Spectral multiplicity and odd K-theory

SPECTRAL MUL TIPLICITY AND ODD K-THEOR Y R ONALD G. DOUGLAS AND JEROME KAMINKER Abstract. In this pa p er we b egin a study of families of un- bo unded self-adjoint F redholm op erators with co mpa ct resolv an t. The go al is to incorp orate the information in the eigenspaces and eigenv alues o f the op erators , par ticularly the ro le that the multi- plicit y of eigenv alues plays, in obtaining top ological inv aria n ts of the families. 1. Intr oduction In t he early sixties, K- theory , a generalized cohomology theory was defined b y A tiyah and Hirzebruc h, [2] based on a construction of Grothendiec k used earlier in algebraic geometry . F ollo wing some spectacular applica- tions in top ology , the dev elopmen t of K-theory was intrins ically related the index theorem of A tiy ah and Singer, [3 ]. One r esult of this en tan- glemen t w as the realization, b y A tiy ah, [1] and J¨ anic h, [8] of elemen ts of the ev en gr o up as homotop y classes of maps in to the F redholm op era- tors. F o r the o dd g roup, Atiy ah and Singer sho w ed in [4 ] that one could use homotop y classes of maps in to the space of self-adjoint F redholm op erators. Singer raised the question, [14], o f describing elemen ts in the coho- mology of the space of self-adjoin t F redholm op erators in a concrete w a y . The generator of the first coho mo lo gy group can b e related to sp ec tral flow a nd the eta in v arian t following the work of Atiy ah-Pato di- Singer, [6], on the index formula for certain elliptic b oundary v alue problems. This notion has prov ed piv otal in a n umber of directions, including ph ysics, where it w as used in the study o f anomalies, [5]. Let us b e a little more precise. In [4] A tiy ah and Singer established homoto p y equiv alences b et wee n v arious realizations of the o dd K- theory group. The pro ofs in v olv ed, among other t hing s, a careful analysis of finite p ortions o f the sp ectrum of the op era t ors. Ho w eve r, t he precise r elatio ns hips of some of these ob jects w ere left unresolv ed. In subseque n t years, there has b een some Date : Octob er 18, 2021. The research of the first author was supp orted in part by the National Science F oundation. 1 2 RONALD G. DOUGLA S AND JEROME KAMIN KER follo w up on t hese ideas but analyzing families has turned to other notions suc h as gerb es whic h can inv olve ancillary structure. Our goa l in this pap er is to return to the framew or k in tro duced in [4] and attempt to relate the o dd classes directly to the b eha vior o f the self-adjoin t op erators. One can sho w that sp ectral flow is determined b y just a knowle dge of the b eha vior o f the eigenv alues of the family , along with their m ul- tiplicities. Ho w eve r, that is not true for the class in K- t he ory . T o o v e rcome this defect, one mu st bring in the b eha vior o f the eigenspaces as well. Hence , w e seek to unrav el this dep en dence and understand how to obtain in v arian ts similar to c haracteristic classes. W e believ e it is lik ely that these relationships will hav e applications to ph ys ics, suc h as “higher anomalies”, and lead to a study of “higher” sp ectral flow and index theory in general. The authors w ould lik e to thank Ryszard Nest for hospitalit y during a vis it to the Univ ersit y of Cop e nhagen where this pro ject b egan. W e w ould also like to thank Alan Carey who to ok part in initial discussions on this t o pic and provided v aluable insights. This pap er should b e view e d as a first step in whic h some ba s ic structure is rev ealed and some results are o btained. One goal here is t o form ulate basic questions and frame critical issues whic h merit further in v estigation. Before prov iding an o v erview of our results w e need to in tro duce some definitions and notation. W e b egin with more details on t he space of self-adjoin t op erators. It is w ell kno wn that the space of b ounded self-adjoint F redholm o perators on a separable Hilb ert space H , with the norm top ology , is a classifying space fo r o dd K -theory . That is, for any compact Hausdorff space, X , one has K 1 ( X ) ∼ = [ X , F sa ] . (1) Here w e are letting F sa denote the comp onen t of the space of b ounded self-adjoin t F redholm op erators whic h hav e b oth p ositiv e and negativ e essen tial sp ectrum. W e will also consider the subspace, F sa 0 , consisting of op erators, T , with k T k ≤ 1 and the essen tial sp ectrum of T equal to {± 1 } . In applications one is often prov ided with a family of un b ounded F r edholm op erators parametrized by a lo cally path connected and con- nected space X . T o b e more precise we will consider the follo wing subset of un bounded self-adjoint F redholm op erators. Definition 1.1. The r e gular unb ounde d self-adjoint F r e dholm op er a- tors, denote d F sa R , c ons i s t s of li n e ar op er ators, T , which satisfy i) T is close d and self-ad joint, ii) ( I + T 2 ) − 1 is c om p act, SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 3 iii) T has infin i tely many p ositive and infinitely man y ne gative eigen- values. Remark 1.1. W e ma y also consider non-self-adj o in t op erators, T , sat- isfying the condition that T ∗ T ∈ F sa R . Statemen ts made ab out F sa R will hold with appropriate mo difications for suc h operators whic h w e will denote b y F R . W e shall study fa milie s of op erators, D = { D x } : X → F sa R . (2) Let C b ( R ) denote the b ounded con tin uous functions on R for whic h the limits at ±∞ exist. W e consider families which are con tin uous in the sense t ha t the function Θ : C b ( R ) × X → B ( H ) (3) defined b y Θ( f , x ) = f ( D x ) is norm con tin uous. There is a topolo gy on the set F sa R for whic h this holds. With the Riesz to pology , which is the one determined by the b ounded transform, D 7→ D ( I + D 2 ) − 1 2 , it has b een shown by L. Nicolaescu in [13] tha t the space of un bounded s elf-adjoin t F r edholms satisfying (i) and (iii) in Definition 1.1, but not necess arily (ii), pro vides a classifying space fo r o dd K -theory . There is a r elat ed result b y M. Joa c him in [9] whic h states that those satisfying (ii) a s w ell also f orm a classifying space. W e will not need to mak e use of these results in the presen t pap er, but they will b