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

0.14 MB·English

by  Le Anh Vu

K-THEORY OF THE LEAF SPACE OF FOLIATIONS
FORMED BY THE GENERIC K-ORBITS OF SOME
INDECOMPOSABLE MD -GROUPS
5
1
1
0 Le Anh Vu* and Duong Quang Hoa**
2
Department of Mathematics and Informatics

n
a University of Pedagogy, Ho Chi Minh City, Vietnam
J
E-mail: (*) [email protected]
1
1
(**) [email protected]

]
T
K
Abstract
.
h
t The paper is a continuation of the authors’ work [18]. In [18], we consider foliations
a formed by the maximal dimensional K-orbits (MD -foliations) of connected MD -
m 5 5
groups such that their Lie algebras have 4-dimensional commutative derived ideals and
[
give a topological classification of the considered foliations. In this paper, we study K-

2 theory of the leaf space of some of these MD -foliations and characterize the Connes’
v 5
3 C*-algebras of the considered foliations by the method of K-functors.
1
3
5
. INTRODUCTION
3
0
0
1 In the decades 1970s 1980s, works of D.N. Diep [4], J. Rosenberg [10], G. G. Kasparov [7],
−
v: V. M. Son and H. H. Viet [12],... have seen that K-functors are well adapted to characterize
i a large class of group C*-algebras. Kirillov’s method of orbits allows to find out the class
X
of Lie groups MD, for which the group C*-algebras can be characterized by means of suit-
r
a
able K-functors (see [5]). In terms of D. N. Diep, an MD-group of dimension n (for short,
an MD -group) is an n-dimensional solvable real Lie group whose orbits in the co-adjoint
n
representation (i.e., the K- representation) are the orbits of zero or maximal dimension. The
Lie algebra of an MD -group is called an MD -algebra (see [5, Section 4.1]).
n n
In 1982, studying foliated manifolds, A. Connes [3] introduced the notion of C*-algebra
associated to a measured foliation. In the case of Reeb foliations (see A. M. Torpe [14]),
the method of K-functors has been proved to be very effective in describing the structure of
Connes’ C*-algebras. For every MD-group G, the family of K-orbits of maximal dimension
forms a measured foliation in terms of Connes [3]. This foliation is called MD-foliation
associated to G.
0Key words: Lie group, Lie algebra, MD5-group, MD5-algebra, K-orbit, Foliation, Measured foliation,
C*-algebra,Connes’ C*-algebra associated to a measured foliation.
2000AMS Mathematics Subject Classification: Primary 22E45,Secondary 46E25,20C20.
1
Combining methods of Kirillov (see [8, Section 15]) and Connes (see[3, Section 2, 5]),
the first author had studied MD -foliations associated with all indecomposable connected
4
MD -groups and characterized Connes’ C*-algebras of these foliations in [16]. Recently, Vu
4
and Shum [17] have classified, up to isomorphism, all the MD -algebras having commutative
5
derived ideals.
In [18], we have given a topological classification of MD -foliations associated to the
5
indecomposable connected and simply connected MD -groups, such that MD -algebras of
5 5
them have 4-dimensional commutative derived ideals. There are exactly 3 topological types
of the considered MD -foliations, denoted by , , . All MD -foliations of type are
5 1 2 3 5 1
F F F F
the trivial fibrations with connected fibre on 3-dimesional sphere S3, so Connes’ C*-algebras
of them are isomorphic to the C*-algebra C(S3) following [3, Section 5], where denotes
⊗K K
the C*-algebra of compact operators on an (infinite dimensional separable) Hilbert space.
The purpose of this paper is to study K-theory of the leaf space and to characterize the
structure of Connes’ C*-algebras C (V, ) of all MD -foliations (V, ) of type by the
∗ 5 2
F F F
method of K-functors. Namely, we will express C (V, ) by two repeated extensions of the
∗
F
form
0 //C (X ) // C (V, ) //B //0,
0 1 ∗ 1
⊗K F
0 // C (X ) // B //C (Y ) // 0,
0 2 1 0 2
⊗K ⊗K
then we will compute the invariant system of C (V, ) with respect to these extensions. If
∗
F
the given C*-algebras are isomorphic to the reduced crossed products of the form C (V)⋊H,
0
where Hisa Lie group, we canuse the Thom-Connes isomorphism to compute theconnecting
map δ ,δ .
0 1
In another paper, we will study the similar problem for all MD -foliations of type .
5 3
F
1 THE FOLIATIONS OF TYPE
MD
5 2
− F
Originally, we will recall geometry of K-orbit of MD -groups which associate with MD -
5 5
foliations of type (see [18]).
2
F
In this section, G will be always an connected and simply connected MD -group such
5
that its Lie algebras is an indecomposable MD -algebra generated by X ,X ,X ,X ,X
5 1 2 3 4 5
G { }
with 1 := [ , ] = R.X R.X R.X R.X = R4, ad End( ) Mat (R). Namely,
G G G 2⊕ 3⊕ 4⊕ 5 ∼ X1 ∈ G ≡ 4
will be one of the following Lie algebras which are studied in [17] and [18].
G
1.
G5,4,11(λ1,λ1,ϕ)
cosϕ sinϕ 0 0
−
 sinϕ cosϕ 0 0 
ad = ;λ ,λ R 0 ,λ = λ ,ϕ (0,π).
X1 0 0 λ 0 1 2 ∈ \{ } 1 6 2 ∈
1
 
