K-Theory for the Leaf Space of Foliations formed by the Generic K-Ornits of some indecomposable $MD_5$-Groups

K-THEOR Y OF THE LEAF SP A CE OF F OLIA TIONS F ORMED BY THE GENERIC K -ORBITS OF SOME INDECOMPOSABLE M D 5 -GR OUPS Le Anh V u* and Duong Quan g Ho a** Department of Mathemat ics and Info rmat ics Universit y of Pedagogy , Ho C hi Minh City , Vietnam E-mai l : (*) vula@m a th.hcmup.edu.vn (**) duongqua nghoa b t @y aho o.com .vn Abstract The pap er is a con tin uation of the authors’ work [18]. In [18], w e consider foliations formed b y the maximal dimensional K-orbits ( M D 5 -foliatio ns) o f connected M D 5 - groups such that their Lie algebras ha v e 4-dimensional comm utativ e derive d ideals and giv e a top ologi cal classification of the considered foliations. In this pap er, we stud y K- theory of the leaf space of some of these M D 5 -foliatio ns and characte rize the Con n es’ C*-algebras of the considered foliations by the method of K -functors. INTR ODUCTION In the decades 1 970s − 1 980s, works of D.N. D iep [4], J. Ro senberg [10], G. G . Kasparov [7], V. M. Son and H. H. Viet [12],... ha v e seen tha t K-functors are we ll adapted to c haracterize a larg e class of group C*-algebras. Kirillo v’s metho d of orbits allows to find out the class of Lie groups MD, for whic h the group C*-a lg ebras can b e c haracterized by means of suit- able K-f unctors (see [5]). In terms of D. N. Diep, an MD- group of dimension n (for short, an M D n -group) is an n-dimensional solv able real Lie gr oup whose orbits in the co- adjoint represen tation (i.e., the K- represen tation) are the orbits of zero or maximal dimension. The Lie algebra of an M D n -group is called an M D n -algebra (see [5, Section 4.1]). In 198 2 , studying foliated manifolds, A. Connes [3] intro duced the no t ion of C*-algebra asso ciated to a measured foliat io n. In the case of Reeb foliatio ns (see A. M. T orp e [14 ]), the metho d of K-f unctor s has b een pro v ed to b e v ery effectiv e in describing the structure of Connes’ C*-algebras. F or ev ery MD-group G, the family of K- o rbits of maximal dimension forms a measured foliation in terms of Connes [3]. This foliat ion is called MD-foliatio n asso ciated to G. 0 Key words : L ie gr oup, Lie algebra , M D 5 -group, M D 5 -algebra , K-orbit, F oliation, Measured folia tion, C*-algebr a, Connes’ C*-alg ebra asso ciated to a measur ed foliation. 