A DESCENT THEOREM IN TOPOLOGICAL K­THEORY
by Max Karoubi
1. Introduction. Let A be a Banach algebra and
k(A) the ``K­theory space'' K 0 (A) x
BGL(A), where GL(A) and BGL(A) have the usual topology and K 0 (A) the discrete topology.
If A' = A
* R C is the complexification of A, the group Z/2 acts on
k(A') by complex
conjugation and we have a natural map
s : k(A) zzc k(A') hZ/2
Here, Y hZ/2 denotes in general the homotopy fixed point set of the Z/2­space Y. More
precisely, Y hZ/2 is the space of (continuous) sections of the Borel fibration
EZ/2 x Z/2 Y
d
BZ/2
It is easy to see that Y hZ/2 is also the space of equivariant maps EZ/2 zzc Y. The purpose
of this paper is to prove the following ``descent theorem'' 1 .
2. THEOREM. The map s : k(A) zzc k(A') hZ/2 defined above is an homotopy
equivalence.
3. COROLLARY. There is a spectral sequence with term E 2
p,q
= H p (Z/2 ; K ­ q (A'))
converging to the graded object associated to a filtration of the group K p­q (A).
4. Strategy of the proof. The first step is to remark the following basic fact : if Y is the
product X x X with the action of Z/2 switching the factors, Y hZ/2 may be identified with the
space of maps from EZ/2 to X, a space which is homotopically equivalent to X, since EZ/2 is
contractible. Therefore, the map
s : X zzc (X x X) hZ/2
1 This theorem is well known if A is the Banach algebra of real numbers : cf. [2], lemma 3.5. Other proofs
have been given by J. Lannes (unpublished) and B. Kahn in a joint work with Hinda Hamraoui (in preparation).
On the other hand, the statement seems new if A is the algebra of quaternions : then we have Bsp = BU hZ/2 for
an involution on BU which is of course different from the usual one given by complex conjugation.
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is an homotopy equivalence. From this remark, we deduce immediately the following
proposition :
5. PROPOSITION. The theorem is true if A is already the complexification of a Banach
algebra B.
Proof. For A = B', it is well known (and quite easy to see) that A' is isomorphic to A x A, the
complex conjugation switching the factors. Therefore the space
k(A') is just the product
k(A) x k(A) and the proposition follows from the considerations in § 4.
All the paper is now devoted to reduce the theorem to this easy case, using in an essential way
the KR­theory of Atiyah [1].
6. KR­theory of Banach algebras. Let us consider a compact space X provided with an
involution x # x, following the notation used by Atiyah. We write A(X) for the Banach algebra
of continuous functions f : X zzc A' such that f(x) = f(x), the complex conjugate of f(x).
If A is the field of real numbers and X an arbitrary Z/2­space, it is easy to see that the K­theory
of A(X) is isomorphic to Atiyah's KR­theory (denoted by him KR(X)). If X is a locally
compact space, we may extend this definition by choosing A(X) to be the space of continuous
functions (with values in A') which go to 0 when x goes to ¥. Note that if the involution on X
is trivial, A(X) is just the usual Banach algebra of continuous functions on X with values in A.
On the other hand, if X is a space with 2 points which are switched by the involution, A(X) is
canonically isomorphic to A'.
7. The role of Clifford algebras. In general, we define S p,q (resp. D p,q ) as the sphere
(resp. the ball) of R p+q with the involution induced by (x 1 , ..., x p , y 1 , ..., y q ) #
(­x 1 , ..., ­x p , y 1 , ..., y q ) on R p+q . For p > q, the locally compact space S p,0 ­ S q,0 is Z/2­
homeomorphic to S p­q,0 x R q,0 and we have therefore the following exact sequence of Banach
algebras
0 zzc A(S p­q,0 )(R q,0 ) zzc A(S p,0 ) zzc A(S q,0 ) zzc 0
On the other hand, using the Clifford algebra definition of the higher K­groups, it has
been proved in [3] and [4] respectively that for any Banach algebra with unit L, we have
natural isomorphisms
K p,0 (L)
-- K(L(R p,0 )) and K p,0 (L)
-- K(S p L)
with the following conventions :
A. K p,q (L) denotes in general the Grothendieck group of the ``restriction of scalars'' functor
p(C p,q+1 * L) zzcp(C p,q * L) , where C p,q is the standard Clifford algebra of R p+q and
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p(B) is the category of finitely generated projective B­modules.