 0 0 0 λ 
2
 
2
2.
5,4,12(λ,ϕ)
G
cosϕ sinϕ 0 0
−
 sinϕ cosϕ 0 0 
ad = ;λ R 0 ,ϕ (0,π).
X1 0 0 λ 0 ∈ \{ } ∈
 
 0 0 0 λ 
 
3.
5,4,13(λ,ϕ)
G
cosϕ sinϕ 0 0
−
 sinϕ cosϕ 0 0 
ad = ;λ R 0 ,ϕ (0,π).
X1 0 0 λ 1 ∈ \{ } ∈
 
 0 0 0 λ 
 
The connected and simply connected Lie groups corresponding to these algebras are
denoted by G , G , G . All of these Lie groups are MD -groups
5,4,11(λ1,λ1,ϕ) 5,4,12(λ,ϕ) 5,4,13(λ,ϕ) 5
(see [17]) and G is one of them. We now recall the geometric description of the K-orbits of
G in the dual space of . Let X ,X ,X ,X ,X be the basis in dual to the basis
G∗ G { 1∗ 2∗ 3∗ 4∗ 5∗} G∗
X ,X ,X ,X ,X in . Denote by Ω the K-orbit of G including F = (α,β +iγ,δ,σ) in
1 2 3 4 5 F
{ } G
= R5.
∗ ∼
G
If β +iγ = δ = σ = 0 then Ω = F (the 0-dimensional orbit).
F
• { }
If β +iγ 2 +δ2 +σ2 = 0 then Ω is the 2-dimensional orbit as follows
F
• | | 6
x,(β +iγ).e(a.e−iϕ),δ.eaλ1,σ.eaλ2 , x,a R
∈
 n(cid:16) (cid:17) o
when G = G , λ ,λ R , ϕ (0;π).
5,4,11(λ1,λ2,ϕ) 1 2 ∈ ∗ ∈
 x,(β +iγ).e(a.e−iϕ),δ.eaλ,σ.eaλ , x,a R
Ω =  ∈
F  n(cid:16) (cid:17) o
 when G = G , λ R , ϕ (0;π).
5,4,12(λ,ϕ) ∗
 ∈ ∈
 x,(β +iγ).e(a.e−iϕ),δ.eaλ,δ.aeaλ +σ.eaλ , x,a R

