Cobordism invariance of the family index

Cob ordism in v ariance of the family index ∗ Catarina Carv alho, Departmen t of Mathematics, Instituto Sup erior T ´ ecnico, T e chnic al Univ ersit y of Lisb on email: ccarv@math.i st.utl.pt Abstract W e give a K -theory pro of of the inv ar iance under cob ordism of the fa mily index. W e consider elliptic p se udo differential families o n a contin uous fibre bundle with smo oth fibres M ֒ → M → B , a nd define a notion of cob or da nt families using K 1 - groups on fibra tions w ith b o undary . W e show that the index of t wo such fa milies is the sa me using pro p e rties o f the push-forward map in K -theory to reduce it to families o n B × R n . In tro du ction The in v ariance under cob ordism of the F redholm index has b een a useful to ol in ind ex theory , b oth as a means t o obtain index f ormulas and as an imp ortant step to w ards so-calle d relativ e index th eorems. The particular case of twisted signature op erators w as crucial in Atiy ah and Singer’s first pro of of the index theorem on closed manifolds [1]. There are no w many pro ofs of cob ord ism inv ariance for Dirac op erators on closed manifolds, see for instance [5, 9, 21, 23], mostly relying on the geometric structure of the Dirac op erator and n ot easily extended to other settings. In [7], we ga v e a pro of of cob ordism in v ariance that applies to general elliptic pseud o different ial op erators, under suitable conditions on their K -theory sym b ol classes defining what we called symbol- cob ordism. See also [22] for an analytic form u lation of this r esult using the calculus of cusp pseudo d ifferen tial op erators on manifolds with b oun dary . In this pap er, w e giv e a K -theory proof of cob ordism in v ariance for families of elliptic pseudo d ifferen tial op erators on closed manifolds along the lines of [7]. In p articular, w e establish conditions on the symbol of a giv en elliptic pseudo d ifferen tial family on a ∗ Researc h supp orted in part by F unda¸ c˜ ao para a Ci ˆ encia e T ecnologia t hrough the grant FCT POCI/MA T/55958 /2004. 2000 Mathematics Subje ct Classific ation : Primary 19K56, Secondary 58J20, 14D99. Keywor ds and phr ases : F amily index. Cob ordism. 1 Cobordism inv ariance f or f amilies 2 b ound ary that yield the v anishing of its ind ex in the K -theory of the base. The crucial p oint is to use push -forw ard maps and functorialit y of th e f amily in dex in K -theory , as pro v ed by A tiy ah and S inger as the main tool in their pro of of the index th eorem for families [4]. W e consider con tin u ous f amilies, in the spirit of [4], so we w ill b e w orking in the s etting of con tinuous fi brations with s mo oth fibre d iffeomorphic to some giv en manifold. W e need a we ll-b eha v ed notion of b ound ary o v er th e base sp ace, in order to define sym b ol-cob ordism for families, and it will b e imp ortan t to constru ct b ound ary p reserving em b eddings int o Euclidean sp ace o v er the b ase. One can then use the indu ced pu sh-forward maps to reduce the p roblem to this setting. Cob ordism in v ariance for families is then a consequence of th e fun ctorialit y of the ind ex map and, moreo v er, it dep end s only on prop erties of the s ym b ol K -theory class of a giv en elliptic family on a b oun dary . Moreo ve r , we r efine here the result in [7], in that we giv e also a condition at the op erator lev el taking symb ols of s up en d ed op erators on manifolds with b oundary Index theory for families h as b een a su b ject of recen t study in the con text of sin- gular spaces, in particular, in the con text of op erators on fib ered manifolds and , more generally , of pseudo d ifferential calculi on group oids [13, 16, 17, 18, 19, 20, 24, 25, 26]. In what regards cob ordism in v ariance of family indices, there is an early result on families of Dirac op erators u sed by S hih [27 ] to yield a weak er v ersion of th e index theorem for families (extending A tiy ah and S in ger’s early cob ordism pr o of ). More recen tly , cob or- dism in v ariance has b een a key tool to study index theory on manifolds with b ound ary . Melrose and Piazza ga ve a p ro of of cobord ism inv ariance for smo oth families of Dirac op erators on a bou n dary , b oth f or th e o d d and ev en cases [16, 17], and used it to obtain an ind ex theorem for families of Dirac op erators on a manifold with b oundary in the con text of the b -calculus. Ev en if their p ro of uses A tiya h and Singer’s ind ex form u la, in that they chec k th at the top ologica l index is cob ordism in v arian t, the to ols used are similar to the ones w e apply here. They u se functorialit y of the top ological index with resp ect to push -forw ard s (whic h r educes to functorialit y of the Thom isomorp hism) and, in the even case, the v anish ing of the index relies on extension prop erties of the sym b ol class of the Dirac family on a b ound ary with resp ect to the K 1 -sym b ol for the self-adjoin t Dirac family on th e fib ered manifold with b oun dary . This extension condition on the symbols can b e seen to b e equiv alen t to th e one established here for general elliptic p s eudo differentia l families (see § 4). An analytic form ulation of cob ord ism inv ariance for pseudo differen tial families can also b e found in [18], in the con text of existence of inv ertible p ertu rbations of cus p pseudo d ifferen tial families on manifolds with b ound ary . On a very d ifferen t line, Hilsum [11] used the ge n eral fr amework of Hilb ert modu les o v er C ∗ -algebras to show cob ordism in v ariance of the index of Dirac op erators on f oliated m anifolds, that is, of contin u ous families of Dirac op erators on the lea v es. In this case, the so-called longitudinal index is an elemen t of K 0 ( C ∗ G ), wh ere G is the holonom y group oid and C ∗ G is the group oid Cobordism inv ariance f or f amilies 3 C ∗ -algebra. The approac h to cobordism inv ariance follo w ed here h ighlights