2000AMS Mathematics Sub ject Classifica tion: P rimary 22 E45, Secondary 46 E25, 20C20 . 1 Com bining metho ds of Kirillov (see [8, Section 15]) a nd Connes ( see[3, Section 2, 5]), the first author had studied M D 4 -foliations asso ciated with all indecomp osable connected M D 4 -groups and c haracterized Connes’ C*-algebras o f these foliatio ns in [16]. Recen tly , V u and Shum [17] ha v e classified, up to isomorphism, all the M D 5 -algebras having commutativ e deriv ed ideals. In [18], w e hav e g iv en a top olo gical classification of M D 5 -foliations asso ciated to the indecomp o sable connected and simply connected M D 5 -groups, suc h that M D 5 -algebras of them hav e 4-dimensional comm utativ e derived ideals. There are exactly 3 top ological ty p es of the considered M D 5 -foliations, denoted b y F 1 , F 2 , F 3 . All M D 5 -foliations o f t yp e F 1 are the trivial fibrations with connected fibre on 3-dimesional sphere S 3 , so Connes’ C*-algebras of them are isomorphic to the C*-a lg ebra C ( S 3 ) ⊗ K fo llo wing [3, Section 5], where K denotes the C*-algebra of compact op erators on a n (infinite dimensional separable) Hilb ert space. The purp ose o f t his pap er is to study K-theory of the leaf space and t o characterize the structure of Connes’ C*- algebras C ∗ ( V , F ) of all M D 5 -foliations ( V , F ) of type F 2 b y the metho d of K-functors. Namely , we will express C ∗ ( V , F ) by tw o rep eated extensions of the form 0 / / C 0 ( X 1 ) ⊗ K / / C ∗ ( V , F ) / / B 1 / / 0 , 0 / / C 0 ( X 2 ) ⊗ K / / B 1 / / C 0 ( Y 2 ) ⊗ K / / 0 , then w e will compute the in v arian t system of C ∗ ( V , F ) with resp ect to these extensions. If the giv en C*-alg ebras are isomorphic to the reduced crossed pro ducts o f the form C 0 ( V ) ⋊ H , where H is a Lie gro up, w e can use the Thom-Connes isomorphism to compute the connecting map δ 0 , δ 1 . In another pap er, w e will study the similar problem for all M D 5 -foliations of type F 3 . 1 THE M D 5 − F OLIA TIONS OF TYPE F 2 Originally , w e will recall geometry of K- orbit of M D 5 -groups whic h asso ciate with M D 5 - foliations of ty p e F 2 (see [18]). In this section, G will b e alw a ys an connected and simply connected M D 5 -group suc h that it s Lie algebras G is an indecomp o sable M D 5 -algebra generated by { X 1 , X 2 , X 3 , X 4 , X 5 } with G 1 := [ G , G ] = R .X 2 ⊕ R .X 3 ⊕ R .X 4 ⊕ R .X 5 ∼ = R 4 , ad X 1 ∈ E nd ( G ) ≡ M at 4 ( R ). Namely , G will b e one of the follo wing Lie algebras whic h a re studied in [17] and [18]. 