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Real Vector Bundles and KR-Theory: Atyiah's Paper Explanation and Significance, Study notes of Mathematics

This document, authored by M.F. Atyiah, introduces the concept of real vector bundles and discusses the significance of KR-theory in the study of real elliptic operators. Real vector bundles are complex vector bundles with a real structure, allowing the involution on the base space to commute with the bundle structure. KR-theory is essential for understanding the index problem of real elliptic operators. The document also covers the homotopy property of real vector bundles and the definition of real vector bundles over real algebraic spaces.

What you will learn

  • What is the significance of real algebraic spaces in the context of real vector bundles?
  • How is KR-theory related to the study of real elliptic operators?
  • What is the difference between a complex vector bundle in the category of real spaces and a real vector bundle?
  • How is the homotopy property of real vector bundles deduced?
  • What is a real vector bundle?

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X-THEORY AND REALITY
By
M. F.
ATIYAH
[Received 9 August 1966]
Introduction
THE
^-theory
of
complex vector bundles (2,
5)
has many variants
and
refinements. Thus there are:
(1) ^-theory
of
real vector bundles, denoted
by KO,
(2) ^"-theory of self-conjugate bundles, denoted by
KC
(1) or
KSC
(7),
(3) JT-theory
of
G-vector bundles over (?-spaces (6), denoted by
KQ.
In this paper we introduce
a
new ^-theory denoted
by KR
which
is,
in a sense, a mixture of these three. Our definition is motivated partly by
analogy with real algebraic geometry and partly
by the
theory
of
real
elliptic operators. In fact, for a thorough treatment of the index problem
for real elliptic operators, our KR-theory is essential. On the other hand,
from
the
purely topological point
of
view, KR-theory
has a
number
of advantages and there is
a
strong case for regarding
it
as the primary
theory and obtaining all the others from it. One of the main purposes of
this paper is in fact to show how i£.R-theory leads to
an
elegant proof of
the periodicity theorem
for
XO-theory, starting essentially from
the
periodicity theorem
for
JT-theory
as
proved
in
(3). On the way we also
encounter,
in a
natural manner,
the
self-conjugate theory
and
various
exact sequences between the different theories. There is here
a
consider-
able overlap with the thesis of Anderson (1) but, from our new vantage
point, the relationship between the various theories is much easier to see.
Recently Karoubi (8) has developed an abstract Z'-theory
for
suitable
categories with involution. Our theory is included in this abstraction but
its particular properties are
not
developed
in
(8), nor
is it
exploited
to
simplify the iTO-periodicity.
The definition and elementary properties of
KR
are given
in
§
1.
The
periodicity theorem
and
general cohomology properties
for KR are
discussed
in
§
2. Then
in
§
3 we
introduce various derived theories
KR with coefficients
in
certain spaces—ending
up
with the periodicity
theorem
for KO. In
§
4 we
discuss briefly
the
relation
of KR
with
Clifford algebras
on the
lines
of
(4),
and in
particular
we
establish
a
lemma which
is
used
in
§
3.
The
significance
of
KR-thsory
for the
topological study of real elliptic operators is then briefly discussed in
§
5.
Quit.
J.
Mmth. Oxford (2), 17 (1966), 367-86.
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14

Partial preview of the text

Download Real Vector Bundles and KR-Theory: Atyiah's Paper Explanation and Significance and more Study notes Mathematics in PDF only on Docsity!