the fact that the v an- ishing of the ind ex for an elliptic family on a b oundary d ep ends only on th e existence of suitable extensions of the sym b ol class to the b oundary . It is close in spirit w ith A tiyah and S inger’s K -theory p ro of of the index formula, in that it relies on functorialit y prop erties of the analytic index w ith resp ect to push -forw ards and on the constru ction of suitable embedd ings into Eu clidean space. In that resp ect, it has the adv antag e of ha ving immediate generalizati ons, namely to the non-compact and equiv ariant case. Moreo v er, it wo u ld b e of in terest to determine if one can relate it with the functorialit y prop erties of th e foliation index, whic h were crucial in the pr o of of the index theorem for foliations, in ord er to fin d K -theoretical conditions en suring the inv ariance of the longitudinal index of pseudo differentia l op erators u nder foliatio n cob ord ism. Giv en a closed smo oth man if old M a n d a compact base space B , w e consider a manifold M ov er B as a fi bre b undle with fi bre M and structure group Diff ( M ); this is a family of smo oth manifolds diffeomorphic to M such that the smo oth structure v aries con tinuously o ver B . A p seudo d ifferential op erator on M is a con tin uous family P = ( P b ) b ∈ B of pseudo d ifferen tial op erators on the fibres M b (Definition 1.2). If eac h P b is elliptic, then there is a well-defined index, ind( P ) := [k er ˜ P ] − [ C k ] ∈ K 0 ( B ) , where ˜ P is a su itable p erturb ation of P (see Definition 1.4). Moreo v er, in d( P ) only dep end s on the symb ol class of P , σ ( P ) ∈ K 0 ( T M ), where T M denotes the tangen t space along the fibres. Let no w X b e a manifold o ver B , with fi b er a manifold X with b oundary , M = ∂ X (w e assume that the structure group of X preserves M ). Then there is a corresp ondent con tin u ou s b undle of b ound aries M with fib er M and structure grou p Diff ( M ) (more precisely , the closed subgroup of those diffeomorphisms that extend to X ). W e call M the b oundary over B of X , M = ∂ B X . Of cours e, if the to tal space of X is smo oth, then ∂ B X = ∂ X . Many results on manifolds with bou n dary carry ov er to the b oun dary ov er B , namely that ∂ B X alwa ys has a collar in X and , in particular, that T X M ∼ = T M × R , where T X M is the restriction of the tangen t s p ace along th e fibr es of X to M . The constructions of [7] can therefore b e generalized to families and w e obtain a map of restriction of s y mb ols u X : K 1 ( T X ) → K 0 ( T M ) defined us in g maps of restriction to the b ound ary and the Bott isomorphism. Our main result states th at: Theorem 0.1. L et P b e an el liptic family of pseudo differ ential op er ators on a manifold M over B , with symb ol σ ( P ) ∈ K 0 ( T M ) . If X is such that ∂ B X = M and σ ( P ) ∈ Im u X , then ind( P ) = 0 . Cobordism inv ariance f or f amilies 4 W e sa y that the pair ( M , σ ( P )) as ab ov e is symbol-cob ordant to zero. Defining sym b ol-cob ordan t families in the ob vious wa y , w e h av e then that the family index is in v arian t u nder such r elation. T o pro ve Theorem 0.1, w e sho w that symb ol-cob ord ism is inv arian t with resp ect to push-forward, as long as we consider b oun dary-preserving emb eddings. W e see that there alw ays exist suc h emb eddings int o Euclidean space o ver B , and then that the relev ant K -group in this case is zero. F u nctorialit y of the family index with resp ect to push-forward then yields the resu lt. As a consequence of Theorem 0.1, we also obtain a condition for cob ordism in v ari- ance at the lev el of symbols of families on X (Corollary 4.6). The k ey p oin t is to identify elemen ts of K 1 ( T X ) with symb ol classes of s usp ended f amilies on X and noting that in this case restriction to the b oundary yields the in dicial op erator. As w e hav e seen in [7], cob ordism in v ariance holds also on non-compact manifolds, considering op erators that are multiplicati on ou tsid e a compact and taking the closure of a suitable ∗ -algebra. In particular, we saw that the f unctorialit y of the index with resp ect to push -forw ards can b e extended to th is class. (Note that in this case symb ol cob ordism is really a condition on the symb ols, since any manifold is cob ordan t to zero through a non-compact cob ordism.) Ev en though we do not p u rsue the non-compact approac h in detail here, one can c hec k that the compatibilit y of the index w ith push- forw ards giv en in [7] can b e extend ed to families of m ultiplication op erators outside a compact, so that an analogue of Theorem 0.1 follo ws in this case as w ell (see Remark 2.2). Note that the closure of th is class of op erators can b e used, through homotop y , to compute the index on large classes of op erators and, moreo ve r, it contai n s w ell-kno wn pseudo d ifferen tial calculi. See [6, 8] for details. It is imp ortan t to mentio n that Moroian u obtained in [22] a result equiv alen t to the one in [7 ] on closed manifolds, using quite differen t tec hniques. He used the calc u lus of cusp op erators to s h o w that an elliptic pseud o different ial op er ator on a b ound ary that has a su itable exte n s ion to a cusp pseu do differentia l op erator has zero index. This approac h has th e ad v ant age of giving an explicit condition for cob ordism in v ariance at the op erator lev el. He also ga ve a K -theory form u lation of this result and sh o we d it is equiv alent to th e one giv en in [7] in the closed case. There is a straigh tforward analogue for families of Moroian u ’s condition at the leve l of K -theory , and, u sing T h eorem 0.1, the same pro of applies to sho w that it yields cob ordism in v ariance for elliptic families. 