Product formula for Atiyah-Patodi-Singer index classes and higher signatures

PR ODUCT F ORMULA F OR A TIY AH-P A TODI-SINGER INDEX CLASSES AND HIGHER SIGNA TURES CHARLOTTE W AHL Abstract. W e deﬁne generalized Atiy ah-Pa todi- Singer boundary conditions of product type for Di rac operators asso ciated to C ∗ -vec tor bundles on the product of a compact manif old with boundary and a closed manifold. W e pro ve a product f ormula for the K -theoretic i ndex classes, which we use to generalize the product formula for the t op ological signature to higher signatures. 1. Introduction It is an elementary fac t f rom algebra ic to po lo gy that the top ologic al signa ture fulﬁlls sign( M ) · sig n( N ) = sign( M × N ) , if M is an oriented compa c t manifold with bo undary and N is an oriented closed manifold. In this pap er we pro ve a s imilar pro duct formula for higher signature s – mor e g enerally: fo r the signature cla sses of the s ignature op erato r t wisted by a ﬂat C ∗ -vector bundle. (In the hig her ca se this bundle is the Mishenko-F o menko bundle.) In the closed case the sig nature cla ss equals the K - theoretic index of the signa ture op erator. There are several deﬁnitions of a higher signature class for a manifold with bo undary , which conjecturally give the same class (see [LP 04, § 13 I]): Two a na lytic ones (whose Chern characters agree), se e [LLP00], and a top ologica l deﬁnition ba sed on L -theory [LLK 0 2]. W e refer to the survey [LP 04] fo r a historica l acco un t. The basis for our c o nsiderations is the deﬁnition o f the signa ture class as the index of the signature op erato r with genera lized Atiy ah-Patodi- Sing er b oundary conditions given by a s ymmetric sp ectral sectio n [LP00][LP03]. The clas s is well-deﬁned only under certain homologica l conditions. W e prove the following generaliza tion of the ab ov e formu la: Let A , B b e unital C ∗ -algebra s. If F M resp. F N is a ﬂat unitary A - res p. B -vector bundle on and ev en-dimensional manifold M resp. N , then σ ( M , F M ) ⊗ σ ( N , F N ) = σ ( M × N , F M ⊠ F N ) ∈ K 0 ( A ⊗ B ) , if b oth sides are deﬁned. Here σ ( M , F M ) ∈ K 0 ( A ) resp. σ ( N , F N ) ∈ K 0 ( B ) ar e the signature classes. If M or N is o dd-dimensional, there is a similar formula, how ever the sig nature dep ends then on the a dditional choice of a La grangia n. The actual result we prov e is slightly more gener a l such that it applies to higher signa tures (see § 7). The pro of o f the s ig nature fo rmula builds on a pr o duct formula for A tiyah-P ato di- Singer index cla sses (Theorem 2.2), whic h is the main result of the ﬁrst part of this pa per ( § 2). W e use a class of b oundary conditions of At iyah-Patodi-Singer t yp e that generaliz es the b o undary conditions in tro duced in [MP97 a][MP97b] for families and adapted in [LP 98][LP03] to higher index theory . In this cla ss we can asso ciate to any bo undary condition for a Dirac op erator on M a ca nonical bo undary condition for a suita ble pro duct Dirac o per ator on the pro duct M × N . The pro o f of the pro duct for m ula is based on K K -theoretical metho ds, in par ticular 1 2 CHARLOTTE W AHL the relative index theorem [Bu95]. It ca rries ov er to family index theory , wher e a pro duct formula might also b e of interest. A sp ecial ca se is the equa lit y b etw een the Dira c o per ator and its Dirac susp ension, which w as deﬁned and established in [MP97 b, § 5] in the family case and adapted to the noncommutativ e con text in [LP03, § 3]. (Note the following subtlety: In [MP97b][LP03] o dd index classes were deﬁned in terms of a susp ensio n map origina lly due to A tiyah and Sing er. Here we use a K K - theoretic a ppr oach, which is makes ca lc ulations mor e s tr aightforw ard and allows to treat the even and o dd case on an equal fo oting. The index clas ses deﬁned by both approaches a gree, see [W07, § 9 ].) The product form ula for Atiy ah-Patodi-Sing er classes has applica tio ns to the study of conco rdance classes of metrics o f po sitive sca la r cur v ature: Stolz de- ﬁned b ordism g roups R n ( π ) for a ﬁnitely pre s en ted group π (in fact, more gen- erally for so-called sup ergr o ups) [St][RS01, § 5]. These groups consist of equiv- alence classes of n -dimensio nal s pin manifolds with b oundary that a re endow ed with a reference map to B π and with a metric of p ositive scala r cur v ature o n the b oundary . T ak ing the index of the Dirac oper ator twisted by the Mishenko- F omenko bundle asso ciated to the maximal gr oup C ∗ -algebra y ie lds a homomor- phism R n ( π ) → K n ( C ∗ max π ) (see [Bu95, § 1.4], with the real reduced C ∗ -algebra used ther e replaced b y C ∗ max π ). F or ﬁnitely pr e sent ed groups π 1 , π 2 the Carte- sian pro duct induces a pro duct R n ( π 1 ) × Ω spin m ( B π 2 ) → R n + m ( π 1 × π 2 ). There is also an index map Ω spin m ( B π 2 ) → K m ( C ∗ max π 2 ). By the pro duct for m ula for A tiyah-P ato di-Singer classes these maps ﬁt into a commut ing diagram R n ( π 1 ) × Ω spin m ( B π 2 ) / /   R n + m ( π 1 × π 2 )   K n ( C ∗ max π 1 ) ⊗ K m ( C ∗ max π 2 ) ⊗ / / K n + m ( C ∗ max ( π 1 × π 2 )) . This can be applied to study the b ehavior of the concorda nce class es under Carte- sian pro duct, see [W e99, Remark 0 .7] for rela ted questions . W e exp ect that our metho ds also work in K O -theory , which should b e used here : The index ma ps in the diagram facto r through K O -theor y of the corres po nding real maximal C ∗ - algebras . A spec ial case o f the analog ue of the above diag ram in K O -theor y is the fact that the homomo rphism lim − → R n +8 j ( π ) → K O n ( C ∗ I R ,max π ) is well-deﬁned: The limit is induced b y taking the pro duct with a particular closed 8 -dimensional manifold (the Bott manifo ld) [St][RS01, § 5 ]. A more g e neral diag ram is given in the pr eprint [St], whic h w as nev er publis he d. Also for the a b ove diag r am (res p. its analogue in K O -theory ) ther e seems to be no published pro of. A nov elty used in the pro of of the pr o duct formula for signature cla sses is a gen- eralization of the deﬁnition o f symmetric b oundary conditions for the sig nature op erator. Symmetric sp ectral sectio ns , as intro duced in [LP00][LP03], are symmet- ric with res p ect to a pa rticular in volution. The c la ss o f b oundary conditions deﬁned by symmetric sp ectral sections is no t clos ed under taking pro ducts. W e consider more general in volutions and study the depe ndence of the in volution. The results allow us to derive the pro duct formula for the signature clas s es from the pr o duct formula for A tiyah-P ato di-Singer classes. It would b e interesting to hav e a simila r pro duct formula esta blished for the top olog- ically deﬁned higher s ignatures. In ge ne r al, the main a dv antage o f the K -theoretica l approach is that it als o works for foliations, as noted in Remark 2 at the end of [LP03]. The metho ds of the pre s en t pap er together with the pro duct for mu la for η -for ms prov en in [W09] also lead to a pro duct formula for the a nalytic higher ρ -inv ariants PRODUCT FORMULA FOR APS-CLASSES 3 for the s ig nature op er ator. (Details will b e given elsewhere.) These were deﬁned in [W09] motiv ated by a sugg estion in [Lo92]. A n alternative deﬁnition based on a diﬀerent r egularizatio n can b e given using the higher η -forms for the signature op erator introduced in [LLP00]. T op olog ical hig her ρ -inv ar ia n ts w ere previo usly int ro duced in [W e99]. Ther e Ca rtesian pr o ducts were the motiv ating exa mples, and a pro duct formula was men tioned. A connectio n to the analytic deﬁnition has not yet been established. Con v e ntions. If not s p eciﬁed, a tensor pro duct b etw e e n C ∗ -algebra s is under- sto o d as the s patial (=minimal) C ∗ -algebra ic tenso r pro duct, a nd a tensor pr o duct betw een Hilbert C ∗ -mo dules is the exterior Hilb ert C ∗ -mo dule tensor pro duct. In the few r emaining ca ses the tenso r pro duct is as sumed to b e algebra ic. A tensor pro duct of graded spaces is gr aded. How ever, for op era to rs we ﬁx the following conv ention: If A resp. B are op era tors on gr aded vector spaces H 1 resp. H 2 , then A ⊗ B is the o per ator on H 1 ⊗ H 2 deﬁned by using the ungr ade d tensor pro duct, hence neglecting the grading. In contrast the op erato r AB on H 1 ⊗ H 2 is deﬁned via the graded tensor pro duct as usual. Thus AB = A ⊗ B + + A z ⊗ B − , where z is the grading op era to r on H 1 and B = B + + B − with B ± even resp. o dd. In this spirit we usually o mit tensor pro ducts when dealing with op erators and write A for A ⊗ 1 resp. B for 1 ⊗ B + + z ⊗ B − . W e also usua lly omit the tensor pro duct when dealing with morphisms b et ween diﬀerent space s. In a g raded context we tacitly endow ungraded spa ces with the trivial Z Z / 2-gra ding (for which all elemen ts are p ositive). In or de r to av oid confusion w e add indices to geo metr ic op erators as the de Rham op erator. W e will omit them sometimes when co nfusion seems unlikely . 2. Product formula for Dirac classes W e ass ume throughout the paper that A , B are unital C ∗ -algebra s. Let M b e an or ient ed Riemannian manifold with b ounda r y ∂ M and pro duct struc- ture near the b oundary . Denote by M cy l the cor resp onding ma nifold with cylin- dric end Z r ⊂ M cy l . That is , we assume that there is ε > 0 and an isometry e : Z r ∼ = ( − ε, ∞ ) × ∂ M such that M cy l \ e − 1 ((0 , ∞ ) × ∂ M ) = M . The co ordina te deﬁned b y the compo sition of e with the pro jection onto ( − ε , ∞ ) is denoted b y x 1 . W e deﬁne Z = I R × ∂ M . W e set U ε = e − 1 (( − ε, 0] × ∂ M ) ⊂ M and deno te by p : U ε → ∂ M the comp osition o f e with the pro jection on to ∂ M . The pro jection Z → ∂ M will b e deno ted by p as well. Dirac o per ators over C ∗ -algebra s are b y now well-studied. It turns out tha t muc h of the class ical theor y car ries over, see for ex ample [ST01] [S05] for relev ant back- ground mater ial. Let E b e a her mitian A -vector bundle on M (the sc a lar pro duct on the ﬁb e r s is assumed to b e A -v alued). Then E is called a Dirac A -bundle if the following conditions ar e fulﬁlled: (1) The bundle E is a Cliﬀord mo dule. This means that there is a left actio n of the Cliﬀord bundle C( T ∗ M ) on E commuting with the right action of A such that the c ( v ) is a sk ewadjoin t endomor phism on E for any v ∈ T ∗ M . If M is e ven-dimensional, then E is assumed to b e Z Z / 2-gra ded a nd c ( v ) is assumed to b e odd for any v ∈ T ∗ M . (2) F urthermore E is endow ed with a connection ∇ E compatible with the her- mitian pr o duct and fulﬁlling c ( ∇ M v ) = [ ∇ E , c ( v )]. Here ∇ M is the Levi- Civit` a connection. 