e relev an t for future w ork. The main examples will b e families of Dirac op erators o n an o dd- dimensional manifold M parametrized b y a compact space, X . A fa mily { D x } , a s ab o v e, determines an elemen t of K 1 ( X ). It is obtained by applying the f unc tion χ ( x ) = x (1 + x 2 ) − 1 / 2 to each op erator D x to obtain the family of b ounded self-adj o in t F redholm op erators { ˜ D x } = { χ ( D x ) } . (4) Then each op erator in the r esulting family { ˜ D x } is a b ounded self- adjoin t F redholm op erator and the ho mot op y class of the fa mily yields an elemen t of K 1 ( X ). The Chern character of suc h a family , view ed in real cohomo lo gy , has comp onen ts only in o dd degrees, ch ( { χ ( D x ) } ) ∈ M i ≥ 0 H 2 i +1 ( X , R ) . The class in H 1 ( X , R ) corresp onds to sp ectral flo w, and the class in H 3 ( X , R ) is determined by the index gerb e, c.f. [10]. One goal o f the 4 RONALD G. DOUGLA S AND JEROME KAMIN KER presen t work is to dev elop a metho d that leads to a different descrip- tion of these classes a nd the higher dimensional classes whic h obstruct the triviality o f the K- theory class asso ciated to the family . These ob- structions are to b e determined explicitly in terms of the sp ectrum and eigenspaces of the op erators in the family . This is in a spirit similar to sp ec tral flo w as w e men tioned earlier. As a first step, in the presen t pa- p er we will consider the ro le that the mu ltiplicit y of eigen v alues play s. 2. The mul tiplicity of e igenv alues W e will recall some basic definitions and facts that w e will use. Prop osition 2.1. L et D b e an unb ounde d self-adjoint op er ator as ab ove. L et λ b e an eig e nvalue of D . L et δ > 0 b e s uch that ther e is no other eigenv alue in [ λ − δ , λ + δ ] . T h en the sp e ctr al pr oje ction onto the eig e nsp ac e fo r λ is P λ ( D ) = 1 2 π i Z | z | = δ dz ( z − D ) . (5) and the multiplicity of λ is given b y m ( λ, D ) = rank( P λ ( D )) . No w, consider a family of op erators, { D x } . W e in tro duce the follow- ing terminolo g y . Definition 2.2. The gr aph (or sp e ctr al gr aph) of the family { D x } , Γ( { D x } ) ⊆ X × R , is Γ( { D x } ) = { ( y , λ ) | λ is an eigenva lue o f D y } . (6) Note that Γ( { D x } ) is a closed subset of X × R . When the sp ecific family is clear fr o m the contex t, w e will simply use Γ and simply call it the g r aph, dropping the term “sp ectral”. Both the sp ectral pro jection and the multiplicit y of eigen v alues define functions on the g r a ph of the family . W e mus t cons ider con tin uit y prop erties of these functions. Let ( x, λ ) ∈ Γ b e a p oin t in the graph of the family . Definition 2.3. A c anonic al neig h b orho o d of ( x, λ ) is one of the form V × ( λ − δ, λ + δ ) , wher e x ∈ V , δ > 0 , such that a) λ is the only eigenvalue of D x in ( λ − δ, λ + δ ) , b) if k = m ( D x , λ ) , then, for e ach y ∈ V , one ha s X ( y, µ ) ∈ ( V × ( λ − δ ,λ + δ )) ∩ Γ m ( D y , µ ) = k for e ach y ∈ V . SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 5 Prop osition 2.4. Every p oint ( x, λ ) in Γ adm its a c a n onic al neighb or- ho o d, V × ( λ − δ , λ + δ ) such that if λ − δ < λ 1 ( y ) ≤ . . . ≤ λ k ( y ) < λ + δ ar e the e igenvalues of D y in the g i v e n interval, then e ach λ j ( y ) is c o n - tinuous o n V . Pr o of. This will follow from a corr e sp onding statemen t for b ounded op erators in [7]. How ev er, w e will need a precise form of this fact so w e recall the steps. Let f ( t ) = t/ √ 1 + t 2 − λ/ √ 1 + λ 2 , and consider the family of b ounded op erators { f ( D x ) } . Cho ose a δ > 0 so tha t there is no o t her eigen v alue of D x in ( λ − δ, λ + δ ). Assume m ( D x , λ ) = k . Con- sider f ( D x ) and f ( δ ) and apply [7, p. 138], to obtain a neighborho o d V of x suc h that for y ∈ V there a re exactly k eigenv alues of f ( D y ) in ( f ( λ − δ ) , f ( λ + δ )), whic h w e will lab el − f ( δ ) < ˜ λ 0 ( y ) ≤ ˜ λ 1 ( y ) ≤ . . . ≤ ˜ λ k ( y ) < f ( δ ) , Moreo v er, | ˜ λ j ( y ) − ˜ λ j ( y ′ ) | < k f ( D y ) − f ( D y ′ ) k . Then λ j ( y ) = f − 1 ( ˜ λ j ( y )), V , a nd δ yield the conclusion.  As a corollary one obtains the followin g refinemen t. Prop osition 2.5. L et λ 0 ( x ) < . . . < λ n ( x ) b e a list of the distinct eigenvalues of sp ec( D x ) which lie in a b ounde d interval of R . Then ther e ar e disjoint c an onic al neighb orho o ds of e ach ( x, λ j ( x )) , al l with the same b ase V . Pr o of. This fo llows easily from the metho d of pro of of Prop osition (2.4)  Note that there can b e no p oin ts of the graph b et w een the standard neigh bor ho o ds obt a ined . It follows easily fro m t his argument that the m ultiplicit y function will b e lo w er semi-contin uous. The next result describes the conditions under whic h it is actually contin uous at a p oin t ( x, λ ) ∈ Γ. Prop osition 2.6. L et U b e a c anonic al neighb orho o d of ( x, λ ) ∈ Γ . The fo l lowing ar e e quivalen t. i) Ther e is a p ositive inte ger k so that the multiplicity function is c onstantly e qual to k on U , ii) Ther e is a δ > 0 and a neighb o rh o o d V o f x so that, for e ach y ∈ V , D y has only on e eigenvalue in the interval [ λ − δ, λ + δ ] , iii) The func tion asso ciating the sp e ctr al pr oje ction to a p oint in the gr aph is norm c ontinuous on Γ ∩ ( V × [ λ − δ, λ + δ ]) . Definition 2.7. The fa mily { D x } h a s c onstant multiplicity k at ( x, λ ) if it satisfies the c onditions in Pr op osition 2. 6. 6 RONALD G. DOUGLA S AND JEROME KAMIN KER W e will next consider criteria for the triviality of the K-t heory class asso ciated to a family of self-adjoint o p erators. Prop osition 2.8. L et { D x } b e a c ontinuous family of sel f - adjoint op- er ators. The fol lowing ar e e q uiva lent. i) The fa mily defines the trivial element in K 1 ( X ) , ii) { D x } is homotopic to a fami l y { D ′ x } for which ther e is a c on t in- uous function, σ : X → R , such that σ ( x ) is not an eigenvalue of D ′ x for e ac h x , iii) { D x } is homotopic to a fa m ily { D ′ x } for w hich ther e is a norm c ontinuous family of pr oje ctions { P ′ x } with r ange the sum of the eigensp ac es for p ositive eigenvalues. iv) Ther e exi sts a sp e ctr al s e ction for the family { D x } , in the sense of Melr ose-Piaz z a , [11] . (i.e. ther e is a no rm c ontinuous fam ily of p r oje ctions which a gr e e with the p r oje ctions o nto the p osi- tive eigensp ac es outside of a close d interval, the interval itself dep end ing c ontinuously on x .) Pr o of. This follows using the steps in the pro of o f Proposition 1 in Melrose-Piazza, [11].  