.9) X n ≥ 0 p n ell orb  S P n ( M /G )  = exp  X r ≥ 1 p r T ( r )  ell orb ( M /G )   , where T ( r ) is the r - th Heck e op erato r a cting on weigh t 0 Jacobi forms: (1.10) T ( r ) h X m ≥ 0 k ∈ Z c ( m, k ) q m y k i = X ad = r 1 a X m ≥ 0 k ∈ Z c ( md, k ) q am y ak . Is there a corres p onding Hecke op era tor in our geometric context? Su ch a Hec ke op erator must assign a certa in spa ce to a giv en space . Our geometr ic decomp osition formula (1.7) s uggests what geometr ic Heck e o pe r ators s ho uld b e. F or each po sitive int eger r , we exp ect the r -th ge ometric He cke op er ator T ( r ) w ould act on a space of the form Map orb (Σ , M /G ), and pro duces a space inv olving all the co nnec ted 4 HIR OT AKA T AMANOI r -s heeted covering spaces of Σ, as fo llows. (1.11) T ( r )  Map orb (Σ , M /G )  def = a [Σ ′ → Σ] | Σ ′ / Σ | = r Map orb (Σ ′ , M /G ) / D (Σ ′ / Σ) . The usual Hec ke op era tors use covering spaces of the torus [9], a nd in [3], they explain the abov e result (1.9) from this point of view. Our form ula (1.11) uses cov ering spaces of Σ whose fundament al gro up is not necessarily ab elian. Thus, in a sense, o ur Heck e op erator ca n b e thought of as a non-a b elian gener alization o f the usual Heck e op era tors. A general discussion o f geo metric Heck e op erato rs in the framework of functors is mor e conv enient and will b e giv en in section 4. Let F be a functor from the category C of top olog ical spac e s and contin uous maps to itself. F or example, for a G -ma nifold M , let F ( M ,G ) be a conrtav arian t functor from C to itself given by F ( M ,G ) (Σ) = Map orb (Σ , M /G ). Then, T ( n ) acts on the functor F by the following formula for a connected space Σ. (1.12)  T ( n ) F  (Σ) def = a [Σ ′ → Σ] conn. | Σ ′ / Σ | = n F (Σ ′ ) / D (Σ ′ / Σ) , where disjoint unio n runs ov er all isomo r phism cla sses of co nnected n -sheeted cov- ering space of Σ. When Σ is not connected, we apply the ab ov e construction for each connected co mp o nent of Σ. In terms of g eometric Heck e ope rators , formu la (1.7) can be simply rewr itten as (1.7 ′ ) X n ≥ 0 p n Map orb  Σ , S P n ( M /G )  ∼ = Y r ≥ 1 S p r h  T ( r ) F ( M ,G )  (Σ) i . It is very sugg estive to compar e this form ula with (1.9). If we re gard the n -th symmetric pro duct S P n ( X ) as X n /n !, since S n has n ! element s, then w e can regar d S p ( X ) as ex p( pX ). F rom this po int of view, the a nalogy b etw een (1.9) and (1.7 ′ ) is rea s onably precise. Ho wev er, see also a remar k after (4.2). The name geometric Heck e op erator seems appr opriate since these op erator s do satisfy the usual identit y when Σ is a genus 1 Riemann s urface. Theorem B (Hec ke Identit y for Geometric Hec ke O pe rators ) . L e t T b e a 2 - dimensional torus. L et F b e a functor fr om the c ate gory C of t op olo gic al sp ac es to itself. Then the ge ometric He cke op er ators T ( n ) , n ≥ 1 , satisfy (1.13)  ( T ( m ) ◦ T ( n )) F  ( T ) = X d | ( m,n ) d ·  T  mn d 2  ◦ R ( d )  F  ( T ) , wher e the op er ator R ( d ) on the functor F is given by (1.14) ( R ( d ) F )( T ) = F  R ( d ) T  D  R ( d ) T /T  , in which R ( d ) T = e T / ( d · L ) if T = e T / L for some lattic e L ⊂ e T ∼ = R 2 . Thu s, R ( d ) T is a d 2 -sheeted c overing spa c e of T . The co efficient d in the right hand side of (1.13) means a disjoint top o lo gical union of d co pie s . Note that (1.13) can b e restated in a more familia r form as follows: (1.13 ′ ) T ( m ) ◦ T ( n ) = T ( mn ) , if ( m, n ) = 1 , T ( p r ) ◦ T ( p ) = T ( p r +1 ) + p · T ( p r − 1 ) ◦ R ( p ) , if p pr ime . INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP A CES 5 As is w ell known in the theory of modular forms, these iden tities are equiv alent to an Euler pro duct decomp ositio n of the Dirichlet ser ies with the ab ove Hecke op erator co efficients. See (4.14). It would b e of interest to inv estigate relations among T ( n )s when Σ is a hig her genus Rie ma nn surfaces, or higher dimensional tori whose fundamental g roup is free ab elian. F or a generaliza tion of o r bifold elliptic genus to the setting of g eneralized coho- mology theory , s ee a pap e r by Ganter [4]. The or ganization of this paper is a s follows. In sec tio n 2, w e prove our ma in geometric decomp osition fo rmula in Theorem A. In section 3, we sp ecialize our in- finite dimensiona l geometric formula to the finite dimensional subspace of constant orbifold maps, and we deduce v arious fo rmulae o f g enerating functions of orbifold inv ar ia nts. In sectio n 4, after discussing so me ge nerality of geometric Heck e op er a- tors on functors, we prov e the Heck e identit y (1.13). The main r esult of this pap er , Theorem A, was first a nnounced a t a workshop at Banff International Research Station in June 20 0 3. 