1. G 5 , 4 , 11( λ 1 ,λ 1 ,ϕ ) ad X 1 =     cos ϕ − sin ϕ 0 0 sin ϕ cos ϕ 0 0 0 0 λ 1 0 0 0 0 λ 2     ; λ 1 , λ 2 ∈ R \ { 0 } , λ 1 6 = λ 2 , ϕ ∈ (0 , π ) . 2 2. G 5 , 4 , 12( λ,ϕ ) ad X 1 =     cos ϕ − sin ϕ 0 0 sin ϕ cos ϕ 0 0 0 0 λ 0 0 0 0 λ     ; λ ∈ R \ { 0 } , ϕ ∈ (0 , π ) . 3. G 5 , 4 , 13( λ,ϕ ) ad X 1 =     cos ϕ − sin ϕ 0 0 sin ϕ cos ϕ 0 0 0 0 λ 1 0 0 0 λ     ; λ ∈ R \ { 0 } , ϕ ∈ (0 , π ) . The connected and simply connected Lie groups corresp onding to these algebras are denoted b y G 5,4,11 ( λ 1 , λ 1 , ϕ ) , G 5,4,12( λ , ϕ ) , G 5,4,13( λ , ϕ ) . All of these Lie groups are M D 5 -groups (see [1 7 ]) and G is one o f them. W e no w recall the geometric description of the K-orbits of G in the dual space G ∗ of G . Let { X ∗ 1 , X ∗ 2 , X ∗ 3 , X ∗ 4 , X ∗ 5 } b e the basis in G ∗ dual to the basis { X 1 , X 2 , X 3 , X 4 , X 5 } in G . Denote by Ω F the K-orbit of G including F = ( α , β + iγ , δ, σ ) in G ∗ ∼ = R 5 . • If β + iγ = δ = σ = 0 then Ω F = { F } (the 0-dimensional orbit). • If | β + iγ | 2 + δ 2 + σ 2 6 = 0 then Ω F is the 2-dimensional orbit as follows Ω F =            n x, ( β + iγ ) .e ( a.e − iϕ ) , δ.e aλ 1 , σ.e aλ 2  , x, a ∈ R o when G = G 5,4,11( λ 1 ,λ 2 ,ϕ ) , λ 1 , λ 2 ∈ R ∗ , ϕ ∈ (0; π ) . n x, ( β + iγ ) .e ( a.e − iϕ ) , δ.e aλ , σ.e aλ  , x, a ∈ R o when G = G 5,4,12( λ,ϕ ) , λ ∈ R ∗ , ϕ ∈ (0; π ) . n x, ( β + iγ ) .e ( a.e − iϕ ) , δ.e aλ , δ.ae aλ + σ .e aλ  , x, a ∈ R o when G = G 5,4,13( λ,ϕ ) , λ ∈ R ∗ , ϕ ∈ (0; π ) . In [18], w e hav e show n that, the family F of maximal-dimensional K-o rbits of G forms measured foliation in terms of Connes on the op en submanifold V =  ( x, y , z , t, s ) ∈ G ∗ : y 2 + z 2 + t 2 + s 2 6 = 0  ∼ = R ×  R 4  ∗ ( ⊂ G ∗ ≡ R 5 ) F urthermore, all foliations  V , F 4 , 11( λ 1 ,λ 2 ,ϕ )  ,  V , F 4 , 12( λ,ϕ )  ,  V , F 4 , 13( λ,ϕ )  are to p ologically equiv alen t to each other ( λ 1 , λ 2 , λ ∈ R \ { 0 } , ϕ ∈ (0; π )). Th us, w e need only choose a en v oy among t hem to describ e the structure of the C*-algebra. In this case, w e ch o ose the fo liation  V , F 4 , 12 ( 1 , π 2 )  . In [18], w e hav e describ ed the f oliation  V , F 4 , 12 ( 1 , π 2 )  b y a suitable action of R 2 . Na mely , w e ha v e the following prop osition. 