3. Spectral exhaustions and spectral flow In t his section we will pro v e the existence and essen t ia l uniqueness of sp ec tral exhaustions. L e t { D x } b e a contin uous family of op erators parametrized b y the compact space X , whic h w e assume fo r now is a simplicial complex. Definition 3.1. A s p e ctr al exhaustion for the family { D x } is a family, of c ontinuous functions µ n : X → R , indexe d by Z , satisfying i) µ n ( x ) is an eigenvalue of D x for e ac h x , ii) { µ n ( x ) : n ∈ Z } exh a ust s the sp e ctrum of D x c ounting multi- plicity, for e a ch x , iii) for e ach x and for e a c h n ∈ Z , µ n ( x ) ≤ µ n +1 ( x ) . Remark 3.1. Note that , if the graphs of functions µ n and µ n − 1 are dis- join t a nd the parameter space X is connected, then µ n ( x ) > µ n − 1 ( x ), for all x , so σ ( x ) = 1 2 ( µ n ( x ) − µ n − 1 ( x )) satisfies condition (ii) of Prop o- sition 2.8 . Th us, the K- theory class o f a family admitting a sp ectral exhaustion with this prop ert y is trivial. Definition 3.2. A n enumer ation of the sp e ctrum of an op er ator D ∈ F sa R is a function e D : Z → R mapping Z onto the sp e ctrum o f D and satisfying SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 7 i) if λ is an eigen v a lue of D of m ultiplic i ty k , then ther e i s an inte ger N such that λ = e D ( N ) = e D ( N + 1) = . . . = e D ( N + k ) , and ii) e D ( n ) ≤ e D ( n + 1) f o r al l n . Our goal in this s ection is to sho w that, if the spectral flo w of the family { D x } is zero, one can construct an en umeration of the the sp ec- trum of D x , for each x , in such a wa y that the functions µ n ( x ) = e D x ( n ) are contin uous. Th us, w e will obtain a sp ectral exhaustion for { D x } . Prop osition 3.3. Given a n op er ator, D , a n enumer ation of the sp e c- trum al w ays e xists and any two differ by tr an s lation by an inte ge r. Pr o of. Choose an eigen v alue, λ , of m ultiplicit y k . W e se t e D (0) = λ and e D ( − k + 1) = . . . = e D (0) = λ . One can now uniquely extend this lab eling to the rest of t he sp ectrum. It is easy to chec k that this pro cess pro vides an enume ration of the sp ectrum of D . No w supp ose that f D is another one. W e will show that there is an N suc h t hat f D ( n + N ) = e D ( n ) f or a ll n . Let λ b e a p oint in t he spectrum and let n 0 , m 0 b e the largest integers so that e D ( n 0 ) = λ = f D ( m 0 ). Le t N = m 0 − n 0 . Then it is easy to c hec k that e D ( n ) = f D ( n + N ) for all n .  Note that the existence of an in teger n suc h that e D ( n ) = f D ( n ) is not sufficien t to guarantee that e D = f D . How ev er, if there is a n in teger N suc h that e D ( N ) = f D ( N ) and e D ( N + 1) > e D ( N ), f D ( N + 1) > f D ( N ), then it is the case that e D = f D . Fix x ∈ X and let λ b e an eigen v alue of D x . Cho ose an en umeration of the sp ec trum of D x satisfying e D x (0) = λ e D x (1) = λ ′ > λ. Find canonical neighborho o ds W = V × ( λ − δ , λ + δ ), W ′ = V × ( λ ′ − δ ′ , λ ′ + δ ′ ) o f ( x, λ ) and ( x, λ ′ ) resp ectiv ely . Let µ 0 ( y ) = max { λ | λ ∈ sp ec( D y ) and ( y , λ ) ∈ W } . Similarly , let µ 1 ( y ) = min { λ | λ ∈ sp e c( D y ) and ( y , λ ) ∈ W ′ } . Prop osition 3.4. The functions µ 0 and µ 1 ar e c ontinuous on V . Pr o of. It will b e sufficien t to consider µ 0 , the case o f µ 1 b eing similar. Let e 0 ( y ) ≤ . . . ≤ e k ( y ) b e the part of the sp ectrum of D y in ( λ − δ, λ + δ ). Then µ 0 ( y ) = e k ( y ), and by the remark after Prop osition 2.4, µ 0 ( y ) is con tin uo us.  Using µ 0 and µ 1 w e define a sp ectral exhaustion ov er V b y taking, for eac h y ∈ V , the unique (not just up to translation) e n umeration 8 RONALD G. DOUGLA S AND JEROME KAMIN KER consisten t with those ch oices. Th us, w e ha v e µ n ( y ) defined for eac h in teger n and eac h y ∈ V . Prop osition 3.5. The functions µ n ar e c ontinuous on V and, henc e, { µ n } is a sp e ctr al exhaustion over V . Pr o of. Choose a p oin t y ∈ V . T aking a p ossibly smaller neighborho o d V ′ of y , w e get n + 1 disjoint canonical neigb orho o ds of the fo r m V ′ × ( ˜ µ j ( y ) − δ j , ˜ µ j ( y ) + δ j ), where ˜ µ j ( y ) are the eigen v alues of D y from µ 0 ( y ) to µ n ( y ) listed m ultiply . No w, for each z ∈ V ′ , µ n ( z ) is in the canonical neighborho o d corresp onding to the gr eat est real in terv al and it corresp onds to one of the eigen v alues λ r ( z ) in it. W e claim it m ust b e the same r for each z in V ′ . T o see this, let N b e the num b er of eigen v alues in t he canonical neighborho o ds b elo w the top one and let r b e the index corresp onding to µ n ( y ). Then n = N + r . If w e lo ok at a p oin t z and µ n ( z ) = λ r ′ ( z ), then w e still mus t hav e n = N + r ′ , so that r = r ′ . Thus , by Prop osition 2.4, µ n ( z ) v aries con tin uously .  Doing this construction in a neighborho o d of eac h p oin t x ∈ X , w e obtain a family of sp ec tral exhaustions, each ov er an elemen t o f an op en cov er, { V i } , where w e ma y assume the op en sets are connected. On the o v erlaps, any t w o exhaustions differ b y an in teger, so we obtain an in teger v alued 1-co c hain relativ e to { V i } b y taking the difference of the partial exaustions, ν ij = µ 0 ,i | V i ∩ V j − µ 0 ,j | V i ∩ V j : V i ∩ V j → Z . It is easily c hec k ed to b e a cocycle and its cohomology class in ˇ H 1 ( X , Z ) will b e defined to b e the sp ectral flo w of the family , Sf ( { D x } ). It is straigh tforw ard to see that this definition agr ee s with other definitions of sp ec tral flo w. (c.f. [7]). Theorem 3.6. A s p e ctr al exhaustion exists for the family { D x } if and only if the sp e ctr al flow of the family is zer o, Sf ( { D x } ) = 0 . Pr o of. If Sf ( { D x } ) = 0 t he n the co cycle , whic h is defined with resp ect to the op en co v er { V i } , is a cob oundary , so that δ ( σ ) = ν for some co c hain σ . Then the 0-co c hain with comp onen ts µ n,i − σ n,i can b e used to define global functions µ n . These µ n ’s pro vide the required exhaustion. F o r the conv erse, if a n exhaustion exists, this determines the c hoices in constructing the co cycle represen ting Sf ( { D x } ), and since the lo cally defined exhaustion functions all piece together to yield glo bal functions, the class is equal to zero.  4. F amilies with s pectr um of cons t ant mul tiplicity In t his section we will obtain the first results relating sp ectral mul- tiplicit y to K -theory . Recall t ha t w e assume that the parameter space SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 9 is a finite simplicial complex. While this assumption is not alwa ys nec- essary , the top ology issues that would arise with additio na l g en eralit y are not fundamen tal ones. Prop osition 4.1. Supp ose that the family { D x } has c ons tant multi- plicity at e ach p oint of a c omp onent, ˜ X , of Γ . Then pr 1 : ˜ X → X is a c overing . Pr o of. W e use Proposition 2.6 (ii) w hic h states that, f o r each x ∈ X and eac h eigenv alue λ of D x there is a neigh b orho od, V and a δ > 0 suc h that for eac h y ∈ V , D y has only one eigen v alue in the in terv al [ λ − δ, λ + δ ]. Then the function, σ x,λ : V → R , whic h sends y to tha t eigen v alue, is con tin uous. It then follows that eac h component of Γ( D x ) is a co v ering of X .  W e defined the sp ectral flow of a family { D x } to b e a 1-dimensional cohomology class, Sf ( { D x } ) ∈ ˇ H 1 ( X , Z ) . This class defines a homomorphism, for whic h we will use the same notation, Sf ( { D x } ) : π 1 ( X ) → Z . (7 ) The follow ing is an easy consequence of the definitions. Prop osition 4.2. I f a c o m p onent, ˜ X , of Γ is a c overing, pr 1 : ˜ X → X , then it c orr esp onds to the h o momorphism Sf ( { D x } ) : π 1 ( X ) → Z . i.e. image( pr 1 ∗ ) = k er(Sf ( { D x } )) . The next results giv e a criterion for the exis tence of a sp ectral ex- haustion with disjoin t g raphs. Theorem 4.3. L et { D x } b e a fami ly with sp e ctrum of c onstant mul- tiplicity; that is, ther e exists an inte ger k such that m ( D x , λ ) = k , for e ach ( x, λ ) in Γ . Assume that the sp e ctr al flow of the fam ily is z er o, Sf ( { D x } ) = 0 . Then a sp e ctr al exha ust ion with functions hav i n g disjoint images, ( ex- c ept for r e p e ate d functions due to m ult iplicity), exis ts. Pr o of. Since t he m ultiplicit y of t he co v ering is constant, eac h comp o- nen t is a co v ering. Moreo v er, each o f these co v erings corresp onds to the homomorphism sf ( { D x } ) : π 1 ( X , x 0 ) → Z , giv en b y sp ectral flo w. Th us, if the sp ectral flow of the family is zero, eac h of the co v erings is a homeomorphism, so that Γ ∼ = X × sp ec( D x 0 ), 10 RONALD G. DOUGLA S AND JEROME KAMIN KER for some p oin t x 0 ∈ X . En umerate the sp ec trum o f D x 0 as { λ n ( x 0 ) } and let ˜ X n b e the compo nent of Γ containing λ n ( x 0 ). Then set µ n ( x ) = pr 2 ◦ ( pr 1 | ˜ X n ) − 1 . These functions satisfy the requiremen ts to b e a sp ec- tral exhaustion, and their graphs, b eing the comp onen ts of Γ, are dis- join t.  W e obtain the follo wing corollary from Remark 3.1. Corollary 4.4. L et { D x } b e a family with sp e ctrum of c o nstant mul- tiplicity. Assume that the sp e ctr al flow of the family is zer o, Sf ( { D x } ) = 0 . Then the family { D x } is trivia l in K-the ory. It is also w orth noting the follo wing result. Corollary 4.5. Supp ose that some c omp onent of Γ is c omp act. Then the K-the ory class of the fam i l y is trivial. Pr o of. Let the comp onen t ˜ X b e compact. Then the num b er of sheets in the co v er is the cardinalit y of π 1 ( ˜ X ) ∼ = image(Sf ( { D x } )), whic h m ust b e finite. How eve r, this is a subgroup of Z , so it will ha v e to b e zero. Th us, the sp ectral flow of the family is zero a nd its class is tr ivial b y Prop osition 4.4.  Finally , w e consider ho w the h ypothesis of constan t m ultiplicit y can b e replaced b y an asymptotic v ersion. Theorem 4.6. L e t { D x } b e a family w i t h Sf ( { D x } ) = 0 Supp ose that ther e exists an inte ger N such that if ( x, λ ) ∈ Γ an d λ > N then the family, { D x } , has c onstant multiplicity at ( x, λ ) . Then the class of the family is trivial in K 1 ( X ) . Pr o of. Let Γ R = { ( x, λ ) : λ > R } . W e will sho w that there is a n R > N so that some comp onen t of Γ R is a cov ering of X . If so, then as in 4.3, Sf ( { D x } ) = 0 will imply that this comp onen t is compact and b y Corollary 4 .5 the K-theory class of the family will b e trivial. Th us, we m ust sho w that there is a pa th comp onen t, ˜ X , of Γ whic h is con tained in Γ R for some R > N . Since Sf ( { D x } ) = 0 a sp ectral exhaustion, µ n , exists. Let Γ n = image µ n . F or each x ∈ X there exists an n x and a ne igh bo rhoo d of x , U x , suc h that µ n ( y ) > N + 1 for all y ∈ U x . Get a finite sub co v er, U x 1 , . . . , U x k , and let m = max { n x i } . Then µ m ( x ) > N + 1 for all x ∈ X . This implies that the image of µ m is a cov er of X and is compact a nd connected.  SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 11 5. F amilies with bounded mul tiplicity In t his section we will cons ider the question of when an elemen t o f o dd K -theory can be represen ted b y a family with uniform b ounded m ultiplicit y . T o this end le t, for n ≥ 1 , F sa R ( n ) denote the o perators with m ultiplicit y less than or equal to n . Then F sa R ( n ) ⊆ F sa R ( n + 1) and w e set F sa R ( ∞ ) = S F sa R ( n ). W e do the same for F R . Througho ut , X will b e a compact space. Recall that in A tiy ah-Singer, [4], the following diag ram w as studied. F sa 0 ˆ F U ( I + K ) U ∞ ˆ F n U ( I + F n ) U n / / exp( i · ) o o o o O O            / / exp( i · ) O O            o o O O            (8) Here, F sa 0 is the b ounded self-adjoin t F redholms with essen tial sp ec- trum on b oth sides o f the origin, while ˆ F is those op erators with norm 1 and essen tia l sp ectrum ± 1. Also, U ( I + F n ) is the unitary op erators of the f orm I + K , with K of rank n , ˆ F n is the o perators in ˆ F with finitely man y eigenv alues in ( − 1 , 1 ) and fo r whic h exp( iT ) ∈ U n . The unlab eled arrow s ar e inclusions. The Atiy ah- Singer result sho ws that the comp osition of the maps on the top ro w and their appropriate ho- motop y inv erses provide