EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 1, 2014, 65-76
ISSN 1307-5543 – www.ejpam.com

Relative Differential K-theory

Adnane Elmrabty∗, Mohamed Maghfoul

Department of Mathematics, Faculty of Sciences, Ibn Tofail University, Kenitra, Morocco

Abstract. Let ρ : Y → X be a smooth map between two smooth compact manifolds. We define the
relative differential K-theory group Ǩ∗(ρ) and show that it fits into a six-term exact sequence. We
define Ǩ∗(ρ,R/Z), the K-theory of ρ with R/Z coefficients. It turns out that Ǩ∗(ρ,R/Z) is isomorphic
to the group of homomorphisms from the relative K-homology of ρ [8] to R/Z up to a degree-shift by
one.
2010 Mathematics Subject Classifications: 19K33, 51H25
Key Words and Phrases: differential K-characters, geometric K-homology, R/Z K-theory

1. Introduction

Differential K-theory is a generalized differential cohomology theory introduced by Freed
and Hopkins [5] as a refinement of topological K-theory for a concrete description of RR-fields
in string theory. This theory encode geometric as well as topological information. Roughly
speaking, differential K-theory combines topological K-theory with differential forms [3–7, 9].
Benameur and Maghfoul [2] pointed out the relevance to differential K-characters of a de-
scription of differential flat K-theory. The group of differential K-characters on a smooth
compact manifold X is defined as the K-theoretic version of the group of Cheeger-Simons
differential characters on X using the (M , E, f )-picture of Baum-Douglas for K-homology. Re-
call that a geometric K-cycle of Baum-Douglas over X is a triple (M , E, f ) such that: M is a
smooth compact Spinc manifold without boundary, E is a Hermitian vector bundle over M
with a fixed Hermitian connection ∇E , and f : M → X is a smooth map. Let C∗(X ) be the
semigroup for the disjoint union of equivalence classes of K-cycles over X generated by di-
rect sum and vector bundle modification [1]. A differential K-character on X is a semigroup
homomorphism h : C∗(X ) → R/Z such that its restriction to the boundaries is given by the
following formula:

h(∂W,ε|∂W , g|∂W ) :=

∫

W

g∗(w)Ch(ε)T d(W ) mod Z,

∗Corresponding author.

Email addresses: elmrabty_adnane@yahoo.fr (A. Elmrabty), mmaghfoul@lycos.com (M. Maghfoul)

http://www.ejpam.com 65 c© 2014 EJPAM All rights reserved.



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 66

where w is a closed differential form on X with integer K-periods [2], Ch(ε) is the Chern form
of the connection ∇ε on ε, and T d(W ) is the Todd form of the tangent bundle of W .

The purpose of this paper is to construct Ǩ∗(ρ), the relative differential K-theory of a
smooth map ρ : Y → X between two smooth compact manifolds. To motivate our construc-
tion, the group Ǩ∗(ρ) must recovers the usual non-relative group of differential K-characters
on X . We define the K-theory of ρ : Y → X with R/Z coefficients and show that it is isomor-
phic to the group of homomorphisms from the relative K-homology of ρ [8] to R/Z up to a
degree-shift by one.

The paper is organized as follows:
In Section 2, we recall the definition of the group of differential K-characters and study some
of its properties. In Section 3, we define the relative differential K-theory of a smooth map
ρ : Y → X between two smooth compact manifolds and show that it fits into a six-term exact
sequence. Finally, section 4 is concerned with the definition of the K-theory of a smooth map
ρ : Y → X with R/Z coefficients and the construction of an isomorphism between this group
and the group of homomorphisms from the relative K-homology of ρ [8] to R/Z.

2. Differential K-characters

In this section, we give the construction of the group of differential K-characters following
[2]. As mentioned in the introduction, in this construction we use the (M , E, f )-picture of
Baum-Douglas for K-homology [1].

Definition 1. Let X be a smooth compact manifold. A K-chain over X is a triple (W,ε, g) such
that

• W is a smooth compact Spinc manifold;

• ε is a Hermitian vector bundle over W with a fixed Hermitian connection ∇ε; and

• g : W → X is a smooth map.

There are no connectedness requirements made upon W, and hence the bundle ε can have different
fibre dimensions on the different connected components of W. It follows that disjoint union

(W,ε, g)t (W ′,ε′, g ′) := (W tW ′,ε t ε′, g t g ′)

is a well-defined operation on the set of K-chains over X .