3 PR OPOSITI ON 1. The fo l i a tion  V , F 4 , 12 ( 1 , π 2 )  c an b e given by an action of the c om mu- tative Lie gr oup R 2 on the man i f o ld V . Pr o of. One needs only to v erify tha t the follow ing action λ of R 2 on V give s the foliatio n  V , F 4 , 12 ( 1 , π 2 )  λ : R 2 × V → V (( r , a ) , ( x, y + iz , t, s )) 7→ ( x + r , ( y + iz ) .e − ia , t.e a , s.e a ) where ( r , a ) ∈ R 2 , ( x, y + iz , t, s ) ∈ V ∼ = R × ( C × R 2 ) ∗ ∼ = R × ( R 4 ) ∗ . Hereafter, f or simplicit y of notation, w e write ( V , F ) instead of  V , F 4 , 12 ( 1 , π 2 )  . It is easy to see that the g r aph o f ( V , F ) is inden tical with V × R 2 , so b y [3, Section 5], it follo ws from Prop osition 1 t ha t: COR OLLAR Y 1 (analytical description of C ∗ ( V , F )) . The Connes C ∗ -algebr a C ∗ ( V , F ) c an b e analytic al ly de scrib e d the r e duc e d cr osse d of C 0 ( V ) by R 2 as fol lows C ∗ ( V , F ) ∼ = C 0 ( V ) ⋊ λ R 2 .  2 C ∗ ( V , F ) A S TW O REPEA TED EXTENSIONS 2.1. Let V 1 , W 1 , V 2 , W 2 b e the fo llo wing submanifolds of V V 1 = { ( x, y , z , t, s ) ∈ V : s 6 = 0 } ∼ = R × R 2 × R × R ∗ , W 1 = V \ V 1 = { ( x, y , z , t, s ) ∈ V : s = 0 } ∼ = R × ( R 3 ) ∗ × { 0 } ∼ = R × ( R 3 ) ∗ , V 2 = { ( x, y , z , t, 0) ∈ W 1 : t 6 = 0 } ∼ = R × R 2 × R ∗ , W 2 = W 1 \ V 2 = { ( x, y , z , t, 0) ∈ W 1 : t = 0 } ∼ = R × ( R 2 ) ∗ . It is easy to see that the action λ in Prop osition 1 preserv es the subsets V 1 , W 1 , V 2 , W 2 . Let i 1 , i 2 , µ 1 , µ 2 b e the inclusions and the restrictions i 1 : C 0 ( V 1 ) → C 0 ( V ) , i 2 : C 0 ( V 2 ) → C 0 ( W 1 ) , µ 1 : C 0 ( V ) → C 0 ( W 1 ) , µ 2 : C 0 ( W 1 ) → C 0 ( W 2 ) where eac h function of C 0 ( V 1 ) (resp. C 0 ( V 2 )) is exten ted to the one of C 0 ( V ) (resp. C 0 ( W 1 )) b y taking the v alue of zero outside V 1 (resp. V 2 ). It is kno wn a fact that i 1 , i 2 , µ 1 , µ 2 are λ - equiv arian t and t he f o llo wing sequences are equiv arian tly exact: (2.1.1) 0 / / C 0 ( V 1 ) i 1 / / C 0 ( V ) µ 1 / / C 0 ( W 1 ) / / 0 (2.1.2) 0 / / C 0 ( V 2 ) i 2 / / C 0 ( W 1 ) µ 2 / / C 0 ( W 2 ) / / 0 . 4 2.2. No w w e denote by ( V 1 , F 1 ) , ( W 1 , F 1 ) , ( V 2 , F 2 ) , ( W 2 , F 2 ) the foliations-restrictions of ( V , F ) on V 1 , W 1 , V 2 , W 2 resp ectiv ely . THEOREM 1. C ∗ ( V , F ) admits the fol lowing c anonic al r ep e a te d extensions ( γ 1 ) 0 / / J 1 b i 1 / / C ∗ ( V , F ) c µ 1 / / B 1 / / 0 , ( γ 2 ) 0 / / J 2 b i 2 / / B 1 c µ 2 / / B 2 / / 0 , wher e J 1 = C ∗ ( V 1 , F 1 ) ∼ = C 0 ( V 1 ) ⋊ λ R 