4 CHARLOTTE W AHL Let E M be a Dira c A -bundle o n M and assume that E M | U ε = p ∗ ( E M | ∂ M ) as (graded, if M is even-dimensional) hermitian A -vector bundle. F urther more the connection on E M | U ε is as sumed to b e of pr o duct type. Let ∂ / M := c ◦ ∇ E M be the asso cia ted Dirac op erato r. The bundle E M is Z Z / 2 - graded if M is even-dimensional. The grading o per ator is denoted by z M . W e write E ∂ M := E + M | ∂ M if M is even-dimensional and E ∂ M = E | ∂ M if M is o dd-dimensiona l. The induced C liﬀo r d mo dule str ucture on E ∂ M is given by c ∂ M ( v ) := c M ( dx 1 ) c M ( v ) for v ∈ T ∗ ∂ M ⊂ T ∗ M (the inclusion b eing de ﬁned via the metric). W e denote the Dirac op erator asso ciated to E ∂ M by ∂ / ∂ M . If M is o dd-dimens io nal, the Dira c bundle E ∂ M is Z Z / 2-g raded with grading op erator z ∂ M := ic M ( dx 1 ) and on U ε ∂ / M = c M ( dx 1 )( ∂ 1 − ∂ / ∂ M ) . (2.1) If M is even-dimensional, we identify E + | U ε with E − | U ε via ic ( dx 1 ) and th us obtain an isomor phism E | U ε ∼ = ( C + ⊕ C − ) ⊗ ( p ∗ E ∂ M ) . Here C ± denotes C w ith grading induced by the gra ding oper ator ± 1. On U ε ∂ / M = c M ( dx 1 )( ∂ 1 − z M ∂ / ∂ M ) . (2.2) Given ∂ / M , the o per ator ∂ / ∂ M is uniquely determined by these formulas and is called the b ounda ry o p er a tor induced b y ∂ / M . In the following the b ounda r y op era tor of a Dirac o p er a tor ∂ / will sometimes b e denoted b y B ( ∂ / ). W rite D ∂ M for the clo sure of ∂ / ∂ M : C ∞ ( ∂ M , E ∂ M ) → L 2 ( ∂ M , E ∂ M ). Now we in tro duce the boundary conditions: Assume ﬁrst that M is even-dimensional. Then a selfadjoint operator A ∈ B ( L 2 ( ∂ M , E ∂ M )) such that D ∂ M + A has a b ounded in verse is called a trivia lizing op erator for D ∂ M on L 2 ( ∂ M , E ∂ M ). Deﬁne D M ( A ) + as the c lo sure of ∂ / + M : { f ∈ C ∞ ( M , E + ) | 1 ≥ 0 ( D ∂ M + A )( f | ∂ M ) = 0 } → L 2 ( M , E − ) . Let D M ( A ) − be the a djoin t of D M ( A ) + . Then D M ( A ) =  0 D M ( A ) − D M ( A ) + 0  is a selfadjoin t op erator on L 2 ( M , E ) = L 2 ( M , E + ) ⊕ L 2 ( M , E − ). If M is odd-dimensio nal, an op erator A as a bove is calle d a trivializing op era tor if in addition it is o dd with resp ect to z ∂ M . Then the o per ator D M ( A ) is deﬁned a s the closur e of ∂ / M : { f ∈ C ∞ ( M , E ) | 1 ≥ 0 ( D ∂ M + A )( f | ∂ M ) = 0 } → L 2 ( M , E ) . The op erato r D M ( A ) is a r egular selfadjoint F r edholm op erato r with c o mpact resol- ven ts. (This c an b e shown as in [W u9 7]). Let i b e the parity of the dimension of M . F rom the Baa j-Julg picture of K K -theory v ia unbo unded Kasparov mo dules [Bl98, § 17.11 ] it follows that there is an induced class [ D M ( A )] ∈ K K i ( C , A ) ∼ = K i ( A ), called the index (class) of D M ( A ). W e also need cylindric index c lasses: Let χ : M cy l → [0 , 1] be a smo oth function with supp o rt in Z r such that χ | { x 1 ≥ − 3 ε/ 4 } = 1. W e deﬁne D cy l M ( A ) as the closure of ∂ / E − c ( dx 1 ) χA : C ∞ c ( M , E ) → L 2 ( M , E ) PRODUCT FORMULA FOR APS-CLASSES 5 if M is o dd-dimensiona l and as the clos ure of ∂ / E − c ( dx 1 ) χ z A : C ∞ c ( M , E ) → L 2 ( M , E ) if M is even-dimensional. Aga in, D cy l M ( A ) is a reg ula r selfadjoint F r edholm op era tor (see for exa mple [W09] for a de ta iled discussion) and thus deﬁnes an element in K K i ( C , A ). Here the res olven ts ar e non-compact, hence the Ba a j-Julg picture do es not a pply . See [W07, Def. 2 .4 ] for the r elev ant deﬁnition o f the Ka sparov cla ss that will b e us e d in the following. The following equa lit y has b een ess en tially established in the even case in [LLP00, § 10] a nd follows in the o dd ca se from [LP03, § 3.3 ] together with [W07, Lemma 9 .2]. W e giv e a diﬀerent pro of her e, whose metho d will a lso b e used in the pro of of the pro duct formula for index cla s ses, Theor em 2.2. It is similar to the pro of of [LLP 00, Theorem 7.2 ]. Prop ositio n 2 . 1. In K K i ( C , A ) [ D M ( A )] = [ D cy l M ( A )] . Pr o of. W e consider the case i = 1. The even case is ana logous with the obvious changes. Recall that p : Z → ∂ M is the pro jection. Endow E Z = p ∗ E ∂ M with the pro duct Dirac bundle s tructure. Let ∂ / Z be the as so ciated Dir ac o per ator and denote b y D Z ( A ) the closure of ∂ / Z − c ( dx 1 ) A : C ∞ c ( Z, E Z ) → L 2 ( Z, E Z ) . F urthermore let Z l = ( −∞ , 0 ] × ∂ M ⊂ Z a nd denote by D Z l ( A ) the closure of ∂ / Z − c ( dx 1 ) A : { f ∈ C ∞ c ( Z l , E Z ) | 1 ≥ 0 ( D ∂ M + A )( f | x 1 =0 ) = 0 } → L 2 ( Z l , E Z ) . The manifolds Z l and M cy l are obtained fr om Z a nd M b y cutting and pas ting along the hypersurfac e s x 1 = − ε/ 2. By the r elative index theorem (which is proven in [Bu95] for manifolds without b oundary and unp erturb ed Dira c o per ators, how ever the pro of works here as w e ll), [ D M ( A ) − χc ( dx 1 ) A ] + [ D Z ( A )] = [ D cy l M ( A )] + [ D Z l ( A )] . The op erato r D Z l ( A ) is in vertible: Set P = 1 ≥ 0 ( D ∂ M + A ) and σ := c ( dx 1 ). Let f ∈ C ∞ c ( Z l , E Z ). W e consider f ( x 1 ) := f | { x 1 }× ∂ M as a n e le men t in C ∞ ( ∂ M , E ∂ M ). Then ( D Z l ( A ) − 1 f )( x 1 ) = − Z x 1 0 e − ( x 1 − y 1 ) D ∂ M ( A ) (1 − P ) σ f ( y 1 ) dy 1 + Z −∞ x 1 e − ( x 1 − y 1 ) D ∂ M ( A ) P σ f ( y 1 ) dy 1 . The op erator D Z ( A ) is invertible as well. Hence [ D Z ( A )] = [ D Z l ( A )] = 0. The assertion follows since [ D M ( A ) − χc ( dx 1 ) A ] = [ D M ( A )].  Next we discuss Cartesia n pro ducts: Let N b e an oriented closed Riema nnia n manifold. Let E N be a Dira c B -bundle on N and let ∂ / N : C ∞ ( N , E N ) → L 2 ( N , E N ) b e the asso ciated Dira c o pe r ator. Its closure D N induces an index c lass [ D N ] ∈ K K j ( C , B ), where j is the par it y o f the dimension of N . In the following we a ssume that M and N ar e ev en-dimensional. The o ther cases will b e dis c ussed b elow. 6 CHARLOTTE W AHL Let z N be the gra ding oper ator on E N . The bundle E M ⊠ E N is an Z Z / 2-gra ded hermitian A ⊗ B -bundle on M × N with grading op erato r z M × N = z M z N = z M ⊗ z N and with connec tion. The pr o duct Dirac op erator ac ting on C ∞ ( M × N , E M ⊠ E N ) is de ﬁned b y ∂ / M × N = ∂ / M + ∂ / N . In order to illustrate our conv ent ion on the notation for tenso r pro ducts we note that this equals ∂ / M ⊗ 1 + z M ⊗ ∂ / N . W e sketc h how one see s that ∂ / M × N is indeed a Dira c op era to r: F or f ∈ C ∞ ( M × N ) set c M × N ( d f ) := [ ∂ / M × N , f ]. Then for v ∈ T M ⊂ T ( M × N ) one has c M × N ( v ) = c M ( v ), and similarly for v ∈ T N . Using this one chec ks ea sily that c M × N is a Cliﬀord multiplication, e ndowed with which E M ⊠ E N bec omes a Dirac A ⊗ B - bundle, and that ∂ / M × N is the asso ciated Dirac op erato r . In particula r c M × N ( dx 1 ) = c M ( dx 1 ). Using the iso morphism ic ( dx 1 ) : E + M | ∂ M ∼ = E − M | ∂ M we get a n isomo r phism Ψ : E ∂ M ⊠ E N ∼ = − → (( E + M ⊠ E + N ) ⊕ ( E − M ⊠ E − N )) | ∂ M = E ∂ ( M × N ) . It holds that ∂ / ∂ ( M × N ) = Ψ( z N ∂ / ∂ M + ∂ / N )Ψ − 1 . (2.3) The op erato r ˆ A := Ψ( z N A )Ψ − 1 = Ψ( A ⊗ z N )Ψ − 1 is a trivializing op era tor fo r ∂ / ∂ ( M × N ) . Hence we get as ab ov e a F redholm op erator D M × N ( ˆ A ), whose index is an element of K K 0 ( C , A ⊗ B ). Our ma in r esult in this s e c tion e x presses this index in ter ms o f the indices o f D M ( A ) and D N via the K asparov pro duct K K ∗ ( C , A ) × K K ∗ (C , B ) → K K ∗ (C , A ⊗ B ) , ( a, b ) 7→ a ⊗ b . W e br ieﬂy r ecall its deﬁnition: Let D 1 resp. D 2 be an o dd selfadjoint op era tor with compact reso lvents on a countably g enerated Z Z / 2- graded Hilb e rt A re s p. B -module H 1 resp. H 2 . Recall [Bl98, § 18 .9 ] that in the B aa j-Julg picture of K K -theor y the Kaspar ov pro duct [ D 1 ] ⊗ [ D 2 ] is repres en ted by the closure of the op erator D 1 + D 2 whose domain (b efore taking closur e) is the alg ebraic tensor pro duct dom D 1 ⊗ dom D 2 . Actually , this formula was the motiv ation for our deﬁnitio n of the pro duct Dirac op erator. Theorem 2. 2 . It holds that [ D M ( A )] ⊗ [ D N ] = [ D M × N ( ˆ A )] . Pr o of. By the comparing the ab ove description of the Kasparov pro duct with the deﬁnition o f the pro duct Dira c op erator one sees that the class o n the left hand side is repres en ted by the closure D pro d M × N ( ˆ A ) of the o dd op erator ∂ / M × N with domain dom D M ( A ) ⊗ dom D N (understo o d as an a lgebraic tensor pro duct). W e use the metho d of the pro of of Pro p. 2.1 in o rder show tha t [ D pro d M × N ( ˆ A )] = [ D cy l M × N ( ˆ A )] . Then the assertion follows from Prop. 2.1. Let D pro d Z l × N ( ˆ A ) b e the closur e of the op erator ∂ / Z × N − c ( dx 1 ) z Z × N ˆ A with domain dom D Z l ( A ) ⊗ do m D N , where dom D Z l ( A ) is deﬁned as in the pro of of Prop. 2.1. The op era tor D pro d Z l × N ( ˆ A ) is in vertible with inv erse D pro d Z l × N ( ˆ A ) − 1 = Z ∞ 0 D pro d Z l × N ( ˆ A ) e − t D Z l ( A ) 2 e −D 2 N dt . PRODUCT FORMULA FOR APS-CLASSES 7 The integral conv erges for t → ∞ since D Z l ( A ) is in vertible, see the pr o of of Prop. 2.1. Deﬁne D Z × N ( ˆ A ) a s the closur e of ∂ / Z × N − c ( dx 1 ) z Z × N ˆ A : C ∞ c ( Z × N , E Z ⊠ E N ) → L 2 ( Z × N , E Z ⊠ E N ) . By the r elative index theo rem [ D pro d M × N ( ˆ A ) − χc ( dx 1 ) z M × N ˆ A ] + [ D Z × N ( ˆ A )] = [ D cy l M × N ( ˆ A )] + [ D pro d Z l × N ( ˆ A )] . Since D Z × N ( ˆ A ) is also in vertible, the assertion follows.  3. Products of unbounded Ka sp aro v mo dules – the remaining cases Before discussing the cases in which M a nd N a re not b oth even-dimensional we derive the gene r al form o f the Kasparov pro duct fo r the remaining pa rities from its description in the even case given above. (It is needed her e that the description remains v alid if w e deal with graded C ∗ -algebra s.) The expressions we g et for the pro duct are the motiv ation for the deﬁnitions of the pro duct Dirac op erato rs in the following sectio n. Let C 1 be the Cliﬀord a lg ebra with one o dd generator σ fulﬁlling σ 2 = 1. The pro duct inv o lving o dd K K -theor y is deﬁned via the isomo rphism K K 1 ( C , A ) ∼ = K K 0 ( C , A ⊗ C 1 ). It maps a class [ D ] represented b y selfa djo int F redholm op era to r D on an ungra ded co untably generated Hilb ert A -mo dule H to the class [ σ D ] ∈ K K 0 ( C , A ⊗ C 1 ), where σD is deﬁned o n the Z Z / 2-g raded Hilber t A ⊗ C 1 -mo dule H ⊗ C 1 . On the other hand g iven an o dd selfadjoint F r edholm op erato r D ′ on a Z Z / 2-g raded Hilb ert A ⊗ C 1 -mo dule H ′ and an o dd inv o lution T on H ′ with T D ′ = D ′ T , then the r estriction of T D ′ to the positive eigens pa ce of T repr esents the pr eimage of [ D ′ ] under the ab ove iso mo rphism. Note that right multiplication by the pro jection 1 2 (1 − σ ) is trivia l o n the p os itiv e eigenspace of T , thus it is endow ed with a canonical Hilb ert A -mo dule structur e. If D ′ = σ D and H ′ = H ⊗ C 1 as befo re, we may choose T = σ to get exactly the K asparov mo dule back we sta rted with. Let D 1 resp. D 2 be a selfa djo int op erato r with compa ct resolven ts on a count ably generated Hilb ert A - resp. B -module H 1 resp. H 2 . 