X-THEORY AND REALITY

By M. F. ATIYAH

[Received 9 August 1966] Introduction THE ^-theory of complex vector bundles (2, 5) has many variants and refinements. Thus there are: (1) ^-theory of real vector bundles, denoted by KO, (2) ^"-theory of self-conjugate bundles, denoted by KC (1) or KSC (7), (3) JT-theory of G-vector bundles over (?-spaces (6), denoted by KQ. In this paper we introduce a new ^-theory denoted by KR which is, in a sense, a mixture of these three. Our definition is motivated partly by analogy with real algebraic geometry and partly by the theory of real elliptic operators. In fact, for a thorough treatment of the index problem for real elliptic operators, our KR-theory is essential. On the other hand, from the purely topological point of view, KR-theory has a number of advantages and there is a strong case for regarding it as the primary theory and obtaining all the others from it. One of the main purposes of this paper is in fact to show how i£.R-theory leads to an elegant proof of the periodicity theorem for XO-theory, starting essentially from the periodicity theorem for JT-theory as proved in (3). On the way we also encounter, in a natural manner, the self-conjugate theory and various exact sequences between the different theories. There is here a consider- able overlap with the thesis of Anderson (1) but, from our new vantage point, the relationship between the various theories is much easier to see. Recently Karoubi (8) has developed an abstract Z'-theory for suitable categories with involution. Our theory is included in this abstraction but its particular properties are not developed in (8), nor is it exploited to simplify the iTO-periodicity. The definition and elementary properties of KR are given in § 1. The periodicity theorem and general cohomology properties for KR are discussed in § 2. Then in § 3 we introduce various derived theories— KR with coefficients in certain spaces—ending up with the periodicity theorem for KO. In § 4 we discuss briefly the relation of KR with Clifford algebras on the lines of (4), and in particular we establish a lemma which is used in § 3. The significance of KR-thsory for the topological study of real elliptic operators is then briefly discussed in § 5. Q u i t. J. Mmth. Oxford (2), 17 (1966), 367-86.

368 M. F. ATIYAH

This paper is essentially a by-product of the author's joint work with

I. M. Singer on the index theorem. Since the results are of independent

topological interest it seemed better to publish them on their own.

1. The real category

By a space with involution, we mean a topological space X together

with a homeomorphism T: X -> X of period 2 (i.e. T* = Identity). The

involution T is regarded as part of the structure of X and is frequently

omitted if there is no possibility of confusion. A space with involution

is just a Z 2 -space in the sense of (6), where Z% is the group of order 2. An

alternative terminology which is more suggestive is to call a space with

involution a real space. This is in analogy with algebraic geometry. In

fact if X is the set of complex points of a real algebraic variety it has a

natural structure of real space in our sense, the involution being given

by complex conjugation. Note that the fixed points are just the real

points of the variety X. In conformity with this example we shall

frequently write the involution T as complex conjugation:

T(X) = x.

By a real vector bundle over the real space X we mean a complex vector

bundle E over X which is also a real space and such that

(i) the projection E ->- X is real (i.e. commutes with the involutions

on E, X) ;

(ii) the map Ex^> Et is anti-linear, i.e. the diagram

commutes, where the vertical arrows denote the involution and

C is given its standard real structure (T(Z) = z).

It is important to notice the difference between a vector bundle in the

category of real spaces (as denned above) and a complex vector bundle

in the category of Zg-spaces. In the definition of the latter the map

is assumed to be complex-linear. On the other hand note that if E is a

real vector bundle in the category of Zj-spaces its complexification can

be given two different structures, depending on whether

Ex-* ET<X)

is extended linearly or anti-linearly. In the first it would be a bundle in

370 M. F. ATIYAH

Suppose now that X is a real algebraic space (i.e. the complex points of

a real algebraic variety) then, as we have already remarked, it defines in

a natural way a real topological space X^ i-»- Xtop. A real algebraic

vector bundle can, for our purposes, be taken as a complex algebraic

vector bundle TT: E -*• X where X, E, n, and the scalar multiplication

C X E -v E are all denned over R (i.e. they are given by equations with

real coefficients). Passing to the underlying topological structure it is

then clear that Etav is a real vector bundle over the real space Xtop.

Consider as a particular example X = P(Cn), (ra—1)-dimensional

complex projective space. The standard line-bundle H over P(Cn) is

a real algebraic bundle. In fact H is denned by the exact sequence of

vector bundles o ^ l ^

where E c XxCn^ consists of all pairs ((z), u) e l x C " satisfying

J,uizi = 0.

Since this equation has real coefficients E is a real bundle and this then

implies that H is also real. Hence H defines a real bundle over the real

space P(Cn).

As another example consider the afifine quadric

Since this is affine a real vector bundle may be denned by projective

modules over the affine ring A+ = 'R[z 1 ,...,zn]l( 2 Z?+1)- Now the

intersection of the quadric with the imaginary plane is the sphere

the involution being just the anti-podal map y v-*- — y. Thus projective

modules over the ring A + define real vector bundles over Sn~x^ with the

anti-podal involution. If instead we had considered the quadric

then its intersection with the real plane would have been the sphere with

trivial involution, so that projective modules over

. Rfo zn]

— (T3=ir

define real vector bundles over S*^1 "^1 with the trivial involution (and so

these are real vector bundles in the usual sense). The significance of S"-^1

in this example is that it is a deformation retract of the quadric in our

category (i.e. the retraction preserving the involution).