2. I nfinite product decomposition of orbifold mapping sp a ces First, we discuss some general facts of orbifold mapping spaces. F or a homomor- phism θ : Γ → G and a θ - equiv ariant ma p α : e Σ → M , let α : Σ → M /G b e the induced map on quo tient spaces . Th us we hav e a cano nical map Map θ ( e Σ , M ) → Map(Σ , M /G ). Let C G ( θ ) be the cen tralizer of the image of θ in G . Note that inv erse ima ges of this map are C G ( θ ) spaces . The a c tion of g ∈ G o n M has the effect g · : Map θ ( e Σ , M ) − → Map g · θ · g − 1 ( e Σ , M ) , and for every α ∈ Map θ ( e Σ , M ), we hav e α = g · α in Map(Σ , M /G ). Th us, we have a cano nica l map (2.1) Map orb (Σ , M /G ) def = a ( θ ) ∈ Hom(Γ ,G ) /G Map θ ( e Σ , M ) /C G ( θ ) − → Map(Σ , M / G ) . This map is in genera l not surjective nor injectiv e. W e consider a necessar y c o ndition for a map f : Σ → M / G to hav e a lift to a θ - equiv ariant map ˜ f : e Σ → M for some θ . Let η b e a n arbitra ry co ntractible lo op in Σ. Since e Σ → Σ is a cov ering, η alwa ys lifts to a cont ra ctible lo op ˜ η in e Σ, and hence ˜ f ( ˜ η ) is also contractible. Th us, for the existence of a lift ˜ f of a giv en map f , it is necessar y that for ev ery contractible lo op η in Σ, f ( η ) ⊂ M /G lifts to a contractible lo o p in M . Next, we discuss a functoria l prop er ty of orbifold mapping spaces. Prop ositi on 2. 1. (i) L et M b e a G -manifold. Any map f : Σ 1 → Σ 2 b etwe en c onne cte d manifolds induc es a wel l-define d map (2.2) f ∗ : Map orb (Σ 2 , M /G ) − → Map orb (Σ 1 , M /G ) . (ii) F o r two maps f 1 : Σ 1 → Σ 2 and f 2 : Σ 2 → Σ 3 , we have ( f 2 ◦ f 1 ) ∗ = f ∗ 1 ◦ f ∗ 2 . Pr o o f. Let Γ i be the group D ( f Σ i / Σ i ) of all deck transforma tions for the universal cov er e Σ i → Σ i for i = 1 , 2. Since an isomorphism D ( f Σ i / Σ i ) ∼ = π 1 (Σ i ) dep ends on the choice o f a base p oint in e Σ i , it is b etter to regar d Γ i as the group o f dec k transformatio ns rather than as the fundamen tal group o f Σ i . W e c ho ose a lift 6 HIR OT AKA T AMANOI ˜ f : e Σ 1 → e Σ 2 of f . Then ˜ f induces a homomorphism ˜ f ∗ : Γ 1 → Γ 2 such that ˜ f ( p · γ 1 ) = ˜ f ( p ) · ˜ f ∗ ( γ 1 ) for all p ∈ e Σ 1 and γ 1 ∈ Γ 1 . F or a map α ∈ Map θ ( e Σ 2 , M ) with θ ∈ Hom(Γ 2 , G ), we hav e α ◦ ˜ f ∈ Ma p θ ◦ ˜ f ∗ ( e Σ 1 , M ). Hence the co mp o sition with ˜ f giv es an induced map (2.3) ˜ f ∗ : a θ ∈ Hom(Γ 2 ,G ) Map θ ( e Σ 2 , M ) → a ρ ∈ Hom(Γ 1 ,G ) Map ρ ( e Σ 1 , M ) . Obviously , this map commutes with the G -a ction on M . Hence by quotienting by G , we hav e a map (2.4) ˜ f ∗ : Map orb (Σ 2 , M /G ) → Map orb (Σ 1 , M /G ) . W e hav e to verify that this map is independent o f the chosen lift ˜ f . Let ˜ f ′ : e Σ 1 → e Σ 2 be another lift of f . By examining the image of one p oint and using the uniquenes s of lifts, we must have that ˜ f ′ = ˜ f · γ 2 , globally on e Σ 1 , for some uniquely determined γ 2 ∈ Γ 2 . Then, ( α ◦ ˜ f ′ )( p 1 ) = α  ˜ f ( p 1 ) · γ 2  = θ ( γ 2 ) − 1 · ( α ◦ ˜ f )( p 1 ) for all p 1 ∈ e Σ 1 . Note that θ ( γ 2 ) ∈ G . Thus fo r a ll pos sible choices o f lifts ˜ f , the collection { α ◦ ˜ f } is contained in a single G -orbit in ` ρ ∈ Hom(Γ 1 ,G ) Map ρ ( e Σ 1 , M ). Th us difference of ˜ f ∗ and ( ˜ f ′ ) ∗ in (2.3) disa ppe ar after dividing by G , and the map (2.4) is indep endent of the choice o f lifts ˜ f . Hence we may simply call it f ∗ as in (2.2). The pr o of of the formula for the induced map o f a co mpo sition is routine.  As a n immediate co nsequence, we hav e Corollary 2.2. L et Σ ′ → Σ b e a c onne cte d c overing sp ac e. Then the gr oup D (Σ ′ / Σ) of al l de ck tr ansformations acts on Map orb (Σ ′ , M /G ) . F or later use, w e giv e deta ils o f this action. As before , let D ( e Σ / Σ) = Γ and Σ ′ = e Σ /H for some H ⊂ Γ. Then D (Σ ′ / Σ) ∼ = N Γ ( H ) /H . F or f ∈ Map ρ ( e Σ ′ , M ), u ∈ N Γ ( H ), and g ∈ G , the ac tio n of u, g on f is given by (2.5) ( u · f )( p ) = f ( pu ) , ( g · f )( p ) = g · f ( p ) , p ∈ e Σ ′ . These actions commute, but they do no t preserve ρ ∈ Hom( H , G ). How ρ trans- forms under these a ctions can b e eas ily computed and we hav e the following com- m utative diagra m: (2.6) Map ρ ( e Σ ′ , M ) u · − − − − → ∼ = Map ρ u − 1 ( e Σ ′ , M ) g ·   y ∼ = g ·   y ∼ = Map g · ρ · g − 1 ( e Σ ′ , M ) u · − − − − → ∼ = Map g · ρ u − 1 · g − 1 ( e Σ ′ , M ) , where ρ u − 1 ( h ) = ρ ( u − 1 hu ) for all h ∈ H . Since C G ( ρ ) = C G ( ρ u − 1 ), commutativit y of this diagram als o implies that for u ∈ N Γ ( H ), (2.7) u · : Map ρ ( e Σ ′ , M ) ∼ = − → Map ρ u − 1 ( e Σ ′ , M ) , C G ( ρ )-equiv ariant . A glo bal statement is the following for u ∈ N Γ ( H ): (2.8) u · : a ρ ∈ Hom( H,G ) Map ρ ( e Σ ′ , M ) ∼ = − → a ρ ∈ Hom( H,G ) Map ρ ( e Σ ′ , M ) , G -equiv arian t . INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP A CES 7 In other w or ds, the group N Γ ( H ) × G acts on ` ρ Map ρ ( e Σ ′ , M ). Also note that the same gr oup N Γ ( H ) × G acts o n the set Ho m( H, G ) b y [( u, g ) · ρ ]( h ) = g · ρ u − 1 ( h ) · g − 1 for h ∈ H . The effect of changing u ∈ N Γ ( H ) by h ∈ H ca n b e computed a s (2.9) ρ ( uh ) − 1 ( · ) = ρ ( h ) − 1 ρ u − 1 ( · ) ρ ( h ) , ρ ( hu ) − 1 ( · ) = ρ u − 1 ( h ) − 1 ρ u − 1 ( · ) ρ u − 1 ( h ) . This shows that mo dification of u by elements in H ha s the same effect as the conjugation action by elements in G . Hence the ma p induced from (2.8) on G - orbits is well defined for u ∈ N Γ ( H ) /H , and we have (2.10) u · : Map orb (Σ ′ , M /G ) ∼ = − → Map orb (Σ ′ , M /G ) . This is the a ction in Corolla ry 2.2. Since the action of D (Σ ′ / Σ) commutes with the pro jection map π : Σ ′ → Σ, the action of D (Σ ′ / Σ) on Map orb (Σ ′ , M /G ) commutes with the induced map π ∗ . In particular, the image o f π ∗ is in the D (Σ ′ / Σ)-fixed p oint subset: (2.11) Map orb (Σ , M /G ) π ∗ − → Map orb (Σ ′ , M /G ) D (Σ ′ / Σ) . W e will need a n iden tit y on nested equiv ariant mapping spaces. Let P → Z be a left Γ-equiv ariant right G -principal bundle over a left Γ-s pa ce Z , where the left Γ-a c tion a nd the right G -action on P c ommute. W e s imply call such a bundle Γ- G bundle [10]. W e studies this co nc e pt in detail in section 3 of [13], where the classification theor e m of such bundles is discussed. Note that Map G ( P, M ) is a left Γ-space when P is a Γ - G bundle. Prop ositi on 2. 3. With notations as ab ove, we have (2.12) Map Γ  e Σ , Map G ( P, M )  = Map G ( e Σ × Γ P, M ) . Pr o o f. Without eq uiv ariance, this identit y is o bvious. So a ll we hav e to chec k is that the c a nonical co rresp o ndence preserves the corr ect equiv ariance prop er ty . Let f : e Σ → Map G ( P, M ), and let u ∈ e Σ. The Γ-equiv ariance of f and G - equiv ariance o f f ( u ) means f ( u γ ) = γ − 1 · f ( u ) = f ( u ) ◦ γ and f ( u )( pg ) = g − 1 f ( u )( p ) for all γ ∈ Γ, g ∈ G , p ∈ P . Le t the ca nonically corr esp onding map ˆ f : e Σ × P → M be defined by ˆ f ( u, p ) = f ( u )( p ). The Γ-equiv ariance o f f implies that ˆ f ( uγ , p ) = ˆ f ( u, γ · p ) fo r all u , γ , p . Hence ˆ f facto rs throug h e Σ × Γ P who se elements we denote by [ u , p ]. Using G -equiv ariance o f f , w e ha ve ˆ f ([ u, p ] g ) = ˆ f ([ u, pg ]) = f ( u )( pg ) = g − 1 · f ( u )( p ) = g − 1 ˆ f ([ u, p ]). Thus, ˆ f is G -equiv ariant. The obvious inv ers e corr esp ondence can be simila r ly chec ked to behave cor rectly with res p e c t to equiv ariance.  W e examine the left hand side of the formula (1.7). F or a po sitive integer n , let n = { 1 , 2 , . . . , n } . Then the wrea th pro duct G n = G ≀ S n is defined by (2.13) G n = G ≀ S n = Map( n , G ) ⋊ S n . When M is a G -manifold, the wrea th pro duct G n naturally a cts on the Cartesia n pro duct M n , and its quotient space M n /G n = S P n ( M /G ) is the n -the symmetr ic 8 HIR OT AKA T AMANOI orbifold of M /G . F or detailed information on wreath pro duct, s ee sec tion 3 of [13]. T o under stand (1.7), first we note that (2.14) Map orb  Σ , S P n ( M /G )  = a ( θ ) ∈ Hom(Γ ,G n ) /G n  Map θ ( e Σ , M n ) /C G n ( θ )  . Let n × G → n b e the trivial G -principal bundle ov er an n -element set n . Since Aut G ( n × G ) ∼ = G n (see [13] Lemma 3- 3 ), the spa ce of G -equiv arian t maps Map G ( n × G, M ) has the structur e of left G n space and we hav e a G n -equiv ariant homeomor- phism (2.15) M n ∼ = Map G ( n × G, M ) . F or a given homomo rphism θ : Γ → G n , b oth of the ab ov e spaces ca n b e thought of as Γ-spaces. Esp ecially , the trivial G -bundle n × G → n acquir es the structure of a Γ-equiv aria nt G -principa l bundle, or simply a Γ- G bundle, via θ . W e denote this by ( n × G ) θ . No w (2.15) a nd Pro p osition 2.3 imply that (2.16) Map θ ( e Σ , M n ) ∼ = Map Γ ( e Σ , Map G  ( n × G ) θ , M )  = Map G  e Σ × Γ ( n × G ) θ , M  . A Γ - G bundle P → Z is ca lled irreducible if Z is a tr ansitive Γ-set. In this case, Γ × G acts transitively on P . In section 3 o f [13], we cla ssified all the is o morphism classes of irr educible Γ- G bundles. W e sho wed that any irreducible Γ- G bundle m ust b e of the form P H,ρ = Γ × ρ G → Γ /H for some subgroup H ⊂ Γ and a homomorphism ρ : H → G . W e also show ed that t wo ir reducible Γ- G bundles corres p o nding to ( H 1 , ρ 1 ) and ( H 2 , ρ 2 ) ar e iso morphic as Γ- G bundles if and only if (i) the subgroups H 1 and H 2 are co njugate in Γ, and (ii) when H 1 = H 2 = H , we m ust have [ ρ 1 ] = [ ρ 2 ] ∈ Hom( H, G ) / ( N Γ ( H ) × G ) ([13], Theo rem E), wher e N Γ ( H ) and G act on Hom( H , G ) by conjuga ting H and G , resp ec tively . F rom now on, an element in Hom( H , G ) / ( N Γ ( H ) × G ) is denoted with a square brack et as in [ ρ ], and an element in Hom( H, G ) /G is denoted b y a round bra ck et as in ( ρ ), to distinguish these tw o kinds of co njugacy cla sses. Let r θ ( H, ρ ) be the num b er of irreducible Γ- G bundles iso mo rphic to P H,ρ → Γ /H in the irreducible de c omp osition o f ( n × G ) θ → n . Th us, (2.17) [( n × G ) θ → n ] ∼ = a [ H ] a [ ρ ] r θ ( H,ρ ) a [ P H,ρ → Γ /H ] . Here [ H ] runs over a ll the conjugacy classes of finite index subgroups of Γ, and for each H , [ ρ ] runs ov er the set Ho m( H, G ) / ( N Γ ( H ) × G ). By examining the decomp osition o f the ba se space