3.1. E v en times o dd. Fir st assume that H 1 is Z Z / 2-gr aded, H 2 is trivia lly g raded, and D 1 is odd. W e write z 1 for the gr ading o per ator on H 1 . The K asparov pro duct of [ D 1 ] ∈ K K 0 ( C , A ) with [ σ D 2 ] ∈ K K 0 (C , B ⊗ C 1 ) is [ D 1 + σ D 2 ] ∈ K K 0 (C , A ⊗ B ⊗ C 1 ). W e set T = σ z 1 . W e hav e that D 1 + σ D 2 = σ z 1 ( σ z 1 D 1 + z 1 D 2 ) and that the p ositive eigenspac e o f T equals H 1 ⊗ H 2 ⊗ C(1 + σ ). The c hoice of the base vector 1 2 (1 + σ ) of C(1 + σ ) deﬁnes an obvious isomor phism to H 1 ⊗ H 2 . Here we c onsider H 1 ⊗ H 2 ungraded. The isomor phism intert wines σ z 1 D 1 + z 1 D 2 with D 1 + z 1 D 2 . Thus [ D 1 ] ⊗ [ D 2 ] = [ D 1 + z 1 D 2 ] ∈ K K 1 ( C , A ⊗ B ) . 3.2. O dd times ev en. Now we a ssume that H 2 is Z Z / 2-g raded, H 1 is trivia lly graded, a nd D 2 is o dd. W e write z 2 for the gra ding o pe r ator on H 2 . The K asparov pro duct of [ σ D 1 ] ∈ K K 0 ( C , A ⊗ C 1 ) with [ D 2 ] ∈ K K 0 (C , B ) is [ σ D 1 + D 2 ] ∈ K K 0 ( C , A ⊗ B ⊗ C 1 ). Then σ D 1 + D 2 = σ z 2 ( z 2 D 1 + σ z 2 D 2 ), and the p ositive eigenspace o f σ z 2 is  H 1 ⊗ C(1 + σ ) ⊗ H + 2  ⊕  H 1 ⊗ C(1 − σ ) ⊗ H − 2  ∼ = H 1 ⊗ H 2 . The last is omorphism intert wines z 2 D 1 + σ z 2 D 2 with z 2 D 1 + D 2 . Thus [ D 1 ] ⊗ [ D 2 ] = [ z 2 D 1 + D 2 ] ∈ K K 1 ( C , A ⊗ B ) . 8 CHARLOTTE W AHL 3.3. O dd time s o dd. Now let H 1 , H 2 be trivially gra ded. W e write C ′ 1 , C ′′ 1 for t wo copies of C 1 with g enerators σ ′ , σ ′′ resp ectively . The clas s [ σ ′ D 1 ] ⊗ [ σ ′′ D 2 ] ∈ K K 0 ( C , A ⊗ B ⊗ C ′ 1 ⊗ C ′′ 1 ) is represe nted by the odd oper ator σ ′ D 1 + σ ′′ D 2 on H 1 ⊗ H 2 ⊗ C ′ 1 ⊗ C ′′ 1 . Note that 1 2 (1 + iσ ′ σ ′′ ) is a rank one pro jection. By Mor ita equiv a lence the homo- morphism p : C → C ′ 1 ⊗ C ′′ 1 , x 7→ 1 2 x (1 + iσ ′ σ ′′ ) induces an is omorphism p ∗ : K K 0 ( C , A ⊗ B ) → K K 0 (C , A ⊗ B ⊗ C ′ 1 ⊗ C ′′ 1 ). W e deﬁne a representative of the preimage of [ σ ′ D 1 + σ ′′ D 2 ] under p ∗ . The a lgebra C ′ 1 ⊗ C ′′ 1 acts on C 2 via the is omorphism C ′ 1 ⊗ C ′′ 1 → M 2 ( C) , σ ′ 7→ Γ 1 :=  1 0 0 − 1  , σ ′′ 7→ Γ 2 :=  0 i − i 0  . The actio n is compatible with the gra ding if on C 2 the gra ding deﬁned by the op erator − i Γ 1 Γ 2 =  0 1 1 0  . In the following we show that the o dd ope rator Γ 1 D 1 + Γ 2 D 2 on H 1 ⊗ H 2 ⊗ C 2 represents the preimage. Deﬁne the Hilb ert C ′ 1 ⊗ C ′′ 1 -mo dule V := 1 2 (1 + i σ ′ σ ′′ )( C ′ 1 ⊗ C ′′ 1 ). The unit vector e 1 := 1 2 (1 + i σ ′ σ ′′ ) spans V + , and the unit vector e 2 := 1 2 ( σ ′ − iσ ′′ ) spans V − . Note that cano nically C 2 ⊗ p ( C ′ 1 ⊗ C ′′ 2 ) ∼ = C 2 ⊗ V . Cho ose a unit vector v 1 ∈ ( C 2 ) + and let v 2 := Γ 1 v 1 ∈ (C 2 ) − . The even iso morphism of Hilbe r t C ′ 1 ⊗ C ′′ 2 -mo dules C 2 ⊗ V → C ′ 1 ⊗ C ′′ 2 , v 1 ⊗ e 1 7→ e 1 , v 1 ⊗ e 2 7→ e 2 , v 2 ⊗ e 1 7→ σ ′ e 1 , v 2 ⊗ e 2 7→ σ ′ e 2 , is c ompatible with the left C ′ 1 ⊗ C ′′ 2 -action o n b oth spaces. Summarizing, we get an isomorphism H 1 ⊗ H 2 ⊗ C 2 ⊗ p ( C ′ 1 ⊗ C ′′ 2 ) ∼ = H 1 ⊗ H 2 ⊗ C ′ 1 ⊗ C ′′ 1 int ertwining Γ 1 D 1 + Γ 2 D 2 and σ ′ D 1 + σ ′′ D 2 . Thu s [ D 1 ] ⊗ [ D 2 ] = [Γ 1 D 1 + Γ 2 D 2 ] ∈ K K 0 ( C , A ⊗ B ) . (This calculation cor rects a similar but ﬂaw ed ar gument in the pr o of of [W07, Lemma 9.2]) PRODUCT FORMULA FOR APS-CLASSES 9 4. Product structures for Dira c opera tors – the remaining cases 4.1. M is e v en-di mensional and N o dd-dimensional. Le t z M be the g rading op erator on E M . The bundle E M ⊠ E N is now considered a n ungraded A ⊗ B -vector bundle. The pro duct Dirac o per ator is deﬁned a s ∂ / M × N = ∂ / M + z M ∂ / N . Hence here also c M × N ( dx 1 ) = c M ( dx 1 ). The isomo rphism ic ( dx 1 ) : E + M | ∂ M ∼ = E − M | ∂ M induces an is omorphism Ψ : ( E ∂ M ⊕ E ∂ M ) ⊠ E N ∼ = − → ( E M ⊠ E N ) | ∂ M = E ∂ ( M × N ) . W e let the matrices Γ 1 , Γ 2 , which w ere deﬁned in § 3, act on ( E ∂ M ⊕ E ∂ M ) ⊠ E N . Then ∂ / ∂ ( M × N ) = Ψ(Γ 1 ∂ / ∂ M + Γ 2 ∂ / N )Ψ − 1 . (4.1) Theorem 2.2 holds in this situation for ˆ A := ΨΓ 1 ( A ⊗ 1 )Ψ − 1 . 4.2. M is o dd-dim ensional and N even-dimensional. In ana logy to the previ- ous ca se the bundle E M ⊠ E N is considered ungraded and the pro duct Dirac op era tor is deﬁned as ∂ / M × N := z N ∂ / M + ∂ / N . It follows that c M × N ( dx 1 ) = z N c M ( dx 1 ). W e have that E ∂ ( M × N ) = ( E M ⊠ E N ) | ∂ M = E ∂ M ⊠ E N , which is a g raded vector bundle w ith grading op erator z ∂ M × N = ic M ( dx 1 ) z N = z ∂ M z N . Then ∂ / ∂ ( M × N ) = ∂ / ∂ M − i z N ∂ / N . (4.2) Theorem 2.2 holds with ˆ A := A ⊗ 1. 4.3. M , N are o dd-dim ensional. C o nsider the bundle E M × N := ( E M ⊕ E M ) ⊠ E N , on which Γ 1 , Γ 2 from § 3 ac t. The ass o ciated pro duct Dirac o per ator is deﬁned by ∂ / M × N = Γ 1 ∂ / M + Γ 2 ∂ / N and the grading is giv e n b y z M × N = − i Γ 1 Γ 2 . W e s ee that c M × N ( dx 1 ) = Γ 1 c M ( dx 1 ). W e ha ve an isomorphism Ψ : E ∂ M ⊠ E N → E ∂ ( M × N ) = E + M × N | ∂ M x ⊗ y 7→ 1 √ 2 ( x, x ) ⊗ y . Then ∂ / ∂ ( M × N ) = Ψ( ∂ / ∂ M + z ∂ M ∂ / N )Ψ − 1 . (4.3) Theorem 2.2 holds with ˆ A := Ψ( A ⊗ 1)Ψ − 1 . 10 CHARLOTTE W AHL 5. Product formula for twisted signa ture cl asses Let F M be a ﬂat her mitian A -v ector bundle on M e ndow ed with a compatible ﬂat connection and let F ∂ M = F M | ∂ M . W e a ssume that F M | U ε = p ∗ F ∂ M as a hermitian vector bundle and that the co nnection is of pro duct type on U ε . Analogously let F N be a ﬂat B -vector bundle on N , also endow ed with a hermitian str ucture and a compatible ﬂat connection. W e deno te by Ω ∗ ( M , F M ) the space of smo o th twisted de Rha m forms w ith de Rham diﬀerential d M . Let Ω ∗ (2) ( M , F M ) the Hilb ert A -module of L 2 -forms. W e endow Λ ∗ T ∗ M with the L e v i-Civit` a connection. Thu s we hav e a n induced connection on Λ ∗ T ∗ M ⊗ F . The bundle Λ ∗ T ∗ M ⊗ F M is a Cliﬀord mo dule with Cliﬀord multiplication c M ( α ) ω = α ∧ ω − ι ( α ) ω . Recall that the induced chiralit y op erator τ M is a selfadjoint inv o lution o n Λ ∗ T ∗ M ⊗ F , see [BGV96, Lemma 3.17]. W e denote by Λ ± T ∗ M ⊗ F M resp. Ω ± ( M , F M ) the eig enspace asso cia ted to the eigenv alue ± 1 of τ M . If M is e ven-dimensional we endow Λ ∗ T ∗ M ⊗ F with the Z Z / 2- grading induced by τ M . With these structures Λ ∗ T ∗ M ⊗ F M is a Dirac bundle. The signature o per ator is deﬁned a s the asso cia ted Dirac op erato r, see [BGV96, § 3.6]. W e ﬁx the isometry Φ M : Λ ∗ T ∗ ∂ M ⊗ F ∂ M → (Λ + T ∗ M | ∂ M ) ⊗ F ∂ M , α 7→ 1 √ 2  dx 1 ∧ α + τ M ( dx 1 ∧ α )  . 5.1. The even case. In the following we assume that M is even-dimensional. F or α ∈ Λ ∗ T ∗ ∂ M τ M ( dx 1 ∧ α ) = τ ∂ M α and τ M ( α ) = dx 1 ∧ τ ∂ M ( α ) . The signa ture oper ator on Ω ∗ ( M , F M ) equals d sign M := d M + d ∗ M = d M − τ M d M τ M . Note that the normalizatio n her e is as in [BGV96, § 3 .6] and diﬀers from [HS92][LLP00]. The corre s po nding index classes agr ee up to sig n, see § 8.2. Ac- cordingly , also our conv e n tion in the o dd ca se is diﬀer en t. It holds that B ( d sign M ) = Φ M ( d ∂ M τ ∂ M + τ ∂ M d ∂ M )Φ − 1 M . (5.1) W e denote the clos ure o f d ∂ M τ ∂ M + τ ∂ M d ∂ M : Ω ∗ ( ∂ M , F ∂ M ) → Ω ∗ (2) ( ∂ M , F ∂ M ) by D bd ∂ M . In orde r to avoid confusion, we p oint out that D bd ∂ M agrees with the odd signature op era tor in the conv ention of s o me authors, but not in co n ven tio n use d here. F o r the precise r elation see § 5 .2 . The following deﬁnition generalizes the b oundar y c onditions cons ide r ed in [LP03, § 6.3]. Deﬁnition 5.1. Assume t hat ther e is an ortho gonal de c omp osition Ω ∗ (2) ( ∂ M , F ∂ M ) = V ⊕ W with r esp e ct to which τ ∂ M and D bd ∂ M ar e diago- nal. F urthermor e assume that D bd ∂ M | V is invertibl e. L et I b e an op er ator on Ω ∗ (2) ( ∂ M , F ∂ M ) t hat vanishes on V , is an involution on W and antic ommutes with τ ∂ M and D bd ∂ M . PRODUCT FORMULA FOR APS-CLASSES 11 We c al l a trivializing op er ator A of B ( d sign M ) symmetric with r esp e ct to I if it is diagonal with r esp e ct to the de c omp osition Φ M ( V ) ⊕ Φ M ( W ) , vanishes on Φ M ( V ) and antic ommutes with Φ M I Φ − 1 M . If A is a symmetric trivializing op er ator, t hen the index class σ I ( M , F M ) := [ D sign M ( A )] ∈ K 0 ( A ) is c al le d the (t w is ted) signature class . We c al l the symmetric t riviali zing op er ator A I := i Φ M I τ ∂ M Φ − 1 M the c a nonical symmetric trivializi ng op er ator of B ( d sign M ) with r esp e ct to I . Since ( D bd ∂ M + i I τ ∂ M ) 2 = ( D bd ∂ M ) 2 + I 2 is in vertible, the oper ator D bd ∂ M + i I τ ∂ M is inv ertible a s well. Hence A I is indeed a trivializing op erator for B ( d sign M ). In the following we ex tend any op erato r on W tac itly to Ω ∗ (2) ( ∂ M , F ∂ M ) by letting it v anish o n V . The following result sharp ens and generalizes similar calc ulations in [LP 00]. Lemma 5. 2. The twiste d signatur e class σ I ( M , F M ) do es not dep end on the choic e of the symm et ric trivializing op er ator. Pr o of. Let A 0 , A 1 be t wo trivializ ing o p er a tors fo r B ( d sign M ) that are symmetric with resp ect to I . Consider the cylinder Z := I R × ∂ M and let D sign Z be the signature op erator o n Ω ∗ (2) ( Z, p ∗ F ∂ M ). Recall that the p ositive and neg ative eig enspace of τ Z are identiﬁed via ic ( dx 1 ). W e get translation inv ariant spa c es ˜ V = L 2 (I R) ⊗ Φ Z ( V ) ⊗ C 2 ˜ W := L 2 (I R) ⊗ Φ Z ( W ) ⊗ C 2 such that Ω ∗ (2) ( Z, p ∗ F ∂ M ) = ˜ V ⊕ ˜ W . The op erators A i and Φ Z I Φ − 1 Z deﬁne trans- lation inv ariant op erator s on Ω ∗ (2) ( Z, p ∗ F ∂ M ). Note that D sign Z is invertible on ˜ V since D bd ∂ M | ∂ M is inv er tible on V . Let χ 0 , χ 1 : Z → [0 , 1] b e smo oth functions such that χ 0 ( x 1 , x 2 ) = 1 if x 1 ≤ 0 and χ 0 ( x 1 , x 2 ) = 0 if x 1 ≥ 1 2 and that χ 1 ( x 1 , x 2 ) = 1 if x 1 ≥ 1 and χ 1 ( x 1 , x 2 ) = 0 if x 1 ≤ 1 2 . Prop. 2.1 and the relative index theo r em [Bu95] imply that [ D sign M ( A 0 )] + [( D sign Z − c ( dx 1 ) τ Z ( χ 0 A 0 + χ 1 A 1 )) | ˜ W ] = [ D sign M ( A 1 )] . Let j : C → C 1 be the uniq ue unital homo morphism. It holds that [ j ] ∈ K K 0 (C , C 1 ) = 0, thus Im( j ∗ : K K 0 ( C 1 , A ) → K K 0 ( C , A )) = 0 . There is an even unital homomorphism C 1 → B ( ˜ W ) mapping σ to ic ( dx 1 ) τ Z (Φ Z I Φ − 1 Z ). Since ( D sign Z − c ( dx 1 ) τ Z ( χ 0 A 0 + χ 1 A 1 )) | ˜ W anticomm utes with ic ( dx 1 ) τ Z (Φ Z I Φ − 1 Z ), we ha ve that [( D sign Z − c ( dx 1 ) τ Z ( χ 0 A 0 + χ 1 A 1 )) | ˜ W ] ∈ Im( j ∗ ) .  