a homotop y equiv alence whic h w e shall denote b y ˆ χ : F sa 0 → U ∞ . There is an obv ious inclusion map of F sa R in to F sa 0 and hence into U ∞ . T o study the ques tion of b ounded m ultiplicity w e make the follow ing definition. Definition 5.1. L et K 1 ( ∞ ) ( X ) b e the s ubse t of K 1 ( X ) c onsisting of classes [ α ] , α : X → F sa R , such that ther e is an n and an α ′ ≃ α with α ′ : X → F sa R ( n ) . L et K 0 ( ∞ ) ( X ) b e defi ne d in an analo gous way using F R . Note that the homotop y b et w een α and α ′ is a llow ed to run throug h all o f F sa 0 . Prop osition 5.2. K ∗ ( ∞ ) ( X ) is a natur al sub gr oup of K ∗ ( X ) . Pr o of. This subgroup is clearly preserv ed b y induced homomorphisms and con tains the identit y elemen t o f K ∗ ( X ). Additio n in K 1 ( X ) is induced b y comp osition of o perators whic h is homoto pic to orthog - onal direct sum. Th us, the sum of classes repres en ted b y b ounded 12 RONALD G. DOUGLA S AND JEROME KAMIN KER m ultiplicit y e lemen ts is also represen ted b y a family of b ounded m ul- tiplicit y . Moreo ver, since t he in ve rse of an elemen t given b y a family α : X → F sa 0 is represen ted b y − α , this op eration preserv es the prop- ert y of havin g b ounded m ultiplicit y . Thus , the result follows.  Prop osition 5.3. K ∗ ( ∞ ) ( X ) is mapp e d to itself under Bott p erio dicity. Pr o of. The Bott p erio dicit y map is giv en by taking the pro duct with the Bott elemen t of ˜ K 0 ( S 2 ). The pro duct op eration can b e realized in the presen t contex t by letting each o perator in the family act o n the Hilb ert space obtained b y tensoring with the L 2 sections of the bundles represen ting the K 0 class. If the bundle is trivial, then the m ultiplicit y will b y m ultiplied by its dimension. If it is not trivial, then it is a summand of a trivial bundle a nd one can see that restricting to the image of the pro jection o n to the sections of the bundle can only low er the m ultiplicit y . Note that in this setting, the op erators in the family will commute with the pro jections on to the sections of the bundle.  W e will no w mak e use of a construction whic h appears in a pap er of Mic k elsson, [12 ]. It will b e used to associate to a map into the finite dimensional unitary group, U n , an explicit family o f unbounded self-adjoin t F redholm op erators with t he m ultiplicity of their sp ectrum uniformly b ounded by n . Let U ∈ U n . Consider the op erator − i d dx : C ∞ ([0 , 1 ] , C n ) → C ∞ ([0 , 1 ] , C n ) with the b oundary condition ξ (1) = U ξ (0) . This yields a self-adjoin t F redholm op erator on L 2 ([0 , 1 ] , C n ) whic h w e will denote D U . It is straigh tforw ard to compute the sp ectrum of D U and the result is as fo llo ws. Let { z 1 , . . . , z n } b e the sp ectrum of U . Let λ j satisfy 0 ≤ λ j < 1 and z j = e 2 π iλ j . Then, sp ec( D U ) = { m + λ j | m ∈ Z , 1 ≤ j ≤ n } . The m ultiplicit y of the eigen v alue m + λ j is the same a s t ha t of the eigen v alue z j of U , and it follows tha t the m ultiplicit y of the sp ectrum of D U is less than or equal to n . Let µ n : U n → F sa R b e defined by µ n ( U ) = D U . W e will refer to µ n as the Mic k els son map. SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 13 Prop osition 5.4. The Mickelsson map yields a map µ : U ∞ = [ n ≥ 1 U n → F sa R → F sa 0 , (9) which induc es an isomorphi s m on homotopy gr oups, µ ∗ : π i ( U ∞ ) → π i ( F sa 0 ) , (10) for al l i . Pr o of. The first statemen t follo ws from the definitions while the second is a consequence of the facts tha t the Mic k elsson map comm utes with p eriodicity and the computatio n f rom [12] that it is an isomorphism for S 3 .  Note tha t if X is a compact space, then µ ∗ : [ X , U ∞ ] → [ X , F sa 0 ] actually maps in to K 1 ∞ ( X ). These three prop ositions yield the follow ing theorem. Theorem 5.5. L et X b e a c omp act m e tric sp ac e. Then one ha s K ∗ ( ∞ ) ( X ) = K ∗ ( X ) . Pr o of. It follow s from Prop ositions 5.2 a nd 5.3 that K ∗ ( ∞ ) ( X ) defines a cohomology theory on compact spaces with a 6-term exact sequence of the same type as that for K ∗ ( X ). The inclusion induces a map of 6-term seq uences. Assume first that X is a finite complex. Then applying the cohomology theories to the sequence of sk eletons, X ( k ) → X ( k +1) → _ S ( k +1) , (11) will yield the result b y induction once one kno ws t ha t it holds f o r spheres. Ho w ev er, for spheres the Mick elsson map comp osed with t he inclusion, π i ( U ∞ ) π i ( F sa 0 ( ∞ )) π i ( F sa 0 ) / / µ ∗ / / i (12) agrees with t he isomorphism from A tiy ah-Singer, [4]. Here, F sa 0 ( ∞ ) denotes the subset o f F sa 0 homotopic to regular op erators of b ounded m ultiplicit y . By Prop osition 5.4, µ ∗ is an isomorphism on spheres , hence so is t he inclusion, i . This prov es t he result for finite complexes . By expressing a compact metric space as a n in v erse limit o f finite com- plexes o ne obtains the desired conclusion.  14 RONALD G. DOUGLA S AND JEROME KAMIN KER Note that the same argumen t show s that the Mick elsson map is a n isomorphism. This result has connections to the paper of Nicolaescu, [1 3], in whic h the relatio n of K 1 ( X ) and homotopy classes of maps in to the space of un bounded self-adjoin t F redholm op erators with essen tial sp ectrum {± 1 } , but p ossibly ha ving some con tin uo us sp ec trum, is addressed. F r o m the v an ta ge of this pap er Theorem 5.5 shows that ev ery elemen t in K 1 ( X ) is represen t e d b y a f amily of regular un bo unde d self-adjoint F r edholm op erators. As a consequence of this fact, one sees that an y family is homotopic to a family with b ounded multiplic it y . One can estimate the b ound on the m ultiplicit y in a rough wa y using the dimension of X . It would b e desirable to get a refined estimate based on the top ology of X . Definition 5.6. L et { D x } b e a family on X . The minimal multiplicit y of the fa m ily is the le ast inte ger n such that { D x } ∼ = { D x } ′ wher e { D x } ′ is a f a m ily w ith multiplicity b ound e d by n . Prop osition 5.7. L et [ α ] ∈ K 1 ( X ) and supp ose the dimension of X is k . Then [ α ] is r epr esente d by a famil y { D x } with minimal m ultiplicit y < [ k +1 2 ] , wher e [ x ] denotes the l a r gest inte g e r less than or e qual to x . Pr o of. Suppose that α : X → U N is giv en. Using the fibrations U n − 1 → U n → S 2 n − 1 one can inductiv ely reduce the dimension of the unitar y group to the least p oss ible, whic h is [ k +1 2 ]. The result follow s up on applying Theorem 5.5.  