A topological invariant for continuous fields of the Cuntz algebras
aa r X i v : . [ m a t h . OA ] F e b A topological invariant for continuous fields of the Cuntzalgebras
Taro Sogabe
Graduate School of Science, Kyoto University, [email protected]
February 18, 2020
Abstract
For a continuous field of the Cuntz algebra over a finite CW complex, we introducea topological invariant, which is an element in Dadarlat–Pennig’s generalized cohomologygroup, and prove that the invariant is trivial if and only if the field comes from a vectorbundle via Pimsner’s construction.
This paper is a continuation of our work [19], and our main interest is continuous fieldsof the Cuntz algebra O n +1 . By M. Dadarlat’s result [10], continuous fields of O n +1 over afinite CW complex X are automatically locally trivial, and the homotopy theory is useful toinvestigate them. He studied continuous fields of O n +1 over X coming from vector bundlesin [9], and proved that the ideals of K -theory ring defined by the vector bundles give thecomplete invariant of these fields. He also showed that every continuous field comes from avector bundle when the cohomology H ∗ ( X ) has no n -torsion (see [9, 19]). This classificationresult is used in the work [20] of M. Izumi and H. Matui for the problem of classifying groupactions on O n +1 . However, in general there exist examples of continuous fields which do notcome from vector bundles. The purpose of this paper is to introduce a topological invariantfor continuous fields of O n +1 , and we show that the invariant is trivial if and only if thecontinuous fields come from vector bundles.To sketch our construction of the invariant, let us review relevant results. The Dixmier–Douady theory classifies the locally trivial bundles over a finite CW complex X , whose fiber isthe algebra of compact operators K , by the third cohomology group H ( X ) (see [30, 32]). Intheir remarkable work, M. Dadarlat and U. Pennig generalized this theory to K ⊗ D for everystrongly self-absorbing C*-algebra D . The strongly self-absorbing C*-algebras introducedin [34] are an important class of C*-algebras containing the Cuntz algebra O , O ∞ and theJang–Su algebra Z . They revealed that the set of the isomorphism classes of the locally trivialcontinuous fields of K ⊗ D is identified with the first group E D of a generalized cohomology E ∗ D (see [7]).In this paper, using the Cuntz–Toeplitz algebra E n +1 , we construct the invariant men-tioned above as an element of the group E D ( X ). Recall that E n +1 ⊗ D has a unique proper deal K ⊗ D and the quotient algebra ( E n +1 ⊗ D ) / ( K ⊗ D ) is isomorphic to O n +1 ⊗ D . Our maintechnical result is as follows : the group homomorphism q : Aut( E n +1 ⊗ D ) → Aut( O n +1 ⊗ D )is a weak homotopy equivalence for every strongly self-absorbing C*-algebra D satisfying theUCT (Corollary 3.15). For the proof, we use the arguments developed in [33] for computingthe homotopy groups of Aut( E n +1 ). Then the map q , the group homomorphism ⊗ id D :Aut( O n +1 ) → Aut( O n +1 ⊗ D ) and the restriction map η n : Aut( E n +1 ⊗ D ) → Aut( K ⊗ D )give the invariant b D : [ X, BAut( O n +1 )] → E D ( X ). We show that the set b − M ( n ) (0) consistsof the continuous fields coming from vector bundles which are classified in [9, 19]. Here wedenote by M ( n ) the UHF algebra of infinite type whose K -group is the localization of Z atthe ideal n Z (Theorem 4.10).It would be desirable to completely determine the image and the inverse images of themap b M ( n ) but we have not been able to do this. It is not known whether the image of b M ( n ) is a group or not. However it is worth pointing out the following two cases. In the caseof dim X ≤
3, the classical obstruction theory tells that [ X, BAut( O n +1 )] is identified with H ( X, Z n ), and b M ( n ) is identified with the Bockstein map H ( X, Z n ) → Tor( H ( X ) , Z n )(Theorem 5.3). For the suspension X = SY , the map b M ( n ) reduces to the Bockstein map˜ K ( X ; Z n ) → Tor( K ( X ) , Z n ). Acknowledgments
The author would like to express his greatest appreciation to his supervisor Prof. MasakiIzumi who informed him of Theorem 5.3 and gave him the idea of the construction of theinvariant and many other insightful comments.
Let A be a unital C*-algebra and let U ( A ) be the group of unitary elements in A . We denoteby U ( A ) the path component of 1 A in U ( A ). We denote the set of positive elements of A by A + and denote the set of positive contractions by ( A + ) . For a non-unital C*-algebra B , we denote its unitization by B ∼ . The K -groups of A are denoted by K i ( A ) , i = 0 , p ] the class of a projection p in K ( A ) and by [ u ] the class of a unitary u in K ( A ). We denote K ( A ) + := { [ p ] ∈ K ( A ) | p ∈ P ( A ⊗ K ) } where P ( A ⊗ K ) is theset of projections. We denote by SA the suspension of a C*-algebra A , and one has theBott periodicity : K ( A ) ∼ = K ( SA ). For a compact Hausdorff space X , we denote by C ( X )the C*-algebra of continuous functions on X and denote by C ( X, x ) the C*-subalgebraconsisting of all functions vanishing at x ∈ X , which is the kernel of the evaluation mapev x : C ( X ) ∋ f f ( x ) ∈ C . We denote K i ( X ) = K i ( C ( X )) and, for connected X , denote˜ K i ( X ) = K i ( C ( X, x )) for short. We denote by M n the n by n matrix algebra. For two unitalC*-algebras A and A , we denote by Hom( A , A ) u the set of unital ∗ -homomorphisms from A to A and denote End( A ) := Hom( A , A ) u . For a topological space Y and two elements y , y ∈ Y , we denote y ∼ h y in Y if there is a continuous path from y to y . A unitalC*-algebra A is called K -injective if the natural map U ( A ) / ∼ h → K ( A ) is injective.We denote by K the algebra of compact operators of the infinite dimensional separableHilbert space H . For an algebra A ⊗ K , we denote its multiplier algebra by M ( A ⊗ K ) nd denote by Q ( A ⊗ K ) the quotient algebra M ( A ⊗ K ) /A ⊗ K with the quotient map π : M ( A ⊗ K ) → Q ( A ⊗ K ). We remark that Q ( A ⊗ K ) is K -injective (see [26, Section 1.13]).The algebra M ( C ( X ) ⊗ A ⊗ K ) is identified with C bs ( X, M ( A ⊗ K )), that is, the algebra of M ( A ⊗ K )-valued bounded continuous functions on X with respect to the strict topology (see[2, Corollary 3.4], [30, Proposition 2.57]). Theorem 2.1 ([16, Theorem 1]) . For a unital C*-algebra A , the unitary group U ( M ( A ⊗ K )) is contractible with respect to the norm topology, and we have K i ( M ( A ⊗ K )) = 0 , i = 0 , . For C*-algebras B and C , an extension C of B by A ⊗ K is an exact sequence0 → A ⊗ K → C → B → , and the Busby invariant of the extension is the induced map τ : B → Q ( A ⊗ K ). We identify anextension with the corresponding Busby invariant. We refer to [4] for the following definitionand the basic facts of the theory of extensions of C*-algebras. Definition 2.2.
Let A be a unital C*-algebra and let B be a C*-algebra. An extension τ : B →Q ( A ⊗ K ) is called trivial if the corresponding exact sequence splits. The extension is calledessential if τ is injective, and called unital if B is unital and τ is a unital ∗ -homomorphism.Two Busby invariants τ i : B → Q ( A ⊗ K ) , i = 1 , are said to be strongly unitarilyequivalent, τ ∼ s.u.e τ , if there is a unitary U ∈ M ( A ⊗ K ) with τ = Ad π ( U ) ◦ τ . Theyare said to be weakly unitarily equivalent, τ ∼ w.u.e τ , if there is a unitary u ∈ Q ( A ⊗ K ) with τ = Ad u ◦ τ . We denote τ ∼ s τ if there exist two trivial extensions ρ and ρ with τ ⊕ ρ ∼ s.u.e τ ⊕ ρ and denote by Ext(
B, A ⊗ K ) the set of the equivalence classes of theextensions with respect to the equivalence relation ∼ s . We remark that τ ∼ w.u.e τ implies τ ∼ s τ (see [4, Proposition 15.6.4]). For B nuclearand separable, the set Ext( B, A ⊗ K ) is a group. Theorem 2.3 ([4, Theorem 23.1.1]) . Let A and B be nuclear separable C*-algebras with A in the bootstrap class. Then there is a short exact sequence → M i =0 , Ext Z ( K i ( B ) , K i ( A )) → Ext(
B, A ⊗ K ) → M i =0 , Hom( K i ( B ) , K i +1 ( A )) → , which splits unnaturally. If L i =0 , Hom( K i ( B ) , K i +1 ( A )) = 0 , we have an isomorphism Ext(
B, A ⊗ K ) → M i =0 , Ext Z ( K i ( B ) , K i ( A )) that sends the class of an extension → A ⊗ K → C → B → to the corresponding groupextension [ K i ( A ) → K i ( C ) → K i ( B )] ∈ Ext Z ( K i ( B ) , K i ( A )) , i = 0 , . Let E n +1 be the universal C*-algebra generated by n + 1 isometries with mutually orthog-onal ranges, called the Cuntz–Toeplitz algebra, and let { T i } n +1 i =1 be the canonical generatorsof E n +1 . The closed two-sided ideal generated by the minimal projection e := 1 − P n +1 i =1 T i T ∗ i is isomorphic to K , which is known to be the only closed non-trivial two-sided ideal. Let π : E n +1 → O n +1 be the quotient map by the ideal K and denote S i := π ( T i ). The quo-tient algebra O n +1 is the universal C*-algebra generated by n + 1 isometries with relation: S ∗ i S j = δ ij , P n +1 i =1 S i S ∗ i . We denote by O ∞ the universal C*-algebra generated by ountably infinite isometries with mutually orthogonal ranges. These algebras O n +1 , O ∞ arecalled the Cuntz algebras with the following K -groups K ( O n +1 ) = Z n , K ( O ∞ ) = Z , K ( O n +1 ) = K ( O ∞ ) = 0(see [5, Theorem 3.7, 3.8, Corollary 3.11]). They are purely infinite and simple (i.e., for everynon-zero element x , there exist two elements y, z with yxz = 1).For a separable unital C*-algebra A , the short exact sequence 0 → K ⊗ A → E n +1 ⊗ A →O n +1 ⊗ A → K ( A ) − n / / K ( A ) / / K ( O n +1 ⊗ A ) exp (cid:15) (cid:15) K ( O n +1 ⊗ A ) δ O O K ( A ) o o K ( A ) , − n o o where δ is the index map, and the image of δ is Tor( K ( A ) , Z n ).Elliott–Kucerovsky’s absorption theorem is the main technical ingredient in Section 3.2. Definition 2.4 ([6, 15]) . An extension → A ⊗ K → C → B → is called purely large iffor every c ∈ C \ A ⊗ K , there is a subalgebra D ⊂ c ( A ⊗ K ) c ∗ satisfying the following :
1) The subalgebra D is not contained in any proper closed two-sided ideal of A ⊗ K ,2) The subalgebra D is σ -unital and satisfies D ∼ = D ⊗ K . Every purely large extension is essential. The following is a special case of [6, p 387].