Isomorphism. Two K-chains (W,ε, g) and (W ′,ε′, g ′) over X are isomorphic if there exists a
diffeomorphism h : W →W ′ such that

• h preserves the Spinc structures;

• h∗ε′ ∼= ε; and



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 67

• the diagram

W

g
��

h // W ′

g ′}}
X

commutes.

A K-cycle is a K-chain (M , E, f ) without boundary; that is ∂M = ;. The boundary ∂ (W,ε, g)
of a K-chain (W,ε, g) is the K-cycle (∂W,ε|∂W , g|∂W ).

We are going to construct an Abelian group from the set of isomorphism classes of K-cycles
over X so as to obtain the geometric K-homology group of X . In order to define the relation in
this group we need to introduce several kinds of relations involving K-cycles.

Vector bundle modification. Let (W,ε, g) be a K-chain over X and let H be a Spinc Euclidean
vector bundle over W with even-dimensional fibers. Let 1W denote the trivial real line bundle over
W. We denote by cW := S(H ⊕ 1W ), the unit sphere bundle of H ⊕ 1W . Let π : cW → W be the
bundle projection. The Spinc structures on TW and H induce a Spinc structure on TcW. Let
S = S− ⊕ S+ be the Z2-graded bundle of Clifford modules over W associated with the Spinc

structure on H. We denote by H0 and H1 the pullbacks of S− and S+, respectively, to H by the
bundle projection H →W. Then H acts on H0 and H1 by Clifford multiplication map: H0

σ→ H1.

The manifold cW can be thought of as formed of two copies, B0(H) and B1(H), of the unit ball
bundle of H (carrying opposite Spinc structures) glued together by the identity map of S(H):

cW = B0(H)∪S(H) B1(H).

The vector bundle bH over cW is obtained by putting H0 over B0(H) and H1 over B1(H) and then
clutching these two vector bundles along S(H) by the isomorphism σ. The process of obtaining
the K-chain (cW , bH ⊗π∗ε, g ◦π) from (W,ε, g) is called vector bundle modification.
Note that

∂ (cW , bH ⊗π∗ε, g ◦π) = (Ô∂W ,×H|∂W ⊗π∗(ε|∂W ), g|∂W ◦π|Ô∂W ).

Definition 2. We define the set C∗(X ) as the quotient of the set of isomorphism classes of K-cycles
over X by the equivalence relation ∼ generated by the relations of

• direct sum: if E = E1⊕ E2, then (M , E1, f )t (M , E2, f )∼ (M , E1⊕ E2, f ); and

• vector bundle modification.

An operation on C∗(X ) is given by disjoint union,

(M , E, f )t (M ′, E′, f ′) := (M tM ′, E t E′, f t f ′).

This operation turns C∗(X ) into an Abelian semigroup. Since the relation ∼ preserves the par-
ity of the dimension of M in K-cycles (M , E, f ), one can define the subsemigroup C0(X ) (resp.



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 68

C1(X )) consisting of classes of K-cycles (M , E, f ) for which all connected components of M are of
even (resp. odd) dimension. Then C∗(X ) = C0(X )⊕ C1(X ) has a natural Z2-grading.

Bordism. Two K-cycles (M , E, f ) and (M ′, E′, f ′) over X are bordant if there exists a K-chain
(W,ε, g) such that the two K-cycles ∂ (W,ε, g) and (M , E, f ) t (−M ′, E′, f ′) are isomorphic,
where −M ′ denotes M ′ with the Spinc structure on its tangent bundle T M ′ reversed [1].

The bordism relation induces a well-defined equivalence relation ∼b on C∗(X ). This rela-
tion is compatible with the semigroup structure, and then the quotient set C∗(X )/ ∼b turns
out to be an Abelian semigroup. The Abelian semigroup C∗(X )/∼b is in fact an Abelian group.
The additive inverse of the class of a K-cycle is obtained by reversing the Spinc structure:

−[M , E, f ] = [−M , E, f ].

The neutral element is represented by the empty manifold, or any K-cycle bordant to the
empty manifold.

Definition 3. The quotient group C∗(X )/ ∼b is denoted by K∗(X ) and called the geometric K-
homology group of X . It has a natural Z2-grading:

K∗(X ) = K0(X )⊕ K1(X ).

The geometric construction of K-homology is functorial. If ρ : Y → X is a smooth map
between two smooth compact manifolds, then the induced homomorphism

ρ∗ : K∗(Y )→ K∗(X )

of Z2-graded Abelian groups is given on classes of K-cycles [M , E, f ] ∈ K∗(Y ) by

ρ∗[M , E, f ] := [M , E,ρ ◦ f ].