ON ^-THEORY AND REALITY 371

The Grothendieck group of the category of real vector bundles over a

real space X is denoted by KR(X). Restricting to the real points of X we

obtain a homomorphism

KR(X) -+ KR(XR) a* KO(XR).

In particular if X = XR we have

KR(X) (^) s KO(X).

For example taking X = P(O) we have XR = P(Rn) and hence a

restriction homomorphism

KR{P(Cn)) -+ KR(P(Rn)) = KO{P(Rn)).

Note that the image of [E] in this homomorphism is just the standard real

Hopf bundle over P(Rn).

The tensor product turns KR(X) into a ring in the usual way.

If we ignore the involution on X we obtain a natural homomorphism

If X = XR then this is just complexification. On the other hand if E is

a complex vector bundle over X, E @ T*E has a natural real structure

and so we obtain a homomorphism

If X = XR then this is just 'realization', i.e. taking the underlying real

space.

2. The periodicity theorem

We come now to the periodicity theorem. Here we shall follow care-

fully the proof in (3) [§ 2] and point out the modifications needed for our

present theory.

If E is a real vector bundle over the real space X then P{E), the projec-

tive bundle of E, is also a real space. Moreover the standard line-bundle H

over P{E) is a real line-bundle. Then the periodicity theorem for KR

asserts:

THBOBEM 2.1. Let Lbea real line-bundle over the real compact space X, E the standard real line-bundle over the real space P(L © 1). Then, as a KR(X)-algebra, KR(P(L © 1)) is generated by H, subject to the single

relation ([ff]-[l])([£][ff]-[l]) = 0- M85.2.17 B b

ON X-THEORY AND REALITY 373 trivial involution, we can also consider R with the involution x -y —x. It is often convenient to regard the first case as the real axis R c C and the second as the imaginary axis i R c C , the complex numbers C always having the standard real structure given by complex conjugation. We use the following notation: RV.Q = Rff©tRp, IP* = unit ball in Sv* = unit sphere in Note that R"* ^ O. Note also that, with this notation, S™ has dimension p--q —1. The relative group KR(X, Y) is defined in the usual way as KR(X/Y) where KR is the kernel of the restriction to base point. We then define the (p, q) suspension groups KRT'"{X, Y) = KR(X X &>*, X x Bv* U 7 x -B™). Thus the usual suspension groups KR-o are given by

As in (2) one then obtains the exact sequence for a real pair (X, Y) ... -* KR-^X) -• KR-^Y) -• KR(X, Y) -• KR(X) -• KR(Y). (2.2) Similarly one has the exact sequence of a real triple (X, T, Z). Taking the triple (X x B"-°, X X £p-° U 7 X BP-°, X X <Sp-°) one then obtains an exact sequence ... -* KR?-\X) -> KE?-\Y) -> KR"-°(X, Y) -+ K&>-^0 (X) -> KR?-°{Y) for each integer p > 0. The ring structure of KR(X) extends in a natural way to give external products KRP-O(X, Y) ® KRP'*(X', Y') -> KRP+^-^(X", Y"), where X" = X x X', Y" = X X 7 ' U X' X 7. By restriction to the diagonal these define internal products. We can reformulate Theorem 2.1 in the usual way. Thus let b = [H]-l eKR^ipoini) = KR(B™, 8™) = Zi2(P(C^2 )) and denote by /9 the homomorphism KBP*[X, Y) -+ KRP+^+^1 (X, Y) given by x i-»- 6.x. Then we have THEOREM 2.3. £: KRf-o(X, Y) -• ZiJP+^+HX, 7) w an isomorphism. Note also that the exact sequence of a real pair is compatible with the periodicity isomorphism. Hence if we define

KRP(X, Y) = KR?-°(X, Y) forp^O

374 M. F. ATIYAH

it follows that the exact sequence (2.2) for (X, Y) can be extended to

infinity in both directions; Moreover we have natural isomorphisms

We consider now the general Thorn isomorphism theorem as proved

for iT-theory in (2) [§2.7]. We recall that the main steps in the proof

proceed as follows:

(i) for a line-bundle we use (2.1),

(ii) for a decomposable vector bundle we proceed by induction using

(iii) for a general vector bundle we use the splitting principle.

An examination of the proof in (2) [§ 2.7] shows that the only point

requiring essential modification is the assertion that a vector bundle is

locally trivial and hence locally decomposable. Now a real vector bundle

has been denned as a vector bundle with a real structure. Thus it has

been assumed locally trivial as a vector bundle in the category of spaces.