6. Mul tiplicity ≤ 2 As a sample of ho w conditions on the m ultiplicit y b ey ond assuming constancy can b e used to study the K-theory class of a family , w e will consider the case when the multiplic it y is less than or equal to 2 . W e will also assume t ha t the sp ectral flow o f the fa mily is ze ro. The main result of this section is that, if w e assume that the space X ha s no torsion in cohomolog y , then suc h a family is trivial in K-theory if a certain 3- dime nsional cohomology class v anis hes. Th us, the index gerb e will b e zero also. Let us assume that we hav e a family with m ult( { D x } ) ≤ 2 and recall the standing assumption that the parameter spac e X is a connected finite simplicial complex. Assume Sf ( { D x } ) = 0 and let { µ n } b e an exhaustion. Our pro cedu re will b e to deform the family inductiv e ly ov er k-sk ele tons for increasing k , so that the e xhaustion for the deformed family , { ˜ µ n } , has ˜ µ 0 ( x ) < ˜ µ 1 ( x ) for all x . The trivialit y will then follo w from Prop osition 2.8. SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 15 Let C i,i +1 = { x | µ i ( x ) = µ i +1 ( x ) } , for an y i ∈ Z . Since m ult( { D x } ) ≤ 2 w e hav e C − 1 , 0 , C 0 , 1 and C 1 , 2 disjoin t closed sets. Let W i,i +1 , i = − 1 , 0 , 1, b e disjoint op en neigh b o rhoo ds of C i,i +1 . W e assume the t ri- angulation of X so fine that a ny closed simplex whic h meets C i,i +1 is con tained in W i,i +1 . Th us, there are a finite n umber of simplices, σ l , suc h tha t C i,i +1 ⊆ in terior( n [ 1 σ l ) ⊆ n [ 1 σ l ⊆ W i,i +1 . Our pro cedure for deforming a family in v o lv es succes siv e application of certain ty p es of “mo v es ”. The first is a preliminary flattening pro cess whic h allo ws one to con trol the geometry of the sets ov er whic h the family has eigen v alues of multiplic it y 2. W e will state things f or C 0 , 1 to simplify notation, but all results hold fo r C i,i +1 with the appropriate mo difications. Prop osition 6.1 (Flattening) . L et K b e a close d subset of C 0 , 1 and le t W 1 , W 2 b e op en sets with c om p act closur es satisfying K ⊆ W 1 ⊆ ¯ W 1 ⊆ W 2 ⊆ ¯ W 2 ⊆ W 0 , 1 . Assume further that K = C 0 , 1 ∩ W 2 . Then ther e exists a family { ˜ D x } with as so ciate d exhaustion { ˜ µ n } , whi c h satisfies i) K ⊆ W 2 ∩ ˜ C 0 , 1 = ¯ W 1 , wher e ˜ C 0 , 1 = { x | ˜ µ 0 ( x ) = ˜ µ 1 ( x ) } , ii) ˜ D x = D x for x ∈ X r W 2 , and iii) { ˜ D x } ≃ { D x } . Pr o of. Let φ : X → [0 , 1] b e a function satisfying φ ( x ) = ( 0 for x ∈ X r W 2 1 for x ∈ ¯ W 1 Define ˜ D x,t = D x + tφ ( x )( h x ( D x ) − D x , where h x : R → C is a con tin uous function satisfying h x ( t ) =      t for t ≤ µ − 1 ( x ) or t ≥ µ 1 ( x ) µ 1 ( x ) for µ 0 ( x ) ≤ t ≤ µ 1 ( x ) λ x ( t ) for µ − 1 ( x ) ≤ t ≤ µ 0 ( x ) , where λ x ( t ) is the linear function with graph connecting ( µ − 1 ( x ) , µ − 1 ( x )) to ( µ 0 ( x ) , µ 1 ( x )) Letting ˜ D x = ˜ D x, 1 , with asso ciated exhaustion ˜ µ n ( x ), o ne chec ks that K ⊆ W 2 ∩ ˜ C 0 , 1 = ¯ W 1 and that the conclusions of the pro p osition hold fo r the family { ˜ D x } .  16 RONALD G. DOUGLA S AND JEROME KAMIN KER Th us, the preceeding deforma t io n allo ws one to determine the set precisely , ( ¯ W 1 ab o v e), on whic h multiplicit y o f ( x, ˜ µ 0 ( x )) is 2. Next, w e will mo dify the family o v er neighborho o ds of these sets. Let X (0) b e the 0- ske leton of the parameter space X . The first step will b e to deform the family { D x } on a neigh b orhoo d of X (0) . Prop osition 6.2. L et y ∈ X (0) . Then ther e exists a c ontr actible neigh- b orho o d V with V ∩ X (0) = { y } and a fa m ily { ˜ D x } satisfying i) ˜ D x = D x for x ∈ X r V , ii) { ˜ D x } ≃ { D x } , and iii) Ther e is a neighb orho o d W o f y such that ˜ µ 0 ( x ) < ˜ µ 1 ( x ) for x ∈ W ⊆ ¯ W ⊆ V . Pr o of. If µ 0 ( y ) < µ 1 ( y ) then this uniquely will hold in a neighborho o d of y and the origina l family w ill satisfy conditio ns (i)-(iii). If, on the other ha nd, y ∈ C 0 , 1 , so that µ − 1 ( y ) < µ 0 ( y ) = µ 1 ( y ) < µ 2 ( y ), then w e apply Prop osition 6.1 to obtain a contractible neigb orho o d with compact closure, V , of y on whic h µ 0 ( x ) = µ 1 ( x ) for x ∈ V . Let E = { ( x, v ) ∈ V ×H | v is in the span of the eigen v ec tors for µ 0 ( x ) and µ 1 ( x ) } . Then E → X is a 2-dimensional v ector bundle on some, p ossibly smaller, neigh borho o d of y whic h w e con tinue to call V . Since V is con- tractible, the bundle is trivial. Thus, there e xists a f r a ming { σ 0 , σ 1 } , where σ j ( x ) is an eigen v ector for µ j ( x ), for j = 0 , 1. Shrink V to get W ⊆ ¯ W ⊆ V a nd, using a bump function φ , w e extend σ i to all of X . Let α ( t, x ) = t µ 1 ( x )+ µ 2 ( x ) 2 , and set ˜ D x,t = D x + α ( t, x ) P σ 1 ( x ) , where P σ 1 ( x ) is the orthogonal pro jection on to the subspace spanned b y σ 1 ( x ). The fa mily { ˜ D x, 1 } satisfies the requiremen ts of the prop osition. W e rep eat this construction, with the obv ious mo difications, for v er- tices in C − 1 , 0 . Alternativ ely , one may observ e that it is p ossible to apply this metho d to the v ertices in b oth C − 1 , 0 and C 0 , 1 sim ultaneously .  