 n(cid:16) (cid:17) ∈ o
 when G = G , λ R , ϕ (0;π).
5,4,13(λ,ϕ) ∗
 ∈ ∈
In [18], we have shown that, the family of maximal-dimensional K-orbits of G forms
F
measured foliation in terms of Connes on the open submanifold
V = (x,y,z,t,s) G∗ : y2+z2 +t2 +s2 = 0 ∼= R R4 ∗( ∗ R5)
∈ 6 × ⊂ G ≡
(cid:8) (cid:9) (cid:0) (cid:1)
Furthermore, all foliations V, , V, , V, are topologically
F4,11(λ1,λ2,ϕ) F4,12(λ,ϕ) F4,13(λ,ϕ)
equivalent to each other (λ ,λ(cid:0),λ R 0 ,ϕ(cid:1) (cid:0)(0;π)). Th(cid:1)us(cid:0), we need on(cid:1)ly choose a envoy
1 2
∈ \{ } ∈
among them to describe the structure of the C*-algebra. In this case, we choose the foliation
V, .
(cid:16) F4,12(1,π2)(cid:17)
In [18], we have described the foliation V, by a suitable action of R2. Namely,
(cid:16) F4,12(1,π2)(cid:17)
we have the following proposition.
3
PROPOSITION 1. The foliation V, can be given by an action of the commu-
(cid:16) F4,12(1,π2)(cid:17)
tative Lie group R2 on the manifold V.
Proof. One needs only to verify that the following action λ of R2 on V gives the foliation
V,
(cid:16) F4,12(1,π2)(cid:17)
λ : R2 V V
× →
((r,a),(x,y +iz,t,s)) (x+r,(y +iz).e ia,t.ea,s.ea)
−
7→
where (r,a) R2, (x,y +iz,t,s) V ∼= R (C R2)∗ ∼= R (R4)∗. Hereafter, for
∈ ∈ × × ×
simplicity of notation, we write (V, ) instead of V, .
F (cid:16) F4,12(1,π2)(cid:17)
It is easy to see that the graph of (V, ) is indentical with V R2, so by [3, Section 5],
F ×
it follows from Proposition 1 that:
COROLLARY 1 (analytical description of C (V, )). The Connes C -algebra C (V, )
∗ ∗ ∗
F F
can be analytically described the reduced crossed of C (V) by R2 as follows
0
C (V, ) = C (V)⋊ R2.
∗ ∼ 0 λ
F
(cid:3)
2 AS TWO REPEATED EXTENSIONS
C (V, )
∗
F
2.1. Let V , W , V , W be the following submanifolds of V
1 1 2 2
V = (x,y,z,t,s) V : s = 0 = R R2 R R ,
1 ∼ ∗
{ ∈ 6 } × × ×
W1 = V V1 = (x,y,z,t,s) V : s = 0 ∼= R (R3)∗ 0 ∼= R (R3)∗,
\ { ∈ } × ×{ } ×
V = (x,y,z,t,0) W : t = 0 = R R2 R ,
2 1 ∼ ∗
{ ∈ 6 } × ×
W2 = W1 V2 = (x,y,z,t,0) W1 : t = 0 ∼= R (R2)∗.
\ { ∈ } ×
It is easy to see that the action λ in Proposition 1 preserves the subsets V ,W ,V ,W .
1 1 2 2
Let i ,i ,µ ,µ be the inclusions and the restrictions
1 2 1 2
i : C (V ) C (V), i : C (V ) C (W ),
1 0 1 0 2 0 2 0 1
→ →
µ : C (V) C (W ), µ : C (W ) C (W )
1 0 0 1 2 0 1 0 2
→ →
where each function of C (V ) (resp. C (V )) is extented to the one of C (V) (resp. C (W ))
0 1 0 2 0 0 1
by taking the value of zero outside V (resp. V ).
1 2
It is known a fact that i ,i ,µ ,µ are λ-equivariant and the following sequences are