n into transitive Γ-sets, we have (2.18) X [ H ] , [ ρ ] r θ ( H, ρ ) | Γ / H | = n. Let P H,ρ = e Σ × Γ P H,ρ and Σ H = e Σ × Γ (Γ /H ) = e Σ /H . Then P H,ρ is a G -bundle ov er a cov ering spa ce Σ H of Σ. Note that in P H,ρ → Σ H → Σ, for each p oint in Σ, fibres of these bundles give P H,ρ → Γ /H . The ab ov e decomp osition now implies (2.19) e Σ × Γ [( n × G ) θ → n ] ∼ = a [ H ] a [ ρ ] r θ ( H,ρ ) a [ P H,ρ → Σ H ] . INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP ACES 9 This isomor phism allows us to rewrite (2 -16) as (2.20) Map θ ( e Σ , M n ) ∼ = Y [ H ] Y [ ρ ] r θ ( H,ρ ) Y Map G ( P H,ρ , M ) ∼ = Y [ H ] Y [ ρ ] r θ ( H,ρ ) Y Map ρ ( e Σ H , M ) . The last isomorphism is because P H,ρ = e Σ H × ρ G . This gives m ultiplicative deco m- po sition of each disjoint summand of the right hand side of (2.14). Nex t, w e need to understand the centralizer C G n ( θ ) of the ima g e of the ho mo morphism θ : Γ → G n in G n . One of the main results of [13] is the description of the s tructure of the centralizer C G n ( θ ). It says that (2.21) C G n ( θ ) ∼ = Y [ H ] Y [ ρ ]  Aut Γ- G ( P H,ρ ) ≀ S r θ ( H,ρ )  , where Aut Γ- G ( P H,ρ ) is the gr oup of Γ-equiv a r iant G -principal bundle a utomor- phisms of P H,ρ → Γ /H . In terms of the G -bundle P H,ρ → Σ H ov er a cov ering space, Aut Γ- G ( P H,ρ ) is isomor phic to the gr o up Aut G ( P H,ρ ) Σ H / Σ of G -bundle iso - morphisms of P H,ρ whose induced map on Σ H is a dec k transforma tion of Σ H → Σ ([13], Pr op osition 7-3 ). Next we describ e the structure of Aut Γ- G ( P H,ρ ). W e recall that the gro up N Γ ( H ) × G acts on the set Hom( H, G ) by ( u, g ) · ρ = g · ρ u − 1 · g − 1 for u ∈ N Γ ( H ), g ∈ G and ρ ∈ Hom( H, G ). Le t T ρ be the isotropy subgroup of this a ction at ρ : (2.22) T ρ = { ( u, g ) ∈ N Γ ( H ) × G | g · ρ u − 1 ( h ) · g − 1 = ρ ( h ) for all h ∈ H } . This gr oup T ρ contains a subgr oup H ρ =  h, ρ ( h )  ∈ T ρ | h ∈ H  ∼ = H . Then Theorem 4- 4 in [1 3] shows that H ρ is a normal subgroup of T ρ and we hav e the following exa ct sequence: (2.23) 1 → H ρ → T ρ → Aut Γ- G ( P H,ρ ) → 1 . Now we are ready to prov e Theorem A. Pr o of of The or em A. Using (2.14), (2.1 8), (2.20), (2.21) , we c a n rewrite the left hand side of (1.8) a s X n ≥ 0 p n Map orb  Σ , S P n ( M /G )  = X n ≥ 0 X [ θ ] Y [ H ] Y [ ρ ] p r θ ( H,ρ ) | Γ /H | h  r θ ( H,ρ ) Y Map ρ ( e Σ H , M )  /  Aut Γ- G ( P H,ρ ) ≀ S r θ ( H,ρ )  i 10 HIR OT AKA T AMANOI Here Aut Γ- G ( P H,ρ ) ∼ = Aut G ( P H,ρ ) Σ H / Σ acts o n Map ρ ( e Σ H , M ) ∼ = Map G ( P H,ρ , M ) by the obvious action. = X n ≥ 0 X [ θ ] Y [ H ] Y [ ρ ] p r θ ( H,ρ ) | Γ /H | S P r θ ( H,ρ )  Map ρ ( e Σ H , M ) / Aut Γ- G ( P H,ρ )  = Y [ H ] Y [ ρ ] h X r ≥ 0 p r | Γ /H | S P r  Map ρ ( e Σ H , M ) / Aut Γ- G ( P H,ρ ) i = Y [ H ] Y [ ρ ] S p | Γ /H |  Map ρ ( e Σ H , M ) / Aut Γ- G ( P H,ρ )  = Y [ H ] S p | Γ /H | h a [ ρ ] Map ρ ( e Σ H , M ) / Aut Γ- G ( P H,ρ ) i . Here in the ab ov e form ulae, [ ρ ] ∈ Hom( H , G ) / ( N Γ ( H ) × G ). On the other hand, since D (Σ H / Σ) ∼ = N Γ ( H ) /H , we have Map orb (Σ H , M /G ) / D (Σ H / Σ) =  a ρ ∈ Hom( H,G ) Map ρ ( e Σ H , M )  G  N Γ ( H ) /H  =  a ρ ∈ Hom( H,G ) Map ρ ( e Σ H , M )  / ( N Γ ( H ) × G ) . Here w e r ecall that the action o f G and N Γ ( H ) commut es, a nd the action o f H ⊂ N Γ ( H ) ca n be abso r b ed into the action of G . See (2.5), (2.6), (2.8) and (2.9) for details on this. In particula r , the action of ( u, g ) ∈ N Γ ( H ) × G is s uch that ( u, g ) : Map ρ ( e Σ H , M ) ∼ = − → Map gρ u − 1 g − 1 ( e Σ H , M ) . Since T ρ in (2.22) is exac tly the subgro up which preserves ρ ∈ Hom( H , G ) under ( N Γ ( H ) × G )-action, in the ab ove identit y , we g et Map orb (Σ H , M /G ) / D (Σ H / Σ) = a [ ρ ]  Map ρ ( e Σ H , M ) /T ρ  , where [ ρ ] runs over the o rbit set Hom( H , G ) / ( N Γ ( H ) × G ). Next o bserve tha t the subgroup H ρ of T ρ acts trivially on Map ρ ( e Σ H , M ). T o see this, let  h, ρ ( h )  ∈ H ρ for h ∈ H , and f ∈ Map ρ ( e Σ H , M ). The n, for any p ∈ e Σ H , we hav e  h, ρ ( h )  f  ( p ) = ρ ( h ) · ( hf )( p ) = ρ ( h ) f ( ph ) = ρ ( h ) ρ ( h ) − 1 f ( p ) = f ( p ) . Thu s, H ρ acts tr ivially on Map ρ ( e Σ H , M ). Hence q uotienting b y T ρ in the ab ove formula ca n b e replaced by quotien ting b y T ρ /H ρ ∼ = Aut Γ- G ( P H,ρ ). Thus, co llecting all the ab ov e calculations, we fina lly hav e X n ≥ 0 p n Map orb  Σ , S P n ( M /G )  = Y [ H ] S p | Γ /H |  Map orb (Σ H , M /G ) / D (Σ H / Σ)  . This completes the pro of.  