Theorem 2.5.
Let A and B be separable C*-algebras, and assume that B is unital andnuclear. Let ρ : B → M ( A ⊗ K ) be a unital ∗ -homomorphism, and let τ : B → Q ( A ⊗ K ) be the Busby invariant of a unital extension → A ⊗ K → C → B → . If the extension ispurely large, then we have τ ⊕ ( π ◦ ρ ) ∼ s.u.e τ . Lemma 2.6.
For a unital separable C*-algebra A , every unital extension → A ⊗ K → C →O n +1 → is purely large.Proof. We denote I := 1 C = 1 M ( A ⊗ K ) . For c ∈ C \ A ⊗ K , we show that there is an element y ∈ M ( A ⊗ K ) with y ∗ c ∗ cy = I . Since O n +1 is purely infinite and simple, there exists x ∈ C satisfying I − x ∗ c ∗ cx ∈ A ⊗ K . Let f, h ∈ ( A ⊗ K ) + be the positive part and the negativepart of the selfadjoint element I − x ∗ c ∗ cx = f − h ∈ A ⊗ K . Since ( I + h ) / is invertible and f h = hf = 0, one has ( I + h ) − / x ∗ c ∗ cx ( I + h ) − / = I − f . One has 0 ≤ f ≤ I because( I + h ) − / x ∗ c ∗ cx ( I + h ) − / is positive. There exists a number N and a positive contraction f ∈ (( A ⊗ M N ) + ) ⊂ A ⊗ K with || f − f || ≤ /
2. For the rank N projection 1 N ∈ M N ⊂ K ,we have ( I − f )( I − (1 A ⊗ N )) = I − (1 A ⊗ N ) = 1 A ⊗ (1 M ( K ) − N ). One can take anisometry V ∈ M ( K ) with (1 A ⊗ V )(1 A ⊗ V ) ∗ = 1 A ⊗ (1 M ( K ) − N ). Direct computation yields(1 A ⊗ V ) ∗ ( I − f )(1 A ⊗ V ) = (1 A ⊗ V ) ∗ ( I − f )( I − (1 A ⊗ N ))(1 A ⊗ V )= (1 A ⊗ V ) ∗ ( I − (1 A ⊗ N ))(1 A ⊗ V )= I. Therefore we have || (1 A ⊗ V ) ∗ (1 − f )(1 A ⊗ V ) − I || = || (1 A ⊗ V ) ∗ (1 − f )(1 A ⊗ V ) − (1 A ⊗ V ) ∗ (1 − f )(1 A ⊗ V ) ||≤ || f − f ||≤ / . he selfadjoint element (1 A ⊗ V ) ∗ (1 − f )(1 A ⊗ V ) is invertible, and y ∗ c ∗ cy = I holds for y := x (1 + h ) − / (1 A ⊗ V ) { (1 A ⊗ V ) ∗ (1 − f )(1 A ⊗ V ) } − / . Now the argument in [24, p 425]proves the statement.The proof of [33, Lemma 2.14] shows the following corollary. Corollary 2.7.
Let A be a unital separable C*-algebra, and let τ i : O n +1 → Q ( A ⊗ K ) , i = 1 , be two unital extensions. If [ τ ] = [ τ ] ∈ Ext( O n +1 , A ⊗ K ) , we have τ ∼ w.u.e τ . We refer to [33, Proposition 2.17] for the proof of the following proposition.
Proposition 2.8.
For a unital C*-algebra A with Tor( K ( A ) , Z n ) = K ( A ) = 0 and a unitalextension σ : O n +1 → Q ( K ⊗ A ) , we have K ( σ ( O n +1 ) ′ ∩ Q ( K ⊗ A )) = 0 . We introduce strongly self-absorbing C*-algebras (see [34]).
Definition 2.9.
A unital separable C*-algebra D = C is called strongly self-absorbing ifthere exists an isomorphism φ : D → D ⊗ D and a sequence of unitaries u n ∈ D ⊗ D with lim n →∞ || φ ( d ) − u n ( d ⊗ D ) u ∗ n || = 0 for every d ∈ D . For the set P n of all prime numbers p with GCD( n, p ) = 1, let M ( n ) be the UHF algebra M ( n ) := O p ∈ P n M p ∞ . C*-algebras M ( n ) and M ( n ) ⊗ O ∞ are strongly self-absorbing. Theorem 2.10.
Let D be a strongly self-absorbing C*-algebra, then the following holds :
1) There is a sequence of unital ∗ -homomorphisms φ n : D → D with lim n →∞ || φ n ( d ) x − xφ n ( d ) || = 0 for every x, d ∈ D (see [34]).2) The algebra D is Z -stable and it is K -injective (see [37, 31]). Furthermore, the map U ( A ⊗ D ) / ∼ h → K ( A ⊗ D ) is bijective for a separable unital C*-algebra A (see [21]).3) Consider α, β ∈ Hom(
D, A ⊗ D ) u for a unital separable C*-algebra A . There is acontinuous path of unitaries { u t } t ∈ [0 , ⊂ A ⊗ D satisfying u = 1 and lim t → || u t α ( d ) u ∗ t − β ( d ) || = 0 for every d ∈ D (see [12]). Let Q N A be the C*-algebra of all bounded sequences of A and let L N A be the subalgebraof the sequences converging to 0. For a unital C*-algebra A , we introduce the central sequencealgebra A ∞ := Y N A/ M N A ! ∩ A ′ , where A is embedded in ( Q N A/ L N A ) by the constant sequence map A ∋ a ( a ) n ∈ Q N A . Theorem 2.11 ([34, Theorem 2.2]) . For A separable unital and D strongly self-absorbing,an isomorphism A ∼ = A ⊗ D holds if and only if there is a unital embedding D → A ∞ . .2 Some notions in the homotopy theory For pointed topological spaces (
X, x ) and ( Y, y ), we denote the set of the continuous mapsfrom X to Y by Map( X, Y ) and denote by Map ( X, Y ) the subset of Map(
X, Y ) consistingof the base point preserving maps. We denote the homotopy set Map(
X, Y ) / ∼ h by [ X, Y ]and denote Map ( X, Y ) / ∼ h by [ X, Y ] . The i -th homotopy group is denoted by π i ( X, x ) :=[ S i , X ] . A pointed space ( Y, y ) is an H -space if there is a continuous map m : Y × Y → Y such that the two maps y m ( y, y ) and y m ( y , y ) are homotopic to id Y by the basepoint preserving homotopy. We remark that the natural map [ X, Y ] → [ X, Y ] is bijectivefor a path connected H -space ( Y, y ) and a pointed space ( X, x ) with a non-degenerate basepoint x , in particular, for a pointed CW complex (see [13, 6.16, p 159]). Definition 2.12 ([13, p 117]) . Let
X, Y and Z be topological spaces, and let π : X → Y be acontinuous map. The map π has the homotopy lifting property (abbreviated to HLP) for Z ,if every commuting diagram { } × Z g / / (cid:15) (cid:15) X π (cid:15) (cid:15) [0 , × Z f / / Y, admits a continuous map ˜ g : [0 , × Z → X satisfying ˜ g (0 , z ) = g ( z ) and π ◦ ˜ g = f . The map π is a Serre fibration, if it has HLP for every n -disc. For example, every locally trivial fiber bundle is a Serre fibration. We refer to [7, Lemma2.8, 2.16, Corollary 2.9] for the proof of the following lemma.
Lemma 2.13.
Let D be a unital strongly self-absorbing C*-algebra with K ( D ) = 0 . Considerthe set U ( E n +1 ⊗ D )((1 − e ) ⊗ D ) := { u ((1 − e ) ⊗ D ) ∈ E n +1 ⊗ D | u ∈ U ( E n +1 ⊗ D ) } . The map p : U ( E n +1 ⊗ D ) ∋ u u ((1 − e ) ⊗ D ) ∈ U ( E n +1 ⊗ D )((1 − e ) ⊗ D ) gives a locally trivial fiber bundle with a fiber homeomorphic to U ( D ) . Since K ( E n +1 ⊗ D ) = 0, Theorem 2.10 shows that the group U ( E n +1 ⊗ D ) and the set U ( E n +1 ⊗ D )((1 E n +1 − e ) ⊗ D ) are path connected. Lemma 2.14.