12 CHARLOTTE W AHL Note that for I opp := − i I τ ∂ M the s ignature class σ I opp ( M , F M ) is well-deﬁned and that A I opp = Φ M I Φ − 1 M . Since I opp I = iτ ∂ M , the second ass ertion of the following Lemma implies tha t σ I ( M , F M ) = σ I opp ( M , F M ) . (5.2) Lemma 5.3. F or j = 0 , 1 let Ω ∗ (2) ( ∂ M , F ∂ M ) = V j ⊕ W j b e an ortho gonal de c om- p osition and let I j b e an involution on W j such that σ I j ( M , F M ) is wel l-deﬁne d. (1) A ssume that W 1 ⊂ W 0 and I 1 = I 0 | W 1 . (2) A ssume that W := W 0 = W 1 . L et E + b e the p ositive and E − the ne gative eigensp ac e of τ ∂ M on W . We identify E − with E + using the isomo rphism I 0 : E − → E + . Then t her e is a unitary u on E + such that with r esp e ct to the de c omp osition W = E + ⊕ E − I 1 =  0 u ∗ u 0  . We assume that t he sp e ctrum of u is not e qual to S 1 . If one of the pr evious two c onditions holds, then σ I 0 ( M , F M ) = σ I 1 ( M , F M ) . Pr o of. In the ﬁrst case we get the equa lit y since any trivializing op era tor that it symmetric with r esp ect to I 1 is als o symmetric with respect to I 0 . Now assume (2). Since the sp ectrum of u is not e q ual to S 1 , there is a selfadjoin t op erator a o n E + such that u = e ia . Set u t = e ita , t ∈ [0 , 1]. The inv olutions I 0 , I 1 are homoto pic to each o ther via the path of inv o lutions I t =  0 u ∗ t u t 0  . Since D bd ∂ M anticomm utes with I 0 and co mmutes with τ ∂ M , we get that D bd ∂ M =  D 0 0 − D  with D = ( D bd ∂ M ) | E + . F ur thermore D bd ∂ M also anticomm utes with I 1 . This implies that D commutes with u and u ∗ . Hence it commutes also with u t and u ∗ t . It follows that D bd ∂ M anticomm utes with I t . Thus σ I t ( M , F M ) is w ell-deﬁned. By the homotopy inv a riance of K K -theo ry classes it do es no t dep end on t .  The following prop osition gener alizes b oth cases of the previous Lemma: Prop ositio n 5.4. F or j = 0 , 1 let Ω ∗ (2) ( ∂ M , F ∂ M ) = V j ⊕ W j b e an ortho gonal de c omp osition and let I j b e an involution on W j such that σ I j ( M , F M ) is wel l- deﬁne d. Assume that V 0 = ( V 0 ∩ V 1 ) ⊕ ( V 0 ∩ W 1 ) and W 0 = ( W 0 ∩ V 1 ) ⊕ ( W 0 ∩ W 1 ) and that I 0 , I 1 r estrict t o involutions on W 0 ∩ W 1 . L et I 0 | W 0 ∩ W 1 and I 1 | W 0 ∩ W 1 fulﬁl l c ondition (2) of the pr evious L emma. Then σ I 0 ( M , F M ) = σ I 1 ( M , F M ) . Pr o of. Set ˜ I j = I j | W 0 ∩ W 1 . Note tha t σ ˜ I j ( M , F M ) is well-deﬁned. By part (1 ) of the pr evious Lemma σ I j ( M , F M ) = σ ˜ I j ( M , F M ) and part (2) implies tha t σ ˜ I 0 ( M , F M ) = σ ˜ I 1 ( M , F M ).  PRODUCT FORMULA FOR APS-CLASSES 13 Now, follo wing [LP03], we in tro duce the particula r inv olution that is used for the deﬁnition of the signature class . F or brevity it w ill be de no ted α M though it dep ends only on the structures on ∂ M . Let m = dim M / 2. Let V M be the closure o f d ∗ Ω m ( ∂ M , F ∂ M ) ⊕ d Ω m − 1 ( ∂ M , F ∂ M ) and W M = V ⊥ M . Deﬁne Ω < F M as the closed subspace of W M spanned by forms o f degr ee s maller than or equal to m − 1 and corresp ondingly deﬁne Ω > F M as the subspace spa nned by forms of degr ee bigger than or equal to m . W e make the following assumption: Assumption 5. 5. Th e closur e of d : Ω m − 1 ( ∂ M , F ∂ M ) → Ω m (2) ( ∂ M , F ∂ M ) has close d r ange. Note that the op erator s τ ∂ M , d ∂ M , d ∗ ∂ M restrict to op erato r s on V M resp. W M and that τ ∂ M : Ω < F M → Ω > F M is an is omorphism. Assumption 5.5 implies that D bd ∂ M is inv er tible on V M and that V M ⊕ W M = Ω ∗ (2) ( ∂ M , F ∂ M ) . Let α M be the inv olution on W M with p ositive eigenspac e Ω < F M and negative eigenspace Ω > F M . Then D bd ∂ M and α M anticomm ute. W e write σ ( M , F M ) := σ α M ( M , F M ). Note that Assumption 5 .5 does not depend on the choice of the Riemannian metric since Ω m (2) ( ∂ M , F ∂ M ) as a top ologic a l vector spa ce do es no t dep end on the Rie- mannian metric. Using the homoto p y inv a riance of K K -theory classe s one a lso shows that σ ( M , F M ) do es not depend on the choice of the Rie ma nnian metric. The following tec hnical lemma will b e needed when w e apply Prop. 5.4. Lemma 5.6. Assum e that N is even-dimensional. L et the de Rham op er ators on Ω ∗ ( ∂ M , F ∂ M ) and on Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ) fulﬁl l Ass u mption 5.5. We have that V M × N = ( V M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N ⊕ ( W M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N W M × N = ( V M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N ⊕ ( W M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N . The op er ator D bd ∂ M × N is diagonal with r esp e ct to t he de c omp ositions on t he right hand side and is invertible on V M ⊗ Ω ∗ (2) ( N , F N ) . Pr o of. Note ﬁrst that d ∂ M × N , d ∗ ∂ M × N and τ ∂ M × N map the s paces V M ⊗ Ω ∗ (2) ( N , F N ) and W M ⊗ Ω ∗ (2) ( N , F N ) to themselves. F or each k Ω k ( ∂ M × N , F ∂ M ⊠ F N ) = Ω k ( ∂ M × N , F ∂ M ⊠ F N ) ∩ ( V M ⊗ Ω ∗ (2) ( N , F N )) ⊕ Ω k ( ∂ M × N , F ∂ M ⊠ F N ) ∩ ( W M ⊗ Ω ∗ (2) ( N , F N )) . Hence we only need to consider the degr ees k := (dim M + dim N ) / 2 and k − 1. W e b egin by proving the ﬁrs t equa tion: Let γ = d ( α ∧ β ) ∈ d Ω k − 1 ( ∂ M × N , F M ⊠ F N ) ⊂ V M × N with α ∈ Ω ∗ ( ∂ M , F ∂ M ) , β ∈ Ω ∗ ( N , F N ). If α ∈ W M , then dα ∈ W M , th us γ ∈ ( W M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N . If α ∈ V M , then dα ∈ V M , hence 14 CHARLOTTE W AHL γ ∈ ( V M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N . An ana logous co nsideration works for d ∗ Ω k ( ∂ M × N , F M ⊠ F N ). This shows the ﬁrst equa tio n. F or the pro of of the s econd equation let γ ∈ Ω k − 1 ( ∂ M × N , F M ⊠ F N ) ∩ W M × N . Hence dγ = 0. W rite γ = γ 1 + γ 2 with γ 1 ∈ W M ⊗ Ω ∗ (2) ( N , F N ), γ 2 ∈ V M ⊗ Ω ∗ (2) ( N , F N ). Then dγ 1 ∈ W M ⊗ Ω ∗ (2) ( N , F N ), dγ 2 ∈ V M ⊗ Ω ∗ (2) ( N , F N ). Since these spaces a re orthog onal to each other, the equation d ( γ 1 + γ 2 ) = 0 implies that dγ 1 = dγ 2 = 0. T hus γ 1 , γ 2 ∈ W M × N . The case γ ∈ Ω k ( ∂ M × N , F M ⊠ F N ) with d ∗ γ = 0 ca n be tr eated analog ously . Now the second equa tio n follows. The o per ator D bd ∂ M × N resp ects the decomp ositions on the right hand side since d and τ ∂ M × N do. It s square is the Laplace o per ator ∆ ∂ M × N = ∆ ∂ M + ∆ N . Since ∆ ∂ M is in vertible on V M , the op erator ∆ ∂ M × N is in vertible on V M ⊗ Ω ∗ (2) ( N , F N ). Hence also D bd ∂ M × N is inv ertible on V M ⊗ Ω ∗ (2) ( N , F N ).  Theorem 5. 7 . L et M , N b e even-dimensional. If Assumption 5.5 ho lds for the de Rham op er ators on Ω ∗ ( ∂ M , F ∂ M ) and on Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ) , then σ ( M , F M ) ⊗ σ ( N , F N ) = σ ( M × N , F M ⊠ F N ) ∈ K 0 ( A ⊗ B ) . Pr o of. W e denote by Γ M the grading o p er a tor with resp ect to the Z Z / 2-g rading determined by the parity of the degr ee of a diﬀerential form o n M . The de Rham oper ator on M × N fulﬁlls d M × N = d M ⊗ 1 + Γ M ⊗ d N . Thu s d M × N + d ∗ M × N = ( d M + d ∗ M ) ⊗ 1 + Γ M ⊗ ( d N + d ∗ N ) . (5.3) Note for later that these tw o equations also hold for M or N o dd-dimensional. W e b egin by proving the theore m for closed M . W e conclude (recall our co nven tion on gra ded tensor products ) that D sign M × N = D sign M ⊗ 1 + Γ M ⊗ D sign N = D sign M + Γ M τ M D sign N . F urthermore τ M × N = τ M τ N . W e ﬁx the following nota tion: Let D be an o dd selfadjoint F r edholm oper ator on a Z Z / 2-graded countably generated Hilb ert A -mo dule H and let I be a unitary on H − . W e deﬁne the sy mmetrized pro duct S ( I , D ) =  0 D − I ∗ I D + 0  . Then S ( I , D ) is a r egular selfadjo int o dd F r edholm o pe r ator and [ S ( I , D )] = [ D ] ∈ K K 0 ( C , A ) by the additivity of the F redho lm index. If I is an even unitary deﬁned on H that commutes with D , then S ( I | H − , D ) = I D . Applying this prope rty twice with I = Γ M τ M yields that in K K 0 ( C , A ⊗ B ) [ D sign M ] ⊗ [ D sign N ] = [Γ M τ M D sign M ] ⊗ [ D sign N ] = [Γ M τ M D sign M + D sign N ] = [Γ M τ M ( D sign M + Γ M τ M D sign N )] = [ D sign M + Γ M τ M D sign N ] = [ D sign M × N ] . PRODUCT FORMULA FOR APS-CLASSES 15 The second equality follows from the description o f the K asparov pro duct b efore Theorem 2.2 . Now we consider the case where M is a ma nifold with b oundary . Deﬁne the involution ˜ α M := (Φ − 1 M × N ◦ Ψ ◦ Φ M ) α M τ N (Φ − 1 M ◦ Ψ − 1 ◦ Φ M × N ) on ˜ W M := (Φ − 1 M × N ◦ Ψ ◦ Φ M )( W M ⊗ Ω ∗ (2) ( N , F N )) and set ˜ V M := (Φ − 1 M × N ◦ Ψ ◦ Φ M )( V M ⊗ Ω ∗ (2) ( N , F N )) . Sublemma 5.8. (1) It holds that ˜ V M = V M ⊗ Ω ∗ (2) ( N , F N ) ˜ W M = W M ⊗ Ω ∗ (2) ( N , F N ) and that ˜ α M = α M . (2) The op er ator ˜ α M antic ommutes with D bd ∂ M × N and τ ∂ M × N and c ommutes with α M × N . Pr o of. F or α ∈ Λ ∗ T ∗ ∂ M ⊗ F M , β ∈ Λ ∗ T ∗ N ⊗ F N (Ψ ◦ Φ M )( α ∧ β ) = 1 √ 2 Ψ( dx 1 ∧ α ∧ β + τ M ( dx 1 ∧ α ) ∧ β ) = 1 2 √ 2  dx 1 ∧ α + τ M ( dx 1 ∧ α )) ∧ ( β + τ N β ) + i ( − α + τ M ( α )) ∧ ( β − τ N β )  = 1 2 √ 2  dx 1 ∧ α ∧ ( β + τ N β ) + idx 1 ∧ τ ∂ M ( α ) ∧ ( β − τ N β )  + τ M × N ( . . . ) . Here the dots represent a r e p etition of the ﬁrst summand, such that the last line is in the positive eigenspace of τ M × N . Thu s (Φ − 1 M × N ◦ Ψ ◦ Φ M )( α ∧ β ) = 1 2  α ∧ ( β + τ N β ) + iτ ∂ M α ∧ ( β − τ N β )  . In particula r (Φ − 1 M × N ◦ Ψ ◦ Φ M )( α ∧ ( β + τ N β )) = α ∧ ( β + τ N β ) and (Φ − 1 M × N ◦ Ψ ◦ Φ M )( τ ∂ M α ∧ ( β − τ N β )) = iα ∧ ( β − τ N β ) . Let ˜ W ± M be the p ositive resp. ne g ative eigenspa ce of ˜ α M . It follows that ˜ W + M = Ω < F M ⊗ Ω ∗ (2) ( N , F N ) ˜ W − M = Ω > F M ⊗ Ω ∗ (2) ( N , F N ) . This shows the seco nd and thir d equality of asser tion (1). The ﬁrst equality follows since ˜ V M is the orthogonal complement of ˜ W M . F urthermore τ ∂ M × N int erchanges the s paces Ω < F M ⊗ Ω ∗ (2) ( N , F N ) and Ω > F M ⊗ Ω ∗ (2) ( N , F N ) wherea s d ∂ M × N preserves them. This implies assertion (2).  