1 The index for families W e consider families of pseudo different ial op erators as in [4] (see also [14] for a detailed accoun t). L et B b e a compact Hausd orff s p ace, M b e a smo oth compact manif old without b oundary , E a smo oth v ector bun dle o v er M . W e denote b y Diff ( M ) the Cobordism inv ariance f or f amilies 5 group of diffeomorphism s of M , endow ed with the top ology of uniform con v ergence on eac h deriv ativ e. Also, Diff ( M ; E ) denotes th e s ubgroup of Diff ( E ) of those d iffeomor- phisms that carry fib res to fib res linearly; c ho osing a connection on E , Diff ( M ; E ) is a top ological group. Definition 1.1. A manifold over B is is a fi bre bund le M ֒ → M → B with fi bre M and structure group Diff ( M ). A ve ctor bundle E o ve r M is said to b e a smo oth ve ctor bund le (along the fibr es) if E defin es a fibre b undle E ֒ → E → B with fib re E and structure group Diff ( E ; M ). A manifold o v er B is then a family of manifolds diffeomorphic to M s u c h that the smo oth structure v aries con tin uously . I f M = B × M , we call it a trivial family; lo cally , ev ery manifold ov er B is of this f orm. Note also th at a smo oth ve ctor bundle is a con tin u ou s family of smo oth vecto r bu ndles o ver M ; m oreo v er, giv en s uc h a bun d le E ֒ → E → B , the map Diff ( E ; M ) → Diff ( M ) induces a manifold M o v er B . The cotangen t and tangent bu ndles along the fi bres, denoted by T ∗ M , T M , resp ectiv ely , are smo oth vecto r bu ndles o v er M , as in Definition 1.1. A submanifold of a manifold M ֒ → M → B is jus t a sub-bun dle. It is clear that if N ⊂ M is closed, then ther e is a su bmanifold N of M with fibre N if, and only if, the structure group of M can b e reduced to the closed su bgroup Diff ( M , N ) of diffeomorphisms that pr eserv e N . Let no w E , F b e v ector bun d les o v er M , with Γ( E ), Γ( F ) the sp aces of smo oth sections of E , F . W e denote by Ψ m ( M ; E , F ) the space of order m classical ps eu do dif- feren tial op erators P : Γ( E ) → Γ( F ), as in [12] . It is a F r ´ ec het space with the top ology induced by the semi-norms of lo cal sym b ols and their deriv ativ es k P k K,α,β :=      ∂ α x ∂ β ξ p ( x, ξ ) (1 + | ξ | ) m −| β |      , where p is the classical sym b ol inducing P on some co ordinate c hart U of M trivializing E and F , K ⊂ U is compact, and for any m u lti-indices α , β . If w e let Diff ( E , F ; M ) b e the subgroup of Diff ( E ⊕ F ; M ) of those diffeomorphisms that map E to E and F to F , th en Diff ( E , F ; M ) ac ts on Ψ m ( M ; E , F ) by P 7→ f − 1 1 P f 2 , for f = ( f 1 , f 2 ) and the action is con tin uou s (see [4]). T o eac h pair of cont inuous families E , F of v ector b u ndles ov er M , with fi bres d iffeomorphic to E , F , we can then asso ciate a fibre bundle Ψ m ( M ; E , F ) → B with fibre Ψ m ( M ; E , F ) and stru cture group Diff ( E , F ; M ). The manifold M o ver B is induced by the map Diff ( E , F ; M ) → Diff ( M ). Definition 1.2. A c ontinuous family of pseudo differ ential op er ators on M is a con- tin uous section P of Ψ m ( M ; E , F ); w e write P = ( P b ) b ∈ B . The family P is s aid to b e el liptic if eac h P b , b ∈ B , is elliptic. Cobordism inv ariance f or f amilies 6 If M = B × M , E = B × E , F = B × F , then a con tin uous family is just a con tin uous map P : B → Ψ m ( M ; E , F ). Ev ery family is lo cally of this f orm. Recall also from [12] that there is a surj ective symb ol map σ : Ψ m ( M ; E , F ) → S m ( M ; E , F ) , where S m ( M ; E , F ) is the space of classical symbols, that is, sm o oth maps on the cotangen t bun dle T ∗ M \ 0 with v alues in E n d( E , F ), that are p ositiv ely homogeneous of d egree m . The map σ is in v arian t u nder the actio n of Diff ( E , F ; M ) and the action is con tin u ou s (end owing S m ( M ; E , F ) with the sup-norm top ology on the sphere b undle S M ), so that w e get a fibre bund le S m ( M ; E , F ) o v er B with fib re S m ( M ; E , F ) and group Diff ( E , F ; M ). Definition 1.3. The symb ol σ B ( P ) of a family P = ( P b ) b ∈ B is the con tin uous section of S m ( M ; E , F ) giv en fibrewise b y σ ( P b ). The family of smo oth symbols σ ( P b ) dep end s contin u ously on b ∈ B . Elemen ts of S m ( M ; E , F ) are maps T ∗ M → End( π ∗ E , π ∗ F ), with T ∗ M the cotangen t bu ndle along the fibres and π : T ∗ M → M the p ro jection. If the family is elliptic, then eac h σ ( P b ) is in vertible outside the zero-section, so that σ B ( P ) is in ve r tib le outside a compact in T ∗ M . Moreo v er, endowing M with a metric, that is, with a con tin uous family of metrics on the fi bres T M , w e can identify T ∗ M w ith T M and (el liptic) symb ols reduce to (in vertible) maps on the sphere bu ndle S M . Finally , we no w define the index of an elliptic family . Let P b e an elliptic family in Ψ m ( M ; E , F ), so that eac h P b , b ∈ B , is F red h olm. If dim k er P b w ere lo cally constant, then the family of v ector spaces k er P b w ould form a v ector bun dle o ver B , and the same for coke r P b ; in th is case, one could define th e index of the family P as the K -theory class [k er P b ] − [cok er P b ] ∈ K 0 ( B ). In the general case, one can define an elli p tic family ˜ P : Γ( E ) ⊕ C k → Γ ( F ) such that ˜ P b is sur jectiv e, for eac h b ∈ B , where ˜ P b ( u ; λ 1 , ..., λ k ) := P b ( u ) + λ 1 w 1 ( b ) + ... + λ k w k ( b ) , (1) for some smo oth sections w i of F , i = 1 , ..., k . Since eac h ˜ P b is surjectiv e, the family k er ˜ P b indeed defines a vec tor bund le o ver B (eac h ˜ P b is F redholm and the F redholm index is lo cally constan t). Moreo v er, th e class [k er ˜ P ] − [ B × C k ] ∈ K 0 ( B ) dep ends only on P (see [4, 14]). Definition 1.4. The analytic index of the el liptic family P is giv en b y ind( P ) := [k er ˜ P ] − [ B × C k ] ∈ K 0 ( B ) , where ˜ P : Γ( E ) ⊕ C k → Γ ( F ) is as in (1). Cobordism inv ariance f or f amilies 7 The in dex defined ab o ve is homotop y in v arian t, hence it d ep ends only on the (h o- motop y class) of the symb ol σ B ( P ). Moreo v er, it can b e seen that the index only dep end s on the v alues of the symbol on S M , that