What we have to show is that it is also locally trivial in the category of re

spaces. To do this we have to consider two cases.

(i) x e X a real point. Then Ex ^ O in our category. Hence by the

extension lemma there exists a real neighbourhood U of x such

that E | U ^ U X Cn^ in the category.

(ii) x =£ x. Take a ccinp'ex isomorphism Ex ^ Cn. This induces an

isomorphism Ex ^ Cn. Hence we have a real isomorphism

where Y = {x,x}. By the extension lemma there exists a real

neighbourhood U of Y so that E\U ^UxCn.

Thus we have

THEOREM 2.4 (Thom Isomorphism Theorem). Let E be a real vector

bundle over the real compact space X. Then :KR{X)-+KR(XE) is an isomorphism where (x) = \B. x and XE is the element of KR{XE defined by the exterior algebra of E.

Among other results of (2) [§ 2.7] we note the following:

KR{ Ix P(O)) ^ KR(X)[t]Jt*-\

We leave the computation of KR for Grassmannians and Flag mani-

folds as exercises for the reader. The determination of KR for quadrics

376 M. F. ATIYAH Remark, fi* is clearly a ^iJ(X)-module homomorphism. Since the same is true of £ this implies that the periodicity isomorphism yp = fi$p>: KB(X X Sp-°) -+ KR-^{X X &-^0 ) is multiplication by the image cp of 1 in the isomorphism KR(S*>-°) - This element cp is given by

For any Y the projection X x Y -> X will give rise to an exact coeffi- cient sequence involving KR and KR with coefficients in Y. When Y is a sphere we get a type of Gysin sequence:

PROPOSITION 3.2. The projection TT: SP-° -+ point induces the following exact sequence

4 ^ KR-*(X X i

where (^) x is the product toith (—IJ)P, and 17 e Z-R-^point) ^ KR(P(R*)) is the reduced real Hopf bundle. Proof. We replace TT by the equivalent inclusion 5p>0^ ->• B*^1 -^0. The relative group is then KR"^(X). To compute x w© use the commutative diagram

Let 6 be the automorphism of Kip'p+<l{X) obtained by interchanging the two factors RP^0 which occur. Then the composition x ^ is just multi- plication by the image of bv^ in Z-R"-P(point) ->- Zit°-P(point). But this is just -q". It remains then to calculate 6. But the usual proof given in (2) [§ 2.4] shows that 6 = (-1)"' = (-1)". We proceed to consider in more detail each of the theories in (3.1). For p = 1, iS"-^0 is just a pair of conjugate points {+1, —1}. A real vector bundle E over I x { + 1 , - 1 ) is entirely determined by the complex vector bundle E+ which is its restriction to X X {+1}. Thus we have

PROPOSITION 3.3. There is a natural isomorphism KR(X X S^1 -^0 ) a* K(X).

ON Z-THEORY AND REALITY 377

Note in particular that this does not depend on the real structure of X T)ut just on the underlying space. The period 2 given by (3.1) confirms what we know about K(X). The exact sequence of (3.2) becomes now

... ->- KR^(X) 4- KR-<{X) £ K-«(X) X KR*^(X)^ ... (3.4)

where x is multiplication by — 77 and 77* = c is complexification. We leave the identification of 8 as an exercise for the reader. This exact sequence is well-known (when the involution on X is trivial) but it is always deduced from the periodicity theorem for the orthogonal group. Our procedure has been different and we could in fact use (3.4) to prove the orthogonal periodicity. Instead we shall deduce this more easily later from the case p = 4 of (3.1). Next we consider p = 2 in (3.1). Then KR~*(X X S^2 -^0 ) has period 4. We propose to identify this with a self-conjugate theory. If X is a real space with involution T a self-conjugate bundle over X will mean a complex vector bundle E together with an isomorphism a: E -v r*E. Consider now the space X x <S*-° and decompose S^° into two halves S\° and S^2_?_ with intersection {±1}.

si-^0

si,o

It is clear that to give a real vector bundle F over X X S^2 -^0 is equivalent to giving a complex vector bundle F+ over X X S^^0 (the restriction of F) together with an isomorphism

But X X { + 1} is a deformation retract of X X S*+° and so [cf. (3) 2.3] we have an isomorphism 8:F+\Xx{-l} -+ F+\Xx{+l} unique up to homotopy. Thus to give is equivalent, up t o homotopy, to giving an isomorphism. ™ -j-w

where E is the bundle over X induced from F+ by x -*- (x, 1) and a (^) i = 0(2,-1) <Ax,l> In other words isomorphism classes of reed bundles over X x Stfi^ corre- spond bijectively to homotopy classes of self-conjugate bundles over X. Moreover this correspondence is clearly compatible with tensor products.