A surjection π ∈ Hom( A , A ) u gives a Serre fibration π : U ( A ) → U ( A ) .Proof. For two continuous maps f : [0 , × D n → U ( A ) and g : D n → U ( A ) with f (0 , x ) = π ( g ( x )) , x ∈ D n , we identify f (resp. g ) with a unitary of U ( C ([0 , × D n ) ⊗ A ) (resp. U ( C ( D n ) ⊗ A )). Since f π (1 C ([0 , ⊗ g ∗ ) ∈ U ( C ([0 , × D n ) ⊗ A ), there is a lift u ∈ U ( C ([0 , × D n ) ⊗ A ) of f π (1 C ([0 , ⊗ g ∗ ) with u (0 , x ) = 1 A , x ∈ D n . The unitary u (1 C ([0 , ⊗ g ) ∈ U ( C ([0 , × D n ) ⊗ A ) gives a lift of f , and the map π has HLP for every n -disc. Theorem 2.15 ([13, Corollary 6.44]) . For a Serre fibration π : ( X, x ) → ( Y, y ) with thefibre F := π − ( y ) , there is a long exact sequence of groups ( i ≥ , and exact sequence ofpointed sets ( i ≥ · · · → π i ( F, x ) → π i ( X, x ) → π i ( Y, y ) → · · · → π ( F, x ) → π ( X, x ) → π ( Y, y ) . or a topological group G , the set of isomorphism classes of principal G -bundles over X is identified with the homotopy set [ X, B G ], where B G is the classifying space, and wedenote by E G → B G the universal principal G -bundle. For example, for H = l , the space H = { f ∈ H | || f || = 1 } is contractible and is a model of ES with free S action bythe scalar multiplication (see [30]). Identifying B S with the set of all minimal projections,the map E S = H ∋ ξ ξ ⊗ ξ ∗ ∈ B S gives the universal bundle where we denoteby ξ ⊗ ξ ∗ the operator H ∋ x
7→ h x, ξ i ξ ∈ H . Fix a base point e ∈ B S . The mapAut( K ) ∋ α α ( e ) ∈ B S is a homotopy equivalence, and the following Dixmier–Douadytheorem holds. Theorem 2.16 ([14, 32, 30]) . For a finite CW complex X , the set [ X, BAut( K )] , identi-fied with the set of isomorphism classes of the locally trivial continuous fields of K , has amultiplication given by the fiber wise tensor product of the fields, and it is isomorphic to H ( X ) . Using the isomorphism φ : D ∼ = D ⊗ D , one gives K ( C ( X ) ⊗ D ) a ring structure whosemultiplication comes from the following map∆ X : ( C ( X ) ⊗ D ) ⊗ ∋ f ( x ) ⊗ f ( y ) φ − ( f ( x ) ⊗ f ( x )) ∈ C ( X ) ⊗ D (see [7, Section 2.4]). We denote the set of the invertible elements by K ( C ( X ) ⊗ D ) × . Theorem 2.17 ([7, Theorem 2.22, 3.8, Lemma 2.8]) . Let D = C be a strongly self-absorbingC*-algebra and let X be a compact metrizable space. We denote by Aut ( K ⊗ D ) the pathcomponent of Aut( K ⊗ D ) containing id K ⊗ D and denote by P ( K ⊗ D ) the set of projectionswhich are homotopy equivalent to e ⊗ D .1) For two continuous maps α, β : X → Aut( K ⊗ D ) , which are identified with the C ( X ) -linear isomorphisms of C ( X ) ⊗ K ⊗ D , one has K (∆ X ) ◦ K ( α ⊗ β )([(1 C ( X ) ⊗ e ⊗ D ) ⊗ ] ) = K ( α ◦ β )([1 C ( X ) ⊗ e ⊗ D ] ) , and the following map is multiplicative :[ X, Aut( K ⊗ D )] ∋ [ α ] [ α ( e ⊗ D )] ∈ K ( C ( X ) ⊗ K ⊗ D ) . The image of the map is K ( C ( X ) ⊗ D ) + ∩ K ( C ( X ) ⊗ D ) × .2) The following map η is a homotopy equivalence and induces an isomorphism η ∗ forconnected X : η : Aut ( K ⊗ D ) ∋ α α ( e ⊗ D ) ∈ P ( K ⊗ D ) ,η ∗ : [ X, Aut ( K ⊗ D )] → K ( C ( X, x ) ⊗ D ) .
3) The homotopy set E D ( X ) := [ X, BAut( K ⊗ D )] has a group structure defined by thefiber wise tensor product of the locally trivial continuous fields of K ⊗ D . .4 The homotopy groups of Aut( O n +1 ) Using the well-known homeomorphism u : End( O n +1 ) ∋ α n +1 X i =1 α ( S i ) S ∗ i ∈ U ( O n +1 )M. Dadarlat computed the homotopy groups of Aut( O n +1 ). Theorem 2.18 ([10, Theorem 1.1]) . The inclusion map
Aut( O n +1 ) → End( O n +1 ) is a weakhomotopy equivalence, and we have π odd (Aut( O n +1 )) = π odd (End( O n +1 )) = Z n , π even (Aut( O n +1 )) = π even (End( O n +1 )) = 0 . Our goal in this section is to prove Corollary 3.15. Throughout this section, we assume that D is a strongly self-absorbing C*-algebra satisfying Tor( K ( D )) = K ( D ) = 0 and O n +1 ⊗ D ∼ = O m +1 for some m ≥
1. Note that the above conditions automatically hold if D satisfiesthe UCT (see [34, Proposition 5.1]). We denote by End ( E n +1 ⊗ D ) the path component ofEnd( E n +1 ⊗ D ) containing id E n +1 ⊗ D . Since β ( e ⊗ D ) ∼ h e ⊗ D for β ∈ End ( E n +1 ⊗ D ),one has β ( K ⊗ D ) ⊂ K ⊗ D , and we have a semi-group homomorphism q : End ( E n +1 ⊗ D ) ∋ β ˜ β ∈ End( O n +1 ⊗ D ) . End ( E n +1 ⊗ D ) Theorem 3.1.
The following map is a weak homotopy equivalence :End ( E n +1 ⊗ D ) ∋ β n +1 X i =1 β ( T i ⊗ D )( T ∗ i ⊗ D ) ∈ U ( E n +1 ⊗ D )((1 E n +1 − e ) ⊗ D ) . Note that the above map is well-defined (see [33, Lemma 2.11]). By the isomorphism l := φ − : D ⊗ → D , it is enough to show that the following map is a weak homotopyequivalence :Π : End ( E n +1 ⊗ D ⊗ ) ∋ β n +1 X i =1 β ( T i ⊗ D ⊗ )( T ∗ i ⊗ D ⊗ ) ∈ U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ ) . By Theorem 2.10, there is a path of unitaries { v t } t ∈ [0 , ⊂ U ( D ⊗ ) satisfying v = 1 D ⊗ andlim t → || Ad v t ( d ) − l ( d ) ⊗ D || = 0 for every d ∈ D ⊗ . Let H t be the map H t : End ( E n +1 ⊗ D ⊗ ) ∋ β (id E n +1 ⊗ (Ad v ∗ t ◦ ( l ⊗ D ))) ◦ β ∈ End ( E n +1 ⊗ D ⊗ )for t ∈ [0 ,
1) and H ( ρ ) := ρ . Let h t be the map h t : U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ ) → U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ )defined by h t ( w ) := (id E n +1 ⊗ (Ad v ∗ t ◦ ( l ⊗ D )))( w ) for t ∈ [0 ,
1) and h ( w ) := w . One can checkthat H t ( β ) converges to β as t tends to 1 in the point-wise norm topology, lim t → || h t ( w ) − || = 0 and Π ◦ H t = h t ◦ Π. For w ∈ U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ ), we define a map ̺ w ∈ Hom( E n +1 , E n +1 ⊗ D ⊗ ) u by ̺ w ( T i ) := w ( T i ⊗ D ⊗ ). Since h ( w ) ∈ E n +1 ⊗ D ⊗ C D ,the image of ̺ h ( w ) is in E n +1 ⊗ D ⊗ C D , and we have a ∗ -endomorphism α w : E n +1 ⊗ D ⊗ ∋ f ⊗ d ̺ h ( w ) ( f )(1 E n +1 ⊗ D ⊗ l ( d )) ∈ ( E n +1 ⊗ D ) ⊗ D. Since U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ ) is path connected, Theorem 2.10 implies id E n +1 ⊗ D ⊗ ∼ h id E n +1 ⊗ D ⊗ l ∼ h α w , and one has α w ∈ End ( E n +1 ⊗ D ⊗ ) and Π( α w ) = h ( w ). Lemma 3.2.
We have h t ( w ) ∈ Im Π for w ∈ U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ ) , t ∈ [0 , .The inclusion map Im Π ֒ → U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ ) is a homotopy equivalence.Proof. Direct computation yields h t ( w ) = Ad(1 E n +1 ⊗ v ∗ t ) ◦ h ( w )= (1 E n +1 ⊗ v ∗ t )Π( α w )(1 E n +1 ⊗ v t )= n +1 X i =1 (1 E n +1 ⊗ v ∗ t ) α w ( T i ⊗ D ⊗ )( T ∗ i ⊗ D ⊗ )(1 E n +1 ⊗ v t )= Π(Ad(1 E n +1 ⊗ v ∗ t ) ◦ α w ) . This shows h t ( w ) ∈ Im Π for t ∈ [0 , E n +1 ⊗ D ⊗ l (resp. (1 E n +1 − e ) ⊗ D ⊗ ) as a base point of End ( E n +1 ⊗ D ⊗ )(resp. U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ )). Since h t fixes the base point, one has [ γ ] =[ h ◦ γ ] ∈ π k ( U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ )), and the following lemma holds. Lemma 3.3.