Since vector bundles over M extend to vector bundles over M × [0,1], it follows by bordism
that K∗(ρ) := ρ∗ depend only on the smooth homotopy classes of ρ.

Let X be a smooth compact manifold. Let L∗(X ) be the quotient of the set of isomorphism
classes of K-chains over X by ∼. Note that the boundary map on the set of K-chains over X
descends to a boundary map

∂ : L∗(X )→ C∗−1(X )⊂ L∗−1(X ).

Let Ω∗(X ) be the graded algebra of real-valued differential forms on X . Let
ϕ : Ω∗(X )→ Hom(L∗(X ),R) be the map defined by

ϕw(W,ε, g) :=

∫

W

g∗(w)Ch(ε)T d(W ),

where Ch(ε) is the Chern form of the connection ∇ε on ε and T d(W ) is the Todd form of the
tangent bundle of W . The set of K-periods of a real-valued differential form w ∈ Ω∗(X ) is the



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 69

subset ϕw(C∗(X )) of R. The Abelian group of closed real-valued differential forms on X with
integer K-periods is denoted by Ω∗0(X ). It has a natural Z2-grading:

Ω∗0(X ) = Ω
even
0 (X )⊕Ωodd

0 (X ).

Example 1. Let X be a smooth compact manifold. Let F be a Hermitian vector bundle over X
with a Hermitian connection ∇. An example of a form with integer K-periods is given by the
Atiyah-Singer index theorem applied to the positive part of the Dirac operator associated to the
Spinc structure on a Spinc compact manifold M in K-cycles (M , E, f ) with coefficients in E⊗ f ∗F:

Ind([D+(E ⊗ f ∗F)]) =

∫

M

f ∗(Ch(∇))Ch(E)T d(M) ∈ Z.

Definition 4. (i) Let X be a smooth compact manifold. A differential K-character on X is a
homomorphism of semigroups

h : C∗(X )→ R/Z

such that its restriction to the boundaries is given by the following formula:

h(∂ (W,ε, g)) :=

∫

W

g∗(w)Ch(ε)T d(W ) mod Z,

where w is a closed real-valued differential form on X with integer K-periods.

(ii) The set of differential K-characters on X is denoted by K̂∗(X ). It is an Abelian group which
has a natural Z2-grading:

K̂∗(X ) = K̂0(X )⊕ K̂1(X ).

The differential form w associated to h, indicated above, is unique. It will be denoted by
δ0(h). Thus we have a homomorphism

δ0 : K̂∗(X )→ Ω∗+1
0 (X ).

Note that a differential form v ∈ Ω∗(X ) determines a differential K-character fϕv on X by
setting

fϕv(M , E, f ) :=

∫

M

f ∗(v)Ch(E)T d(M) mod Z.

It is easy to see that δ0(fϕv) = dv.

We can measure the size of K̂∗ by inserting it in a certain exact sequence.
We have the short exact sequence

0→ Hom(K∗(X ),R/Z) ,→ K̂∗(X )
δ0→ Ω∗+1

0 (X )→ 0.

This, together with the fact that the only K-cycles on pt are (pt,Ck, idpt), implies that

K̂0(pt)∼= R/Z and K̂1(pt)∼= Z.



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 70

The construction of K̂∗(X ) is functorial. If ρ : Y → X is a smooth map between two smooth
compact manifolds, then the induced homomorphism

ρ∗ : K̂∗(X )→ K̂∗(Y )

of Z2-graded Abelian groups is given on differential K-characters on X by

ρ∗(h)(M , E, f ) := h(ρ∗(M , E, f )) for all (M , E, f ) ∈ C∗(X ).

It is obvious that δ0(ρ∗(h)) = ρ∗(δ0(h)).

Let X be a smooth compact manifold. Let i be the inclusion pt ,→ X . Set

K̃∗(X ) := ker[K̂∗(X )
i∗→ K̂∗(pt)].

Since the short exact sequence

0→ K̃∗(X ) ,→ K̂∗(X )
i∗→ K̂∗(pt)→ 0

is split, we obtain isomorphisms

K̂0(X )∼= K̃0(X )⊕R/Z and K̂1(X )∼= K̃1(X )⊕Z.

3. Relative Differential K-theory

In this section, we define the relative differential K-theory of a smooth map between two
smooth compact manifolds and show that it fits into a six-term exact sequence.