is, the in d ex of a family P do es n ot dep end on the order of P . Therefore, as far as computing ind ices go es, it is just as go o d taking op erators of order 0. No w, the sym b ol of an elliptic family σ B ( P ) is in vertible outside a compact, hence it defin es a class [ σ B ( P )] := [ π ∗ E , π ∗ F , σ B ( P )] ∈ K 0 ( T M ) (2) (where we identify T ∗ M with T M ). Since K 0 ( T M ) is exhausted by symb ol classes, just as in the op erator case, we ha ve then a well-defined family index m ap ind : K 0 ( T M ) → K 0 ( B ) , [ π ∗ E , π ∗ F , σ B ( P )] 7→ in d( P ) , (3) whic h is, moreo ver, a group homomorphism. W e will show in this pap er that the family index is cob ordism inv arian t und er a su itable notion of cob ordant families that dep ends only on the manifolds in vo lved and on the K -theory class of the f amily sym b ol. T o do this, w e will mak e use of pu sh-forward maps. 2 Push-forw ard maps W e consider no w a manifold X ov er B with fib er some sm o oth manifold X p ossibly with b oundary . In this case, w e assume that the stru cture group of X is Diff ( X, ∂ X ), the sub group of Diff ( X ) of th ose diffeomorphisms that lea ve ∂ X in v arian t. Also, the structure group of a smo oth bund le E o ve r X is give n in this case by Diff ( E , X ; ∂ E ) of those elemen ts of Diff ( E , X ) that preserve ∂ E = E | ∂ X . Recall that an emb edding i : X → Y is neat, follo win g [10], if ∂ X = X ∩ ∂ Y and if X in tersects ∂ Y transve r s ally; in this case, the b oundaries are also em b edded and X alw a ys has an op en tubu lar neigh b orho o d in Y . Let i : X → B × V b e a morp hism, w here V d enotes either R m or R m + , the p ositiv e half-space in R m , in case X has a b oundary . W e sa y that i is a ne at emb e dding of manifolds over B if i restricts to neat emb eddings i b : X b → V on eac h fib er of X . Note that in this case, i is a homeomorphism on to i ( X ), smo oth on eac h fib er. W e iden tify here the op en tubular neigh b ourh o o d of X b in V with the normal b undle N b = N . W e obtain then a con tin uous family N of vect or b undles o ver X with fib re N ; moreov er, N is op en in B × V . The in duced embedd ing T X → B × T V is suc h that the fib erwise em b eddings are also neat. The normal bu ndle of T X b in T V is giv en by T N b ∼ = π ∗ X ( N b ⊕ N b ), where π X : T X → X is the pr o jection, so that the v ector bu ndle T N can b e iden tified with π ∗ X ( N ⊕ N ), π X : T X → X , and th erefore it has a complex structure. W e h a v e then a Thom isomorphism ρ : K 0 ( T X ) → K 0 ( T N ) . (4) Cobordism inv ariance f or f amilies 8 Comp osing with the K -theory map indu ced by the op en inclusion T N → B × T V h : K 0 ( T N ) → K 0 ( B × T V ) , (5) one obtains the push-forwar d map i ! = h ◦ ρ : K 0 ( T X ) → K 0 ( B × T V ) . (6) One can also easily define a p u sh-forward map b et w een K 1 -groups: simply tak e the induced em b edding T X × R → B × T V × R to get i ! : K 1 ( T X ) → K 1 ( B × T V ). All the constructions ab o ve w ork in the same w a y for maps i : X → Y , for some manifold Y o ve r B , as long as i r estricts to neat em b eddings on the fi b ers. H ere w e only need the case Y = B × V ; we will see later (Corollary 4.1) that such an em b edding in to a trivial f amily of manifolds alw ays exists, and in fact it can b e chosen so as to yield an embed ding M → R n of the b oundaries o ve r B . One of the crucial p rop erties of the pus h -forw ard m ap is its functorialit y with resp ect to the ind ex. In their pro of of the index theorem for families [4], At iyah and S inger sho w th at: Theorem 2.1 (A tiy ah-Sin ger) . L et M b e a manifold over B , with fibr e M c omp act without b oundary. Then the fol lowing diagr am c ommutes K 0 ( T M ) i ! − − − − → K 0 ( B × T R m ) ind   y   y ind K 0 ( B ) K 0 ( B ) (7) wher e ind stands for the analytic i ndex for families, as in (3). The map on the righ t-hand s ide computes the index of families of op erators that are the identit y outside a compact of R m (see also Remark 2.2 b elo w). The families index th eorem in this setting states that it coincides with the Bott isomorphism. Th e pro of of Th eorem 2.1 go es m u c h as in the case of the classic index theorem, taking extra care with the added sections in the defin ition of the family ind ex. Remark 2.2 (Op erators that are multiplicat ion at infi nit y) . Let M b e a non-compact manifold, E , F v ector bundles on M . In [7], we considered the class Ψ mult ( M ; E , F ) of pseudo d ifferen tial op erators on M that are m ultiplication at infi nit y . F or an op er ator P : C ∞ c ( M ; E ) → C ∞ ( M ; F ), w e ha ve P ∈ Ψ mult ( M ; E , F ) if, and only if, there is a compact K ⊂ M and p : E → F su c h that P = P 1 + p, (8) where P 1 : C ∞ c ( M ; E ) → C ∞ ( M ; F ) is a pseudo differentia l op erator on M suc h Im P 1 ⊂ C ∞ K ( M ; F ), Im P ∗ 1 ⊂ C ∞ K ( M ; E ), that is, P 1 has compactly su pp orted Sc hw artz k ernel. Cobordism inv ariance f or f amilies 9 This means that if sup p u ⊂ M \ K , then P u = pu , P ∗ u = p ∗ u . In particular, σ ( P ) is indep en d en t of ξ for x / ∈ K . Let now M , E , F b e as b efore, with n on-compact M , o ver some co mp act b ase space B , and consider the class Ψ mult ( M ; E , F ) of con tin uous families of pseudo differen tial op erators that are multiplicatio n at infinity , with b ound ed symb ols. S ince B is co mp act, giv en such a family P = ( P b ), w e can pic k a compact K ⊂ M b , indep endent of b ∈ B , in the ab ov e definition. W e sa y that P ∈ Ψ mult ( M ; E , F ) is ful ly el liptic if, for eac h b ∈ B , σ ( P b ) is inv ertible on T ∗ M \ 0 and the inv erse is b ounded , which means that it also indep enden t of ξ for x / ∈ K . S ince σ ( P b ) is constan t on fi b ers ov er M \ K , it is in vertible outside the (fixed) compact π − 1 ( K ) ⊂ T ∗ M , with π : B ∗ M → M the ball cotangen t bundle. Hence, if P is a fully elliptic family , then the sy mb ol σ B ( P ) is inv ertible outside a compact in T M and it d efi nes a K -theory class. Moreo ver, the family index (as in Definition 1.4) is well defined, since we ha v e in this case k er P b ⊂ C ∞ K ( M ; E ), cok er P b ⊂ C ∞ K ( M ; F ), and it dep end s only on σ ( P ). The excision p rop erty of th e ind ex giv en in [4] h olds in this setting and one can use it to extend the compatibilit y of the family index with pu s h-forwa rd s, exactly as in [7] (Theorem 3.9), since all the constructions carry o ver nicely to families. W e obtain then, using the notation of Th eorem 2.1, with M not n ecessarily compact, ind ◦ i ! = ind (9) where no w we are computing the index of families of op erators that are multiplic ation at infinity on M . 