The following map is surjective :Π ∗ : π k (End ( E n +1 ⊗ D ⊗ )) → π k ( U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ )) . For Γ ∈ Map ( S k , End ( E n +1 ⊗ D ⊗ )) with Π ∗ ([Γ]) = 0, we define another map Γ ′ byΓ ′ x := α Π(Γ x ) . Lemma 3.4.
We have Γ ∼ h Γ ′ in Map( S k , End ( E n +1 ⊗ D ⊗ )) .Proof. Note that Γ is homotopic to H ◦ Γ. Regarding ̺ h (Π(Γ x )) , x ∈ S k as a C ( S k )-linearmap in Hom( E n +1 , C ( S k ) ⊗ E n +1 ⊗ D ⊗ ) u and identifying it by ̺ h (Π(Γ)) , we define a unitalseparable C*-algebra P Γ := (Im ̺ h (Π(Γ)) ) ′ ∩ C ( S k ) ⊗ E n +1 ⊗ D ⊗ . Since 1 C ( S k ) ⊗ E n +1 ⊗ D ⊗ D ⊂ P Γ , Theorem 2.10, 2.11 show an isomorphism P Γ ∼ = P Γ ⊗ D . By Theorem 2.10, there is apath { θ t } t ∈ [0 , ⊂ Hom( D ⊗ , P Γ ) u with θ ( d ) = 1 C ( S k ) ⊗ E n +1 ⊗ D ⊗ l ( d ), θ ( d ) : S k ∋ x H ◦ Γ x (1 E n +1 ⊗ d ) , d ∈ D ⊗ . The following map gives a homotopy to prove Γ ′ ∼ h H ◦ Γ :Γ t : E n +1 ⊗ D ⊗ ∋ f ⊗ d ̺ h (Π(Γ)) ( f ) θ t ( d ) ∈ C ( S k ) ⊗ E n +1 ⊗ D ⊗ , t ∈ [0 , . roof of Theorem 3.1. By Lemma 3.3, it is enough to show the injectivity of the map Π ∗ .Since End ( E n +1 ⊗ D ⊗ ) is a path connected H -space, the map π k (End ( E n +1 ⊗ D ⊗ )) → [ S k , End ( E n +1 ⊗ D ⊗ )] is bijective as mentioned before, and we show that Γ ′ is homotopicto a constant map of id E n +1 ⊗ D ⊗ l . By the assumption of Γ, there is a path { w t } t ∈ [0 , ⊂ Map( S k , ( U ( E n +1 ⊗ D ⊗ )((1 E n +1 − e ) ⊗ D ⊗ )) with w = Π ◦ Γ, w ≡ (1 E n +1 − e ) ⊗ D ⊗ . Ahomotopy from Γ ′ to the constant map is given by α w t , t ∈ [0 , u : End( O n +1 ) ∋ σ P n +1 i =1 σ ( S i ) S ∗ i ∈ U ( O n +1 ). Lemma 3.5.
The following map is a weak homotopy equivalence : u ′ : End( O n +1 ⊗ D ) ∋ σ n +1 X i =1 σ ( S i ⊗ D )( S ∗ i ⊗ D ) ∈ U ( O n +1 ⊗ D ) . Proof.
The unital ∗ -homomorphism ⊗ D : O n +1 ∋ a a ⊗ D ∈ O n +1 ⊗ D ∼ = O m +1 inducesa surjection K k − ( O n +1 ) → K k − ( O n +1 ⊗ D ), and the map u ′∗ is surjective by the followingdiagram π k (End( O n +1 ⊗ D )) u ′∗ / / π k ( U ( O n +1 ⊗ D )) K ( C ( S k ) ⊗ O n +1 ⊗ D ) π k (End( O n +1 )) ⊗ id D ∗ O O u ∗ π k ( U ( O n +1 )) ⊗ D ∗ O O K ( C ( S k ) ⊗ O n +1 ) . K ( ⊗ D ) O O O O The equation | π k (End( O n +1 ⊗ D )) | = | π k ( U ( O n +1 ⊗ D )) | (= m,
0) shows the injectivity.
Theorem 3.6.
The map q : End ( E n +1 ⊗ D ) → End( O n +1 ⊗ D ) is a weak homotopy equiv-alence.Proof. The following diagram commutes p − ((1 E n +1 − e ) ⊗ D ) / / i (cid:15) (cid:15) U ( E n +1 ⊗ D ) p / / U ( E n +1 ⊗ D )((1 E n +1 − e ) ⊗ D ) π (cid:15) (cid:15) End ( E n +1 ⊗ D ) Thm . o o q (cid:15) (cid:15) π − (1 O n +1 ⊗ D ) / / U ( E n +1 ⊗ D ) π / / U ( O n +1 ⊗ D ) End( O n +1 ⊗ D ) u ′∗ o o where two horizontal sequences contain the Serre fibrations in Lemma 2.14, 2.13. By 2) ofTheorem 2.10, the inclusion i is a weak homotopy equivalence, and the vertical map π is a weakhomotopy equivalence by 5-lemma. Theorem 3.1 and Lemma 3.5 prove the statement. Aut( E n +1 ⊗ D ) Lemma 3.7.
The image of the restriction map η n : Aut( E n +1 ⊗ D ) → Aut( K ⊗ D ) iscontained in Aut ( K ⊗ D ) .Proof. Note that Tor( K ( D )) = 0 by our assumption. Since K ( α ) = id K ( E n +1 ⊗ D ) for α ∈ Aut( E n +1 ⊗ D ) and the map K ( K ⊗ D ) − n −−→ K ( E n +1 ⊗ D ) is injective, one has[ α ( e ⊗ D )] = [ e ⊗ D ] ∈ K ( K ⊗ D ), and Theorem 2.17 proves the statement. Lemma 3.8.
For a finite CW complex X with Tor( K ( X ) , Z n ) = 0 , we have Im η n ∗ = { } ⊂ [ X, Aut ( K ⊗ D )] for the map η n ∗ : [ X, Aut( E n +1 ⊗ D )] → [ X, Aut ( K ⊗ D )] . roof. We identify a continuous map α : X → Aut( E n +1 ⊗ D ) with a C ( X )-linear ∗ -isomorphism of C ( X ) ⊗ E n +1 ⊗ D . For u ′ ◦ ˜ α = n +1 X i =1 ˜ α (1 C ( X ) ⊗ S i ⊗ D )(1 C ( X ) ⊗ S ∗ i ⊗ D ) ∈ U ( C ( X ) ⊗ O n +1 ⊗ D ) , we have the following lift of the unitary u ′ ◦ ˜ α ⊕ ( u ′ ◦ ˜ α ) ∗ : (cid:16) P n +1 i =1 α (1 ⊗ T i ⊗ D )(1 ⊗ T ∗ i ⊗ D ) α (1 ⊗ e ⊗ D )1 ⊗ e ⊗ D P n +1 i =1 (1 ⊗ T i ⊗ D ) α (1 ⊗ T ∗ i ⊗ D ) (cid:17) ∈ M ( C ( X ) ⊗ E n +1 ⊗ D ) , and direct computation yields[1 C ( X ) ⊗ e ⊗ D ] − [ α (1 C ( X ) ⊗ e ⊗ D )] = δ ([ u ′ ◦ ˜ α ] ) ∈ Tor( K ( X ) ⊗ K ( D ) , Z n ) = 0 . Therefore Theorem 2.17 proves the statement.For an arbitrary β ∈ Aut( E n +1 ⊗ D ), we consider the following C ( S )-algebras C β := { F ∈ C [0 , ⊗ K ⊗ D | F (0) = β ( F (1)) } ,M β := { F ∈ C [0 , ⊗ E n +1 ⊗ D | F (0) = β ( F (1)) } ,A ˜ β := { a ∈ C [0 , ⊗ O n +1 ⊗ D | a (0) = ˜ β ( a (1)) } ,C ( S ) ⊗ O n +1 ⊗ D = { a ∈ C [0 , ⊗ O n +1 ⊗ D | a (0) = a (1) } ,C ( S ) ⊗ K ⊗ D = { F ∈ C [0 , ⊗ K ⊗ D | F (0) = F (1) } . By Lemma 3.7, there is a path Ξ : [0 , → Aut ( K ⊗ D ) from Ξ = id K ⊗ D to Ξ = η n ( β ). Soone has an isomorphism θ β : C β ∋ F ( t ) Ξ t ( F ( t )) ∈ C ( S ) ⊗ K ⊗ D. Since C β ⊂ M β is an essential ideal, the map θ β induces an essential unital extension τ β : A ˜ β → Q ( C ( S ) ⊗ K ⊗ D ). There is a path ξ : [0 , → Aut( O n +1 ⊗ D ) with ξ = id O n +1 ⊗ D , ξ =˜ β , and one has a C ( S )-linear isomorphism φ ˜ β : A ˜ β ∋ a ( t ) ξ t ( a ( t )) ∈ C ( S ) ⊗ O n +1 ⊗ D .We define a unital embedding by σ β : O n +1 ⊗ D ∋ d τ β ( φ − β (1 C ( S ) ⊗ d )) ∈ Q ( C ( S ) ⊗ K ⊗ D ) . By definition, two Busby invariants ev ◦ σ β and ev ◦ σ id En +1 ⊗ D are equal where we denoteby ev : Q ( C ( S ) ⊗ K ⊗ D ) → Q ( K ⊗ D ) the induced map by the evaluation at 0. Lemma 3.9.