1 2 1 2
equivariantly exact:
(2.1.1) 0 //C (V ) i1 // C (V) µ1 //C (W ) // 0
0 1 0 0 1
(2.1.2) 0 //C (V ) i2 // C (W ) µ2 // C (W ) // 0 .
0 2 0 1 0 2
4
2.2. Now we denote by (V , ),(W , ),(V , ),(W , ) the foliations-restrictions of
1 1 1 1 2 2 2 2
F F F F
(V, ) on V ,W ,V ,W respectively.
1 1 2 2
F
THEOREM 1. C (V, ) admits the following canonical repeated extensions
∗
F
(γ ) 0 //J ib1 //C (V,F) µc1 //B // 0,
1 1 ∗ 1
(γ ) 0 //J ib2 //B µc2 // B //0,
2 2 1 2
where
J = C (V , ) = C (V )⋊ R2 = C (R3 R3) K,
1 ∗ 1 1 ∼ 0 1 λ ∼ 0
F ∪ ⊗
J = C (V , ) = C (V )⋊ R2 = C (R2 R2) K,
2 ∗ 2 2 ∼ 0 2 λ ∼ 0
F ∪ ⊗
B = C (W , ) = C (W )⋊ R2 = C (R ) K,
2 ∗ 2 2 ∼ 0 2 λ ∼ 0 +
F ⊗
B = C (W , ) = C (W )⋊ R2, and the homomorphismes i ,i ,µ ,µ are defined by
1 ∗ 1 1 ∼ 0 1 λ 1 2 1 2
F
i f (r,s) = i f (r,s), k = 1,2
k k b b b b
(cid:16) (cid:17)
(µ f)(r,s) = µ f (r,s), k = 1,2
bk k
Proocf. We note that the graph of (V , ) is indentical with V R2, so by [3, section 5],
1 1 1
F ×
J = C (V , ) = C (V )⋊ R2 . Similarly, we have
1 ∗ 1 1 ∼ 0 1 λ
F
B = C (W )⋊ R2,
1 ∼ 0 1 λ
J = C (V )⋊ R2,
2 ∼ 0 2 λ
B = C (W )⋊ R2,
2 ∼ 0 2 λ
From the equivariantly exact sequences in 2.1 and by [2, Lemma 1.1] we obtain the
repeated extensions (γ ) and (γ ).
1 2
Furthermore, the foliation (V , ) can be derived from the submersion
1 1
F
p : V R R2 R R R3 R3
1 1 ∗
≈ × × × → ∪
p (x,y,z,t,s) = (y,z,t,signs).
1
Hence, by a result of [3, p.562], we get J = C (R3 R3) K. The same argument shows
1 ∼ 0
∪ ⊗
that
J = C R2 R2 K, B = C (R ) K.
2 ∼ 0 2 ∼ 0 +
∪ ⊗ ⊗
(cid:0) (cid:1)
5
3 COMPUTING THE INVARIANT SYSTEM OF
C (V, )
∗
F
DEFINITION. The set of elements γ ,γ corresponding to the repeated extensions (γ ),
1 2 1
{ }
(γ ) in the Kasparov groups Ext (B ,J ), i = 1,2 is called the system of invariants of
2 i i
C (V, ) and denoted by Index C (V, ).
∗ ∗
F F
REMARK. Index C (V, ) determines the so-called stable type of C (V, ) in the set of all
∗ ∗
F F
repeated extensions
0 // J // E //B // 0 ,
1 1
0 // J //B // B //0.
2 1 2
The main result of the paper is the following.
THEOREM 2. Index C (V, ) = γ ,γ , where
∗ 1 2
F { }
0 1
γ = in the group Ext (B ,J ) = Hom(Z2,Z2);
1 (cid:18) 0 1 (cid:19) 1 1
γ = (1,1) in the group Ext (B ,J ) = Hom(Z,Z2).
2 2 2
To prove this theorem, we need some lemmas as follows.
LEMMA 1. Set I2 = C0(R2 R∗) and A2 = C0 (R2)∗
×
The following diagram is commutative (cid:0) (cid:1)
... //K (I ) //K C (R3) //K (A ) //K (I ) //...
j 2 j 0 ∗ j 2 j+1 2
(cid:0) (cid:1)
β1 β1 β1 β1
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