16 CHARLOTTE W AHL W e set I = ˜ α M . By the Sublemma σ I ( M × N , F M ⊠ F N ) is well-deﬁned. One chec ks easily that ˜ α M and α M × N restrict to involutions on ˜ W M ∩ W M × N . Since on that spa ce ( ˜ α M α M × N ) 2 = 1 , the sp ectrum o f the r estriction of ˜ α M α M × N to ˜ W M ∩ W M × N is contained in {− 1 , 1 } . Hence, by Lemma 5.6 a nd Prop. 5.4, σ I ( M × N , F M ⊠ F N ) = σ ( M × N , F M ⊠ F N ) . Let A M be the ca nonical symmetric trivializing op erato r for B ( d sign M ) with resp ect to α M . By deﬁnition ˆ A M = i (Ψ ◦ Φ M )( α M τ ∂ M τ N )(Ψ ◦ Φ M ) − 1 . Hence (Φ M × N ) − 1 ˆ A M Φ M × N = i ˜ α M τ ∂ M = iα M τ ∂ M . Thu s ˆ A M = A I and σ I ( M × N , F M ⊠ F N ) = [ D sign M × N ( ˆ A M )] . Note that Φ − 1 M × N Γ M Φ M × N = − Γ ∂ M and Φ − 1 M × N τ M Φ M × N = Φ − 1 M × N τ N Φ M × N = τ N . Therefore, in contrast to the close d case, I = Γ M τ M commutes neither with D sign M ( A M ) nor with ( D sign M + Γ M τ M D sign N ) cy l ( ˆ A M ). This was the motiv ation for int ro ducing the symmetrized pro duct. W e have that σ ( M , F M ) ⊗ σ ( N , F N ) = [ D sign M ( A M )] ⊗ [ D sign N ] = [ S (Γ M τ M , D sign M ( A M ))] ⊗ [ D sign N ] = [ S  Γ M τ M , ( D sign M + Γ M τ M D sign N ) cy l ( ˆ A M )  ] = [( D sign M + Γ M τ M D sign N ) cy l ( ˆ A M )] = [ D sign,cy l M × N ( ˆ A M )] . The third equality do es not follow directly fro m Theo rem 2.2, but its pro of is analogo us. This concludes the pro of of the theorem.  5.2. The signature class i n the o dd case. Now let M be o dd-dimens io nal. Then for α ∈ Λ ∗ T ∗ ∂ M ⊗ F M τ M α = idx 1 ∧ τ ∂ M α and τ M ( dx 1 ∧ α ) = − iτ ∂ M α . Since Γ M anticomm utes with τ M , it induces an isomo rphism Γ M : Λ ± T ∗ M ⊗ F M → Λ ∓ T ∗ M ⊗ F M . The op erator d M + d ∗ M = d M + τ M d M τ M commutes with τ M and a n ticommutes with Γ M . The (odd twisted) signature oper ator d sign M is deﬁned as the r estriction of d M + τ M d M τ M to Ω + ( M , F M ). Then B ( d sign M ) = i Φ M ( d ∂ M τ ∂ M − τ ∂ M d ∂ M )Φ − 1 M . PRODUCT FORMULA FOR APS-CLASSES 17 Deﬁne the is ometric isomor phism Π : Λ ev T ∗ M ⊗ F M → Λ + T ∗ M ⊗ F M , α 7→ 1 √ 2 ( α + τ M α ) Note the connection of Π − 1 d sign M Π = d M τ M + τ M d M with the b ounda r y op erator in eq. 5.1. How in turn is the b ounda r y op erator of the o dd s ig nature op erator related to the even sig nature op erator? Consider the isometr ic isomorphism Ξ : Λ ∗ T ∗ ∂ M ⊗ F ∂ M → (Λ ev T ∗ M ⊗ F M ) | ∂ M , Ξ( α ) := 1 2 (1 + Γ M )( α + dx 1 ∧ α ) . Deﬁne Σ M = Π ◦ Ξ. Hence Σ M ( α ) = 1 √ 2 ( α + τ M α ) if α ∈ Λ ev T ∗ ∂ M ⊗ F ∂ M is even, and Σ M ( α ) = 1 √ 2 ( dx 1 ∧ α − iτ ∂ M α ) if α ∈ Λ od T ∗ ∂ M ⊗ F ∂ M . W e have that B ( d sign M ) = Σ M d sign ∂ M Σ − 1 M . F urthermore one c hecks that z ∂ M = ic M ( dx 1 ) = Σ M τ ∂ M Σ − 1 M . F or the sake of conformity with [LP03], w e will use this expression for the bounda ry op erator in the following. The following deﬁnition is motiv ated by the b oundar y conditions consider ed in [LP03, § 6.4]. Deﬁnition 5.9. Assume given a ort ho gonal de c omp osition Ω ∗ (2) ( ∂ M , F ∂ M ) = V ⊕ W with r esp e ct to which τ ∂ M , Γ ∂ M and D sign ∂ M ar e diagonal. F urthermor e assume that D sign ∂ M | V is invertible. L et I b e a b ounde d op er ator on Ω ∗ (2) ( ∂ M , F ∂ M ) vanishing on V and whose r estriction to W is an involution antic ommuting with τ ∂ M and c ommuting with D sign ∂ M and Γ ∂ M . We c al l a trivializing op er ator A of B ( d sign M ) symmetric with r esp e ct to I if it is diagonal with r esp e ct to the de c omp osition Σ M ( V ) ⊕ Σ M ( W ) , vanishes on Σ M ( V ) and c ommutes with Σ M I Σ − 1 M on Σ M ( W ) . If A is a symmetric trivializing op er ator, t hen the index class σ I ( M , F M ) := [ D sign M ( A )] ∈ K 1 ( A ) is c al le d the (t w is ted) signature class . We c al l the symmetric t rivializi ng op er ator A I := Σ M I Γ ∂ M Σ − 1 M the canonica l symmetric trivializi ng op er ator of B ( d sign M ) with r esp e ct to I . Note that any symmetric b ounded op erator that is diagonal with resp ect to the decomp osition Σ M ( V ) ⊕ Σ M ( W ), v anishes on Σ M ( V ), anticomm utes with D sign ∂ M and commutes with I is a symmetric trivializing op erator . As in Lemma 5.2 one shows: Lemma 5.1 0. T he twiste d signatur e class σ I ( M , F M ) do es not dep end on the choic e of the symmetric triviali zing op er ator. Pr o of. First we outline the general v anishing arg umen t we a re using: Co ns ider a selfadjoint F redho lm op erator D o n a countably generated ungra ded Hilb e r t C ∗ - mo dule H . Assume given a unital ho mo morphism ρ : C 1 → B ( H ) such that 18 CHARLOTTE W AHL ρ ( σ ) anticomm utes with D . W e deﬁne the even homo mo rphism ρ : C 1 → B ( H ⊗ C 1 ) , ρ ( σ ) = ρ ( σ ) σ . T hen ρ ( σ ) a n ticommutes with σD . Hence [ σ D ] ∈ Im  j ∗ : K K 0 ( C 1 , A ⊗ C 1 ) → K K 0 ( C , A ⊗ C 1 )  = 0 . Thu s [ D ] = 0. (Note tha t the deﬁnition of K asparov mo dules for un b ounded F redholm opera tors in [W07, Def. 2.4], which is the basis for our discus - sion, co n tains a s ig n err or: In the o dd case, ins tead of [ D , ρ ( b )] it should read ( D ρ ( b ) − ( − 1 ) deg b ρ ( b ) D ) for b homogeneous.) W e consider the o per ator D := ( D sign Z − c ( dx 1 )( χ 0 A 0 + χ 1 A 1 )) | ˜ W on H := ˜ W deﬁned as in Lemma 5.2 with the obvious changes. Let ρ : C 1 → B ( ˜ W ) be the unital homomo r phism deﬁned by ρ ( σ ) = Σ M I Σ − 1 M . Since ρ ( σ ) anticomm utes with c M ( dx 1 ) = − i Σ M τ ∂ M Σ − 1 M , it also anticomm utes with D . Th us the class [ D ] ∈ K K 1 ( C , A ) v anis hes .  Lemma 5 . 11. Le t Ω ∗ (2) ( ∂ M , F ∂ M ) = V ⊕ W b e an ortho gonal de c omp osition and let I j , j = 0 , 1 b e an involution on W such that σ I j ( M , F M ) is wel l-deﬁne d. L et E + b e the p ositive and E − the ne gative eigensp ac e of τ ∂ M on W . We identify E − with E + using the isomorphism I 0 : E − → E + . Ther e is a unitary u on E + such that with r esp e ct to the de c omp osition W = E + ⊕ E − I 1 =  0 u ∗ u 0  . Assume that union of the sp e ctr a of u and u ∗ is not e qual to S 1 . Then σ I 0 ( M , F M ) = σ I 1 ( M , F M ) . Pr o of. Since D sign ∂ M commutes with I 0 and anticomm utes with τ ∂ M , it holds that D sign ∂ M =  0 D D 0  with D = ( I 0 D sign ∂ M ) | E + . F ur thermore D sign ∂ M also comm utes with I 1 . This implies that D u = u ∗ D . Let C b e a lo op in the intersection of the resolvent sets o f u and u ∗ . W e assume that C has winding num b er one with resp ect to any p oint in the sp ectra of u and u ∗ and that there is a path fro m the origin to inﬁnity not int ersecting the lo op. W e can deﬁne a = − i log( u ) = − 1 2 π Z C log( λ )( u − λ ) − 1 dλ using any branch of the logarithm. Then Da = − aD . Deﬁne u t = e ita and I t =  0 u ∗ t u t 0  . W e get that Du t = u ∗ t D . This in turn implies that D sign ∂ M commutes with I t . Analogously Γ ∂ M commutes with I t . Th us the class σ I t ( M × N , F M ⊠ F N ) is well-deﬁned. By homo to p y inv ar iance it do es not depend on t .  As in the ev en cas e o ne gets: Prop ositio n 5 . 12. F or j = 0 , 1 let Ω ∗ (2) ( ∂ M , F ∂ M ) = V j ⊕ W j b e an ortho gonal de c omp osition and let I j b e an involution on W j such that σ I j ( M , F M ) is wel l- deﬁne d. Ass u me that V 0 = V 0 ∩ V 1 ⊕ V 0 ∩ W 1 and W 0 = W 0 ∩ V 1 ⊕ W 0 ∩ W 1 and that I 0 and I 1 r estrict t o involutions on W 0 ∩ W 1 . L et I 0 | W 0 ∩ W 1 and I 1 | W 0 ∩ W 1 fulﬁl l the c ondition of the pr evious L emma. Then σ I 0 ( M , F M ) = σ I 1 ( M , F M ) . PRODUCT FORMULA FOR APS-CLASSES 19 The b oundary conditions introduced in the following a re a special ca se o f those in [LP03, § 6.4]. Let m = (dim M − 1) / 2. Let V M be the closure of d ∗ Ω m ( ∂ M , F ∂ M ) ⊕ d Ω m − 1 ( ∂ M , F ∂ M ) ⊕ d ∗ Ω m +1 ( ∂ M , F ∂ M ) ⊕ d Ω m ( ∂ M , F ∂ M ) in Ω ∗ (2) ( ∂ M , F ∂ M ) and let W M = V ⊥ M . The op era tors d, d ∗ , τ ∂ M act on V M and W M . W e make the following assumption: Assumption 5.13. The closur e of d : Ω m − 1 ( ∂ M , F ∂ M ) → Ω m (2) ( ∂ M , F ∂ M ) has close d r ange. It follows that Ω ∗ (2) ( ∂ M , F ∂ M ) = V M ⊕ W M and that D sign ∂ M is inv er tible on V M . Let H ∂ M ⊂ W M be the kernel o f the Laplacian ∆ ∂ M restricted to Ω m (2) ( ∂ M , F ∂ M ). The Assumption implies that H ∂ M is a pro jective A -mo dule. In particular it has an orthog onal complement. Denote b y H ± ∂ M the pos itiv e resp. negative eigenspac e of τ ∂ M restricted to H ∂ M . W e also make the following assumption, which is not present in [LP 03]. In some of the situations w e consider it will be a uto matically fulﬁlled. F urthermor e it ca n alwa y s be enforced by a stabilization pro cedure, se e § 8.1 for a disc us sion. Assumption 5. 14. The sp ac es H ± ∂ M ar e isomorphic A -mo dules. This a ssumption is equiv alent to the assumption that ther e is a submo dule L ⊂ H ∂ M that is Lagrangian with re spec t to the skewhermitian form on H ∂ M induces by iτ ∂ M . Let L ⊥ be its or thogonal co mplemen t in H ∂ M . Recall that the deﬁnition of a Lagra ng ian includes the condition L ⊕ L ⊥ = H ∂ M , which is nontrivial for C ∗ -mo dules. Let Ω < F M be the clos e d subs pace of W M spanned by forms of degree smaller than m , and deﬁne Ω > F M as the s ubspace spanned by forms o f degree bigge r than m . Let α L M be the inv olution on W M with pos itiv e eigenspace Ω < F M ⊕ L and negative eigenspace Ω > F M ⊕ L ⊥ . Then α L M commutes with D sign ∂ M and Γ ∂ M and an ticommutes with τ ∂ M . Thus σ L ( M , F M ) := σ α L M ( M , F M ) is w e ll-deﬁned. If L 1 , L 2 ∈ H ∂ M are tw o Lagra ngians, then there is a diﬀerence element [ L 1 − L 2 ] ∈ K 1 ( A ) and it holds that σ L 1 ( M , F M ) − σ L 2 ( M , F M ) = [ L 1 − L 2 ] . The diﬀerence e le ment was describ ed and the statement prov en in [LP03, § 6.4] using a diﬀerent deﬁnition of odd index classes (via susp ensio n). F or the deﬁnition used here the result follows fr om [W07, § 7-8]. The diﬀerence e le ment v anishes for example if L 1 and L 2 are homotopic through a path of La grangia ns . 6. Product formula for twisted signa ture cl asses – the remaining cases In this s ection we do not make any a prior i a ssumption on the dimens io ns of M and N . W e assume that the de Rham ope rators on Ω ∗ ( ∂ M , F ∂ M ) and on Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ) fulﬁll Assumption 5.5 o r 5.1 3, dep e nding on the dimension of ∂ M resp. ∂ M × N . 