We have σ β ∼ s.u.e σ id En +1 ⊗ D .Proof. Using Theorem 2.3, the isomorphism ev ∗ : Ext( O n +1 ⊗ D, C ( S ) ⊗ K ⊗ D ) ∼ =Ext( O n +1 ⊗ D, K ⊗ D ) yields [ σ β ] = [ σ id En +1 ⊗ D ] ∈ Ext( O n +1 ⊗ D, C ( S ) ⊗ K ⊗ D ). ByCorollary 2.7, there is a unitary w ∈ U ( Q ( C ( S ) ⊗ K ⊗ D )) with σ β = Ad w ◦ σ id En +1 ⊗ D . Onehas ev ◦ σ id En +1 ⊗ D = Ad ev ( w ) ◦ (ev ◦ σ id En +1 ⊗ D ) , and Proposition 2.8 shows [ev ( w )] = 0 ∈ K ( Q ( K ⊗ D )) that implies [ w ] = 0 ∈ K ( Q ( C ( S ) ⊗ K ⊗ D )). Now K -injectivity of Q ( C ( S ) ⊗ K ⊗ D ) proves the statement. heorem 3.10. The group
Aut( E n +1 ⊗ D ) is path connected.Proof. By Lemma 3.9, there is a C ( S )-linear isomorphism M β ∼ = C ( S ) ⊗ E n +1 ⊗ D , andthis gives a homotopy from β to id E n +1 ⊗ D .To prove our main technical result, it remains to show that the inclusion Aut( E n +1 ⊗ D ) ֒ → End ( E n +1 ⊗ D ) is a weak homotopy equivalence. Lemma 3.11.
The map π m − (Aut( E n +1 ⊗ D )) → π m − (End ( E n +1 ⊗ D )) is surjective.Proof. By [33, Lemma 3.4, Theorem 3.14] and [10, Theorem 7.4], the map π m − (End ( E n +1 )) → π m − (End( O n +1 )) is surjective. By the diagram in the proof of Lemma 3.5, the map ⊗ id D ∗ : π m − (End( O n +1 )) → π m − (End( O n +1 ⊗ D )) is surjective. Now Theorem 3.6 and[33, Theorem 3.14] yield the following commutative diagram π m − (Aut( E n +1 ⊗ D )) / / π m − (End ( E n +1 ⊗ D )) T hm . π m − (End( O n +1 ⊗ D )) π m − (Aut( E n +1 )) ⊗ id D ∗ O O π m − (End ( E n +1 )) ⊗ id D ∗ O O / / / / π m − (End( O n +1 )) , Lem . O O O O and diagram chasing proves the statement.Since Theorem 3.6 yields π m (End ( E n +1 ⊗ D )) = 0, the above lemma shows the surjec-tivity of the map π k (Aut( E n +1 ⊗ D )) → π k (End ( E n +1 ⊗ D )) , k ≥
0. We refer to [7, Lemma2.8, 2.16, Corollary 2.9] for the proof of the following lemma.
Lemma 3.12.
The map
End ( E n +1 ⊗ D ) ∋ β β ( e ⊗ D ) ∈ P ( K ⊗ D ) , gives two fibrations Aut e ⊗ D ( E n +1 ⊗ D ) → Aut( E n +1 ⊗ D ) → P ( K ⊗ D ) , End e ⊗ D ( E n +1 ⊗ D ) → End ( E n +1 ⊗ D ) → P ( K ⊗ D ) where Aut e ⊗ D ( E n +1 ⊗ D ) ( resp. End e ⊗ D ( E n +1 ⊗ D )) is a subset of Aut( E n +1 ⊗ D ) ( resp. End ( E n +1 ⊗ D )) consisting of ∗ -automorphisms fixing e ⊗ D . Lemma 3.13.
For α : S k → Aut( E n +1 ⊗ D ) with [ α ] = 0 ∈ π k (End ( E n +1 ⊗ D )) , there existsa map α ′ satisfying [ α ] = [ α ′ ] ∈ π k (Aut( E n +1 ⊗ D )) and [ α ′ ] = 0 ∈ π k (End e ⊗ D ( E n +1 ⊗ D )) .Proof. Lemma 3.12 and Theorem 3.6 yield two long exact sequences of the homotopy groupsand the following commutative diagram π m (Aut( E n +1 ⊗ D )) (cid:15) (cid:15) / / K ( D ) / / π m − (Aut e ⊗ D ( E n +1 ⊗ D )) (cid:15) (cid:15) / / π m − (Aut( E n +1 ⊗ D )) → (cid:15) (cid:15) / / K ( D ) / / π m − (End e ⊗ D ( E n +1 ⊗ D )) / / π m − (End ( E n +1 ⊗ D )) → . One has a map α ′′ : S m − → Aut e ⊗ D ( E n +1 ⊗ D ) with [ α ] = [ α ′′ ] ∈ π m − (Aut( E n +1 ⊗ D )). There is an element a ∈ K ( D ) which is sent to [ α ′′ ] ∈ π m − (End e ⊗ D ( E n +1 ⊗ D )),and one can find a map β : S m − → Aut e ⊗ D ( E n +1 ⊗ D ) to which − a is sent. Since[ β ] = 0 ∈ π m − (Aut( E n +1 ⊗ D )), one has [ α ] = [ α ′′ ] = [ α ′ ] ∈ π m − (Aut( E n +1 ⊗ D ))for the map α ′ : S m − ∋ x β x ◦ α ′′ x ∈ Aut e ⊗ D ( E n +1 ⊗ D ). It is also checked that[ α ′ ] = − a + a = 0 ∈ π m − (End e ⊗ D ( E n +1 ⊗ D )). The map π m (Aut e ⊗ D ( E n +1 ⊗ D )) → π m (Aut( E n +1 ⊗ D )) is surjective by diagram chasing, and we can prove the statement because π m (End e ⊗ D ( E n +1 ⊗ D )) = 0. he proof of [33, Lemma 3.3] shows π even (Aut( E n +1 ⊗ D )) = 0. Following the argument in[33, Lemma 3.7], we can determine the weak unitary equivalence class of the Busby invariantassociated with α ′ and the same computation as in [33, Lemma 3.8, 3.9, 3.10, Theorem 3.11] proves [ α ′ ] = 0 ∈ π odd (Aut( E n +1 ⊗ D )). Therefore the following theorem holds. Theorem 3.14.
The inclusion
Aut( E n +1 ⊗ D ) ֒ → End ( E n +1 ⊗ D ) is a weak homotopyequivalence. Combining Theorem 2.18, 3.6, 3.14, we have the following.
Corollary 3.15.
The group homomorphism q : Aut( E n +1 ⊗ D ) → Aut( O n +1 ⊗ D ) is a weakhomotopy equivalence. In particular, the map B( q ) : BAut( E n +1 ⊗ D ) → BAut( O n +1 ⊗ D ) isa weak homotopy equivalence and gives the bijection B( q ) ∗ of the homotopy sets. By [10, Theorem 1.1], all the continuous fields over finite CW-complexes are locally trivial,and we identify them with the principal Aut( O n +1 ) bundles. b D Definition 4.1.
For a finite CW complex X , we define b D : [ X, BAut( O n +1 )] → E D ( X ) by [ X, BAut( O n +1 )] B( ⊗ id D ) ∗ −−−−−−−→ [ X, BAut( O n +1 ⊗ D )] B( q ) − ∗ −−−−−→ [ X, BAut( E n +1 ⊗ D )] B( η n ) ∗ −−−−−→ [ X, BAut ( K ⊗ D )] ⊂ E D ( X ) . Theorem 4.2.
The image of b D consists of n k -torsion elements of E D ( X ) for some k ≥ .Proof. Since Aut( O ) is contractible (see [7]) and O n +1 ⊗ M n ∞ ∼ = O , the compositionB( ⊗ id M n ∞ ) ∗ ◦ b D = b D ⊗ M n ∞ is the 0 map. Therefore the statement follows from [8, Theorem2.11].If dim X ≤
3, or X = SY , the map b D is given by the Bockstein map, and hence theinvariant is non-trivial (see Section 5). b ( n ) We denote b ( n ) := b M ( n ) for short. For r ≥
1, with GCD( n, r ) = 1, the Kirchberg–Phillipstheorem yields an isomorphism ϕ r : O n +1 → O nr +1 ⊗ M ( n ) , and one has the following map b r : [ X, BAut( O n +1 )] B(Ad ϕ r ) ∗ −−−−−−−→ [ X, BAut( O nr +1 ⊗ M ( n ) )] B( q ) − ∗ −−−−−→ [ X, BAut( E nr +1 ⊗ M ( n ) )] B( η nr ) ∗ −−−−−−→ [ X, BAut( K ⊗ M ( n ) )] . Theorem 4.3.
The map b r is equal to b . We identify E nr +1 ⊗ M ( n ) with a subalgebra of M ( K ⊗ M ( n ) ) by a unital embedding l r : E nr +1 ⊗ M ( n ) ֒ → M ( K ⊗ M ( n ) ) which sends e ⊗ M ( n ) into P ( K ⊗ M ( n ) ). We need thefollowing proposition to prove Theorem 4.3. Proposition 4.4.