... //K C (V ) //K C (W ) // K C (W ) // K C (V ) //...
j+1 0 2 j+1 0 1 j+1 0 2 j 0 2
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
where β is the isomorphism defined in [13, Theorem 9.7] or in [2, corollary VI.3], j Z/2Z.
1
∈
Proof. Let
k2 : I2 = C0 R2 R∗ C0 R3 ∗
× −→
(cid:0) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
v : C R3 ∗ A = C R2 ∗
2 0 2 0
−→
(cid:0)(cid:0) (cid:1) (cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
be the inclusion and restriction defined similarly as in 2.1.
One gets the exact sequence
0 //I k2 // C (R3) v2 //A // 0
2 0 ∗ 2
(cid:0) (cid:1)
Note that
C (V ) = C R R2 R = C (R) I ,
0 2 ∼ 0 ∗ ∼ 0 2
× × ⊗
(cid:0) (cid:1)
C (W ) = C R R2 ∗ = C (R) A ,
0 2 ∼ 0 ∼ 0 2
× ⊗
(cid:0) (cid:0) (cid:1) (cid:1)
C (W ) = C R R3 ∗ = C (R) C R3 ∗.
0 1 ∼ 0 ∼ 0 0
× ⊗
(cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1)
6
The extension (2.1.2) thus can be identified to the following one
0 //C0(R) I2 id⊗k2//C0(R) C0(R3)∗ id⊗v2//C0(R) A2 //0 .
⊗ ⊗ ⊗
Now, using [13, Theorem 9.7; Corollary 9.8] we obtain the assertion of Lemma 1.
LEMMA 2. Set I = C (R2 R ) and A = C(S2)
1 0 ∗ 1
×
The following diagram is commutative
... //K (I ) // K C(S3) // K (A ) //K (I ) // ...
j 1 j j 1 j+1 1
(cid:0) (cid:1)
β2 β2 β2 β2
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
... //K (C (V )) // K (C (V)) //K (C (W )) // K (C (V )) // ...
j 0 1 j 0 j 0 1 j+1 0 1
where β is the Bott isomorphism, j Z/2Z.
2
∈
Proof. The proof is similar to that of lemma 1, by using the exact sequence (2.1.1) and
diffeomorphisms: V ∼= R (R4)∗ ∼= R R+ S3, W1 ∼= R (R3)∗ ∼= R R+ S2.
× × × × × ×
Before computing the K-groups, we need the following notations. Let u : R S1 be the
→
map
u(z) = e2πi(z/√1+z2), z R
∈
Denote by u (resp. u ) the restriction of u on R (resp. R ). Note that the class [u ]
+ + +
(resp. [u ]) is the canon−ical generator of K (C (R )) = Z (resp−. K (C (R )) = Z). Let us
1 0 + ∼ 1 0 ∼
consider−the matrix valued function p : (R2)∗ ∼= S1 R+ M2(C) (resp. p−: S2 ∼= D/S1
× → →
M (C)) defined by:
2
1 cosπ x2 +y2 x+iy sinπ x2 +y2
1 − √x2+y2
p(x;y)(resp. p(x,y)) =  p p .
2 x iy sinπ x2 +y2 1+cosπ x2 +y2
−
√x2+y2
 p p 
Then p (resp. p) is an idempotent of rank 1 for each (x;y) (R2)∗ (resp. (x;y) D/S1).
∈ ∈
Let [b] K (C (R2)) be the Bott element, [1] be the generator of K (C(S1)) = Z.
0 0 0 ∼
∈
LEMMA 3 (See [15, p.234]).
(i) K (B ) = Z2, K (B ) = 0,
0 1 ∼ 1 1
(ii) K (J ) = Z2 is generated by ϕ β [b]⊠[u ] and ϕ β [b]⊠[u ] ; K (J ) = 0,
0 2 ∼ 0 1 + 0 1 1 2
(iii) K (B ) = Z isgeneratedby ϕ β [(cid:0)1]⊠[u ] ;(cid:1)K (B ) = Z(cid:0) isgene−ra(cid:1)tedbyϕ β [p] [ε ] ,