Let X be a smooth compact manifold. Let A⊆ R be a subring of the reals. A K-cochain over
X with coefficients in A is a semigroup homomorphism from L∗(X ) to A. The set of K-cochains
over X with coefficients in A is denoted by L∗(X , A). The set L∗(X , A) is an Abelian group and
a coboundary map on L∗(X , A) is defined by transposition:

δh(W,ε, g) := h(∂ (W,ε, g)).

We set
Ľ∗(X ) = L∗(X ,Z)× L∗−1(X ,R)×Ω∗0(X ),

and define a coboundary map δ̂ : Ľ∗(X )→ Ľ∗+1(X ) by the formula:

δ̂(c, h, w) := (−δc,−ϕw + c+δh, 0).

Let ρ : Y → X be a smooth map between two smooth compact manifolds. We define the
set of relative K-cochains Ľ∗(ρ) as the direct product Ľ∗(X )× Ľ∗−1(Y ). A coboundary map
δ̌ : Ľ∗(ρ)→ Ľ∗+1(ρ) is given by setting

δ̌(S, T ) := (δ̂S,ρ∗S− δ̂T ).

Elements of ker[ Ľ∗(ρ)
δ̌→ Ľ∗+1(ρ)] are called K-cocycles and those of img[ Ľ∗−1(ρ)

δ̌→ Ľ∗(ρ)]
are called K-coboundaries. Let Ž∗(ρ) be the set of K-cocycles and B̌∗(ρ) the set of K-coboundaries.



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 71

Definition 5. We define the relative differential K-theory group Ǩ∗(ρ) as the quotient group
Ž∗(ρ)/B̌∗(ρ).

The construction of relative differential K-theory is functorial. If

Y ′
ρ′ //

g
��
�

X ′

f
��

Y
ρ // X

is a commutative diagram of smooth maps between smooth compact manifolds, then the
homomorphism

( f , g)∗ : Ǩ∗(ρ)→ Ǩ∗(ρ′)

of Z2-graded Abelian groups is given on classes of K-cocycles [S, T] ∈ Ǩ∗(ρ′) by

( f , g)∗([S, T]) := [ f ∗S, g∗T].

Exact Sequence

Let (S, T ) ∈ Ž∗(ρ). If we set S = (cx , hx , wx) and T = (cy , hy , w y), then the equality
δ̌(S, T ) = 0 implies that:

¨

δcx = 0
ϕwx
= δhx + cx

and







ρ∗cx =−δcy
ρ∗hx =−ϕw y

+δhy + cy

ρ∗wx = 0

It follows that the natural homomorphism R ∼→ R/Z composed with the restriction of hx to
C∗−1(X ), denoted by hx , is a differential K-character on X . Let j : Ž∗(ρ) → K̂∗−1(X ) be the
map given by j(S, T ) := hx . It is obvious that j(δ̌(S, T )) = 0. Then we obtain a homomor-
phism from Ǩ∗(ρ) to K̂∗−1(X ), also denoted by j.

Now, let h ∈ K̂∗(Y ). Since R is divisible, there is a real K-cochain h′ with h′ = h. Set

uh′ = ϕδ0(h)−δh′.

It is obvious that uh′ ∈ L∗−1(Y,Z). On the other hand, we have

δuh′ = ϕdδ0(h)− (δ ◦δ)h
′ = 0.

Therefore, [0, (uh′ , h′,δ0(h))] ∈ Ǩ∗(ρ). We claim that [0, (uh′ , h′,δ0(h))] is independent of the
choice of h′. In fact if h′′ is another lift of h, then h′′−h′ = 0 so that h′′ = h′+ c+δγ for same
c ∈ L∗(Y,Z) and γ ∈ L∗−1(Y,R). Thus we finally get

(0, (uh′′ , h′′,δ0(h))) = (0, (uh′ , h′,δ0(h)))− δ̌(0, (c,γ, 0)).

We define a homomorphism θ : K̂∗(Y )→ Ǩ∗(ρ) by setting

θ(h) := [0, (uh′ , h′,δ0(h))].



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 72

Theorem 1. The following six-term sequence

K̂0(X )
ρ∗ // K̂0(Y ) θ // Ǩ0(ρ)

j
��

Ǩ1(ρ)

j

OO

K̂1(Y )θoo K̂1(X )
ρ∗oo

is exact.