There is an automorphism β r ∈ Aut( K ⊗ M ( n ) ) inducing an automorphismof M ( K ⊗ M ( n ) ) that restricts to an isomorphism β r : E n +1 ⊗ M ( n ) → E nr +1 ⊗ M ( n ) . o show the proposition, we need the following two lemmas. The embedding l r gives theBusby invariant i r : O nr +1 ⊗ M ( n ) → Q ( K ⊗ M ( n ) ). By the Kirchberg–Phillips theorem, theisomorphism ϕ r is determined uniquely upto homotopy, and the Busby invariant τ r := i r ◦ ϕ r is defined. Lemma 4.5. In Ext( O n +1 , K ⊗ M ( n ) ) ∼ = Ext Z ( Z n , Z ( n ) ) , we have [ τ r ] = r [ τ ] .Proof. By Theorem 2.3, the element [ τ r ] corresponds to the extension of groups [ Z ( n ) − nr −−→ Z ( n ) → Z n ]. If we identify [ Z ( n ) n −→ Z ( n ) → Z n ] with ¯1 ⊗ ∈ Z n ⊗ Z ( n ) , then one has[ τ r ] = ¯1 ⊗ − r .By Theorem 2.17, there is an automorphism α r ∈ Aut( K ⊗ M ( n ) ) with [ α r ( e ⊗ M ( n ) )] = r ∈ K ( M ( n ) ). For the induced automorphism ˜ α r ∈ Aut( Q ( K ⊗ M ( n ) )), one has[ ˜ α r ◦ τ ] = α r ∗ ([ τ ]) = 1 r [ τ ] = [ τ r ] . By Corollary 2.7, we have a unitary v ∈ Q ( K ⊗ M ( n ) ) with ˜ α r ◦ τ = Ad v ◦ τ r . We would liketo show that v lifts to an element in U ( M ( K ⊗ M ( n ) )). For τ r , we denote Paschke’s unitaryby V τ r := (cid:18) τ r ( S ) π ( w ) O n +1 (cid:19) ∈ M n +2 ( Q ( K ⊗ M ( n ) )) , where w ∈ M n +2 ( M ( K ⊗ M ( n ) )) is a partial isometry with ww ∗ = 0 ⊕ n +1 , w ∗ w = 1 ⊕ n +1 and S := ( S , · · · , S n +1 ) (see [27]). Lemma 4.6.
We have rδ ([ V τ r ] ) = δ ([ V τ ] ) = − ∈ K ( M ( n ) ) .Proof. For τ ⊕ rr := 1 r ⊗ τ r : O n +1 → M r ⊗ Q ( K ⊗ M ( n ) ), one has 1 r ⊗ τ r = (id M r ⊗ i r ) ◦ (1 r ⊗ ϕ r )by definition, and we denote σ := 1 r ⊗ ϕ r : O n +1 → M r ⊗ O nr +1 ⊗ M ( n ) .Let T ( k )1 , · · · , T ( k ) r , ( k = 1 , · · · , n ) , T nr +1 be the canonical nr + 1 generators of E nr +1 . Wedefine the following n + 1 isometries with mutually orthogonal ranges : T k := T ( k )1 · · · T ( k ) r O ∈ M r ( E nr +1 ) , ( k = 1 , · · · , n ) , T n +1 := T nr +1 ⊕ r − . Direct computation yields P n +1 i =1 T i T ∗ i = (1 − e ) ⊕ r − , and we have a unital ∗ -homomorphism σ : O n +1 ∋ S i π ( T i ) ⊗ M ( n ) ∈ M r ( O nr +1 ⊗ M ( n ) ) . By the Kirchberg–Phillips Theorem, one has σ ∼ h σ in Hom( O n +1 , M r ( O nr +1 ⊗ M ( n ) )) u .Therefore we have τ ⊕ rr = (id M r ⊗ i r ) ◦ σ ∼ h (id M r ⊗ i r ) ◦ σ and [ V τ ⊕ rr ] = [ V (id M r ⊗ i r ) ◦ σ ] .The following unitary is a lift of V (id M r ⊗ i r ) ◦ σ ⊕ V ∗ (id M r ⊗ i r ) ◦ σ T ( e ⊗ M ( n ) ) ⊕ r − r ⊗ w O n +1 n +1 r ⊗ w ∗ O n +2 T ∗ O n +1 ∈ M n +4 ( M r ⊗ E nr +1 ⊗ M ( n ) ) , where we denote T := ( T ⊗ M ( n ) , · · · , T n +1 ⊗ M ( n ) ) for simplicity. By the definition of theindex map, we have rδ ([ V τ r ] ) = δ ([ V (id M r ⊗ i r ) ◦ σ ] ) = − [ e ⊗ M ( n ) ] = − ∈ K ( M ( n ) ). roof of Proposition 4.4. Direct computation yields − nδ ([ v ] ) = δ ([ V Ad v ◦ τ r ] ) − δ ([ V τ r ] )= δ ([ V ˜ α r ◦ τ ] ) − δ ([ V τ r ] )= δ ◦ K ( ˜ α r )([ V τ ] ) − δ ([ V τ r ] )= 1 r δ ([ V τ ] ) − δ ([ V τ r ] ) = 0 . So we have a unitary V with π ( V ) = v , and the map β r := Ad V ∗ ◦ α r is the isomorphism. Proof of Theorem 4.3.
Every arrow in the following diagram is a group homomorphism :Aut( O n +1 ) Ad ϕ r / / Aut( O nr +1 ⊗ M ( n ) ) Aut( E nr +1 ⊗ M ( n ) ) o o η nr / / Aut( K ⊗ M ( n ) )Aut( O n +1 ) Ad ϕ / / Aut( O n +1 ⊗ M ( n ) ) Ad ˜ β r O O Aut( E n +1 ⊗ M ( n ) ) o o Ad β r O O η n / / Aut( K ⊗ M ( n ) ) . Ad β r O O Since K ( O nr +1 ⊗ M ( n ) ) = 0, the Kirchberg–Phillips theorem gives a path of unitaries { u t } t ∈ [0 , ⊂ O nr +1 ⊗ M ( n ) satisfying u = 1 and that Ad u t ◦ ϕ r converges to ˜ β r ◦ ϕ as t tends to 1. So the left hand square of the above diagram commutes upto homotopyand the homotopy is given by a path of group homomorphisms. In particular, the follow-ing diagram commutes for a finite CW complex X , where B(Ad β r ) ∗ = id E M ( n ) ( X ) because β r ∈ Aut( K ⊗ M ( n ) ) : [ X, BAut( O n +1 )] B(Ad ϕ r ) ∗ / / [ X, BAut( O nr +1 ⊗ M ( n ) )] [ X, BAut( E nr +1 ⊗ M ( n ) )] Cor . B( η r ) ∗ / / [ X, BAut( K ⊗ M ( n ) )][ X, BAut( O n +1 )] B(Ad ϕ ) ∗ / / [ X, BAut( O n +1 ⊗ M ( n ) )] B(Ad ˜ β r ) ∗ O O [ X, BAut( E n +1 ⊗ M ( n ) )] Cor . B(Ad β r ) ∗ O O B( η ) ∗ / / [ X, BAut( K ⊗ M ( n ) )] . B(Ad β r ) ∗ O O Theorem 4.7.
Two maps
B(Ad ϕ ) ∗ and B( ⊗ id M ( n ) ) ∗ are equal, and we have b r = b ( n ) .Proof. For the isomorphism φ : M ( n ) → M ⊗ n ) , l = φ − , we have a path of unitaries { v t } t ∈ [0 , ⊂ U ( M ⊗ n ) ) satisfying v = 1 and lim t → || Ad v t ( d ) − l ( d ) ⊗ M ( n ) || = 0 for every d ∈ M ⊗ n ) . Iden-tifying O n +1 (resp. O n +1 ⊗ M ( n ) ) with O n +1 ⊗ M ( n ) (resp. ( O M ( n ) ⊗ M ( n ) ) ⊗ M ( n ) ), we identify ϕ with id O n +1 ⊗ φ and define the following automorphism for α ∈ Aut( O n +1 ⊗ M ( n ) )Ψ t ( α ) := Ad(1 O n +1 ⊗ v ∗ t ) ◦ ( α ⊗ id M ( n ) ) ◦ Ad(1 O n +1 ⊗ v t ) ∈ Aut(( O n +1 ⊗ M ( n ) ) ⊗ M ( n ) ) . The map Ψ t is a group homomorphism for every t ∈ [0 ,
1) and Ψ t ( α ) converges to (id O n +1 ⊗ φ ) ◦ α ◦ (id O n +1 ⊗ φ ) − as t tends to 1 in the point norm topology. So Ψ t gives the homotopyof group homomorphisms, and one has B(Ad ϕ ) ∗ = B( ⊗ id M ( n ) ) ∗ . .3 The inverse image b − n ) (0) In this section we assume that X is a finite connected CW complex. Let us recall thecontinuous fields of O n +1 coming from vector bundles (see [9, 19]). For a compact Hausdorffspace X , we denote by Vect m ( X ) the set of the vector bundles of rank m . M. Dadarlatinvestigated continuous fields of O n +1 over X arising from E ∈ Vect n +1 ( X ), which are Cuntz–Pimsner algebras. We refer to [23] and [29] for Cuntz–Pimsner algebras. Fixing a Hermitianstructure of E , we get a Hilbert C ( X )-module from E , which we regard as a C ( X )- C ( X )-bimodule. Then the Pimsner construction gives the Cuntz–Pimsner algebra O E and theexact sequence 0 → K E j E −→ T E → O E → . Since K E has a projection whose range isthe 1-dimensional subspace of the vacuum vector, the Dixmier–Douady theory yields K E ∼ = C ( X ) ⊗ K . The algebra O E (resp. T E ) is a continuous field of O n +1 (resp. E n +1 ) over X with the natural unital inclusion θ E : C ( X ) → O E . Theorem 4.8 ([29, Theorem 4.8]) . The exact sequence → K E j E −→ T E → O E → gives theexact sequence of K -groups : K ( X ) − [ E ] −−−→ K ( X ) K ( θ E ) −−−−→ K ( O E ) . M. Dadarlat found an invariant to classify the C ( X )-linear isomorphism classes of O E . Theorem 4.9 ([9, Theorem 1.1]) . Let X be a compact metrizable space, and let E and F bevector bundles of rank ≥ over X . Then there is a unital ∗ -homomorphism ϕ : O E → O F with ϕ ◦ θ E = θ F if and only if (1 − [ E ]) · K ( X ) ⊂ (1 − [ F ]) · K ( X ) . Moreover we can take ϕ to be an isomorphism if and only if (1 − [ E ]) · K ( X ) = (1 − [ F ]) · K ( X ) . We show the following theorem.