When G = { 1 } , we have Map orb (Σ , M ) = Map(Σ , M ), and for mula (1-8) b e- comes (2.24) X n ≥ 0 p n Map orb  Σ , S P n ( M )  ∼ = Y [Σ ′ → Σ] conn. S p | Σ ′ / Σ |  Map(Σ ′ , M ) / D (Σ ′ / Σ)  . INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP ACES 11 3. Genera ting functions of finite orbifold inv ariants W e s p e c ialize our main deco mpo sition formula of infinite dimensional orbifold mapping spaces to the finite dimensional subspace of constant o r bifold maps. Most of the r esults in [12, 13] follow from this restric ted formula, and we r epro duce some of the main r esults in these pap ers as c o rollar ies to Theo rem A. Since Map θ ( e Σ , M ) const. ∼ = M h θ i , where M h θ i denotes the fixed p oint subse t of θ , we have (3.1) Map orb (Σ , M /G ) const. = a ( θ ) ∈ Hom(Γ ,G ) /G  M h θ i /C ( θ )  def = C Γ ( M /G ) . As an immediate cons e quence of Theore m A, w e hav e the following decomp osition formula for constant orbifold maps. Prop ositi on 3. 1. L et M b e a G -sp ac e and let Γ b e an arbitr ary gr oup. Then, (3.2) X n ≥ 0 p n C Γ  S P n ( M /G )  = Y [ H ] S p | Γ /H |  C H ( M /G ) / ( N Γ ( H ) /H )  = Y [ H ] S p | Γ /H |  a [ ρ ] ( M h ρ i /T ρ )  , wher e [ H ] runs over al l the c onjugacy classes of finite index sub gr oups of Γ , and for e ach [ H ] , [ ρ ] runs over the set Hom( H, G ) /  N Γ ( H ) × G  . Note that in Theorem A, Γ is the fundamental group o f the manifold Σ. But after eliminating Σ by considering constant o rbifold maps, Γ can be an arbitrar y (discrete) gr o up in Prop os ition 3.1. Here we comment o n the a ction of N Γ ( H ) /H on C H ( M /G ) = ( ` ρ M h ρ i ) /G in (3.2), wher e ρ ∈ Hom( H , G ). In view o f (2.5), the actio n of N Γ ( H ) commutes with the action of G , and for an y u ∈ N Γ ( H ) a nd any x ∈ M h ρ i , the actio n of u o n x is suc h that u · x = x , as can b e easily verified by (2.5). How ever, this do es not mean that the action o f N Γ ( H ) on C H ( M /G ) is tr ivial. in fact, it is not trivial in genera l. What happ ens is that the a ction of u sends M h ρ i to M h ρ u − 1 i , where G -conjugacy cla sses ( ρ ) and ( ρ u − 1 ) ca n b e distinct, a ltho ugh these tw o spaces are identical subspac e s of M , since h ρ i = h ρ u − 1 i as subgr oups of G . F or a g iven ( ρ ) ∈ Hom( H, G ) /G , let N ρ Γ ( H ) b e the isotropy subgro up of N Γ ( H ) at ( ρ ). Reca ll that we hav e an exac t seq uenc e of gr oups [[13], for mula (4 -6)]: 1 → C G ( ρ ) → T ρ → N ρ Γ ( H ) → 1 . Thu s, M h ρ i /T ρ =  M h ρ i /C ( ρ )  / N ρ Γ ( H ). W e examine the action of u ∈ N ρ Γ ( H ) on M h ρ i /C ( ρ ). By definition, for any u ∈ N ρ Γ ( H ), ρ and ρ u − 1 are G -co njugate, a nd th us there exists g ∈ G such that ρ u − 1 ( h ) = g − 1 ρ ( h ) g for a ll h ∈ H . This means that ( u, g ) ∈ T ρ . W e have M h ρ i /C ( ρ ) u · =Id − − − − → M h ρ u − 1 i /C ( ρ u − 1 ) = M h g − 1 ρg i /C ( g − 1 ρg ) g · − → ∼ = M h ρ i /C ( ρ ) , by (2 .6). This mea ns that when we apply u · , ρ mov es within the same G -conjuga c y class to ρ u − 1 . T o bring it back to ρ , we then apply g ∈ G . Thus, for u ∈ N ρ Γ ( H ) and x ∈ M h ρ i /C ( ρ ), the a ction of u on x is given by u · x = g · x where g ∈ G is an arbitrar y element such that ( u, g ) ∈ T ρ . 12 HIR OT AKA T AMANOI Let χ ( X ) b e the top olog ical Euler c har acteristic for a topo logical space X . In [13], we introduced a notion of an orbifo ld Euler characteristic asso ciated to a gro up Γ defined for a G -manifold M : (3.3) χ Γ ( M ; G ) def = χ  C Γ ( M /G )  = X ( θ ) ∈ Hom(Γ ,G ) /G χ  M h θ i /C ( θ )  . W e observe that when Γ = Z , (3.4) χ Z ( M ; G ) = X ( g ) ∈ G ∗ χ  M h g i /C ( g )  = 1 | G | X gh = hg χ  M h g,h i  is the physicist’s orbifold Euler characteristic e orb ( M /G ) [2]. Here in the last sum- mation, the pa ir ( g , h ) runs o ver the set of commuting pairs of elements. The s econd ident ity is due to L e fschetz Fixed Point F o rmula. F ormula (3.3) gives the correct generaliza tion of e orb ( M /G ) since it comes from a very natural g eometry o f o rbifold mapping space s (3.1). In [1 3], we introduced a notion of o rbifold Euler c harac teristic of M /G a s so ciated to a Γ-se t X , denoted by χ [ X ] ( M ; G ). When X is a transitive Γ-set of the form X = Γ /H , it is given by (3.5) χ [Γ /H ] ( M ; G ) = χ  C H ( M /G ) / ( N Γ ( H ) /H )  where C H ( M /G ) / ( N Γ ( H ) /H ) = a [ ρ ] ∈ Hom( H,G ) / ( N Γ ( H ) × G ) M h ρ i / Aut Γ- G ( P H,ρ ) = a [ ρ ] M h ρ i /T ρ . The second identit y ab ove can b e proved on top olo gical space level b y an argument similar to the la st part o f the pro o f of Theorem A. Now w e compute the top o logical Euler characteris tic of b oth sides of (3.2). W e recall that χ  S p ( X )  = (1 − p ) − χ ( X ) . Corollary 3. 