Proof. Exactness at Ǩ1(ρ). It is evident that j ◦ θ = 0.
Let [S, T] ∈ Ǩ1(ρ) with S = (cx , hx , wx) and T = (cy , hy , w y). Assume that j[S, T] = 0.
Then we have wx = 0, cx = −δhx , and there exist g ∈ L1(X ,R) and u ∈ L0(X ,Z) such that
hx = δg + u. Since

(S, T ) = (0, (cy −ρ∗u, hy −ρ∗g, w y)) + δ̌((u, g, 0), 0)

and [0, (cy −ρ∗u, hy −ρ∗g, w y)] lies in the image of θ , we get [S, T] ∈ img(θ).

Exactness at K̂1(X ). For any [S, T] ∈ Ǩ0(ρ) with (S, T ) = ((cx , hx , wx), (cy , hy , w y)), the
equality δ̌(S, T ) = 0, together with the fact that w y ∈ Ωodd

0 (Y ), implies that

ρ∗ ◦ j[S, T](σ) =−ϕw y
(σ) + hy(∂ σ) = 0 for all σ ∈ C1(Y ).

Now, let h ∈ ker[K̂0(X )
ρ∗

→ K̂0(Y )]. First, we have ρ∗(δ0(h)) = 0. Furthermore, we can find
f ∈ L1(Y,R) and c ∈ L0(Y,Z) such that

ρ∗h′ = δ f + c and ρ∗uh′ =−δc.

It is easy to check that R := ((uh′ , h′,δ0(h)), (c, f , 0)) defines an element in Ǩ1(ρ) with
j([R]) = h.

Exactness at K̂0(Y ). For every h ∈ K̂0(X ),

θ ◦ρ∗(h) = [0, (uρ∗h′ ,ρ
∗h′,ρ∗δ0(h))] = [δ̌((uh′ , h′,δ0(h)), 0)] = 0.

If f ∈ K̂0(Y ) such that θ( f ) = 0, then there exists ((cx , hx , wx), (cy , hy , w y)) ∈ Ľ1(ρ) with
coboundary (0, (u f ′ , f ′,δ0( f ))). Therefore, we have the equations

¨

δcx = 0
ϕwx
= δhx + cx

and







ρ∗cx +δcy = u f ′

ρ∗hx +ϕw y
−δhy − cy = f ′

ρ∗wx = δ0( f )

which imply that hx is a differential K-character on X with δ0(hx) = wx and ρ∗(hx) = f .

Remark 1. Let X be a smooth compact manifold. Let i be the inclusion pt ,→ X . The above
exact sequence, together with the fact that i∗ : K̂∗−1(X ) → K̂∗−1(pt) is surjective, implies that
j : Ǩ∗(i)→ K̂∗−1(X ) is injective with img( j) = ker(i∗). Thus we get an isomorphism
Ǩ∗(i)∼= K̃∗−1(X ).



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 73

4. R/Z Relative K-theory

This section is concerned with the definition of the K-theory of a smooth map ρ : Y → X
with R/Z coefficients and the construction of an isomorphism between this group and the
group of homomorphisms from the relative K-homology of ρ [8] to R/Z.

Let ρ : Y → X be a smooth map between two smooth compact manifolds. We write
Ľ∗(ρ,R/Z) for the set of relative K-cochains of the form ((cx , hx , 0), (cy , hy , 0)). The set
Ľ∗(ρ,R/Z) is in fact an Abelian subgroup of Ľ∗(ρ). Note that the image of the restriction
of δ̌ : Ľ∗(ρ)→ Ľ∗+1(ρ) to Ľ∗(ρ,R/Z) is included in Ľ∗+1(ρ,R/Z).
The Kernel of δ̌ : Ľ∗(ρ,R/Z) → Ľ∗+1(ρ,R/Z) is denoted by Ž∗(ρ,R/Z) and the image of
δ̌ : Ľ∗−1(ρ,R/Z)→ Ľ∗(ρ,R/Z) is denoted by B̌∗(ρ,R/Z).

Definition 6. We define the relative K-theory of ρ with R/Z coefficients, denoted by Ǩ∗(ρ,R/Z),
as the quotient group Ž∗(ρ,R/Z)/B̌∗(ρ,R/Z).

It is obvious that Ǩ∗(ρ,R/Z) is an Abelian subgroup of Ǩ∗(ρ).