20 CHARLOTTE W AHL A warning ab out gr adings: W e co nsider the grading s as they a rise in § 2. In particular v ector bundles can only b e graded if the underlying manifold is even- dimensional. This implies that the chirality op erator τ need not b e a gra ding op erator. Also the grading on the pro duct is a s deﬁned in § 2 . The pro o f of the following Lemma is ana lo gous to the pr o of of Lemma 5 .6: Lemma 6.1 . It holds that V M × N = ( V M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N ⊕ ( W M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N W M × N = ( V M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N ⊕ ( W M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N . If the dimension of M × N is even, then the op er ator D bd ∂ M × N r esp e cts the de c om- p ositions on the right hand side and is invertible on V M ⊗ Ω ∗ (2) ( N , F N ) . If the dimension of M × N is o dd, then an analo gous statement holds for the op er ator D sign ∂ M × N . The deﬁnition of the space V N , whic h app ears in the sta temen t of the follow- ing lemma, is the analog ue of the deﬁnition of V M for the de Rham op erato r on Ω ∗ ( N , F N ). Lemma 6.2 . (1) If M × N is o dd-dimensional, then H ∂ M × N ⊂ W M ⊗ Ω ∗ (2) ( N , F N ) . (2) If M is o dd-dimensional, then H ∂ M ⊗ Ω ∗ (2) ( N , F N ) = ( H ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N ⊕ ( H ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N . F urt hermor e ( H ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N = H ∂ M ⊗ V N . In p articular V M × N = ( H ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N ⊕ ( H ⊥ ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ V M × N , W M × N = ( H ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N ⊕ ( H ⊥ ∂ M ⊗ Ω ∗ (2) ( N , F N )) ∩ W M × N . Pr o of. 1) It is str aight-forw ard to chec k that V M ⊗ Ω ∗ (2) ( N , F N ) is orthogo nal to H ∂ M × N . 2) Let α ∈ H ∂ M , β ∈ Ω k ( N , F N ). W e only co nsider the case where N is even- dimensional and k = dim N / 2 and le ave the o ther cases to the reader. By the pre- vious Lemma α ∧ β = dω 1 + d ∗ ω 2 + ω 3 , where ω 1 , ω 2 ∈ V M × N and ω 3 ∈ W M × N . Note that d ∗ ω 1 = 0 , dω 2 = 0 . It follows that d ( α ∧ β ) = dd ∗ ω 2 = ∆ ω 2 ∈ V M × N . Th us ω 2 = ( − 1) dim ∂ M / 2 ∆ − 1 ( α ∧ dβ ). It holds that ∆ ∂ M ω 2 = ( − 1) dim ∂ M / 2 ∆ − 1 (∆ ∂ M α ∧ dβ ) = 0. Th us ω 2 ∈ Ker ∆ ∂ M = H ∂ M ⊗ Ω ∗ (2) ( N , F N ). In a similar wa y one con- cludes that ω 1 ∈ H ∂ M ⊗ Ω ∗ (2) ( N , F N ). This implies the ﬁrs t equality . Clearly dω 1 ∈ H ∂ M ⊗ d Ω k − 1 ( N , F N ) a nd d ∗ ω 1 ∈ H ∂ M ⊗ d ∗ Ω k +1 ( N , F N ). Thus if ω 3 = 0 , then α ∧ β ∈ H ∂ M ⊗ V N . In order to show that H ∂ M ⊗ V N ⊂ V M × N it is enough to chec k that H ∂ M ⊗ V N is orthogo nal to W M × N , which is straight-forward. The last tw o equa tio ns follo w from the ﬁrst in a n elementary w ay .  Now we pr ov e the pro duct formula in the remaining three cases. The general strategy is as in the pro of o f Theorem 5 .7. PRODUCT FORMULA FOR APS-CLASSES 21 6.1. M is e v en-di mensional and N is o dd-dime nsional. W e r equire that As- sumption 5 .5 holds for the de Rham op erator on Ω ∗ ( ∂ M , F ∂ M ) and Assumption 5.13 holds for the de Rham oper a tor on Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ). Lemma 6.2 implies that the in volution α M τ ∂ M ⊗ Γ N τ N restricts to a n inv olution o n H ∂ M × N . F ur thermore it anticomm utes with τ ∂ M × N = − iτ ∂ M Γ ∂ M τ N . W e deﬁne the Lag r angian L ⊂ H ∂ M × N to b e its p ositive eigenspace. (Thus Assumption 5.14 is fulﬁlled as w ell.) Prop ositio n 6 . 3. It holds that σ ( M , F M ) ⊗ σ ( N , F N ) = σ L ( M × N , F M ⊠ F N ) ∈ K 1 ( A ⊗ B ) . Pr o of. W e have that τ M × N = τ M Γ M τ N = Γ M τ M τ N . First assume that M is closed. By the desc r iption of the Kasparov pro duct in § 3, [ D sign M + τ M D sign N ] = [ D sign M ] ⊗ [ D sign N ] ∈ K K 1 (C , A ⊗ B ) . The op era tor D sign M + τ M D sign N acts on Ω ∗ (2) ( M , F M ) ⊗ Ω + (2) ( N , F N ). Let Θ : Ω ∗ (2) ( M , F M ) ⊗ Ω + (2) ( N , F N ) → Ω + (2) ( M × N , F M ⊠ F N ) be the iso morphism that equals 1 ⊗ Γ N from (1 − Γ M τ M )Ω ∗ ( M , F M ) ⊗ Ω + ( N , F N ) to (1 − Γ M τ M )Ω ∗ ( M , F M ) ⊗ Ω − ( N , F N ) and the iden tit y on (1 + Γ M τ M )Ω ∗ ( M , F M ) ⊗ Ω + ( N , F N ). Then [ D sign M + τ M D sign N ] = [Θ( D sign M + τ M D sign N )Θ − 1 ] = [ D sign M + τ M Θ D sign N Θ − 1 ] . F or α ∈ (1 − Γ M τ M )Ω ∗ ( M , F M ) , β ∈ Ω − ( N , F N ) Θ D sign N Θ − 1 ( α ∧ β ) = − α ∧ ( d N + τ N d N τ N ) β . Note that the r estrictions of Γ M τ M and τ N to Ω + ( M × N , F M ⊠ F N ) ag ree. W e hav e that τ M Θ D sign N Θ − 1 = 1 2 Γ M τ N  (1 + τ N )(1 ⊗ ( d N + τ N d N τ N )) − (1 − τ N )(1 ⊗ ( d N + τ N d N τ N ))  = Γ M ⊗ ( d N + τ N d N τ N ) . Thu s D sign M × N = D sign M + τ M Θ D sign N Θ − 1 and therefore [ D sign M ] ⊗ [ D sign N ] = [ D sign M × N ] . Now let M b e a ma nifo ld with b oundary . Recall the quantities indexed by M , as α M , V M , W M , which were deﬁned in § 5.1. F urthermore Ψ , Γ 2 are as in § 4.1. Deﬁne the involution ˜ α M := (Σ − 1 M × N ◦ Θ ◦ Ψ)(Γ 2 Φ M α M Φ − 1 M )(Σ − 1 M × N ◦ Θ ◦ Ψ) − 1 on ˜ W M = (Σ − 1 M × N ◦ Θ ◦ Ψ)  (Φ M ( W M ) ⊕ Φ M ( W M )) ⊗ Ω + (2) ( N , F N )  . 22 CHARLOTTE W AHL Set ˜ V M = (Σ − 1 M × N ◦ Θ ◦ Ψ)  (Φ M ( V M ) ⊕ Φ M ( V M )) ⊗ Ω + (2) ( N , F N )  and let ˜ W ± M ⊂ ˜ W M be the p ositive resp. nega tive eigenspace of ˜ α M . Sublemma 6.4. (1) It holds that ˜ V M = V M ⊗ Ω ∗ (2) ( N , F N ) ˜ W M = W M ⊗ Ω ∗ (2) ( N , F N ) and that ˜ α M = α M τ ∂ M ⊗ Γ N τ N . (2) The op er ator ˜ α M c ommutes with D sign ∂ M × N and Γ ∂ M × N and antic ommu t es with τ ∂ M × N and α M × N . Pr o of. Let α 1 , α 2 ∈ Λ ∗ T ∗ ∂ M ⊗ F ∂ M and β ∈ Λ ev T ∗ N ⊗ F N . W e have that (Σ − 1 M × N ◦ Θ ◦ Ψ)  (Φ M ( α 1 ) , Φ M ( α 2 )) ∧ ( β + τ N β )  = 1 √ 2 (Σ − 1 M × N ◦ Θ)  ( dx 1 ∧ α 1 + τ M ( dx 1 ∧ α 1 ) − iα 2 + iτ M ( α 2 )) ∧ ( β + τ N β )  . Assume now tha t α 1 , α 2 ∈ Λ ev T ∗ ∂ M ⊗ F ∂ M . Then the previous expressio n e q uals 1 √ 2 Σ − 1 M × N  ( dx 1 ∧ α 1 + τ M ( dx 1 ∧ α 1 ) − iα 2 + iτ M ( α 2 )) ∧ ( β − τ N β )  = − ( τ ∂ M α 1 + α 1 ) ∧ τ N β + i ( τ ∂ M α 2 − α 2 ) ∧ β . If α 1 , α 2 ∈ Λ od T ∗ ∂ M ⊗ F ∂ M , then it e quals 1 √ 2 Σ − 1 M × N  ( dx 1 ∧ α 1 + τ M ( dx 1 ∧ α 1 ) − iα 2 + iτ M ( α 2 )) ∧ ( β + τ N β )  = ( α 1 + τ ∂ M α 1 ) ∧ β + i ( τ ∂ M α 2 − α 2 ) ∧ τ N β . Thu s the image of ( α, − iτ ∂ M α ) ∧ ( β + τ N β ) under Σ − 1 M × N ◦ Θ ◦ Ψ ◦ (Φ M ⊕ Φ M ) equals − 2 τ ∂ M α ∧ τ N β if α ∈ Λ ev T ∗ ∂ M ⊗ F ∂ M , and 2 α ∧ β if α ∈ Λ od T ∗ ∂ M ⊗ F ∂ M . The image of ( α, iτ ∂ M α ) ∧ ( β + τ N β ) equals − 2 α ∧ τ N β if α ∈ Λ ev T ∗ ∂ M ⊗ F ∂ M , and 2 τ ∂ M α ∧ β if α ∈ Λ od T ∗ ∂ M ⊗ F ∂ M . The ﬁrst pa rt of Sublemma 6 .4 follows. W e deﬁne v 1 ( α, β ) as the imag e of ( α, − iα ) ∧ ( β + τ N β ) under Σ − 1 M × N ◦ Θ ◦ Ψ ◦ (Φ M ⊕ Φ M ), and v 2 ( α, β ) as the image of ( α, iα ) ∧ ( β + τ N β ). The spac e Σ − 1 M × N ˜ W + M is spanned by the set { v 1 ( α, β ) , v 2 ( τ ∂ M α, β ) | α ∈ Ω < F M , β ∈ Ω ev (2) ( N , F N ) } . F or Σ − 1 M × N ˜ W − M an analog ous statement holds with > instead o f < . If α ∈ Λ ev T ∗ ∂ M ⊗ F ∂ M , then v 1 ( α, β ) = − ( τ ∂ M α + α ) ∧ τ N β − ( α − τ ∂ M α ) ∧ β v 2 ( τ ∂ M α, β ) = ( τ ∂ M α + α ) ∧ β + ( τ ∂ M α − α ) ∧ τ N β . If α ∈ Λ od T ∗ ∂ M ⊗ F ∂ M , then v 1 ( α, β ) = ( α + τ ∂ M α ) ∧ β − ( α − τ ∂ M α ) ∧ τ N β v 2 ( τ ∂ M α, β ) = − ( α + τ ∂ M α ) ∧ τ N β + ( τ ∂ M α − α ) ∧ β . PRODUCT FORMULA FOR APS-CLASSES 23 Using these equations one chec ks that ˜ W ± M is the pos itive resp. nega tive eigenspa ce of the inv olutio n α M τ ∂ M ⊗ Γ N τ N . Thus we get the third equation. F r om eq. 5.3 it follows that D sign ∂ M × N commutes with ˜ α M .  W e set I := ˜ α M . By the Sublemma σ I ( M × N , F M ⊠ F N ) is w e ll-deﬁned. The inv olutions α M × N and ˜ α M restrict to inv o lutio ns o n ˜ W M ∩ W M × N . By Lemma 6.1 we can apply Pro p. 5.12 , which yields σ L ( M × N , F M ⊠ F N ) = σ I ( M × N , F M ⊠ F N ) . The canonical symmetric trivializing op erator of B ( d sign M ) with resp ect to α M is A M = i Φ M ( α M τ ∂ M )Φ − 1 M . Then ˆ A M = Ψ( i Γ 1 (Φ M α M τ ∂ M Φ − 1 M ))Ψ − 1 . Since ˆ A M commutes with Ψ(Γ 2 (Φ M α M Φ − 1 M ))Ψ − 1 , the op erator Θ ˆ A M Θ − 1 is a symmetric triv- ializing op era tor for ˜ α M . W e get that [ D sign M ( A M )] ⊗ [ D N ] = [( D sign M + τ M D sign N ) cy l ( ˆ A M )] = [ D sign,cy l M × N (Θ ˆ A M Θ − 1 )] = σ I ( M × N , F M ⊠ F N ) .  6.2. M is o dd-dimensional and N is ev en-dim e nsional. W e requir e tha t As- sumptions 5.13 and 5.14 hold for the de Rham op era tor o n Ω ∗ ( ∂ M , F ∂ M ). The de Rham opera tor o n Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ) is only required to fulﬁll Assumption 5.13. The mo dule H ∂ M × N decomp oses into a direct sum of the pro jective A ⊗ B -mo dules H k,l ∂ M × N := H ∂ M × N ∩ (Ω k ( ∂ M , F ∂ M ) ⊗ Ω l ( N , F N )) . The mo dule H k,l ∂ M × N is only nontrivial if k + l = (dim ∂ M + dim N ) / 2. Let k = (dim ∂ M ) / 2 , l = (dim ∂ N ) / 2. Then H k,l ∂ M × N ∼ = H ∂ M ⊗ (Ker ∆ N ∩ Ω l ( N , F N )) . Thu s any Lagr angian L ∈ H ∂ M deﬁnes a Lagrang ian in H k,l ∂ M × N . F rom this and Lemma 6.2 (1) it follows that the in volution α L M ⊗ Γ N restricts to an inv olution on H ∂ M × N . Deﬁne the Lagr angian L ⊗ ⊂ H ∂ M × N as the po sitive eigenspace of the involution α L M ⊗ Γ N restricted to H ∂ M × N . (Note that the ex is tence o f this Lag rangian implies that Assumption 5.14 is fulﬁlled.) By constructio n α L ⊗ M × N | H ∂ M × N = α L M ⊗ Γ N | H ∂ M × N . Prop ositio n 6 . 5. It holds that σ L ( M , F M ) ⊗ σ ( N , F N ) = σ L ⊗ ( M × N , F M ⊠ F N ) ∈ K 1 ( A ⊗ B ) . Pr o of. W e have that τ M × N = τ M τ N . First assume that M is closed. By § 3 [ τ N D sign M + D sign N ] = [ D sign M ] ⊗ [ D sign N ] ∈ K K 1 ( C , A ⊗ B ) . The op era tor τ N D sign M + D sign N acts on Ω + (2) ( M , F M ) ⊗ Ω ∗ (2) ( N , F N ). 