2 ∼ = C 0 ( R 3 ∪ R 3 ) ⊗ K, J 2 = C ∗ ( V 2 , F 2 ) ∼ = C 0 ( V 2 ) ⋊ λ R 2 ∼ = C 0 ( R 2 ∪ R 2 ) ⊗ K, B 2 = C ∗ ( W 2 , F 2 ) ∼ = C 0 ( W 2 ) ⋊ λ R 2 ∼ = C 0 ( R + ) ⊗ K , B 1 = C ∗ ( W 1 , F 1 ) ∼ = C 0 ( W 1 ) ⋊ λ R 2 , and the homom orphismes b i 1 , b i 2 , b µ 1 , b µ 2 ar e define d by  b i k f  ( r , s ) = i k f ( r , s ) , k = 1 , 2 ( c µ k f ) ( r , s ) = µ k f ( r , s ) , k = 1 , 2 Pr o of. W e note that the graph of ( V 1 , F 1 ) is indentical with V 1 × R 2 , so by [3, section 5], J 1 = C ∗ ( V 1 , F 1 ) ∼ = C 0 ( V 1 ) ⋊ λ R 2 . Similarly , we hav e B 1 ∼ = C 0 ( W 1 ) ⋊ λ R 2 , J 2 ∼ = C 0 ( V 2 ) ⋊ λ R 2 , B 2 ∼ = C 0 ( W 2 ) ⋊ λ R 2 , F rom the equiv arian tly exact sequences in 2.1 and by [2, Lemma 1.1] w e obtain the rep eated extensions ( γ 1 ) and ( γ 2 ). F urthermore, the foliation ( V 1 , F 1 ) can b e derive d from t he submersion p 1 : V 1 ≈ R × R 2 × R × R ∗ → R 3 ∪ R 3 p 1 ( x, y , z , t, s ) = ( y , z , t, sign s ) . Hence, b y a result o f [3, p.562], w e get J 1 ∼ = C 0 ( R 3 ∪ R 3 ) ⊗ K . The same argument sho ws that J 2 ∼ = C 0  R 2 ∪ R 2  ⊗ K, B 2 ∼ = C 0 ( R + ) ⊗ K. 5 3 COMPUTING THE INV ARIANT SYSTEM OF C ∗ ( V , F ) DEFINITION . The set of elemen ts { γ 1 , γ 2 } corresp onding to the rep eated extensions ( γ 1 ), ( γ 2 ) in the Kasparo v groups Ext ( B i , J i ) , i = 1 , 2 is called the system of inv arian ts of C ∗ ( V , F ) and denoted by Index C ∗ ( V , F ). REMARK. Index C ∗ ( V , F ) determines the so-called stable t yp e of C ∗ ( V , F ) in the set of all rep eated extensions 0 / / J 1 / / E / / B 1 / / 0 , 0 / / J 2 / / B 1 / / B 2 / / 0 . The main result o f the pap er is the following. THEOREM 2. Index C ∗ ( V , F ) = { γ 1 , γ 2 } , wher e γ 1 =  0 1 0 1  in the gr oup Ext ( B 1 , J 1 ) = H om ( Z 2 , Z 2 ) ; γ 2 = (1 , 1) in the gr oup Ext ( B 2 , J 2 ) = H om ( Z , Z 2 ) . T o prov e this t heorem, w e need some lemmas as follows . LEMMA 1. Set I 2 = C 0 ( R 2 × R ∗ ) a n d A 2 = C 0  ( R 2 ) ∗  The fol lowing diagr am is c ommutative . . . / / K j ( I 2 ) / / β 1   K j  C 0 ( R 3 ) ∗  / / β 1   K j ( A 2 ) / / β 1   K j +1 ( I 2 ) / / β 1   . . . . . . / / K j +1  C 0 ( V 2 )  / / K j +1  C 0 ( W 1 )  / / K j +1  C 0 ( W 2 )  / / K j  C 0 ( V 2 )  / / . . . wher e β 1 is the isomo rp h ism define d in [1 3 , The or em 9.7] or in [2, c or ol lary VI.3], j ∈ Z / 2 Z . Pr o of. Let k 2 : I 2 = C 0  R 2 × R ∗  − → C 0  R 3  ∗  v 2 : C 0  R 3  ∗  − → A 2 = C 0  R 2  ∗  b e the inclusion a nd restriction defined similarly as in 2 .1. One gets the exact sequence 0 / / I 2 k 2 / / C 0  ( R 3 ) ∗  v 2 / / A 2 / / 0 Note that C 0 ( V 2 ) ∼ = C 0  R × R 2 × R ∗  ∼ = C 0 ( R ) ⊗ I 2 , C 0 ( W 2 ) ∼ = C 0  R ×  R 2  ∗  ∼ = C 0 ( R ) ⊗ A 2 , C 0 ( W 1 ) ∼ = C 0  R ×  R 3  ∗  ∼ = C 0 ( R ) ⊗ C 0  R 3  ∗ . 