3 Cob ordant families W e define h ere the notion of symb ol-cob ordism for families of pseud o differenti al op er- ators, generalizing [7]; on e of its main features is that it is pr eserved b y push -forw ard maps. Let X b e a manifold with b oundary and M = ∂ X . If X is a manifold o ver B with fibre X , the inclusion M → X in d uces a sub manifold M o v er B , with fibr e M and structure group Diff ( M ). W e call M th e b oundary over B of X , and write M = ∂ B X . If X ֒ → X → B is giv en by a sm o oth fibration, then ∂ B X = ∂ X , the indu ced b oun d ary fibration. Note that if M = ∂ B X , then the stru cture grou p of M can b e red uced to the closed subgroup of diffeomorphisms that can b e extended to X . Conv ers ely , giv en M and X suc h that ∂ X = M , then there exists X with ∂ B X = M if, and only if, the structur e group of M can b e reduced to the sub group of diffeomophisms that can b e extended to X . Cobordism inv ariance f or f amilies 10 Before pro ceeding to giv e our main definition, we sh o w that a con tinuous f amily of b ound aries alw a ys has a collar n eighb orh o o d o ver B . Prop osition 3.1. L et M , X b e manifolds over B , with fibr es M , X , su c h that M = ∂ B X . Then, ther e is an isomorp hism Φ : M × [0 , 1) → W ⊂ X , wher e W is a submanifold of X with fibr e W ⊂ X , op e n, and structur e gr oup Diff ( W , M ) , such that Φ r estricts to a c ol lar neighb orho o d of the b oundary on e ach fibr e. Pro of. Let φ : M × [0 , 1] → W ⊂ X b e a collar neigh b orh o o d of M in X . W e first note that the structure group Diff ( X, M ) of X can b e r educed to Diff ( X , W ): for eac h ϕ ∈ Diff ( X , M ), it is easy to c hec k that there are 0 < α, β ≤ 1 suc h th at ϕ ( W α ) = W β , with W α ∼ = M × [0 , α ], W β ∼ = M × [0 , β ]; p ic k ψ α , ψ β ∈ Diff ( X , M ) su ch that ψ α ( W ) = W α , ψ β ( W ) = W β . W e h a v e then ψ − 1 β ϕψ α ∈ Diff ( X , W ). W e conclude that there is a subm an if old W of X , with fibr e W . T o sh o w that there is a globally defined map Φ : M × [0 , 1] → W ⊂ X , we need to c hec k compatibilit y with tran s ition fun ctions. Note that the transition functions for M and W are giv en b y the r estriction of transition functions of X , so that if ϕ ∈ Diff ( X , W ) is a transition function for X , we w an t to see that ϕ ◦ φ = φ ◦ ( ϕ | M , 1) . (10) Noting that ϕ − 1 ◦ φ ◦ ( ϕ | M , 1) : M × [0 , 1] → W is also a collar for M in X , it follo ws from [10] (Th eorem 8.1.8) that there is ϕ ′ diffeotopic to ϕ , with ϕ ′ = ϕ on M and on X \ W , suc h that φ = ( ϕ ′ ) − 1 φ ◦ ( ϕ ′ | M , 1) (11) on M × [0 , α ], for some 0 < α ≤ 1. Iden tifying M × [0 , α ] with M × [0 , 1] through diffeomorphisms of X , w e can assu me that (11) h olds on M × [0 , 1]. Since the lo cal smo oth structure of X remains inv arian t un der diffeot opies, it follo ws that there is a map Φ : M × [0 , 1] → W , with W a m anifold ov er B with fibre W and structure group Diff ( W , M ), suc h that Φ restricts to φ on eac h fib r e. Moreo v er, Φ is an isomorphism. Restricting Φ to M × [0 , 1) w e get the result. Let th en M and X b e m anifolds ov er B with M = ∂ B X . It is easy to c hec k that T X M , the r estriction of the v ertical tangent bund le T X to M , is also a su b- bund le of T X , with fib er T X | M (and structure grou p Diff ( T X | M , M )) . The inclus ion T X M → T X indu ces a map of restriction to the b oundary in K -theory r X , M : K j ( T X ) → K j ( T X M ) . (12) As in [7], it is not hard to sh o w that pu s h-forwa rd maps b eha ve w ell with resp ect to restriction to the b ound ary . Cobordism inv ariance f or f amilies 11 Prop osition 3.2. L et M = ∂ B X . If i : X → B × V is an emb e dding of manifolds over B that r estricts to an emb e dding M → B × W , wher e W = ∂ V , then the fol lowing diagr am c ommutes K j ( T X ) i ! − − − − → K j ( B × T V ) r X , M   y   y r V ,W K j ( T X M ) i ! − − − − → K j ( B × T V W ) . Pro of. F unctorialit y of th e Thom isomorphism yields that it comm utes with re- striction m ap s . T o s ee that the same happ ens with resp ect to the K -theory maps h X : K j ( T N ) → K j ( B × T V ) and h M : K j ( T N M ) → K j ( B × T V W ), as in (5), we just c hec k that r V , W ◦ h X and h M ◦ r X , M can b e written as maps in K -theory induced b y th e same morph ism. No w, it is we ll kn own th at there is a smo oth isomorphism T X | M ∼ = T M × R , for M = ∂ X . It follo ws from Prop osition 3.1 that the same holds for tangen t bund les o v er B : if Φ : M × [0 , 1) → W is a collar neigh b orho o d o v er B , W ⊂ X with op en fib er W , then T X M = T W M ∼ = T ( M × [0 , 1)) M× { 0 } = T M × R , where T M × R is the v ector b u ndle o v er M giv en by T M ⊕ ( M × R ), so that the fib er is T M × R and the structure group acts trivially on R . T h e restriction map b ecomes no w r X , M : K 1 ( T X ) → K 1 ( T M × R ) = K 0 ( T M × R 2 ) . T aking the Bott isomorphism β : K 0 ( T M ) → K 0 ( T M × R 2 ), we consider the map u X , M := β − 1 r X , M : K 1 ( T X ) → K 0 ( T M ) , (13) whic h in [7] w as referred to as restriction of s y mb ols. W e will see later that elemen ts of K 1 ( T X ) can b e r egarded as symb ol classes of su sp end ed op erators. Let n o w, for i = 1 , 2, M i b e a manifold