Let us recall the six-term exact sequence in section 3:

K̂0(X )
ρ∗ // K̂0(Y ) θ // Ǩ0(ρ)

j
��

Ǩ1(ρ)

j

OO

K̂1(Y )θoo K̂1(X )
ρ∗oo

Note that the image of the restriction of j to Ǩ∗(ρ,R/Z) is included in Hom(K∗−1(X ),R/Z),
and the image of the restriction of θ to Hom(K∗(Y ),R/Z) is included in Ǩ∗(ρ,R/Z).
Let K∗(X ,R/Z), the K-theory of X with R/Z coefficients. We have the six-term exact sequence

K0(X ,R/Z)
ρ∗ // K0(Y,R/Z) θ // Ǩ0(ρ,R/Z)

j
��

Ǩ1(ρ,R/Z)

j

OO

K1(Y,R/Z)θoo K1(X ,R/Z)
ρ∗oo

obtained from the above exact sequence and after identification of the groups K∗(X ,R/Z) and
Hom(K∗(X ),R/Z) following [2].

Now, we show that the group Ǩ∗(ρ,R/Z) can be identified with the group of homomor-
phisms from the relative K-homology group K∗(ρ) [8] to R/Z.

Let us recall the construction of the group K∗(ρ) following [8].
We set

L∗(ρ) := L∗(X )× L∗−1(Y )



A. Elmrabty, M. Maghfoul / Eur. J. Pure Appl. Math, 7 (2014), 65-76 74

and define a boundary map ∂̂ : L∗(ρ)→ L∗−1(ρ) by the formula:

∂̂ (α,β) := (∂ α+ρ∗β ,−∂ β).

Let C∗(ρ) denote the kernel of ∂̂ . There is a well-defined operation on C∗(ρ) given by disjoint
union of K-chains,

(α,β) + (α′,β ′) := (αtα′,β t β ′).

Bordism. Two elements (α,β) and (α′,β ′) in C∗(ρ) are bordant if there exists
(σ,τ) ∈ L∗+1(ρ) such that (α,β) + (−α′,−β ′) = ∂̂ (σ,τ).

Definition 7. We define the relative K-homology group K∗(ρ) as the group obtained from quoti-
enting C∗(ρ) by the equivalence relation of bordism.

We denote by K̂∗(ρ,R/Z) the group of homomorphisms from K∗(ρ) to R/Z.

For every K-cocycle (S, T ) in Ž∗(ρ,R/Z) with (S, T ) = ((cx , hx , 0), (cy , hy , 0)), we set

µ(S, T )(α,β) :=fhx(α) +fhy(β) for all (α,β) ∈ C∗−1(ρ).

If (S, T ) ∈ Ľ∗(ρ,R/Z) with (S, T ) = ((cx , hx , 0), (cy , hy , 0)), then for all (σ,τ) ∈ L∗(ρ),

µ(S, T )(∂̂ (σ,τ)) =µ(S, T )(∂ σ+ρ∗τ,−∂ τ)

=fhx(∂ σ) +fhx(ρ∗τ)−fhy(∂ τ)

=Þδhx(σ) + (àρ∗hx −Þδhy)(τ)

=µ(δ̂(S, T ))(σ,τ).

It follows that µ induces a well-defined homomorphism

Ǩ∗(ρ,R/Z)→ K̂∗−1(ρ,R/Z),

also denoted by µ.

Proposition 1. The homomorphism µ : Ǩ∗(ρ,R/Z)→ K̂∗−1(ρ,R/Z) turns out to be an isomor-
phism.

Proof. Let us recall the six-term exact sequence in [8, p. 8]:

K̂1(ρ,R/Z)
ζ // K1(X ,R/Z)

ρ∗ // K1(Y,R/Z)

∂
��

K0(Y,R/Z)

∂

OO

K0(X ,R/Z)
ρ∗oo K̂0(ρ,R/Z)

ζoo



REFERENCES 75

If we combine this six-term exact sequence with that given by Theorem 1, then we obtain the
following commutative diagram

K0(X ,R/Z)
ρ∗ // K0(Y,R/Z) θ // Ǩ0(ρ,R/Z)

j

''
µ

��

Ǩ1(ρ,R/Z)
j

gg

µ

��

K1(Y,R/Z)
θ

oo K1(X ,R/Z)
ρ∗

oo

K0(X ,R/Z)
ρ∗ // K0(Y,R/Z) ∂ // K̂1(ρ,R/Z)

ζ

''
K̂0(ρ,R/Z)

ζ

gg

K1(Y,R/Z)
∂

oo K1(X ,R/Z)
ρ∗

oo

in which the rows are exact sequences. It follows from the five lemma that the homomorphism
µ is an isomorphism.

ACKNOWLEDGEMENTS We thank the referee for various comments and corrections which
have helped to improve the material presented herein.

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