0 2 ∼ 0 1 + 1 2 ∼ 1 1 1
−
where ϕ ,j Z/2Z, is the Thom-C(cid:0)onnes iso(cid:1)morphism (see[2]), β is the isomo(cid:0)rphism i(cid:1)n
j 1
∈
1 0
Lemma 1, ε is the constant matrix and ⊠ is the external tensor product (see, for
1 (cid:18)0 0(cid:19)
example, [2,VI.2]).
LEMMA 4.
(i) K C (V, ) = Z, K C (V, ) = Z,
0 ∗ ∼ 1 ∗ ∼
F F
(cid:0) (cid:1) (cid:0) (cid:1)
7
(ii) K (J ) = 0; K (J ) = Z2 is generated by ϕ β [b]⊠[u ] and ϕ β [b]⊠[u ] ,
0 1 1 1 ∼ 1 2 + 1 2
(iii) K (B ) = 0; K (B ) = Z2 is generated by ϕ β(cid:0)[¯1] and ϕ(cid:1)β [p¯] [ε(cid:0)] , − (cid:1)
1 1 0 1 ∼ 0 2 0 2 1
−
where ¯1 is unit element in C(S2), ϕ is the Thom-Connes is(cid:0)omorphism(cid:1), β is the Bott
0 2
isomorphism.
Proof.
(i) K C (V, ) = K C(S3) = Z, i = 0,1.
i ∗ ∼ i ∼
F
(ii) Th(cid:0)e proof is(cid:1)similar(cid:0)to (ii)(cid:1)of lemma 3.
(iii) By [9, p.206], we have
K C(S2) = Z[¯1]+Z[q], where q P C(S2) .
0 2
∈
(cid:0) (cid:1) (cid:0) (cid:1)
Otherwise, in [9, p.48,53,56]; [13, p.162], one has shown that the map
dim : K C(S2) Z
0
→
(cid:0) (cid:1)
is a surjective group homomorphism which satisfied dim[¯1] = 1, ker(dim) = Z and non-
zero element q P C(S2) in the kernel of the map dim has the form [q] = [p¯] [ε ].
2 1
∈ −
Hence, the result is(cid:0)derived(cid:1)straight away because β and ϕ are isomorphisms.
2 1
Proof of theorem 2
1. Computation of(γ ). Recall that theextension (γ )intheorem1 gives risetoasix-term
1 1
exact sequence
0 = K (J ) // K C (V,F) // K (B )
0 1 0 ∗ 0 1
OO
(cid:0) (cid:1)
δ1 δ0
(cid:15)(cid:15)
0 = K1(B1)oo K1 C∗(V,F) oo K1(J1)
(cid:0) (cid:1)
By [11, Theorem 4.14], the isomorphisms
Ext(B ,J ) = Hom (K (B ),K (J ) = Hom(Z2,Z2)
1 1 ∼ 0 1 1 1 ∼
(cid:0) (cid:1)
associates the invariant γ Ext(B ,J ) to the connecting map δ : K (B ) K (J ).
1 1 1 0 0 1 1 1
∈ →
Since the Thom-Connes isomorphism commutes with K theoretical exact sequence
−
(see[14, Lemma 3.4.3]), we have the following commutative diagram (j Z/2Z):
∈
... // K (J ) //K C (V,F) //K (B ) // K (J ) //...
j 1 j ∗ j 1 j+1 1
OO OO OO OO
(cid:0) (cid:1)
ϕj ϕj ϕj ϕj+1
... //K C (V ) // K C (V) //K C (W ) // K C (V ) // ...
j 0 1 j 0 j 0 1 j+1 0 1
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
In view of Lemma 2, the following diagram is commutative
... //K C (V ) // K C (V) // K C (W ) //K C (V ) // ...
j 0 1 1 0 j 0 1 j+1 1 1
OO OO OO OO
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
β2 β2 β2 β2
... // K (I ) //K C(S3) // K (A ) //K (I ) //...
j 1 j j 1 j+1 1
(cid:0) (cid:1)
8