6 The extension (2.1 .2) th us can b e iden tified to the followin g one 0 / / C 0 ( R ) ⊗ I 2 id ⊗ k 2 / / C 0 ( R ) ⊗ C 0 ( R 3 ) ∗ id ⊗ v 2 / / C 0 ( R ) ⊗ A 2 / / 0 . No w, using [13, Theorem 9.7; Corolla ry 9.8] w e obtain the assertion of Lemma 1. LEMMA 2. Set I 1 = C 0 ( R 2 × R ∗ ) a n d A 1 = C ( S 2 ) The fol lowing diagr am is c ommutative . . . / / K j ( I 1 ) / / β 2   K j  C ( S 3 )  / / β 2   K j ( A 1 ) / / β 2   K j +1 ( I 1 ) / / β 2   . . . . . . / / K j ( C 0 ( V 1 )) / / K j ( C 0 ( V )) / / K j ( C 0 ( W 1 )) / / K j +1 ( C 0 ( V 1 )) / / . . . wher e β 2 is the Bott iso morphism, j ∈ Z / 2 Z . Pr o of. The pro of is similar to that of lemma 1 , by using the exact sequence (2.1.1) and diffeomorphisms: V ∼ = R × ( R 4 ) ∗ ∼ = R × R + × S 3 , W 1 ∼ = R × ( R 3 ) ∗ ∼ = R × R + × S 2 . Before computing the K-gro ups, w e need the f o llo wing notat io ns. Let u : R → S 1 b e the map u ( z ) = e 2 π i ( z / √ 1+ z 2 ) , z ∈ R Denote b y u + (resp. u − ) the r estriction of u on R + (resp. R − ). Note that the class [ u + ] (resp. [ u − ]) is the canonical generator of K 1 ( C 0 ( R + )) ∼ = Z (resp. K 1 ( C 0 ( R − )) ∼ = Z ). Let us consider the matrix v a lued function p : ( R 2 ) ∗ ∼ = S 1 × R + → M 2 ( C ) (resp. p : S 2 ∼ = D /S 1 → M 2 ( C )) defined by : p ( x ; y ) ( r esp. p ( x, y )) = 1 2   1 − cos π p x 2 + y 2 x + iy √ x 2 + y 2 sin π p x 2 + y 2 x − iy √ x 2 + y 2 sin π p x 2 + y 2 1 + cos π p x 2 + y 2   . Then p (resp. p ) is an idemp o ten t of ra nk 1 for each ( x ; y ) ∈ ( R 2 ) ∗ (resp. ( x ; y ) ∈ D /S 1 ). Let [ b ] ∈ K 0 ( C 0 ( R 2 )) b e the Bo t t elemen t, [1 ] b e the g enerato r of K 0 ( C ( S 1 )) ∼ = Z . LEMMA 3 (See [1 5, p.234]) . (i) K 0 ( B 1 ) ∼ = Z 2 , K 1 ( B 1 ) = 0 , (ii) K 0 ( J 2 ) ∼ = Z 2 is gener a te d by ϕ 0 β 1  [ b ] ⊠ [ u + ]  and ϕ 0 β 1  [ b ] ⊠ [ u − ]  ; K 1 ( J 2 ) = 0 , (iii) K 0 ( B 2 ) ∼ = Z is gener ate d by ϕ 0 β 1  [1] ⊠ [ u + ]  ; K 1 ( B 2 ) ∼ = Z is gener ate d by ϕ 1 β 1  [ p ] − [ ε 1 ]  , wher e ϕ j , j ∈ Z / 2 Z , is the Thom -Connes isomorphism (s e e[2]), β 1 is the is