without b oundary and M i b e a manifold o v er B w ith fib ers M i . I f X is suc h that ∂ B X = M 1 ⊔ M 2 , then th e structure group of X can b e r educed to th e subgroup of d iffeomorph isms that leav e b oth M 1 and M 2 in v arian t, so that M 1 , M 2 are s ubmanifolds of X , and it is easy to chec k that there are wel l-defin ed maps of restriction of symbols u X , M i : K 1 ( T X ) → K 0 ( T M i ) , (14) for i = 1 , 2, and that u X , M = X , M 1 ⊕ u X , M 2 . Let P i b e an elliptic family of pseudo d- ifferen tial op erators on M i with symb ol σ i = σ ( P i ) ∈ K 0 ( T M i ), i = 0 , 1. W e give the follo wing definition: Cobordism inv ariance f or f amilies 12 Definition 3.3. W e sa y th at ( M 1 , P 1 ) and ( M 2 , P 2 ) are symb ol-c ob or dant families if there is a manifold X o v er B , w ith fib er X , and ω ∈ K 1 ( T X ), suc h that (i) ∂ B X = M 1 ⊔ M 2 (in particular, ∂ X = M 1 ⊔ M 2 ); (ii) σ 1 = u X , M 1 ( ω ), σ 2 = − u X , M 2 ( ω ). W e write ( M 1 , σ 1 ) ∼ ( M 2 , σ 2 ). Note that ( M 1 , P 1 ) and ( M 2 , P 2 ) are cob ordant if, and on ly if, ( P 1 ⊕ P ∗ 2 , M 1 ⊔ M 2 ) is cob ordant to zero, wh ere P ∗ 2 is the adjoin t family and P 1 ⊕ P 2 is a family on M 1 ⊔ M 2 (w e h a v e − σ 2 = σ ( P ∗ 2 )). As in [7], the defin ition ab o ve defines an equiv alence r elation on p airs ( M , P ) within manifolds with the same fib er dimen s ion and the collection of equiv alence classes is an ab elian group. W e no w c hec k that the notion of cob ordan t families is preserved by push -forw ard maps. Prop osition 3.4. L et X , M b e manifold s over B with ∂ B X = M and let i : X → B × R m +1 + b e an emb e dding of manifolds over B that induc es an emb e dding i : M → B × R m . Then the fol lowing diagr am c ommutes K 1 ( T X ) i ! − − − − → K 1 ( B × T R m +1 + ) u X , M   y   y u m K 0 ( T M ) − − − − → i ! K 0 ( B × T R m ) , wher e u X , M and u m ar e maps of r estriction of symb ols, as in (13). Pro of. It follo ws directly from (3. 2 ) and the fact that pu sh-forwa r d maps commute with the Bott isomorp hism. Corollary 3.5. In the c onditions of Pr op osition 3.4, if σ ∈ K 0 ( T M ) and ( M , σ ) ∼ 0 then ( B × R m , i ! ( σ )) ∼ 0 . Pro of. Straigh tforw ard from the definition of symbol-cob ordism and Prop osition 3.4. 4 Cob ordism in v ariance W e no w show that if a giv en elliptic family is symb ol-cob ord an t to zero, then its index is zero. The main idea is to use C orollary 3.5 to reduce the pro of to Eu clidean space where it w ill b e trivial. W e start with sho wing that an embedd ing suc h as in Prop osition 3.4 indeed exists. Let then X b e a manifold ov er B with ∂ B X = M , X , M with fi bres X , M , resp ectiv ely . Also, let i : X → R k b e an em b ed ding, whic h alw ays exists b y Whitney’s theorem. It is easy to sh o w that, for compact B , one can defin e an em b edding ˜ i : X → B × R m , m ≥ k , (15) Cobordism inv ariance f or f amilies 13 whic h restricts to an embedd ing on eac h fib er X b of X (see [4]). W e n eed h o w ever an em b edd ing that restricts to an em b edding of th e b ound ary M o v er B . In [7], we constructed suc h an em b edding using a collar neigh b orho o d of M in X , which is what w e shall do here. W e use the collar neigh b orho o d o v er B giv en in Prop osition 3.1 to define a map ˜ α : X → [0 , 1] , M = α − 1 (0) , (16) suc h that on eac h fib er, ˜ α restricts to a defining fu nction of th e b oundary ˜ α b = α : X b → [0 , 1], with α − 1 (0) = M and dα 6 = 0 on M . Prop osition 4.1. L et X b e a manifold over B with ∂ B X = M . Ther e exists a ne at emb e dding over B h : X → B × R m +1 + , which induc es an emb e dding M → B × R m . Pro of. Let ˜ i : X → B × R m b e an embed ding o v er B , as in (15) and ˜ α : X → [0 , 1) as in (16). Define h := ( ˜ i, ˜ α ) : X → B × R m +1 + . Then h restricts to neat emb eddings h b = ( i, α ), w h ere i : X → R m is an em b edd ing and α a b ound ary defining function. W e u se the emb ed ding giv en ab o v e, together with Prop osition 3.4 to redu ce the pro of of cob ordism inv ariance to B × R m . It is trivial in this case, as the follo wing lemma will s h o w. Lemma 4.2. K 0 ([0 , 1) × U ) = 0 , f or any lo c al ly c omp act sp ac e U . Pro of. W e ha ve K 0 ([0 , 1) × U ) = K 0 ( C 0 ([0 , 1) × U )), where C 0 ([0 , 1) × U ) is the C ∗ -algebra of con tin uous f unctions on [0 , 1) × U ) that v anish outside a compact set. Since C 0 ([0 , 1) × U ) ∼ = C 0 ([0 , 1) , C 0 ( U )) , whic h is jus t the cone of the C ∗ -algebra C 0 ( U ) and hence, it is homotop y equiv alen t to 0, th e r esult follo ws for K 0 , and also for K 1 , writing K 1 ([0 , 1) × U ) = K 0 ([0 , 1) × U × R ) = 0. In particular, it follo ws from the lemma ab o ve that, for any (lo cally) compact b ase space B and m ∈ N , w e ha ve K 1 ( B × T R m +1 + ) = K 0 ( B × R 2 m +1 × [0 , 1)) = 0 . (17) W e are no w ready to prov e: Theorem 4.3. L et M b e a manifold over B and P ∈ Ψ( M ; E , F ) b e an el liptic family of pseudo differ ential op er ators over B with symb ol σ = σ B ( P ) ∈ K 0 ( T M ) . If ( M , σ ) is symb ol-c ob or dant to zer o, then ind( P ) = 0 . Cobordism inv ariance f or f amilies 14 Pro of. F rom the defin ition of sym b ol-cob ordism, w e kno w th at there is a manifold X o ver B , with fib er X , s uc h that ∂ B X = M and a class ω ∈ K 1 ( T X ) suc h that σ = u X , M ( ω ). Considering an em b edd ing h : X → B × R m +1 + , as in C orollary 4.1, we ha v e from Corollary 3.5 th at ( B × R m , h ! ( σ )) is also cob ordant to zero, so that h ! ( σ ) = u m ( h ! ( ω )) (w e are using the notations of Prop osition 3.4). But h ! ( ω ) ∈ K 1 ( B × T R m +1 + ) = 0, therefore h ! ( σ ) = 0. Since the in d ex is in v arian t und er push -forw ard maps (Theorem 2.1) we ha v e finally ind( P ) = ind( σ ) = ind( h ! ( σ )) = 0 . Corollary 4.4. L et P 1 ∈ Ψ( M 1 ; E 1 , F 1 ) , P 2 ∈ Ψ( M 2 ; E 2 , F 2 ) b e el liptic families of pseudo differ ential op er ators over B with symb ols σ 1 , σ 2 , r esp e