Th us, w e hav e deformed our f amily so that µ 0 ( x ) < µ 1 ( x ) on a neigh bor ho o d of the 0- s k eleton. W e will now pro cee d inductiv ely t o extend this separation to all of X . It is w orth noting that the previous deformations and a ll f ut ure ones ha v e the prop ert y that they are monotone, in the sense tha t µ i ( x ) ≤ ˜ µ i ( x ) for all i and x ∈ X . F r o m now o n, to simplify notation, w e will drop the tilde and rename the family resulting from a deformation as { D x } . Assume that our family has the prop ert y that µ 0 ( x ) < µ 1 ( x ) on a neigh bor ho o d, W , of the (k-1) - sk eleton. Let C 2 → E → W b e the 2-dimensional vec tor bundle whose fiber ov er the point x is the s pan of the eigenspaces for µ 0 ( x ) and µ 1 ( x ). F o r x ∈ W , define σ 0 ( x ) to b e SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 17 the o rthogonal pro j ec tion on to the eigenspace for µ 0 ( x ). This defines a contin uous field of pro jections ov er W . The next result sho ws that one can obtain the desired deformation o f { D x } if the field σ 0 can b e extended to the k-sk eleton. W e will construct the deformation simplex b y simplex. Prop osition 6.3. L et { D x } b e a family of op er ators with µ 0 ( x ) < µ 1 ( x ) on a neighb orho o d, W , of the (k-1)-skel e t on. L et ∆ k b e a k- simplex. L et ′ ∆ k b e a sm al ler k - simplex with C 0 , 1 ∩ ∆ k ⊆ in terior( ′ ∆ k ) ⊆ ′ ∆ k ⊆ in terior (∆ k ) . I f σ 0 | ∂ ( ′ ∆ k ) extends to a field of pr oje ctions onto the eigensp a c es for µ 0 ( x ) over ∆ k , then ther e ar e neigh b orho o ds, V an d W ′ , with X ( k − 1) ∪ ∆ k ⊆ W ′ ⊆ ¯ W ′ ⊆ V and a fa m ily { ˜ D x } satisfying i) ˜ D x = D x for x ∈ X r V , ii) { ˜ D x } ≃ { D x } , and iii) ˜ µ 0 ( x ) < ˜ µ 1 ( x ) for x ∈ W ′ . Pr o of. Let φ ( x ) b e a bump function whic h is 1 on W ′ and 0 on X \ V . Let ˜ D x,t = D x + tφ ( x )( µ 1 ( x )+ µ 2 ( x ) 2 ) σ 0 ( x ). Then it is straig htforw ard t o c hec k t ha t the family D x, 1 satisfies conditions (i) – (iii).  In order to use this construction to deform a family whic h is sep- arated o v er a neigh bo rhoo d of X ( k − 1) , we apply Prop osition 6.1. F or this, note that C 0 , 1 is contained in the interior of k-simplices. Con- sider one suc h simplex and find a sub-simplex with para llel sides whic h con tains its interse ction with C 0 , 1 in its in terior. W e may assume that the b oundary of the smaller simplex is con tained in the op en set ov er whic h µ 0 ( x ) < µ 1 ( x ). Next one applies the flattening lemma t o obtain a new family whic h has the smaller simplex as exactly C 0 , 1 ∩ ∆ k . The pro jection field is defined on ∂ ( ′ ∆ k ). Th us, if one can alw a y extend these pro jection fields from t he b oundary of a k-simple x to the inte- rior, then we can accomplish the deformation of the family to one for whic h µ 0 ( x ) < µ 1 ( x ) o v er a neighborho o d of the k-sk ele ton. Note that, since ∆ k is con tractible, the bundle E → X is trivial o v e r ∆ k . Thus , the existe nce of the extension is equiv alent to the map σ 0 : ∂ ∆ k → Gr 2 ( C 2 ) b eing nu ll-homotopic. Here Gr 2 ( C 2 ) is the Grassmannian of lines in C 2 , whic h is homeomorphic to S 2 . Since π 0 ( S 2 ) = π 1 ( S 2 ) = 0, one can alwa ys obtain a deformed family for whic h µ 0 ( x ) < µ 1 ( x ) on a neighborho o d of the 2- sk eleton, X (2) . How - ev er, since π 2 ( S 2 ) = Z , it is not clear that one can pro ceed. W e shall address this in the application b elo w. Putting these facts together w e briefly presen t a sample of the type of result obta inable using these methods. Note that the res ult b elo w 18 RONALD G. DOUGLA S AND JEROME KAMIN KER mak es the case that the m ultiplicit y of eigen v alues has a strong effect on the classification of families of op erators. Theorem 6.4. L et { D x } b e a family on S 2 n +1 with n ≥ 2 . If the multiplicity is less than or e qual to 2, then the family is r ational ly trivial in K-the ory. Pr o of. W e sk etc h the argumen t. Note that giv en a separation ov er a n op en set, w e get a contin uous eigenpro jection field ov er tha t set. Th us, as ab o v e , we ma y obtain an eigenpro jection field ov er a neigh b orho od of the 2-sk eleton. F or the induction step, let k < 2 n + 1 and assume that in a neigh b orhoo d of the k -sk eleton we ha v e µ − 1 ( x ) < µ 0 ( x ) < µ 1 ( x ). Th us there is a rank 1 eigenpro jection field ov er this neigh borho o d. W e will deform the family so that this prop ert y holds ov er the k -sk eleton, hence on a neighborho o d of it. W e now try to extend the eigenpro jection field ov er the 3-sk eleton. Pro ceedin g as ab o v e b y flattening and deforming, w e obtain an eigen- pro jection field o v er the b oundaries of 3-simplices. W e tak e a slightly differen t approach to c omplete the deformation pro cess. W e first tr y to extend to simply a general pro jection field. This can b e done if the eigenpro jection fields defined on the b oundaries of the 3-simplices, S 2 → Gr 1 ( H ), are nu ll-homotopic. Since Gr 1 ( H ) = C P ∞ , this is not automatic. Ho w e v er, obstruction theory applies a nd there is a class in H 3 ( S 2 n +1 , π 2 ( Gr 1 ( H )) whic h mus t v anish in order for the extension to exist (after going ba ck and redefining o v er lo w e r ske leta). Since w e are w orking with a sphere of dimension greater than 5, this group v anishe s and the extension exists. W e no w pro ceed b y induction. W e ha v e a field ov er all of S 2 n +1 whic h is an eigenpro jection field ov er a neighborho o d of the k - sk eleton. W e next t r y to push this field dow n to b e an eigenpro jection field o v e r the ( k + 1)-sk eleton. W e consider the set of ( k + 1)-simplices and their b oundaries. There are t w o cases. If a ( k + 1)-simplex do esn’t con tain an y singular p oin ts, then w e ha v e t w o pro jection fields ov er it– an eigenpro jection field a nd the general one w e j us t obtained. An y t w o are homotopic relative to the b oundary b ecause only the sec- ond homotop y group of Gr 1 ( H ) is non- ze ro. W e will assign 0 to suc h a simplex . F o r the others, using the flattening pro cedure de- scrib ed ab ov e, w e will get an elemen t in the relativ e homotop y group, π k ( Gr 1 ( H ) , Gr 1 ( C 2 )) ∼ = π k − 1 ( Gr 1 ( C 2 )) = π k − 1 ( S 2 ). This will define an obstruction class in H k +1 ( S 2 n +1 , ( S 2 n +1 ) ( k ) ; π k − 1 ( S 2 )). W e not e that, for a n y complex X , one has H j ( X , X ( k ) ) = 0 if j ≤ k , and H j ( X , X k ) ∼ = H j ( X ) if j > k . Thu s, the only group whic h can b e non-zero is H 2 n +1 ( S 2 n +1 , ( S 2 n +1 ) (2 n ) ); π 2 n − 1 ( S 2 )), a torsion group. F or SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 19 k < 2 n + 1 w e extend the restriction of the eigenpro jection field to the ( k − 1)-sk eleton to the ( k + 1)-sk eleton and use this to deform the family so it is separated there. F or the case of the to p dimension the follo wing pro cedure take s care of this obstacle. W e ha v e a p ossibly non- trivial, obstruction class whic h w ould b elong to H 2 n +1 ( S 2 n +1 ; π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 ))). Since n ≥ 