omorphism in L emm a 1, ε 1 is the c onstant matrix  1 0 0 0  and ⊠ is the external tensor pr o duct ( s e e, for example, [2,VI.2]). LEMMA 4. (i) K 0  C ∗ ( V , F )  ∼ = Z , K 1  C ∗ ( V , F )  ∼ = Z , 7 (ii) K 0 ( J 1 ) = 0; K 1 ( J 1 ) ∼ = Z 2 is gener ate d by ϕ 1 β 2  [ b ] ⊠ [ u + ]  and ϕ 1 β 2  [ b ] ⊠ [ u − ]  , (iii) K 1 ( B 1 ) = 0; K 0 ( B 1 ) ∼ = Z 2 is gener ate d by ϕ 0 β 2 [ ¯ 1] and ϕ 0 β 2  [ ¯ p ] − [ ε 1 ]  , wher e ¯ 1 is unit element in C ( S 2 ) , ϕ 0 is the Thom-Connes isomo rp h ism, β 2 is the Bott isomorphism. Pr o of. (i) K i  C ∗ ( V , F )  ∼ = K i  C ( S 3 )  ∼ = Z , i = 0 , 1. (ii) The pro o f is similar to ( ii) of lemma 3. (iii) By [9, p.206], w e hav e K 0  C ( S 2 )  = Z [ ¯ 1] + Z [ q ] , where q ∈ P 2  C ( S 2 )  . Otherwise, in [9, p.48,5 3,56]; [13, p.162], one has shown that the map dim : K 0  C ( S 2 )  → Z is a surjectiv e group homomorphism whic h satisfied dim[ ¯ 1] = 1 , ker(dim) = Z and non- zero elemen t q ∈ P 2  C ( S 2 )  in the k ernel of the map dim has the form [ q ] = [ ¯ p ] − [ ε 1 ]. Hence, the result is derive d straight a w a y b ecause β 2 and ϕ 1 are isomorphisms. Pro of of theorem 2 1. Computation of ( γ 1 ). Recall that the extension ( γ 1 ) in theorem 1 gives rise to a six-term exact sequence 0 = K 0 ( J 1 ) / / K 0  C ∗ ( V , F )  / / K 0 ( B 1 ) δ 0   0 = K 1 ( B 1 ) δ 1 O O K 1  C ∗ ( V , F )  o o K 1 ( J 1 ) o o By [11, Theorem 4.14], the isomorphisms Ext( B 1 , J 1 ) ∼ = Hom  ( K 0 ( B 1 ) , K 1 ( J 1 )  ∼ = Hom( Z 2 , Z 2 ) asso ciates the in v aria n t γ 1 ∈ Ext( B 1 , J 1 ) to the connecting map δ 0 : K 0 ( B 1 ) → K 1 ( J 1 ). Since the Thom-Connes isomorphism comm utes with K − theoretical exact sequence (see[14, Lemma 3 .4.3]), w e hav e the following comm utativ e diagra m ( j ∈ Z / 2 Z ): . . . / / K j ( J 1 ) / / K j  C ∗ ( V , F )  / / K j ( B 1 ) / / K j +1 ( J 1 ) / / . . . . . . / / K j  C 0 ( V 1 )  / / ϕ j O O K j  C 0 ( V )  / / ϕ j O O K j  C 0 ( W 1 )  / / ϕ j O O K j +1  C 0 ( V 1 )  / / ϕ j +1 O O . . . In view of Lemma 2, the following diagram is comm utativ e . . . / / K j  C 0 ( V 1 )  / / K 1  C 0 ( V )  / / K j  C 0 ( W 1 )  / / K j +1  C 1 ( V 1 )  / / . . . . . . / / K j ( I 1 ) / / β 2 O O K j  C ( S 3 )  / / β 2 O O K j ( A 1 ) / / β 2 O O K j +1 ( I 1 ) / / β 2 O O . . . 