B. S p L denotes the p­topological suspension of the Banach algebra L (which defines
``negative'' K­theory as in [4]).
In more modern and accurate homotopical terms, one may say alternatively that the homotopy
fiber of the map
k(C p,1 * L) zzc k(C p,0 * L)
is also the homotopy fiber
f of the map
k(L) --k(L(D p,0 ) zzc k(L(S p,0 ))
One basic theorem proved in [3] § 3.4 is now the following : one has an homotopy equivalence
W(k(L(S p,0 ))
-- fx W(k(L)) if p ³ 3. In other words, using Bott periodicity, we have a
canonical homotopy equivalence
k(L(S p,0 ))
-- k(L) x
k(S p+1 L)
if p ³ 3.
8. Proof of the descent theorem. We first prove by induction on p (1£ p£ 3) that s
induces an homotopy equivalence
s p :
k(A(S p,0 )) zzc k(A'(S p,0 )) hZ/2
For p = 1, this has already been shown in § 5 since A(S 1,0 )
-- A' and A''
-- A' x A'. For p =
2, we have the following commutative diagram of homotopy fibrations (cf. § 7) :
k(SA(S 1,0 ))
-- k(A(S 1,0 )(R 1,0 )) zzc k(A(S 2,0 )) zzc k(A(S 1,0 ))
d d d
k(SA'(S 1,0 )) hZ/2 -- k(A'(S 1,0 )(R 1,0 )) hZ/2 zzc k(A'(S 2,0 )) hZ/2 zzc k(A'(S 1,0 )) hZ/2
Since the two extreme vertical maps are homotopy equivalences for any Banach algebra A, it
follows that the second vertical map is also an homotopy equivalence.
For p = 3, the same argument shows that s 3 is also an homotopy equivalence.
Now, according to § 7 again, we have a commutative diagram of canonically split
homotopy fibrations
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* zc k(A) zzc k(A(S 3,0 )) zzc k(S 4 A) zc *
d d d
* zc k(A') hZ/2 zzc k(A'(S 3,0 )) hZ/2 zzc k(S 4 A') hZ/2 zc *
Since the middle map is an homotopy equivalence by our previous induction, the descent
theorem follows immediately.
9. Generalization. The scheme of the previous proof may be used in a variety of contexts
(for example to prove Thom isomorphism in real K­theory, starting from Thom isomorphism
in complex K­theory) and it might be useful for the future to extract its main ideas. For this, we
should consider two functors from Banach algebras to spaces, called for instance F(A) and
G(A) (in our example
k(A) and
k(A') hZ/2 respectively), and a natural transformation
a : F(A) zc G(A)
The claim is now the following : under suitable hypothesis, if F(A')
-- G(A') by this natural
transformation, then F(A)
-- G(A) (isomorphisms are taken in the homotopy category).
Following our previous arguments (§ 7 and 8), we see by inspection that it is enough to verify
the following three conditions :
1. F and G satisfy the Mayer­Vietoris axiom (cartesian squares of Banach algebras give rise by
F and G to homotopy cartesian squares).
2. F(A(S 1,0 )) [resp. G(A(S 1,0 ))] is naturally isomorphic to F(A') [resp. G(A')] in a way
compatible with a.
3. For p large enough, and in a way compatible with a, F(A) [resp. G(A)] is a natural direct
summand of F(A(S p,0 )) [resp. G(A(S p,0 ))] through the map induced by the obvious ring
homomorphism A zc A(S p,0 ).
REFERENCES
[1] M.­F. Atiyah. K­theory and Reality. Quart. J. Math. Oxford (2) 17, 367­386 (1966)
[2] M. Hopkins, M. Mahowald et H. Sadofsky. Construction of elements in Picard
groups. Contemporary Math. 158 (1992).
[3] M. Karoubi. Algèbres de Clifford et K­théorie. Ann. Sci. Ec. Norm. Sup. 4e sér. 1,
161­270 (1968).
[4] M. Karoubi. Foncteurs dérivés et K­théorie. Springer Lecture Notes N° 136, p. 107­186
( 1970).
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