24 CHARLOTTE W AHL Let Θ : Ω + (2) ( M , F M ) ⊗ Ω ∗ (2) ( N , F N ) → Ω + (2) ( M × N , F M ⊠ F N ) be the isomor phism that eq uals Γ M from Ω + ( M , F M ) ⊗ Ω − ( N , F N ) to Ω − ( M , F M ) ⊗ Ω − ( N , F N ) and the identit y on Ω + ( M , F M ) ⊗ Ω + ( N , F N ). Note that Γ M D sign N = Θ D sign N Θ − 1 . The signa ture oper ator on Ω + (2) ( M × N , F M ⊠ F N ) fulﬁlls D sign M × N = τ M Θ D sign M Θ − 1 + Γ M D sign N = Θ( τ N D sign M + D sign N )Θ − 1 . Thu s [ τ N D sign M + D sign N ] = [ D sign M × N ] . Now let M b e a ma nifo ld with b oundary . Deﬁne the involution ˜ α L M = (Σ − 1 M × N ◦ Θ ◦ Σ M )( α L M ⊗ τ N )(Σ − 1 M × N ◦ Θ ◦ Σ M ) − 1 on ˜ W M = (Σ − 1 M × N ◦ Θ ◦ Σ M )( W M ⊗ Ω ∗ (2) ( N , F N )) and set ˜ V M = (Σ − 1 M × N ⊗ Θ ◦ Σ M )( V M ⊗ Ω ∗ (2) ( N , F N )) . F urthermore let ˜ W ± M be the p ositive resp. ne g ative eigenspa ce of ˜ α L M . Compare the following sublemma with Sublemma 6.4. Sublemma 6.6. (1) It holds that ˜ V M = V M ⊗ Ω ∗ (2) ( N , F N ) ˜ W M = W M ⊗ Ω ∗ (2) ( N , F N ) and ˜ α L M = α L M ⊗ Γ N . (2) The op er ator ˜ α L M c ommutes with Γ ∂ M × N , α L ⊗ M × N and D sign ∂ M × N and antic om- mutes with τ ∂ M × N . Pr o of. F or α ∈ Ω ev ( ∂ M , F M ) , β ∈ Ω ∗ ( N , F N ) (Σ − 1 M × N ◦ Θ)(Σ M ( α ) ∧ ( β ± τ N β )) = 1 √ 2 (Σ − 1 M × N ◦ Θ)(( α + τ M α ) ∧ ( β ± τ N β )) = 1 √ 2 Σ − 1 M × N (( α ± τ M α ) ∧ ( β ± τ N β )) . F or α ∈ Ω od ( ∂ M , F M ) , β ∈ Ω ∗ ( N , F N ) (Σ − 1 M × N ◦ Θ)(Σ M ( α ) ∧ ( β ± τ N β )) = 1 √ 2 (Σ − 1 M × N ◦ Θ)(( dx 1 ∧ α − iτ ∂ M α ) ∧ ( β ± τ N β )) = 1 √ 2 Σ − 1 M × N (( dx 1 ∧ α ∓ iτ ∂ M α ) ∧ ( β ± τ N β )) . In b oth ca ses this equals α ∧ ( β ± τ N β ) if β ∈ Ω ev ( N , F N ) and ± iτ ∂ M α ∧ ( β ± τ N β ) if β ∈ Ω od ( N , F N ). (These statements hold true if w e c ho ose the sign ab ov e resp. below everywhere.) Thu s ˜ W + M = (Ω < F M ⊕ L ) ⊗ Ω ev (2) ( N , F N ) ⊕ (Ω > F M ⊕ L ⊥ ) ⊗ Ω od (2) ( N , F N ) ˜ W − M = (Ω < F M ⊕ L ) ⊗ Ω od (2) ( N , F N ) ⊕ (Ω > F M ⊕ L ⊥ ) ⊗ Ω ev (2) ( N , F N ) . PRODUCT FORMULA FOR APS-CLASSES 25 It follows that ˜ W ± M is the p ositive resp. negative eige ns pace of the inv o lution α L M ⊗ Γ N . Eq. 5 .3 implies that the inv olution co mm utes with the sig nature o per ator on the bounda ry D sign ∂ M × N . It clearly comm utes with α L ⊗ M × N and an ticommutes w ith τ ∂ M × N = τ ∂ M τ N .  W e write I = ˜ α L M . By the Sublemma σ I ( M × N , F M ⊠ F N ) is well-deﬁned. Using Lemma 6.1 and Lemma 6.2 one chec ks that α M × N and ˜ α L M restrict to inv olutions on ˜ W M ∩ W M × N . By Prop. 5.1 2 σ I ( M × N , F M ⊠ F N ) = σ ( M × N , F M ⊠ F N ) . Let A M be the ca nonical symmetric trivializing op erato r for B ( d sign M ) with resp ect to α L M . Then ˆ A M = Σ M ( α L M Γ ∂ M )Σ − 1 M . F rom the calculations in the pro of of the Sublemma it also follo ws that (Σ − 1 M × N ⊗ Θ ◦ Σ M )Γ ∂ M τ N (Σ − 1 M × N ⊗ Θ ◦ Σ M ) − 1 = Γ ∂ M τ N . Hence (Σ − 1 M × N ◦ Θ) ˆ A M (Σ − 1 M × N ◦ Θ) − 1 = ˜ α L M Γ ∂ M τ N . Since ˜ α L M Γ ∂ M τ N anticomm utes with D sign ∂ M × N and commutes with ˜ α L M , the o per ator Θ ˆ A M Θ − 1 is a symmetric trivia lizing op era tor o f B ( d sign M × N ) with resp ect to I = ˜ α L M . Thu s σ I ( M × N , F M ⊠ F N ) = [ D sign M × N ( τ N Θ ˆ A M Θ − 1 )] . Arguing as in the closed case we hav e that σ L ( M , F M ) ⊗ σ ( N , F N ) = [ D sign M ( A M )] ⊗ [ D sign N ] = [( τ N D sign M + D sign N ) cy l ( ˆ A M )] = [( D sign M × N ) cy l (Θ ˆ A M Θ − 1 )] .  6.3. M , N are o dd- dimensio nal. Let Assumptions 5.13 and 5.14 ho ld fo r the de Rham op erator on Ω ∗ ( ∂ M , F ∂ M ) and Assumption 5.5 for the de Rha m op erator on Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ). Let L ⊂ H ∂ M be a La g rangian. Prop ositio n 6 . 7. It holds that 2 σ L ( M , F M ) ⊗ σ ( N , F N ) = σ ( M × N , F M ⊠ F N ) ∈ K 0 ( A ⊗ B ) . Pr o of. W e have that τ M × N = − iτ M Γ M τ N . First let M b e closed. In the following we will de no te by D sign M | X the c lo sure of d M + τ M d M τ M acting on X ⊂ Ω ∗ (2) ( M , F M ). Without spe c iﬁcation D sign M is understo o d to a ct on the s pace Ω + (2) ( M , F M ), as b efore. The same applies to D sign N . W e may ident ify Ω + (2) ( M , F M ) ⊕ Ω + (2) ( M , F M ) with Ω ∗ (2) ( M , F M ) by apply ing the isomorphism Γ M : Ω + (2) ( M , F M ) → Ω − (2) ( M , F M ) to the second summand. F rom § 4.3 w e get that Γ 1 = τ M , Γ 2 = iτ M Γ M and then fro m § 3 [ D sign M ] ⊗ [ D sign N ] = [( D sign M + iτ M Γ M D sign N ) | Ω ∗ (2) ( M , F M ) ⊗ Ω + (2) ( N , F N ) ] . Here the g rading op erator on Ω ∗ (2) ( M , F M ) ⊗ Ω + (2) ( N , F N ) is − i Γ 1 Γ 2 = Γ M . 26 CHARLOTTE W AHL Deﬁne the is ometric isomor phism Θ :  Ω ∗ (2) ( M , F M ) ⊗ Ω + (2) ( N , F N )  2 → Ω ∗ (2) ( M × N , F M ⊠ F N ) , Θ( ω 1 , ω 2 ) = ω 1 + Γ N ω 2 . On Ω ∗ (2) ( M × N , F M ⊠ F N ) Θ( D sign M + iτ M Γ M D sign N )Θ − 1 = D sign M + iτ M Γ M τ N D sign N = D sign M − τ M × N D sign N . Hence 2[ D sign M ] ⊗ [ D sign N ] = [( D sign M − τ M × N D sign N ) | Ω ∗ (2) ( M × N , F M ⊠ F N ) ] . In or der to co mpare the latter class with the signature class we de ﬁne a unitar y op erator Z on Ω ∗ (2) ( M × N , F M ⊠ F N ) by Z ( α ∧ β ) = 1 √ 2 ( α ∧ β + τ M × N (Γ M α ∧ β )) for α ∈ Ω ∗ (2) ( M , F M ) , β ∈ Ω ∗ (2) ( N , F N ). Then Z Γ M Z − 1 = τ M × N . F urthermore for α ∈ Ω ev ( M , F M ) , β ∈ Ω ∗ ( N , F N ) 1 √ 2 Z ( d sign M − τ M × N d sign N ) Z − 1 ( α ∧ β + τ M × N ( α ∧ β )) = Z  ( d sign M − τ M × N d sign N )( α ∧ β )  = 1 √ 2  d sign M α ∧ β − τ M × N ( d sign M α ∧ β ) + d sign N ( α ∧ β ) − τ M × N d sign N ( α ∧ β )  = 1 √ 2  d sign M α ∧ β − id sign M ( τ M α ∧ τ N β ) + d sign N ( α ∧ β ) + iτ M α ∧ d sign N τ N β  = 1 √ 2 ( d sign M + Γ M d sign N )  α ∧ β + τ M × N ( α ∧ β )  = 1 √ 2 d sign M × N ( α ∧ β + τ M × N ( α ∧ β )) . The last e q uation follows fro m eq. 5.3. It follo ws that d sign M × N = Z ( d sign M − τ M × N d sign N ) Z − 1 as an op er a tor fr om Ω + ( M × N , F M ⊠ F N ) to Ω − ( M × N , F M ⊠ F N ). Since b oth sides o f the equa tion ar e essentially selfadjoint, the equation holds o n Ω ∗ ( M × N , F M ⊠ F N ). Hence in the closed case 2[ D sign M ] ⊗ [ D sign N ] = [ D sign M × N ] ∈ K 0 ( A ⊗ B ) . Now let M b e a manifold with bo undary . The isomor phism Ψ deﬁned in § 4.3 is here a map from (Λ + T ∗ M ⊗ F M ) | ∂ M ⊠ (Λ + T ∗ N ⊗ F N ) to (Λ ev T ∗ M ⊗ F M ) | ∂ M ⊠ (Λ + T ∗ N ⊗ F N ) given b y Ψ( ω ) = 1 √ 2 ( ω + Γ M ω ) . PRODUCT FORMULA FOR APS-CLASSES 27 W e deﬁne the isomorphism Ψ fro m  (Λ ∗ T ∗ ∂ M ⊗ F ∂ M ) ⊠ (Λ + T ∗ N ⊗ F N )  2 to  Λ ∗ T ∗ ( M × N ) ⊗ ( F M ⊠ F N )  | ∂ M × N by Ψ( ω 1 , ω 2 ) = Θ  (Ψ ◦ Σ M )( ω 1 ) , (Ψ ◦ Σ M )( ω 2 )  = 1 √ 2  Σ M ( ω 1 ) + Γ M Σ M ( ω 1 ) + Γ N Σ M ( ω 2 ) + Γ M × N Σ M ( ω 2 )  . W e set ˜ W M = (Φ − 1 M × N ◦ Z ◦ Ψ)  ( W M ⊕ W M ) ⊗ Ω + (2) ( N , F N )  , ˜ V M = (Φ − 1 M × N ◦ Z ◦ Ψ)  ( V M ⊕ V M ) ⊗ Ω + (2) ( N , F N )  and ˜ α L M = (Φ − 1 M × N ◦ Z ◦ Ψ)( α L M Γ ∂ M ⊕ α L M Γ ∂ M )(Φ − 1 M × N ◦ Z ◦ Ψ) − 1 . The following sublemma is simila r to Sublemma 5.8. Sublemma 6.8. (1) It holds that ˜ V M = V M ⊗ Ω ∗ (2) ( N , F N ) ˜ W M = W M ⊗ Ω ∗ (2) ( N , F N ) an t hat ˜ α L M = − α L M . (2) The op er ator ˜ α L M antic ommutes with D bd ∂ M × N and τ ∂ M × N and c ommutes with α M × N . Pr o of. Let α 1 , α 2 ∈ Ω ev ( ∂ M , F M ) , β 1 , β 2 ∈ Ω + ( N , F N ) and set ω := α 1 ∧ β 1 + α 2 ∧ Γ N β 2 . W e hav e that (Φ − 1 M × N ◦ Z ◦ Ψ)( α 1 ∧ β 1 , α 2 ∧ β 2 ) . = (Φ − 1 M × N ◦ Z )( ω ) = 1 √ 2 Φ − 1 M × N ( ω + τ M × N ( ω )) = τ ∂ M × N ( ω ) . F or (dim ∂ M ) / 2 ev en L ⊂ Ω ev ( ∂ M , F M ). If α 1 , α 2 ∈ L , then τ ∂ M × N ( ω ) ∈ L ⊥ ⊗ Ω ∗ ( N , F N ). F or α 1 , α 2 ∈ Ω od ( ∂ M , F M ) , β 1 , β 2 ∈ Ω + ( N , F N ) and ω as be fore we hav e that (Φ − 1 M × N ◦ Z ◦ Ψ)( α 1 ∧ β 1 , α 2 ∧ β 2 ) . = (Φ − 1 M × N ◦ Z )( dx 1 ∧ ω ) = 1 √ 2 Φ − 1 M × N ( dx 1 ∧ ω + τ M × N ( dx 1 ∧ ω )) = ω . If (dim ∂ M ) / 2 is o dd and α 1 , α 2 ∈ L ⊂ Ω od ( ∂ M , F M ), then ω ∈ L ⊗ Ω ∗ ( N , F N ). F rom this o ne deduces (1). Hence ˜ α L M anticomm utes with τ ∂ M × N = Γ ∂ M τ ∂ M τ N and commutes with d ∂ M × N . By Lemma 6.2 the o p er ator α M × N is diagonal with resp ect to the deco mpo s ition H ∂ M ⊗ Ω ∗ (2) ( N , F N ) ⊕ H ⊥ ∂ M ⊗ Ω ∗ (2) ( N , F N ). Using this one gets (2).  W e set I = ˜ α L M . By the Sublemma σ I ( M × N , F M ⊠ F N ) is well-deﬁned. Lemma 6.1 and Lemma 6.2 imply tha t α M × N and ˜ α L M restrict to involutions on ˜ W M ∩ W M × N . By Pro p. 5.4 we get that σ I ( M × N , F M ⊠ F N ) = σ ( M × N , F M ⊠ F N ) . 28 CHARLOTTE W AHL Let A L M be the ca nonical symmetric trivializing op erato r for B ( d sign M ) with resp ect to α L M . By deﬁnition ˆ A L M = (Ψ ◦ Σ M )( α L M Γ ∂ M )(Ψ ◦ Σ M ) − 1 . Hence (Φ − 1 M × N ◦ Z ◦ Θ)( ˆ A L M ⊕ ˆ A L M )(Φ − 1 M × N ◦ Z ◦ Θ) = ˜ α L M . Thu s 2[ D sign M ( A L M )] ⊗ [ D sign N ] = [ D sign M × N  ( Z ◦ Θ)( ˆ A L M ⊕ ˆ A L M )( Z ◦ Θ) − 1  ] = [ D sign M × N (Φ M × N ˜ α L M Φ − 1 M × N )] = σ I opp ( M × N , F M ⊠ F N ) = σ I ( M × N , F M ⊠ F N ) . The ﬁrst equation follows from the pro duct formula in § 4.3. The last equation follows from eq. 5.2.  7. Product f ormula for higher signa tures In the following we g ive a slight gener alization o f the previous pro duct formulas, which also applies to higher signatures. Let C b e unital C ∗ -algebra and let ϕ : A ⊗ B → C b e a unital C ∗ -homomorphis m. There is an induced map ϕ ∗ : K ∗ ( A ⊗ B ) → K ∗ ( C ). The bundle ( F M ⊠ F N ) ⊗ ϕ C is a ﬂa t C -vector bundle on M × N . The pro of of the following theo r em is nearly literally as b efore, if at the rig h t places one plugs in tenso r pro ducts ⊗ ϕ C . Also as b efore, Assumption 5.14 in the statement of the theorem will b e automatically fulﬁlled for the de Rham o p er ator on Ω ∗ ( ∂ M × N , ( F ∂ M ⊠ F N ) ⊗ ϕ C ) if M × N is odd-dimensiona l. Theorem 7. 1. In the fol lowing we assume that the r esp e ctive assumptions (i.e. Assumption 5.5 r esp. Assumptions 5.13 and 5.14, dep en ding on the dimensions of M and N ) hold for t he de Rham op er ators on Ω ∗ ( ∂ M , F ∂ M ) and on Ω ∗ ( ∂ M × N , ( F ∂ M ⊠ F N ) ⊗ ϕ C ) . (1) If M and N ar e even-dimensional, then ϕ ∗  σ ( M , F M ) ⊗ σ ( N , F N )  = σ ( M × N , ( F M ⊠ F N ) ⊗ ϕ C ) . (2) If M is even-dimensional and N is o dd-dimensional and L is the p osi- tive eigensp ac e of the involution α M τ ∂ M ⊗ Γ N τ N on H ∂ M × N ⊂ Ω ∗ ( ∂ M × N , ( F ∂ M ⊠ F N ) ⊗ ϕ C ) , then ϕ ∗  σ ( M , F M ) ⊗ σ ( N , F N )  = σ L ( M × N , ( F M ⊠ F N ) ⊗ ϕ C ) . (3) If M is o dd-dimensional and N i s even-dimensional and L ⊂ H ∂ M is a L agr angian, then we c an deﬁne L ⊗ ⊂ H ∂ M × N as b efor e and get ϕ ∗  σ L ( M , F M ) ⊗ σ ( N , F N )  = σ L ⊗ ( M × N , ( F M ⊠ F N ) ⊗ ϕ C ) . (4) If M and N ar e o dd-dimensional and L ⊂ H ∂ M is a L agr angian, then 2 ϕ ∗  σ L ( M , F M ) ⊗ σ ( N , F N )  = σ ( M × N , ( F M ⊠ F N ) ⊗ ϕ C ) . This result applies to higher sig natures: Let ˜ M resp. ˜ N b e a Galois covering of M re sp. N and let π M resp. π N be the gro up o f deck transfo rmations. By deﬁnition P M = ˜ M × π M C ∗ r π M is the asso ciated Mishenko-F omenko bundle and σ ( M , P M ) is the higher signature cla ss of M asso ciated to the co vering. PRODUCT FORMULA FOR APS-CLASSES 29 The group π M × N := π M × π N is the decktransformatio n group with res pect to the cov er ing ˜ M × ˜ N → M × N . Let P M × N be the corres po nding Mishenko-F omenko bundle. Ther e is a canonical unital C ∗ -homomorphis m ϕ : C ∗ r π M ⊗ C ∗ r π N → C ∗ r ( π M × N ) , and it holds that P M × N = ( P M ⊠ P N ) ⊗ ϕ C ∗ r ( π M × N ) . Thu s from the previous prop osition one gets a pr o duct for m ula for higher sig nature classes. In this case (see [LLK02, Lemma 3.1]) Assumption 5.5 is equiv alent to the m -th Noviko v Sh ubin inv ariant α m ( ∂ ˜ M ) b eing ∞ + , wherea s Assumption 5.13 is equiv