8 Consequen tly , instead of computing δ 0 : K 0 ( B 1 ) → K 1 ( J 1 ), it is sufficien t to compute δ 0 : K 0 ( A 1 ) → K 1 ( I 1 ). Th us, b y the pro of of Lemma 4, we hav e to define δ 0  [ ¯ p ] − [ ε 1 ]  = δ 0  [ ¯ p ]  (b ecause δ 0  [ ε 1 ]  = (0; 0) a nd δ 0  [ ¯ 1]  = (0; 0)). By the usual definition (see[13, p.170]), for [ ¯ p ] ∈ K 0 ( A 1 ) , δ 0  [ ¯ p ]  =  e 2 π i ˜ p  ∈ K 1 ( I 1 ) where ˜ p is a preimage of ¯ p in (a matrix algebra o v er) C ( S 3 ), i.e. v 1 ˜ p = ¯ p . W e can c ho ose ˜ p ( x, y , z ) = z √ 1 + z 2 ¯ p ( x, y ) , ( x, y , z ) ∈ S 3 . Let ˜ p + (resp. ˜ p − ) b e the restriction of ˜ p on R 2 × R + (resp. R 2 × R − ). Then w e hav e δ 0  [ ¯ p ]  =  e 2 π i ˜ p  =  e 2 π i ˜ p +  +  e 2 π i ˜ p −  ∈ K 1  C 0  R 2  ⊗ C 0  R +   ⊕ K 1  C 0  R 2  ⊗ C 0  R −   = K 1 ( I 1 ) By [13, Section 4], for eac h function f : R ± → Q n ^ C 0  R 2  suc h that lim x →± 0 f ( t ) = lim x →±∞ f ( t ), where Q n ^ C 0  R 2  = n a ∈ M n ^ C 0  R 2  , e 2 π ia = I d o , the class [ f ] ∈ K 1  C 0 ( R 2 ) ⊗ C 0 ( R ± )  can b e determined b y [ f ] = W f . [ b ] ⊠ [ u ± ], where W f = 1 2 π i Z R ± T r  f ′ ( z ) f − 1 ( z )  dz is the winding n um b er of f . By simple computation, we get δ 0  [ p ]  = [ b ] ⊠ [ u + ] + [ b ] ⊠ [ u − ]. Thus γ 1 =  0 1 0 1  ∈ Hom Z ( Z 2 , Z 2 ). 2. Computation of ( γ 2 ). The extension ( γ 2 ) giv es rise to a six-term exact sequence K 0 ( J 2 ) / / K 0 ( B 1 ) / / K 0 ( B 2 ) δ 0   K 1 ( B 2 ) δ 1 O O K 1 ( B 1 ) o o K 1 ( J 2 ) = 0 o o By [11, Theorem 4.14], γ 2 = δ 1 ∈ Hom  K 1 ( B 2 ) , K 0 ( J 2 )  = Hom Z ( Z , Z 2 ). Similarly to part 1, taking accoun t of Lemma 1 and 3, we hav e the follo wing comm utativ e diagram ( j ∈ Z / 2 Z ) . . . / / K j ( J 2 ) / / K j ( B 1 ) / / K j ( B 2 ) / / K j +1 ( J 2 ) / / . . . . . . / / K j  C 0 ( V 2 )  / / ϕ j O O K j  C 0 ( W 1 )  / / ϕ j O O K j  C 0 ( W 2 )  / / ϕ j O O K j +1  C 0 ( V 2 )  / / ϕ j +1 O O . . . . . . / / K j − 1 ( I 2 ) / / β 1 O O K j − 1  C 0 ( R 3 ) ∗  / / β 1 O O K j − 1 ( A 2 ) / / β 1 O O K j ( I 2 ) / / β 1 O O . . . Th us w e can compute δ 0 : K 0 ( A 2 ) → K 1 ( I 2 ) instead o f δ 1 : K 1 ( B 2 ) → K 0 ( J 2 ). By the pro of of Lemma 3, w e hav e to define δ 0  [ p ] − [ ǫ 1 ]  = δ 0  [ p ]  (b ecause δ 0  [ ǫ 1 ]  = (0 , 0)). The same a r gumen t as ab o v e, we get δ 0  [ p ]  = [ b ] ⊠ [ u + ] + [ b ] ⊠ [ u − ]. Th us γ 2 = (1 , 1) ∈ Hom Z ( Z , Z 2 ) ∼ = Z 2 . The pro of is completed.  9 REFERENCES 1. BR O WN, L . G .; DO UG LAS, R . G .; FILLMORE, P . 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