ctively. If ( M 1 , σ 1 ) is symb ol-c ob or dant to ( M 2 , σ 2 ) , then in d( P 1 ) = ind( P 2 ) . Pro of. It follo ws fr om ( P 1 ⊕ P ∗ 2 , M 1 ⊔ M 2 ) b eing cob ordant to ze r o, so that ind( P 1 ⊕ P ∗ 2 ) = 0, and th e additivit y of the family ind ex. In the spirit of Remark 2.2, w e note that the definition of sym b ol-cob ord ism and the results giv en th ereafter are pur ely K -theoretica l and d o not dep end in any w a y on the compacit y of the fi b res M of M . Hence, Theorem 4.3 holds for non-compact M in the setting of pseudo differential op er ators that are multiplicat ion at infin ity and we state it h ere as a corollary . Corollary 4.5. L et M b e a man if old over B with non-c omp act fibr es and P = ( P b ) b e a family of ful ly el liptic op er ators in Ψ mult ( M ; E , F ) ,with symb ol σ = σ B ( P ) ∈ K 0 ( T M ) . If ( M , σ ) ∼ 0 , we have ind( P ) = 0 . In particular, we obtain straigh ta w ay the inv ariance of the index for families that are homotopic to multiplicatio n families outside a compact, for instance for families of scattering op er ators; see [8] for d etails. Let X ֒ → X → B b e a manifold o ver B ,with X a manifold with b oundary . W e no w identify classes in K 1 ( T X ) with sym b ols of susp ended operators. W e work here in the setting of the b -calculus (see [15] for an extended treatmen t). W e assume that X is endo wed with a con tin uous family of exact b -metrics and consid er the class Ψ b ( X , G ) of families of b -pseud o differenti al op erators Ψ b ( X, G ) on the fib ers, as in the b ound- aryless case. Th e symb ol of s uc h a f amily is defined on the b -tangen t space along the fib ers b T X ; since there is a (non canonical) bundle isomorp h ism b T X ∼ = T X , at th e K -theory lev el we can wo rk with T X . T o eac h op er ator Q b ∈ Ψ b ( X, G ), b ∈ B , one can asso ciate a translation inv arian t ps eu do differentia l op erator on ∂ X × R , that is, a 1-parameter family of pseudo differential op erators on M = ∂ X , the in d icial family I M ( Q b ) ∈ Ψ sus ( M , G M ) (w h ere the conv olution kernels are assu m ed to v anish r apidly Cobordism inv ariance f or f amilies 15 with all deriv ativ es at infi nit y). It f ollo ws from the definition th at there is a compati- bilit y cond ition with r esp ect to th e p rincipal symbol σ ( I M ( Q )) = σ B ( Q ) |M . (18) In p articular, if Q is elliptic then I M ( Q ) also is and [ σ ( I M ( Q ))] ∈ K 0 ( T ( ∂ X ) × R ). If w e no w tak e s u sp en d ed elliptic f amilies of b -op erators on X , that is, a family Q = ( Q t ) on X ֒ → X ′ = X × [0 , 1] → B ′ = B × [0 , 1], dep ending sm o othly on t , and assu me that Q 0 , Q 1 are in v ertible, then the sym b ol σ B ′ ( Q ) defines a class in K 0 ( T X × (0 , 1)) = K 1 ( T X ). Moreo v er, K 1 ( T X ) is exhausted by suc h sym b ol classes. Note that I M ( Q ) is no w d efined on the d ouble susp ension of M . Corollary 4.6. L et M b e a manifold over B and P ∈ Ψ( M ; E , F ) b e an el liptic family with symb ol [ σ B ( P )] ∈ K 0 ( T M ) . If M = ∂ B X , for some manifold X over B and ther e is an el liptic family Q ∈ Ψ b ( X × [0 , 1]; G ) , i nv e rtible at t = 0 , 1 , with i ndicial family I M ( Q ) , such that [ σ ( I M ( Q ))] = [ σ B ( P )] ∪ β with β the Bott c lass, then ind( P ) = 0 . Pro of. It follo ws f rom (18) that [ σ ( I M ( Q ))] = r M [ σ B ′ ( Q )], with r M the map of restriction to the b oundary . Hence, [ σ B ( P )] = u M [ σ B ′ ( Q )], with u M the sym b ol re- striction map (13 ), whic h giv es that ( M , [ σ B ( P )]) is sym b ol-cob ordan t to zero. Let n o w Q ∈ Ψ b ( X , G ) b e a family of elliptic self-adjoin t op erators. As in [3], w e consider the su sp en d ed f amily Q sus := cos( πt ) + iQ sin( π t ) , t ∈ [0 , 1] . (19) It is an elliptic family o ve r B ′ = B × [0 , 1], in vertible at t = 0 , 1, and ca n b e rega r d ed as an elemen t of Ψ b ( X ′ , G ), with X ′ = X × [0 , 1] fib ered o ver B ′ . Moreov er, Q is h omotopic to Q ′ , through elliptic self-adjoin t op erators, if and only if Q sus is homotopic to Q ′ sus , through elliptic op erators. Hence one can asso ciate to Q the class [ σ B ( Q )] 1 := [ σ B ′ ( Q sus )] ∈ K 0 ( T X × (0 , 1)) = K 1 ( T X ) . (2 0) If r M : K 1 ( T X ) → K 1 ( T M X ) is the map of restriction to the b oundary , w e ha v e that r M [ σ B ( Q )] 1 coincides with the symb ol of the indicial family of Q sus . In th is setting, the sufficien t condition for cob ord ism inv ariance giv en in Corollary 4.6 b ecomes r M [ σ B ( Q )] 1 = β [ σ B ( P )] . (21) for some self-adjoint, elliptic family of b -pseudo differential op erators on X . Cobordism inv ariance f or f amilies 16 The ab ov e construction app lies to families of Dirac op erators (see [15 , 17], and also [16] f or the o dd case). T ak e smo oth fib ered manifolds M , X o ver B , with compact, orien ted fib ers M = ∂ X , with M ev en-dimensional. W e assu me th at M and X are also orien ted, in that the structure group redu ces to orien tation preserving diffeomorphisms. Assume also, as b efore, that X is endo we d with smo othly v arying b -metrics. Let E b e an Hermitian bund le of Clifford mo dules ov er b T ∗ X , end o we d with a smo oth unitary Clifford connection on the fi bres. Let ð b e the asso ciated family of Dirac op erators on X ; then ð is an elliptic, self-adjoint family of b -op erators. Since on X = M w e hav e b T ∗ X ∼ = R  dx x  ⊕ T ∗ M , for s ome b oundary defi n ing function x , there is an induced Clifford action of T ∗ M on E M . Moreo v er, at the b ound ary , E decomp oses as E M = E + 0 ⊕ E − 0 . If we d enote b y ð 0 the induced family of Dirac op erators on M , then ð 0 is self-adjoin t, o dd and elliptic. It w as noted b y Melrose and Piazza in [17] th at ð and ð + 0 v erify (21) and, in fact, this co n d ition was crucial to sho w that th e top ological family index - defined essen tially through the Th om isomorphism - of ð + 0 v anish es, so that cob ordism inv ariance for a family of