Consequently, instead of computing δ : K (B ) K (J ), it is sufficient to compute
0 0 1 1 1
→
δ : K (A ) K (I ). Thus, bythe proofof Lemma 4, we have todefine δ [p¯] [ε ] =
0 0 1 1 1 0 1
→ −
δ [p¯] (because δ [ε ] = (0;0) and δ [¯1] = (0;0)). By the usual defini(cid:0)tion (see[(cid:1)13,
0 0 1 0
p.1(cid:0)70](cid:1)), for [p¯] K(cid:0) (A(cid:1)), δ [p¯] = e2(cid:0)πip˜ (cid:1) K (I ) where p˜ is a preimage of p¯ in (a
0 1 0 1 1
∈ ∈
matrix algebra over) C(S3), i(cid:0).e. (cid:1)v p˜=(cid:2) p¯. (cid:3)
1
z
We can choose p˜(x,y,z) = p¯(x,y), (x,y,z) S3.
√1+z2 ∈
Let p˜ (resp. p˜ ) be the restriction of p˜on R2 R (resp. R2 R ). Then we have
+ +
− × × −
δ [p¯] = e2πip˜ = e2πip˜+ + e2πip˜− K C R2 C R K C R2
0 1 0 0 + 1 0
∈ ⊗ ⊕ ⊗
(cid:16) (cid:17) (cid:16)
(cid:0) (cid:1) (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
C R = K (I )
0 1 1
− (cid:17)
(cid:0) (cid:1) ^
By [13, Section 4], for each function f : R Q C R2 such that lim f(t) =
n 0
± → x 0
^ ^ (cid:0) (cid:1) →±
lim f(t),whereQ C R2 = a M C R2 ,e2πia = Id ,theclass[f] K C (R2)
n 0 n 0 1 0
x n ∈ o ∈ ⊗
→±∞ (cid:0) (cid:1) (cid:0) (cid:1) 1 (cid:0)
C (R ) canbedeterminedby[f] = W .[b]⊠[u ],whereW = Tr f (z)f 1(z) dz
0 f f ′ −
± ± 2πi ZR
(cid:1) ± (cid:0) (cid:1)
is the winding number of f.
By simple computation, we get δ [p] = [b] ⊠ [u ] + [b] ⊠ [u ]. Thus γ = 0 1
0 + − 1 (cid:16)0 1(cid:17) ∈
HomZ(Z2,Z2). (cid:0) (cid:1)
2. Computation of (γ ). The extension (γ ) gives rise to a six-term exact sequence
2 2
K (J ) // K (B ) // K (B )
0 2 0 1 0 2
OO
δ1 δ0
(cid:15)(cid:15)
K (B ) oo K (B ) oo K (J ) = 0
1 2 1 1 1 2
By [11, Theorem 4.14], γ2 = δ1 Hom K1(B2),K0(J2) = HomZ(Z,Z2). Similarly to
∈
part 1, taking account of Lemma 1 and(cid:0)3, we have the f(cid:1)ollowing commutative diagram
(j Z/2Z)
∈
... // K (J ) //K (B ) //K (B ) //K (J ) //...
j 2 j 1 j 2 j+1 2
OO OO OO OO
ϕj ϕj ϕj ϕj+1
... //K C (V ) // K C (W ) // K C (W ) // K C (V ) //...
j 0 2 j 0 1 j 0 2 j+1 0 2
OO OO OO OO
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
β1 β1 β1 β1
... // Kj 1(I2) //Kj 1 C0(R3)∗ // Kj 1(A2) //Kj(I2) //...
− − −
(cid:0) (cid:1)
Thus we can compute δ : K (A ) K (I ) instead of δ : K (B ) K (J ). By the
0 0 2 1 2 1 1 2 0 2
→ →
proof of Lemma 3, we have to define δ [p] [ǫ ] = δ [p] (because δ [ǫ ] = (0,0)).
0 1 0 0 1
−
The same argument as above, we get δ (cid:0)[p] = [b](cid:1)⊠[u (cid:0)]+[(cid:1)b]⊠[u ]. Thu(cid:0)s γ(cid:1)= (1,1)
0 + 2
HomZ(Z,Z2) ∼= Z2. The proof is comple(cid:0)ted(cid:1). − (cid:3)∈
9
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