alent to α m ( ∂ ˜ M ) = α m +1 ( ∂ ˜ M ) = ∞ + . If the m - th Betti num b er b m ( ∂ ˜ M ) v anishes, then H ∂ M = 0, thus Assumption 5.1 4 is fulﬁlled. F or pr o ducts these con- ditions can be chec ked by us ing the pro duct formulas for Noviko v - Sh ubin inv a riants [L ¨ u02, Theo rem 2.5 5 (3)] and L 2 -Betti num b ers [L ¨ u02, Theorem 1.35 (4)]. Ex amples for which the conditions are fulﬁlled can b e found in [LLP0 0, p. 563]. In a similar wa y the pro duct formula applies to twisted higher signatures as studied in [LP9 9]. In [LP99, § 2] examples were giv en wher e the Laplacian on Ω ∗ ( ∂ M , F ∂ M ⊗ ϕ C ) is inv er tible. This implies that also the Laplacian on Ω ∗ ( ∂ M × N , ( F ∂ M ⊠ F N ) ⊗ ϕ C ) is inv ertible, thus the conditions of the theorem are fulﬁlled. Pro duct for m ulas fo r g eometric inv ar iants are relev ant for the following question: Assume that M 1 , M 2 are non-isomo r phic ele ments in a suitable categor y (top o- logical spa c e s up to homotopy/homeomorphism, manifolds up to diﬀeomor phism, manifolds with b ounda r y up to homotopy/homeomorphism/ diﬀeomorphism etc.). Under which co nditions on a clos ed manifold N do es it fo llow that ar e M 1 × N , M 2 × N no t isomo rphic? See the motiv a ting ex amples for the deﬁnition o f the higher ρ -inv a riants given in [W e99]. By a pplying the homotopy inv a riance r e sult of [LLP 00] (which was prov en ther e using diﬀerent b ounda r y conditions ; see the end of § 8 .2 for the justiﬁcation of using it here) w e o btain the fo llowing corollar y , whic h for simplicit y w e only formulate in the even-dimensional case and o nly for universal cov erings: Corollary 7.2. L et M 1 , M 2 b e orientable even-dimensional manifolds with b ound- ary having the same fundamental gr oup π M . L et N b e an orientable eve n- dimensional close d manifold with fundamental gr oup π N . Assume that the higher signatur e classes of M 1 , M 2 , M 1 × N , M 2 × N ar e wel l-deﬁne d (with r esp e ct t o the universal c overings). If the higher signatur e classes of M 1 , M 2 do not agr e e u p to sign in K 0 ( C ∗ r π M ) ⊗ Q and the higher signatur e class of N do es not vanish in K 0 ( C ∗ r π N ) ⊗ Q , then M 1 × N is not homotopic t o M 2 × N as a manifold with b oundary. The non-v anishing of higher s ignature classes for manifolds with b oundar y can be prov en by using the higher A tiyah-P ato di-Singer index theorem of Leich tnam- Piazza, see [LLP 00] and r e fer ences therein. The example in [LLP00, p. 62 4 f.] illustrates the cor ollary . While no detailed argu- men t was given there, for the ca lculation of the relev ant higher s ig natures a pro duct formula for Chern c haracter s a nd η -for ms might hav e been used. Alterna tiv ely one may use the ab ov e pro duct form ula. 30 CHARLOTTE W AHL 8. Fur ther remark s 8.1. Stabil ization in the o dd case. Let dim M = 2 m + 1. W e s ket ch the stabilization tr ic k a nd derive pro duct formulas if Assumption 5 .13 holds for the de Rham o pe r ator on Ω ∗ ( ∂ M , F ∂ M ) but Assumption 5.14 do es not hold. The stabilizatio n comes at a pric e: W e ne e d to r equire that Assumption 5.5 ho lds for the de Rham o per ator on Ω ∗ ( N , F N ) if N is o dd-dimensiona l and Assumption 5.13 if N is even-dimensional. If N is o dd-dimensio nal, it follows that the sig na ture class σ ( N , F N ) v anis hes. This is alr eady suggested b y the pro duct formula in Prop. 6.7 , where the left hand side dep ends on a Lag r angian while the right ha nd side is not. If N is even-dimensional, it follows that H N := K e r ∆ N ∩ Ω (dim N ) / 2 ( N , F N ) is a pro jectiv e B -mo dule. The construction relies on the concept of “ stable” L a grangia ns [LLP00, § 3]. While clearly inspired by it, our sta bilization pro cedure diﬀers from the o ne in [LP 03] a nd av o ids the additional choice of a submo dule a s sp eciﬁed in [LP03, Prop. 11]. Let X be an o dd-dimensio nal manifold with bo undary and as sume that the middle degree homology H o f ∂ X is non-z e r o. Let F X = A k . Then H ∂ X = H ⊗ A k . If L 0 ∈ H is a La grangia n, then σ L 0 ⊗A k ( X, F X ) = 0. F or the following c ho ose a trivialization H = C 2 n , wher e C 2 n is endow ed with the sta ndard s kewhermitian form (which is induced b y the standard symplectic fo rm on I R 2 n ). Consider the disjoint union M ∪ X . Let k be la rge enough s uc h that there is a Lagra ng ian L ⊂ H ∂ M ⊕ A 2 nk . The existence follow from [LLP00, Lemma 3.4] since [ D sign ∂ M ] = 0 b y the bo r dism inv aria nce of the index. W e de ﬁne σ L ( M , F M ) := σ L ( M ∪ X , F M ∪ F X ) . It is often useful to choose X with dimensio n diﬀerent fro m M , see b elow. Th us the right hand side requires a stra ight forward extension of the deﬁnition of the signature class to accommo date for comp onents of diﬀering dimension. One c hecks that the deﬁnition makes sense. It do e s not dep end on the choice of X nor of the trivializa tio n H ∼ = C 2 n , only on the c hoice of L in H ∂ M ⊕ A 2 nk . There is a stabilization argument (see [LLP00, § 3 ]) which allows to make the co nstruction independent of the choice o f n, k as well. In order to g e t the pro duct formula we use X := [0 , 1] for the deﬁnition of the signature class o f ( M , F M ). The homo logy of ∂ X is isomor phic to C 2 , which we endow with the standard skewhermitian form. W e identify C 2 ⊗ F ∂ X with A 2 k . Now w e a pply the pro duct fo rmula to σ L ( M ∪ X , F M ∪ F X ). The additiona l as- sumption on N implies that the de Rham o per ator on Ω ∗ ( ∂ ( X × N ) , F ∂ X ⊠ F N ) fulﬁlls Ass umption 5 .5 if N is odd- dimensional a nd Ass umption 5 .13 if N is ev en- dimensional. This is clea rly necessar y for the applicatio n of the pro duct formula. W e used that ∂ ( X × N ) = N ∪ N . Before we c a n formulate the result we need an additional deﬁnition for N ev en- dimensional: Let L ⊂ H ∂ M ⊕ A 2 k be a La g rangian. Then L ⊗ ∈ H ∂ M × N ⊕ A 2 k ⊗ H N . Since H N is pro jective, w e may em b ed H N int o B j for j lar ge enough. Let V b e the orthogona l co mplemen t of H N in B j and let L 0 ∈ C 2 be a Lag rangian. W e deﬁne the Lagrangian ˜ L ⊗ = L ⊗ ⊕ L 0 ⊗ A k ⊗ V ⊂ H ∂ M × N ⊕ A 2 k ⊗ B j . Prop ositio n 8.1. L et M b e o dd-dimensional and let Assu mption 5.13 ho ld for the de R ham op er ator on Ω ∗ ( ∂ M , F ∂ M ) . PRODUCT FORMULA FOR APS-CLASSES 31 (1) Le t N b e o dd-dimensional, and let Assumption 5.5 hold for the de Rham op er ator on Ω ∗ ( N , F N ) . Then σ ( N , F N ) = 0 and σ ( M × N , F M ⊠ F N ) = 0 . (2) Le t N b e even-dimensional, and let Assu mption 5.13 hol d for the de Rham op er ator on Ω ∗ ( ∂ M × N , F ∂ M ⊠ F N ) and for the de Rham op er ator on Ω ∗ ( N , F N ) . Then σ L ( M , F M ) ⊗ σ ( N , F N ) = σ ˜ L ⊗ ( M × N , F M ⊠ F N ) ∈ K 1 ( A ⊗ B ) . W e leav e it to the reader to fo r mu late a g eneralization of the pro po sition inv olving tensor pro ducts a s in § 7 . Pr o of. (1) If Assumption 5.5 holds, then there is a trivia liz ing opera tor for D sign N . Thu s its index v anishes. Cho ose a La g rangian L ⊂ H ∂ M ⊕ A 2 k . W e get that 0 = σ L ( M ∪ X , F M ∪ F X ) ⊗ σ ( N , F N ) = σ (( M ∪ X ) × N , ( F M ∪ F X ) ⊠ F N ) = σ ( M × N , F M ⊠ F N ) + σ ( X × N , F X ⊠ F N ) = σ ( M × N , F M ⊠ F N ) . Here the se c ond equality follo ws from P rop. 6 .7 . (2) F ro m the deﬁnition of σ L ( M , F M ) from a b ove and P rop. 6 .5 w e get that σ L ( M , F M ) ⊗ σ ( N , F N ) = σ L ⊗ (( M ∪ X ) × N , ( F M ∪ F X ) ⊠ F N ) . W e consider the ma nifo ld Y := ( M × N ) ∪ ( X × N ) ∪ X and the bundle F Y := ( F M ⊠ F N ) ∪ ( F X ⊠ F N ) ∪ ( F X ⊗ B j ) on Y . W e deﬁne Lagrangia ns L 1 , L 2 ⊂ H ∂ M × N ⊕ ( A 2 k ⊗ H N ) ⊕ ( A 2 k ⊗ B j ) by L 1 = { ( x, y , z ) | ( x, y ) ∈ L ⊗ , z ∈ L 0 ⊗ A k ⊗ B j } L 2 = { ( x, y , z ) | ( x, z ) ∈ ˜ L ⊗ , y ∈ L 0 ⊗ A k ⊗ H N } It holds that σ L 1 ( Y , F Y ) = σ L ⊗ (( M ∪ X ) × N , ( F M ∪ F X ) ⊠ F N ) and, by deﬁnition, that σ L 2 ( Y , F Y ) = σ ˜ L ⊗ ( M × N , F M ⊠ F N ) . It remains to calculate [ L 1 − L 2 ]. Note that L 2 is constructed from L 1 by in terchanging the last t wo co o r dinates o n the subspace H ∂ M × N ⊕ ( A 2 k ⊗ H N ) ⊕ ( A 2 k ⊗ H N ). Let U ( t ) b e the unitary whic h equals the identit y on H ∂ M × N ⊕ ( A 2 k ⊗ H N ) ⊕ ( A 2 k ⊗ V ) a nd on H ∂ M × N ⊕ ( A 2 k ⊗ H N ) ⊕ ( A 2 k ⊗ H N ) equals   1 0 0 0 e 2 it cos( t ) sin( t ) 0 − e 2 it sin( t ) cos( t )   . Then U ( t ) L 1 is a path of Lagra ngians with U (0) L 1 = L 1 and U ( π 2 ) L 1 = L 2 . Thus [ L 1 − L 2 ] = 0.  32 CHARLOTTE W AHL 8.2. N ormalization and homo top y in v ariance for higher si gnatures. F or the pro of of the homotopy inv aria nce of the higher signatures in [LLP0 0] the nor - malization of [HS92] of the s ig nature op erator and the gr ading was used. W e recall it here: In the even ca s e deﬁne D α = i p dα for α ∈ Ω p ( M , P M ) and set d sign,H S M := D + D ∗ . Deﬁne the gra ding op erator τ H S M by τ H S M α = i − p ( n − p ) ∗ α , where ∗ is the (standard) Hodg e duality op erator . (It diﬀers fro m the one in [BGV96, Def. 3.57 ], which agrees with our τ M .) Let U be the unitary deﬁned by U α = i p ( p − 1) / 2 α . Then U dU ∗ = D , hence U d sign M U ∗ = d sign,H S M . F ur- thermore U τ M U ∗ = ( − 1) n/ 2 τ H S M with n = dim M . The Cliﬀord op eratio ns are also unitarily eq uiv alent, since they are determined b y sig nature oper ator. Hence for the ca nonical s ymmetric trivia lizing o p er a tor A with resp ect to α M we g et that [ D sign,H S M ( U ∗ AU )] = ( − 1 ) n/ 2 [ D sign M (( − 1) n/ 2 A )]. The right hand side equals ( − 1) n/ 2 σ ( M , P M ) and the left hand side equals the signature clas s deﬁned in [LLP00]. Thus b oth cla sses agr ee up to sign and in particular ag ree in the classic a l case, when the dimension is divisible b y four. The homotop y inv ariance of the Chern c haracter of [ D sign,H S M ( U ∗ AU )], and hence of the Chern character of σ ( M , P M ), follows from the equa lit y established in the Appendix of [LLP00]. The equality was proven there under slightly str onger condi- tions. How ever it seems that the pr o of c a n b e adapted a s ne e de d here. It als o seems to the author that the pr o of a lready s hows the e quality on the level o f K -theo ry classes. W e leave the cons ideration of the odd ca se to the in terested rea der. References [BGV96] N. Berline, E. Getzler and M. V ergne, Hea t Kernels and D i r ac Op er ators (Grundlehren der mathematisc hen Wissensc haften 298), Springer, 1996 5 , 5. 1, 8.2 [Bl98] B. 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Product formula for Atiyah-Patodi-Singer index classes and higher signatures

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