2 ([13] Theor em C) . L et M b e a G -s et and let Γ b e an arbitr ary gr oup. The the gener ating function of orbifold Euler char acteristic asso ciate d to Γ of symmetric orbifolds is given by (3.6) X n ≥ 0 p n χ Γ ( M n ; G n ) = Y [ H ] (1 − p | Γ /H | ) − χ [Γ /H ] ( M ; G ) , wher e [ H ] runs over al l c onjugacy classes of finite index sub gr oups of Γ . W e can rewrite (3.6) in ter ms of Hecke op erator s a s follows. F or a G -manifold, let χ ( M ; G ) be an integer v alued function o n the s et o f discrete groups g iven by (3.7) χ ( M ; G ) (Γ) def = χ  C Γ ( M /G )  = χ Γ ( M ; G ) . F or an integer n ≥ 1, let a Heck e op era tor T ( n ) act on the function χ ( M ; G ) by (3.8)  T ( n ) χ ( M ; G )  (Γ) def = X [ H ] | Γ /H | = n χ  C H ( M /G )  ( N Γ ( H ) /H )  , INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP ACES 13 so that T ( n ) χ ( M ; G ) is a no ther integral function on the set o f discre te groups . Then as functions o n the set o f gr oups, (3.6) means (3.9) X n ≥ 0 p n χ ( M n ; G n ) = Y n ≥ 1 (1 − p n ) − T ( n ) χ ( M ; G ) . Now w e consider the c ase in which Γ is ab elian . In this cas e, the actio n of N Γ ( H ) = Γ on H ⊂ Γ is trivial and so div iding by N Γ ( H ) /H has no effect. Thus, we have C H ( M /G ) / ( N Γ ( H ) /H ) = C H ( M /G ) and conse quently , Corollary 3.3. L et Γ b e an arbitr ary ab elian gr oup. F or any G - s p ac e M , we have (3.10) X n ≥ 0 p n χ Γ ( M n ; G n ) = Y H (1 − p | Γ /H | ) − χ H ( M ; G ) , wher e the pr o duct is over al l finite index sub gr oups H of Γ . In par ticular, when Γ = Z , the fo rmula (3.7) reduces to (3.11) X n ≥ 0 p n e orb  S P n ( M /G )  = Y r ≥ 1 (1 − p r ) − e orb ( M /G ) . This is the for mula proven in [7 ] whe n G is trivia l, and for genera l G in [15]. Instead of Euler character istic, we can consider other numerical inv a riants such as signature, spin index, χ y -characteristic, e tc., in suitable categ ories of manifolds. The for mula (3.2) will then provide us with infinite pro duct for mula of the co rre- sp onding generating functions of or bifold inv ariants o f symmetric orbifolds. What is mor e interesting in this context is that, since we hav e a deco mpo sition on the space level, we ca n apply v ario us (g eneralized) homolo g y and co homolog y functors to obta in infinite pro duct decomp ositio n formulae. This will be discus sed in future pap ers. 4. Geometric Hecke opera tors for functors In this section, we prove the Heck e identit y (1.13) for 2 -dimensional tori. Let C be the categor y of top olog ical spaces and co ntin uous maps. Let F : C → C be a cov a riant (or contra v ar iant) functor. Then it formally follows that whenever f : X → Y is a homeomo rphism, the corres p o nding map F ( f ) : F ( X ) → F ( Y ) (or F ( Y ) → F ( X ) in the contrav ar iant case) is also a homeomorphism. In particular, this implies that when X is a G -spa ce, it automa tica lly follows that F ( X ) is a lso a G -space. The geometric Heck e op erato r T ( n ), n ≥ 1, acts on a functor F as follows. F or any connected space X ∈ C , (4.1)  T ( n ) F  ( X ) def = a [ X ′ → X ] conn. | X ′ /X | = n F ( X ′ ) / D ( X ′ /X ) , where the disjo int union runs ov er the isomor phis m cla s ses of connected n -sheeted cov ering spaces X ′ of X , and D ( X ′ /X ) is the gro up of a ll deck transforma tions of X ′ → X . When X is not connected, we apply the a b ov e construction to each of the connec ted comp onent. In general, we do not expect T ( n ) F : C → C to b e a functor. Ho wev er, see Prop os itio n 4.1 where such a situation do es o ccur. 14 HIR OT AKA T AMANOI F or the purp ose of this paper , the main example o f the functor F is of course the orbifold mapping space functor. Namely , for any G -spa ce M , and a ny co nnec ted space Σ, we let F ( M ; G ) (Σ) = Map orb (Σ , M /G ) . Prop os itio n 2- 1 shows that this is indeed a contrav ar iant functor in Σ. In terms of this notation, Theorem A can b e r estated as a for ma l power series of functors as (4.2) X n ≥ 0 p n F ( M n ; G n ) = Y n ≥ 1 S p n  T ( n ) F ( M ; G )  . How e ver, in some context, fo r example in the Grothendieck ring of v ar ieties, it can make sense and can b e justified to write S p ( X ) = (1 − p ) − X using powers whose exp onents a re spaces [5]. F or the purp ose of our present pap er , we can regard S p ( X ) as the definition of (1 − p ) − X . This is mor e a ppr opriate for o ur purp ose since, for example, for E uler characteristic, we have χ  S p ( X (  = (1 − p ) − χ ( X ) for any space X . In this p o int of view, Theorem A has the following form: (4.3) X n ≥ 0 p n F ( M n ; G n ) (Σ) = Y n ≥ 1 (1 − p n ) − ( T ( n ) F ( M ; G ) )(Σ) . By P rop ositio n 4.1 b elow, this formu la can be reg arded as a gener ating function of functors from the category C π 1 to C , where C π 1 is the ca tegory of top o logical spaces whose morphisms are restricted to those contin uous maps inducing isomorphis ms on fundamental gro ups. By considering constant orbifold maps, we have F ( M ; G ) (Σ) const. = C π 1 (Σ) ( M /G ). Then, by taking to p o logical Euler characteristic of (4.3) restricted to constant orb- ifold maps, we recover the for mula (3.9). Notice that factor s (1 − p n ) in (3.9) ar e already pres ent in (4.3) o n space level. T o define a co mp o sition of g e ometric Heck e op er ators, we need to hav e functo- riality of geometr ic Hec ke op erator s in a certain spe cial situation. Prop ositi on 4.1. L et F : C → C b e a c ovariant funct or. L et X and Y b e c on- ne ct e d sp ac es, and let f : X → Y b e a map such that f ∗ : π 1 ( X ) → π 1 ( Y ) is an isomorphi sm. Then for every p ositive inte ger n , f induc es a map (4.4) f ∗ : ( T ( n ) F )( X ) → ( T ( n ) F )( Y ) , such that for X f − → Y g − → Z , we have ( g ◦ f ) ∗ = g ∗ ◦ f ∗ . A similar statement holds for c ontr avariant functors. Pr o of. W e fix a base p o int x 0 of X . Let p : X ′ → X b e a c onnected n -sheeted cov ering space. F or eac h choice of a base p oint x ′ 0 of X ′ ov er x 0 , the subgroup H = p ∗  π 1 ( X ′ , x ′ 0 )  has index n in π 1 ( X, x 0 ). Since f ∗ : π 1 ( X, x 0 ) → π 1 ( Y , y 0 ), where y 0 = f ( x 0 ), is an isomorphism by hypothesis, the subgro up f ∗ ( H ) has index n in π 1 ( Y , y 0 ). Let ( Y ′ , y ′ 0 ) be a connected n -sheeted covering space with base p oint corres p o nding to f ∗ ( H ). The c hoice of y ′ 0 is unique up to the action o f the group D ( Y ′ / Y ) of deck tr a nsformations . Note that since f ∗ : π 1 ( X, x 0 ) → π 1 ( Y , y 0 ) is a n isomor phism, f ∗ induces a n isomorphism b etw een the co rresp onding deck transformatio ns D ( X ′ /X ) f ∗ − → ∼ = D ( Y ′ / Y ). By the Lifting Theorem in c overing space theory , there exists a unique D ( X ′ /X )-equiv ariant map ˜ f : X ′ → Y ′ such that INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP ACES 15 ˜ f ( x ′ 0 ) = y ′ 0 . By the functor ial proper ty , w e see that F ( ˜ f ) : F ( X ′ ) → F ( Y ′ ) is D ( X ′ /X ) ∼ = D ( Y ′ / Y )- e q uiv ar ia nt. Hence it induces a map on the quotient: F ( ˜ f ) : F ( X ′ ) / D ( X ′ /X ) → F ( Y ′ ) / D ( Y ′ / Y ) . Different choices of the lift ˜ f are related by the action of deck transfor mations. Hence the map F ( ˜ f ) on the orbit s pa ce dep ends only on f . Rep eating the above constructions for each isomorphism class of c o nnected n -sheeted cov ering space s of X , we obtain a ma p (4.5) f ∗ : a [ X ′ → X ] conn. | X ′ /X | = n F ( X ′ ) / D ( X ′ /X ) → a [ Y ′ → Y ] conn. | Y ′ / Y | = n F ( Y ′ ) / D ( Y ′ / Y ) . This is the map (4.4). The b ehavior under the c omp osition of tw o ma ps can b e easily verified. The ar gument for contra v ariant functors is similar.  As a specia l case, let f : X → X b e a homeomorphism. There is one po int which we have to be car eful ab out in the ab ov e construction o f f ∗ . F o r a c o nnected n - sheeted cov ering spa ce p : ( X ′ , x ′ 0 ) → ( X , x 0 ), the based cov ering s pa ce ( X ′′ , y 0 ) → ( X, f ( x 0 )) corr esp onding to the s ubgroup f ∗  p ∗  π 1 ( X ′ , x ′ 0 )  ⊂ π 1  X , f ( x 0 )  may not b e isomorphic to X ′ → X a s a cov ering spa ce ov er X , although X ′ and X ′′ are homeomorphic via a lift ˜ f : X ′ ∼ = − → X ′′ of f . Thus, in gener al, the induced map (4.6) f ∗ : a [ X ′ → X ] conn. | X ′ /X | = n F ( X ′ ) / D ( X ′ /X ) ∼ = − → a [ X ′ → X ] conn. | X ′ /X | = n F ( X ′ ) / D ( X ′ /X ) shuffles connected co mpo nents, and it is no t ea sy to control this shuffling. This is a n obstacle in studying comp ositio ns of Heck e o p erator s given in (4.7 ) b elow. How ever, when f : X → X is a deck tra nsformation of some cov ering X → X 0 , the situation can be completely clarified. In particular, when π 1 ( X 0 ) is ab elian, it turns out that the action of D ( X/ X 0 ) o n ( T ( n ) F )( X ) do es pre serve connected comp onents, a nd there is a simple relation among v arious g r oups of deck tr ansformatio ns inv olved. An yway , as a formal cons equence of Prop ositio n 4.1, we hav e Corollary 4. 2. L et F : C → C b e an arbitr ary c ovariant or c ont r avariant functor. If X is G -sp ac e, then for every p ositive int e ger n , the sp ac e ( T ( n ) F )( X ) is also a G -sp ac e. Next, we consider comp ositions o f Heck e op er a tors given as follows. (4.7)  ( T ( m ) ◦ T ( n )) F  ( X ) = T ( m )  T ( n ) F  ( X ) = a [ X ′ ] m  T ( n ) F  ( X ′ )  D ( X ′ /X ) = a [ X ′ ] m h a [ X ′′ ] n F ( X ′′ ) / D ( X ′′ /X ′ ) i. D ( X ′ /X ) , where [ X ′ ] m runs o ver the set of isomorphism class es of connected m -sheeted cov er- ing spa ces of X , and for a given X ′ , [ X ′′ ] n runs ov er the set of is o morphism cla sses of connected n -sheeted c ov ering spaces o f X ′ . As rema rked earlier conce r ning formula (4.6), the actio n of the g roup of deck transformatio ns D ( X ′ /X ) on  T ( n ) F  ( X ′ ) p ermutes its connected co mpo nents. W e now cla rify what happ ens. 16 HIR OT AKA T AMANOI Let e X → X be the universal cov er of X and let Γ = D ( e X /X ) ∼ = π 1 ( X ) be its group o f deck tra nsformations. W e rega rd e X → X as the rig ht Γ-pr incipal bundle ov er X . Let K ⊂ H ⊂ Γ b e subgroups s uch that | Γ /H | = m and | H/K | = n . W e put X K = e X / K and X H = e X / H . Then X H → X is a connected m -shee ted cov ering of X with D ( X H /X ) ∼ = N Γ ( H ) /H , and X K → X H is a connected n - sheeted cov ering o f X H with D ( X K /X H ) ∼ = N H ( K ) / K . Le t g ∈ N Γ ( H ) ⊂ Γ. T he n the right mult iplication by g induces the following diagr am o f homeomorphisms and cov ering spaces : (4.8) e X − − − − → X K − − − − → X H − − − − → X · g   y ∼ = · g   y ∼ = · g   y ∼ =    e X − − − − → X g − 1 K g − − − − → X H − − − − → X . Since g ∈ N Γ ( H ), the map · g : X H ∼ = − → X H is a deck transformation of X H ov er X . How e ver, since g may not b e