Dirac op erators on a b ound ary then follo ws f rom the families ind ex theorem. (See also [16] for the o d d case and the pr o of ). No w w e consider families of signature op erators and c hec k that Theorem 4.3 applies to sho w that the index of suc h a family on a b oundary v anishes, using a differen t approac h. Once we chec k that the relev ant ob jects are w ell d efined o v er the base, the pro of in K -theory go es m uc h as in the case of the single signature op erator (Prop osition 2.8 in [7]). Let M ֒ → M → B b e a s mo oth fibration with fibre an ev en-dimensional, oriented manifold M . The b undle Λ ∗ ( T ∗ M ) of forms along the fibres is a smo oth bu ndle ov er M with fi b re diffeomorph ic to Λ( T ∗ M ). There is a smo oth action of Λ ∗ ( T ∗ M ) on itself, Λ ∗ ( T ∗ M ) × Λ ∗ ( T ∗ M ) → Λ ∗ ( T ∗ M ) , ( ξ , η ) 7→ c ( ξ ) η , (22) suc h that for ξ ∈ T ∗ M , c ( ξ ) η = ξ ∧ η − i ξ ( η ). W e also ha ve a Z 2 -grading Λ ∗ ( T ∗ M ) = Λ + ( T ∗ M ) ⊕ Λ − ( T ∗ M ), where Λ ± ( T ∗ M ) := (1 ± Γ)Λ ∗ ( T ∗ M ), with Γ := i n/ 2 c ( e 1 ) ...c ( e n ) and e 1 , ..., e n an orien ted basis for T x M , x ∈ M . In particular, c ( ξ )Λ ± ( T ∗ M ) ⊂ Λ ∓ ( T ∗ M ), for all ξ ∈ T ∗ M . Let d denote the exterior deriv ativ e along the fibres and d ∗ its adjoin t. W e define the signatur e family on M as D = d + d ∗ : C ∞ ( M ; Λ + ( T ∗ M )) → C ∞ ( M ; Λ − ( T ∗ M )) . (23) Clearly , D restricts to signature op erators on the fi bres. Th e sym b ol of D is giv en b y the m ap σ ( D ) : π ∗ Λ + ( T ∗ M ) → π ∗ Λ − ( T ∗ M ), with π : T ∗ M → M the pr o jection, suc h that σ ( D )( ξ ) = c ( ξ ). Hence, σ ( D ) is inv ertible outside th e zero-section, that is, D is elliptic. If M is compact, then it defines a K -theory class [ σ ( D )] = [ π ∗ Λ + ( T ∗ M ) , π ∗ Λ − ( T ∗ M ) , c ] ∈ K 0 ( T ∗ M ) . (24) Cobordism inv ariance f or f amilies 17 Note that in this case k er D b and co ker D b are v ector bun dles o ver B , hence th e signatur e index class is giv en by ind( D ) = [ker D ] − [cok er D ] ∈ K 0 ( B ) . (25) W e chec k that it is an inv arian t of cob ordism: Corollary 4.7. L et D b e the signatur e family on a fib e r e d manifold M ֒ → M → B , with M an even- dimensional, oriente d c omp act manifold. If ther e is a c omp act, oriente d fib er e d manifold X ֒ → X → B with ∂ X = M , then ind( D ) = 0 . Pro of. The pro of follo ws the lines of the p r o of of Prop osition 2.8 in [7]. W e sk etc h the main p oin ts. First consider the signature family D 1 on X × R ֒ → X × R → B . Restricting its symb ol to X × { 0 } , we get a K -theory class ω := [( σ ( D 1 ) | T X × R ] ∈ K 0 ( T X × R ) = K 1 ( T X ) . W e h av e th at r ( ω ) = [( σ ( D 2 )) |M×{ 0 } ], w here r : K 1 ( T X ) → K 0 ( T M × R 2 ) is th e restriction map and D 2 is the signature family on M × R 2 . No w, if β denotes a represent ativ e for the Bott class in K 0 ( R 2 ), then σ ( D 2 ) = σ ( D ) ∪ ( π ∗ β ⊕ π ∗ β ), so that r ( ω ) = ( σ ( D ) ∪ β ) ⊕ ( σ ( D ) ∪ β ) = β T M ( σ ( D ) ⊕ σ ( D )) . W e conclude th at ( M , σ ( D ) ⊕ σ ( D )) ∼ 0. It follo ws from Theorem 4.3 that ind( D ⊕ D ) = 0. F rom the additivit y of the family index, w e ha v e then ind( D ) = ind( D ∗ ) ⇔ [k er D ] − [cok er D ] = [k er D ∗ ] − [cok er D ∗ ], so that there exist trivial b undles θ m and θ n suc h that k er D ⊕ θ n ∼ = k er D ∗ ⊕ θ m and cok er D ⊕ θ n ∼ = cok er D ∗ ⊕ θ m . Sin ce k er D ∗ ∼ = cok er D , w e ha v e n = m and therefore [k er D ] = [cok er D ]. Hence, in d( D ) = 0. A similar result holds for signature families t wisted b y s ome sm o oth bundle W o ver M with a sm o oth fibr e conn ection, as long as W can b e extended to the b oun dary . T o finish , w e giv e an alternativ e K -theory form ulation of cob ordism in v ariance, follo wing Moroian u in [22 ]. By a metric on M we mean a con tin uous family of smo oth metrics on T M b , b ∈ B . T he unit sphere bundle S ∗ M = S ( T ∗ M ) is w ell defin ed and it is a manifold o ver B . W e let T ∗ sus M := T ∗ M × R , as b efore, and S ∗ sus M = S ( T ∗ sus M ). Then, as Moroian u noted, there is an isomorph ism d : K 0 ( T ∗ M ) → K 0 ( S ∗ sus M ) /π ∗ K 0 ( M ) , (26) where π : S ∗ sus M → M is the pro j ection, suc h that give n v ector b undles E + , E − o v er T ∗ M (not necessarily sm o oth on fib ers) and a map σ : E + → E − , wh ic h is an isomorphism outside the unit ball, we ha v e [ E + , E − , σ ] ∈ K 0 ( T ∗ M ) 7→ ( E + , on S ∗ M ∩ { ξ ≥ 0 } , E − , on S ∗ M ∩ { ξ < 0 } , (27) Cobordism inv ariance f or f amilies 18 with E + , E − iden tified via σ on S ∗ sus M ∩ { ξ = 0 } = S ∗ M . Moreo v er, taking the b ound ary maps for the relativ e pairs ( B ∗ X , S ∗ X ) and ( B ∗ sus M , S ∗ sus M ), where M = ∂ B X , and the maps of restriction to the b ound ary , the follo wing diagram commutes: K 0 ( S ∗ X ) ∂ − − − − → K 1 ( T ∗ X ) r   y   y r K 0 ( S ∗ sus M ) ∂ − − − − → K 1 ( T ∗ sus M ) q   y   y β − 1 K 0 ( S ∗ sus M ) /π ∗ K 0 ( M ) − − − − → d − 1 K 0 ( T ∗ M ) . Since β − 1 ◦ r = u , the map of restriction of symbols (13), w e ha ve the follo wing analo gue of Moroian u’s K -theory formulati on of cob ordism inv ariance for families. Theorem 4.8. L et M , X b e manifolds over B with M = ∂ B X , and P b e an el liptic family of pseudo differ ential op er ators over B with princip al symb ol σ ∈ K 0 ( T ∗ M ) . If d ( σ ) ∈ r ( K 0 ( S ∗ X )) + π ∗ K 0 ( M ) , then ind( P ) = 0 . Pro of. If d ( σ ) = r ( ω ) + π ∗ K 0 ( M ), with ω ∈ K 0 ( S ∗ X ), then u ( ∂ ω ) = σ ; hence ( M , σ ) ∼ 0. A ck now le dgments . I w ould lik e to th ank Victo r Nistor and Sergiu Moroi anu for v ery helpf ul discussions. I would also like to th ank Mic hel Hilsum and Radu P op escu and the referees for useful commen ts w hic h h elp ed imp ro ve th e pap er. References [1] M. Atiy ah and I . Singer, The index of elliptic op erators on compact manifolds . Bul l. Amer. Math. So c. 69 (1963), 422–433. [2] M. A tiy ah and I. 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