2, π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 )) ∼ = π 2 n − 1 ( S 2 ) is a finite group, sa y of o r de r N . W e will show tha t N { D x } is trivial, and hence { D x } is r a tionally trivial. Let ∆ be an 2 n + 1-simplex whic h meets C 0 , 1 in its interior and let ∆ ′ b e a sub-simplex o bta ined b y flattening. Th us, we may assume that the family has m ultiplicit y tw o at eac h p oint o f ∆ ′ and there is a r a nk 1 eigenpro jection field along the b oundary . No w, consider the family N { D x } . W e trivialize each 2-dimensional eigen bundle separately and get a map c (∆) : (∆ ′ , ∂ ∆ ′ ) → ( Gr 1 ( H ) × . . . × Gr 1 ( H ) , Gr 1 ( C 2 ) × . . . × Gr 1 ( C 2 )) defining a class in π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 )) ⊕ . . . ⊕ π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 )). The map induced b y a ddition on homotop y groups se nds the class of this map to zero in π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 )). F rom the comm utativ e diagram, π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 )) ⊕ . . . ⊕ π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 )) π 2 n ( Gr 1 ( H ) , Gr N ( C 2 N )) π 2 n ( Gr 1 ( H ) , Gr 1 ( C 2 ))           + / / inc ∗ 3 3 g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g inc ∗ w e see that w e may deform the map c (∆) : ∆ ′ → Gr 1 ( H ) to one mapping in to Gr N ( C 2 N ). Using the same deformation as in the rank 1 case, w e obtain a new family for whic h µ 0 ( x ) < µ 1 ( x ) for all x ∈ ∆ ′ . D oing this pro cess ov er each n -simplex yields the required trivial family .  7. Concluding remarks The in ten t io n of the presen t pap er w as to b egin a study of the manner in whic h the v ariation of the eigen v alues and eigenspaces o f a family of self-adjoin t F redholm op erators effects the K-theory class of the family . While w e sho w ed that t he b eha vior of the multiplicit y function can effect the topo logy of the family , there is muc h y et to b e resolv ed. Some questions which seem essen tial to making further progress are listed b elo w. • Ho w is the 3-dimensional in tegral cohomology class whic h arises when applying obstruction theory related to the index gerb e, 20 RONALD G. DOUGLA S AND JEROME KAMIN KER c.f. Lott, [10 ]. If they determine eac h other, can one obtain all the comp onen ts of the Chern c haracter of the family using these metho ds? • Supp ose [ α ] ∈ K 1 ( X ) and there is an α ′ with α ≃ α ′ and with the multiplicit y of α ′ b ounded b y n . Let M ([ α ]) b e the least suc h n . If [ α ] 6 = 0 then M ([ α ]) > 1. How are the to polog ical in v arian ts of [ α ] related to M ([ α ])? • The equiv alence relation generated b y the “mo v es” w e are using to deform the families is p ossibly stronger than homot op y . Is one obtaining a more refined t yp e of K- t he ory in this w a y? • It w ould b e in teresting to know in what sense the K-theory class of a family is determined b y a finite part of the sp ec trum. T o b e more precise, supp ose { D x } is a fa mily with multiplicit y b ounded by n . Is there a n in teger N ( n ) so that the part of the graph of the family , S | k | < N ( n ) µ k ( X ), along with the corre- sp onding eigenspaces, determines whether the family is trivial (or rationally trivial) in K- theory? • Although v arious partial results similar to those in the last sec- tion are k no wn to the authors, the appropriate general state- men t has no t y et b een obtained. W e exp ect that the following will hold. Assume the para meter space o f the family , X , is an n- dimensional finite complex and the sp ectral flow of the family is zero. F urther, supp ose there is an elemen t o f the exhaustion, µ k suc h that the multiplicit y at any p oin t of µ k ( X ) is N or N + 1, where N > n . Then if the 3-dimensional obstruction obtained ab o v e is zero, the family is rationally trivial in K-theory . Reference s [1] M. F. Atiy ah, K - the ory , Benjamin Press , New Y ork, 1 967. [2] M. F. Atiy ah and F. Hirzebruch, V e ctor bu nd les and homo gene ous sp ac es , Pro c. Sympo s. P ure Math., V ol. I II, American Mathematica l So ciety , Pr ovidence, R.I., 1961 , pp. 7–38 . [3] M. F. Atiy ah a nd I. M. Singer, The i ndex of el liptic op er ators I , Annals of Mathematics 87 (196 8), 484– 5 30. [4] M. F. Atiy ah and I. M. Singer, Index the ory for skew-adjoint F r e dholm op er ators , Inst. Hautes ´ Etudes Sci. Publ. Ma th. (1969), no . 37, 5–26 . MR MR02850 33 (44 #2257) [5] , Dir ac op er ators c ou ple d t o ve ctor p otentials , Pro c. Nat. Aca d. Sci. U.S.A. 81 (1984 ), no. 8, Phys. Sci., 25 9 7–2600. MR MR7 42394 (86 g :58127) [6] M. F. Atiy ah, I. M. Singer, and V. K . Pato di, S p e ct r al asymmetry and Rie- mannian ge ometry, I , Ma th. Pro c. Camb. Phil. So c. 77 (1975), 43–69 . [7] B. Bo os-Bavn bek and K. P . W o jciecho wski, El liptic b oundary pr oblems for Dir ac op er ators , Bir kh¨ auser, 1993. SPECTRAL MUL TIPLICITY A ND ODD K-THEOR Y 21 [8] Klaus J ¨ anich, V ektorr aumb ¨ undel und der Raum der Fr e dholm-Op er ator en , Math. Ann. 161 (1965 ), 129 –142. [9] Michael Jo ac him, Unb ounde d F r e dholm op er ators and K -the ory , High- dimensional manifold top ology , W orld Sci. Publ., River Edge, NJ, 2 0 03, pp. 177–1 99. MR MR204 8722 (2005c:4 6104) [10] John Lo tt , Higher-de gr e e analo gs of the determinant line bund le , Co mm. Math. Phys. 23 0 (200 2), no. 1, 41– 69. MR MR1930 571 (2003 j:58052) [11] Ric hard B. Melr ose and Paolo Pia zza, F amilie s of D ir ac op er ators, b ound- aries and the b -c alculus , J. Differen tial Geom. 46 (1997), no. 1, 9 9 –180. MR MR14728 95 (99a:58 144) [12] J. Mick elsson, Gerb es and quantum field the ory , a rXiv: math-ph/0 60303. [13] L. Nicolaescu, O n the s p ac e of Fr e dholm self-adjo int op er ators , An. Sti. Univ. Iasi 53 (2007), 209 – 227. [14] I. M. Singer , F amilies of Dir ac op er ators with applic ations to physics , Ast ´ erisq ue (198 5), no. Numero Hor s Serie, 3 2 3–340, The mathematical her - itage of ´ Elie Cartan (Lyon, 1 984). Dep ar tment of Ma thema tics, Texas A &M University, Coll eg e St a- tion, TX E-mail addr ess : rdouglas@ math.tamu.edu Dep ar tment of Ma thema tical Sciences, IUP UI, Indianapolis, I N Curr ent addr ess : Departmen t of Mathematics, UC Da vis, Da vis, CA E-mail addr ess : kaminker@ math.ucdavis.edu