in N Γ ( K ), · g : X K ∼ = − → X g − 1 K g is only an isomorphism of cov e ring spa ces ov er X . If g ∈ N Γ ( K ), then X g − 1 K g = X K and · g induces a deck transformatio n of X K ov er X . F or the middle s quare, when g ∈ H ⊂ N Γ ( H ), · g induces an isomo rphism of tw o co verings X K and X g − 1 K g ov er X H . If, furthermore, we hav e g ∈ N H ( K ) ⊂ H , then · g induces a deck transfor ma tion of X K ov er X H . This clar ifies the action of D ( X ′ /X ) o n ( T ( n ) F )( X ′ ) where X ′ = X H and X ′′ = X K . The ab ove s itua tion simplifies when the fundament al g roup of X is ab elian. In this case, every elemen t g ∈ Γ induces a dec k tra nsformation · g : X H ∼ = − → X H whose lift · g : X K ∼ = − → X K preserves X K . Also we hav e D ( X K /X H ) ∼ = H/ K for any tw o subgroups K ⊂ H ⊂ Γ. The formula (4.7) now simplifies a s follows. Prop ositi on 4.3. L et X b e a c onne cte d sp ac e whose fundamental gr oup is ab elian. Then, the c omp osition of t wo ge ometric He cke op er ators is given by (4.9)  T ( m )( T ( n ) F )  ( X ) = a H ⊂ Γ | Γ /H | = m h a K ⊂ H | H/K | = n  F ( X K ) / D ( X K /X )  i . Pr o of. By (4.7 ), we hav e  T ( m )( T ( n ) F )  ( X )  = a H ⊂ Γ | Γ /H | = m h a K ⊂ H | H/K | = n  F ( X K ) / D ( X K /X H )  i. D ( X H /X ) Since Γ is ab elian, D ( X H /X ) prese r ves F ( X K ) / D ( X K /X H ) for each K ⊂ H , = a H ⊂ Γ | Γ /H | = m a K ⊂ H | H/K | = n h  F ( X K ) / D ( X K /X H )  . D ( X H /X ) i since D ( X K /X H ) = H/K , D ( X H /X ) = Γ / H , and D ( X K /X ) = Γ / K , we have = a H ⊂ Γ | Γ /H | = m h a K ⊂ H | H/K | = n  F ( X K ) / D ( X K /X )  i . This completes the pro of.  INFINITE PRODUCT DE COMPOSITION OF ORBIFOLD M APPING SP ACES 17 W e contin ue to ass ume that the fundamental group of X is ab elian. F or an int eger d ≥ 1, let R ( d ) X be the covering space of X c o rresp o nding to d · π 1 ( X ) ⊂ π 1 ( X ). Let R ( d ) a ct o n a functor F b y (4.10) ( R ( d ) F )( X ) def = F  R ( d ) X  D  ( R ( d ) X ) /X  . As in P rop ositio n 4.1, we can show that a ny map f : X → X inducing an isomor - phism on fundament al gr o ups gives r is e to a map (4.11) f ∗ : ( R ( d ) F )( X ) → ( R ( d ) F )( X ) . In par ticular, if X is a G -space, then not only F ( X ) is a G -space, but also ( R ( d ) F )( X )  is a G -space for all d ≥ 1. The main result in this section is the following Hecke identit y for geometric Heck e op erator s for 2-dimensio nal tori T . Theorem 4 .4. L et F : C → C b e an arbitr ary c ont r avariant or c ovariant functor. L et T b e a 2 - dimensional torus. Then for every p air of p ositive inte gers m and n , the c omp osition of two ge ometric He cke op er ators satisfy (4.12)  T ( m )( T ( n ) F )  ( T ) = X d | ( m,n ) d ·  T  mn d 2  ( R ( d ) F )  ( T ) . In p articular, T ( m ) and T ( n ) c ommu te. In the right hand side of (4.9), the summation sym b ol means disjoin t top ologica l union, and the fac to r d mea ns a disjoint unio n o f d co pies. F or the pro of, we fir st r ecall the or dinary Heck e identit y for lattices. F or details, see ([9 ], p.16 ). Let A b e the free abe lian group generated b y rank 2 lattices L of C . The the Heck e o p erator T ( n ) for n ≥ 1 is a ma p T ( n ) : A → A defined by T ( n )( L ) = X [ L : L ′ ]= n L ′ ∈ A . Let R ( n ) : A → A be defined b y R ( n ) L = nL consisting of elements { n · ℓ } ℓ ∈ L ⊂ L . Then Heck e identit y s ays (4.13) T ( m ) ◦ T ( n )( L ) = X d | ( m,n ) d · R ( d ) ◦ T  mn d 2  ( L ) , for a ny lattice L . F rom this form ula, it is clear that T ( m ) and T ( n ) comm ute. Also, it is easy to chec k that R ( d ) and T ( n ) commute. Now we ar e ready to prov e Theore m 4.4. Pr o of of The or em 4.4. Since Γ ∼ = π 1 ( T ) ∼ = Z 2 is free abelia n o f r ank 2 , any subg r oup of Γ of finite index is also free a b elian of rank 2. Applying (4.9) in our cont ext, we obtain  T ( m )( T ( n ) F )  ( T ) = a H ⊂ Γ | Γ /H | = m h a K ⊂ H | H/K | = n  F ( T K ) / (Γ /K )  i , where T K is a covering to rus corres po nding to a n index mn s ublattice K ⊂ Γ. By the or dina ry Heck e identit y (4.1 3), any index mn s ublattice K of Γ arising in the ab ov e dis joint union is of the for m d · L for some integer d dividing ( m, n ), a nd for some lattice L of index ( mn ) /d 2 in Γ, and fur thermore, there ar e exactly d such 18 HIR OT AKA T AMANOI sublattices in the ab ove dis joint union. Hence the righ t hand side of the above expression ca n b e r ewritten as  T ( m )( T ( n ) F )  ( T ) = a d | ( m,n ) d a a L ⊂ Γ | Γ /L | =( mn ) /d 2 F ( T d · L )  Γ / ( d · L )  . On the other hand,  T  mn d 2  ( R ( d ) F )  ( T ) = h a L ⊂ Γ | Γ /L | =( mn ) /d 2 ( R ( d ) F )( T L ) / (Γ /L ) i = a L ⊂ Γ | Γ /L | =( mn ) /d 2  F ( T d · L )  L/ ( d · L )  (Γ /L ) = a L ⊂ Γ | Γ /L | =( mn ) /d 2 F ( T d · L )  Γ / ( d · L )  . Thu s combining the ab ove calculations , we ha ve o ur for m ula (4-12).  Theorem B is a sp ecial case of Theor e m 4.4 when F (Σ) = Map orb (Σ , M /G ). 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