Theorem 4.10.
For a finite connected CW complex X , the inverse image b − n ) (0) consistsof the C ( X ) -linear isomorphism classes of the continuous fields of the form O E ⊗ M ( n ) for E ∈ Vect nr +1 ( X ) with r ≥ , GCD( n, r ) = 1 . All necessary arguments are already in [9]. We use Kasparov’s parametrized KK -groups KK X ( · , · ) for C ( X )-algebras (see [22]). Consider two unital separable continuous C ( X )-algebras A, B with the maps θ A , θ B that determine the C ( X )-linear structure. By [11, The-orem 1.1, Theorem 2.7], a KK X -equivalence σ ∈ KK X ( A, B ) with [1 A ] ⊗ σ = [1 B ] liftsto a C ( X )-linear isomorphism A ∼ = B where we identifies KK X ( C ( X ) , A ) ∋ KK X ( θ A ) with KK ( C , A ) ∋ [1 A ] . Note that for two continuous fields C ( X ) ⊗ C, B , one has the isomorphismof groups KK X ( C ( X ) ⊗ C, B ) = KK ( C, B ) (see [11, Proof of Corollary 2.8]). If there is a KK X -equivalence µ ∈ KK X ( SA, SB ) with KK X ( Sθ A ) ⊗ µ = KK X ( Sθ B ), the suspensionisomorphism KK X ( A, B ) ∼ = KK X ( SA, SB ) gives the above σ ∈ KK X ( A, B ). Since [17]implies M ( n ) ⊗ O ∞ -stability of the continuous fields, the proposition below implies Theorem4.10. Proposition 4.11.
Let
O ∈ b − n ) (0) be a continuous field with the map θ : C ( X ) → O thatdetermines the C ( X ) -algebra structure. Then there is a number r ≥ , GCD( n, r ) = 1 and arank nr + 1 vector bundle F , and we have a KK X -equivalence µ ∈ KK X ( S O ⊗ ( M ( n ) ⊗ O ∞ ) , S O F ⊗ M ( n ) ⊗ ( M ( n ) ⊗ O ∞ )) with KK X ( S ( θ ⊗ ⊗ µ = KK X ( S ( θ F ⊗ . or two C ( X )-algebras A, B and a C ( X )-linear ∗ -homomorphism ϕ : A → B , one has amapping cone algebra C ϕ := { ( f, a ) ∈ ( C (0 , ⊗ B ) ⊕ A | f (1) = ϕ ( a ) } , and the following Puppe sequence holds (see [4]) :0 → SB ֒ → C ϕ → A → . If A ⊂ B is an ideal and ϕ is the inclusion, one has the exact sequence0 → C (0 , ⊗ A → C ϕ q ϕ −→ S ( B/A ) → , and the quotient map q ϕ is a KK X -equivalence because C (0 , ⊗ A is C ( X )-linearly con-tractible. We need the following lemma. Lemma 4.12 ([25, Appendix A]) . Let
A, A ′ , B, B ′ be C ( X ) -algebras and let ϕ : A → B, ϕ ′ : A ′ → B ′ be C ( X ) -linear ∗ -homomorphisms. Consider an additive category KK X of C ( X ) -algebras whose morphism Mor(
A, B ) is given by KK X ( A, B ) . For α ∈ KK X ( A, A ′ ) , β ∈ KK X ( B, B ′ ) , we consider the following diagram SB / / Sβ (cid:15) (cid:15) C ϕ / / A ϕ / / α (cid:15) (cid:15) B β (cid:15) (cid:15) SB ′ / / C ϕ ′ / / A ′ ϕ ′ / / B ′ . If the diagram commutes in the category KK X , then there exists γ ∈ KK X ( C ϕ , C ϕ ′ ) thatmakes the diagram commutes. Furthermore, if α, β are KK X -equivalences, one can chose γ to be a KK X -equivalence.Proof of Proposition 4.11. For every
O ∈ b − n ) (0), there is a locally trivial continuous C ( X )-algebra E of E n +1 ⊗ M ( n ) , and the following exact sequence of C ( X )-algebras holds0 → C ( X ) ⊗ K ⊗ M ( n ) j −→ E → O → . The evaluation map ev x at x ∈ X gives an extension 0 → K ⊗ M ( n ) → E n +1 ⊗ M ( n ) →O n +1 ⊗ M ( n ) →
0. We also denote by θ : C ( X ) → E the map that determines the C ( X )-linearstructure. By [11, Theorem 1.1], the map θ ⊗ id : C ( X ) ⊗ ( M ( n ) ⊗ O ∞ ) → E ⊗ ( M ( n ) ⊗ O ∞ )gives a KK X -equivalence. Since K (ev x )([( j ⊗ id M ( n ) ⊗O ∞ )(1 C ( X ) ⊗ e ⊗ M ⊗ n ) ⊗O ∞ )] ) = − n ∈ K ( E n +1 ⊗ M ⊗ n ) ⊗ O ∞ ) , there exist r ≥ , GCD( n, r ) = 1 and y ∈ ˜ K ( X ) satisfying[( j ⊗ id M ( n ) ⊗O ∞ )(1 C ( X ) ⊗ e ⊗ M ⊗ n ) ⊗O ∞ )] = − ( n + yr ) ∈ K ( E ⊗ M ( n ) ⊗ O ∞ ) = K ( X ) ⊗ Z ( n ) . The Kirchiberg–Phillips theorem gives a ∗ -homomorphism M ( n ) ⊗ O ∞ ∋ p ∈ C ( X ) ⊗ M ( n ) ⊗ O ∞ with [ p ] = − ( n + yr ) providing a C ( X )-linear ∗ -homomorphism α − ( n + yr ) : C ( X ) ⊗ M ( n ) ⊗ O ∞ → C ( X ) ⊗ M ( n ) ⊗ O ∞ with [ α − ( n + yr ) (1)] = − ( n + yr ). Let e ⊗ be the map e ⊗ : C ( X ) ⊗ M ( n ) ⊗ O ∞ ∋ f ( e ⊗ M ( n ) ) ⊗ f ∈ ( C ( X ) ⊗ K ⊗ M ( n ) ) ⊗ M ( n ) ⊗ O ∞ hat gives a KK X -equivalence. By the identification KK X ( C ( X ) ⊗ ( M ( n ) ⊗ O ∞ ) , E ⊗ M ( n ) ⊗ O ∞ ) ∼ = KK ( M ( n ) ⊗ O ∞ , E ⊗ M ( n ) ⊗ O ∞ )= Hom( Z ( n ) , K ( X ) ⊗ Z ( n ) ) , one has KK X (( j ⊗ id) ◦ ( e ⊗ )) = − ( n + yr ) = KK X (( θ ⊗ id) ◦ ( α − ( n + yr ) )), and the followingdiagram commutes in KK X ( C ( X ) ⊗ K ⊗ M ( n ) ) ⊗ M ( n ) ⊗ O ∞ j ⊗ id / / E ⊗ M ( n ) ⊗ O ∞ C ( X ) ⊗ M ( n ) ⊗ O ∞ e ⊗ O O α − ( n + yr ) / / C ( X ) ⊗ M ( n ) ⊗ O ∞ . θ ⊗ id O O There is a vector bundle F ∈ Vect nrR +1 ( X ) satisfying [ F ] − ( nrR + 1) = yR ∈ ˜ K ( X ) forsufficiently large R ≥ , GCD( n, R ) = 1. Let α rR (resp. α − ( nrR + yR ) ) be a ∗ -endomorphism of C ( X ) ⊗ M ( n ) ⊗ O ∞ with [ α rR (1)] = rR (resp. [ α − ( nrR + yR ) (1)] = − ( nrR + yR ) = 1 − [ F ]).Now we have the following diagram commuting in KK X S O ⊗ DS E ⊗ D / / C j ⊗ id / / q j ⊗ id O O C ( X ) ⊗ K ⊗ M ( n ) ⊗ D j ⊗ id / / E ⊗
DSC ( X ) ⊗ D S ( θ ⊗ id) O O / / C α − ( n + yr ) O O (cid:15) (cid:15) / / C ( X ) ⊗ D α rR (cid:15) (cid:15) e ⊗ O O α − ( n + yr ) / / C ( X ) ⊗ D θ ⊗ id O O SC ( X ) ⊗ D S ( θ F ⊗ M ( n ) ) ⊗ id (cid:15) (cid:15) / / C α − ( nrR + yR ) (cid:15) (cid:15) / / C ( X ) ⊗ D e ⊗ (cid:15) (cid:15) α − ( nrR + yR ) / / C ( X ) ⊗ D ( θ F ⊗ M ( n ) ) ⊗ id (cid:15) (cid:15) S T F ⊗ M ( n ) ⊗ D / / C j F ⊗ id / / q jF ⊗ id (cid:15) (cid:15) C ( X ) ⊗ K ⊗ M ( n ) ⊗ D − [ F ] / / T F ⊗ M ( n ) ⊗ DS O F ⊗ M ( n ) ⊗ D where we denote D = M ( n ) ⊗ O ∞ for simplicity. By Lemma 4.12, the same argument as in[9, Proof of Theorem 1.1] proves the statement. Remark 4.13.
For two vector bundles F ∈ Vect nR +1 ( X ) and F ′ ∈ Vect nR ′ +1 ( X ) with R, R ′ ≥ , GCD( n, R ) = GCD( n, R ′ ) = 1 , the same argument in the above proof proves that O F ⊗ M ( n ) ∼ = O F ′ ⊗ M ( n ) if and only if (1 − [ F ]) K ( X ) ⊗ Z ( n ) = (1 − [ F ′ ]) K ( X ) ⊗ Z ( n ) .Therefore we can identify b − n ) (0) with the set of equivalence classes ˜ K ( X ) ⊗ Z ( n ) / ∼ n wherethe equivalence relation is defined as follows : a ∼ n b ⇔ ( n + a )(1 + z ) = n + b for some z ∈ ˜ K ( X ) ⊗ Z ( n ) (see [19, Lemma 4.7]). Theorem 4.14.
For a finite CW complex X with Tor( H k +1 ( X ) , Z n ) = 0 for every k ≥ ,we have Im b ( n ) = 0 and [ X, BAut( O n +1 )] = ˜ K ( X ) ⊗ Z ( n ) / ∼ n . roof. By Remark 4.13 and Theorem 4.2, it is enough to show Tor( ¯ E M ( n ) ( X ) , Z n ) = 0. Theassumption yields that the group H k +1 ( X, Z ( n ) ) is torsion free for every k ≥
1, and the gradedsubgroups of ¯ E M ( n ) ( X ) associated with the Atiyah–Hirzebruch spectral sequence are all tor-sion free by the argument of [7, Corollary 4.4]. In particular, we have Tor( ¯ E M ( n ) ( X ) , Z n ) =0. This generalizes [9, Theorem 1.6] and [19, Theorem 4.12], and one can classify the contin-uous fields over the space whose even cohomology admits n -torsion. For example, Theorem4.14 applies to the real projective spaces R P m for m ≥ Remark 4.15.
Following the argument in [8, Proof of Theorem 2.11], we can identify b Z ⊗ X, BAut( O n +1 )] → ¯ E Z ( X ) ⊗ Z ( n ) with b ( n ) = b Z⊗ M ( n ) and we have b − Z (0) = b − n ) (0) . It alsofollows that b − O ∞ (0) = b − n ) (0) . We give some examples of computation of the invariant b ( n ) . dim X ≤ By Corollary 3.15 and [33, Proof of Lemma 3.20], the group homomorphism ⊗ id M ( n ) :Aut( E n +1 ) → Aut( E n +1 ⊗ M ( n ) ) is 3-connected, and the map B( ⊗ id M ( n ) ) is 4-connected. Con-struction of the Postnikov tower yields another 4-connected map P : BAut( E n +1 ) → K ( Z n , η : Aut( E n +1 ) → Aut( K ). There is a map β : K ( Z n , → K ( Z ,
3) = BAut( K ) that gives the Bockstein map H ( X, Z n ) → H ( X ) (see[32]). By [33, Lemma 3.20], one has two elements [ β ◦ P ] , [B( η )] ∈ [BAut( E n +1 ) , K ( Z , H (BAut( E n +1 )) = Z n . Applying the Bockstein exact sequence and [33, Theorem 3.15] forthe reduced suspension of the Moore space M n (see [19, Section 2]), the above elements bothgenerate Z n , and there is a number r ≥ , GCD( n, r ) = 1 with r [ β ◦ P ] = [B( η )] ∈ Z n .For X with dim X ≤
3, we have the following commutative diagram[ X, BAut( E n +1 ⊗ M ( n ) )] B( η ) ∗ / / ¯ E M ( n ) ( X ) = H ( X ) ⊗ Z ( n ) [ X, BAut( E n +1 )] P ∗ (cid:15) (cid:15) B( ⊗ id M ( n ) ) ∗ O O B( η ) ∗ / / Tor( H ( X ) , Z n ) , O O H ( X, Z n ) r β ∗ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ and the left vertical maps are bijective by [36, p 182], where ¯ E M ( n ) ( X ) := [ X, BAut ( K ⊗ M ( n ) )] ⊂ E M ( n ) ( X ) is the reduced group (see [8]). Therefore we have | b − n ) (0) | = | H ( X ) ⊗ Z n | by the Bockstein exact sequence0 → H ( X ) ⊗ Z n → H ( X, Z n ) β ∗ −→ Tor( H ( X ) , Z n ) → . emark 5.1. By [18, Theorem 1.2, p 112], two maps H ( X ) ∋ [ L ] [ L ] − ∈ ˜ K ( X ) and H ( X ) ∋ [ L ] [ L ⊕ C ⊕ n ] ∈ [ X, B U ( n + 1)] are bijective in the case of dim X ≤ . Theinverse map is given by the determinant map det : U ( n + 1) → S . Lemma 5.2.
For a finite CW complex X with dim X ≤ , we have ([ L ] − H ] −
1) = 0 ∈ ˜ K ( X ) , L, H ∈ Vect ( X ) .Proof. By Remark 5.1, the map B(det) ∗ : [ X, B U (2)] → [ X, B S ] is bijective. Since B(det) ∗ ([ L ⊕ H ]) = [ L ⊗ H ] = B(det) ∗ ([( L ⊗ H ) ⊕ C ]), we have [ L ][ H ] + 1 = [ L ] + [ H ] ∈ K ( X ). Theorem 5.3.
For a finite connected CW complex X with dim X ≤ , the following holds : [ X, BAut( O n +1 )] = H ( X, Z n ) ,2) Im b ( n ) = Tor( H ( X ) , Z n ) ⊂ E M ( n ) ( X ) , | b − n ) (0) | = | H ( X ) ⊗ Z n | ,3) b − n ) (0) = O := {O E | E = L ⊕ C ⊕ n , L ∈ Vect ( X ) } / ∼ isom .Proof. Since 1) and 2) are already proved, we show 3). By Lemma 5.2, we have ( n + ([ L ] − H ] = n + ([ L ⊗ H ⊗ n ] − ∈ K ( X ) for L, H ∈ Vect ( X ), and Theorem 4.9 implies O L ⊕ C n ∼ = O L ′ ⊕ C n ⇔ [ L ] − [ L ′ ] ∈ nH ( X ) . Thus we have | b − n ) (0) | = | H ( X ) ⊗ Z n | = | O | < ∞ , and 3) is proved. X = SY Since the group Aut( O n +1 ) is path connected, we have [ SY,
BAut( O n +1 )] = [ Y, Aut( O n +1 )] = K ( C ( Y ) ⊗ O n +1 ) where SY is the non-reduced suspension of Y . The invariant b ( n ) is givenby η n ∗ : [ Y, Aut( E n +1 ⊗ M ( n ) )] → [ Y, Aut ( K ⊗ M ( n ) )] ⊂ K ( Y ) ⊗ Z ( n ) . The group structureof [ Y, Aut( O n +1 )] was determined in [19, Theorem 3.1], and there is a group homomorphism1 − δ : [ Y, Aut( O n +1 )] → K ( Y ) × . The proof of Lemma 3.8 yields the following corollary. Corollary 5.4.
The invariant b ( n ) is identified with the index map δ : K ( C ( Y ) ⊗ O n +1 ⊗ M ( n ) ) → Tor( K ( Y ) ⊗ Z ( n ) , Z n ) , and for [ α ] ∈ [ Y, Aut( O n +1 )] , we have b ( n ) ([ α ]) = 1 − δ ([ u ′ ◦ α ] ) ∈ K ( Y ) ⊗ Z ( n ) , Z n ) ⊂ E M ( n ) ( SY ) . Two rings K ( R P ) = Z [ ν ] / ( ν + 2 ν, ν ) and K ( R P ) = Z [ ν ′ ] / ( ν ′ + 2 ν ′ , ν ′ ) are well-known (see [1, Theorem 7.3]). By the following commutative diagram H ( R P ) / / K ( R P ) (cid:15) (cid:15) Z ⊕ Z Z H ( R P ) / / K ( R P ) Z ⊕ Z the generator of H ( R P ) is sent to one of 1 ± ν ∈ K ( R P ). Since 1 ± ν K ( R P ) , Z ),the image of b ( n ) : [ S R P , BAut( O n +1 )] → E M ( n ) ( S R P ) dose not contain H ( S R P ) byCorollary 5.4. However the invariant b ( n ) is not zero because 1+ 2 ν ∈
1+ Tor( K ( R P ) , Z ) =0, and this example suggests that the third cohomology alone is not enough to investigatethe invariant. .2 Questions We summarize some open problems.
Problem 5.5.
Does the equation | b − n ) ( z ) | = | b − n ) (0) | hold for non-trivial z = 0 ∈ Im b ( n ) ? Isthere any operator algebraic realization, like Cuntz–Pimsner construction, of the elements in b − n ) ( z ) ? Since [19] shows that BAut( O n +1 ) has no H -space structure, it is not obvious that theimage of b D is a subgroup of E D or not. By [8, Example 3.5], it might be worth pointing outthe following questions. Problem 5.6.
Is the set Im b D ⊂ E D closed with respect to taking inverse? Is the set Im b D a subgroup of E D ? Problem 5.7.
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