ON X-THEORY AND REALITY 37»

is therefore postponed until § 4 where we shall be discussing Clifford algebras in more detail. Using (3.9) we are now ready to establish THEOEEM 3.10. Let A e 4 8 , a(A) e iLR-^point) be as above. Then multiplication by a(X) induces an isomorphism KR(X) -+ KRS(X) Proof. Multiplying the exact sequence of (3.8) by a(A) we get a commu- tative diagram of exact sequences 0 -». KR^(X) -> KR-*{X x S*-^0 ) -* KR^(X) -> 0 _\ \ _ 0 -»• KR--{X) -> KR^-B(X x S^1 -^0 ) -*• KR~^3 ^(X) -• 0. By (3.9) we know that tpg coincides with the periodicity isomorphism y 4. Hence q is a monomorphism for all q. Hence ^ ^ in the above diagram ia a monomorphism, and this, together with the fact that if/Q is an iso- morphism, implies that g is an epimorphism. Thus g is an isomorphism as required. Remark. If the involution on X is trivial, BO that KR(X) = K0{X), this is the usual 'real periodicity theorem'. By considering the various inclusions Sqfi^ -*• Sp-° we obtain interesting exact sequences. For the identification of the relative group we need

LEMMA 3.11. The real space (with base point) SvfijS'^1 '^0 is isomorphic to

Proof. SP'°—5«'^0 is isomorphic to 5p-«'oxii«-^0. Now compactify. COBOLLARY 3.12. We have natural isomorphisms: KR(X x £p-°, X x <S«-°) ^ KR°*(X x Sp-<>-^0 ). In view of (3.8) the only interesting cases are for low values of p, q. Of particular interest is the case p = 2, q = 1. This gives the exact sequence [cf. (1)] ... -• K~\X) - • KSC(X) -v K{X) -> K(X) -> .... The exact sequence of (3.8) does in fact split canonically, so that (for p > 3) KR-«(X x S»-°) ^ KR-*(X) ®KRP+1-«(X). (3.13) To prove this it is sufficient to consider the case p = 3, because the general case then follows from the commutative diagram (p ^ 4) 0 -• KR(X) -». KR(X x i I 0 -y KR(X) -• KR(X x

380 M. F. ATIYAH

obtained by restriction. Now S^3 -^0 is the 2-sphere with the anti-podal t involution and this may be regarded as the conic J z\ = 0 in P(C^8 ). o In § 5 we shall give, without proof, a general proposition which will imply that, when T is a quadric, KR(X)-+KR(XxY) has a canonical left inverse. This will establish (3.13).

4. Relation with Clifford algebras Let Cliff( •#"•«) denote the Clifford algebra (over R) of the quadratic form

on Rp*. The involution (y,x) h-v (— y,x) of RP* induces an involutory automorphism of Cliff(i?^) denoted byf a ->d. Let M = ©Jtf^1 be a complex Z 2 -graded Cliff(.RP-«)-module. We shall say that M is a real Z 2 -graded Cliff(.R^-module if M has a real structure (i.e. an anti-linear involution m -*-m) such that (i) the Z 2 -grading is compatible with the real structure, i.e. M< = Mi^ (i = 0,1), (ii) am = dm for a e Cliff(.R^^0 ) and m e M. Note that if p = 0, so that the involution on Cliff(J?p'<?) is trivial, then MR = M°R QMR = {m e M\rh = m} is a real Z 2 -graded module for the Clifford algebra in the usual sense [a C 8 -module in the notation of (4)]. The basic construction of (4) carries over to this new situation. Thus a real graded Cliff(i2P-o)-module M = M°@M^1 defines a triple ( M°, M^1 , a) where o: S"-" X M ° -v Sp-Q^ X M^1 is a real isomorphism given by a(a,m) = (8, am). In this way we obtain a homomorphism h: M(p,q) -• Z-RP-^point) where M(p,q) is the Grothendieck group of real graded Cliff(.Rp'<?)- modules. If M is the restriction of a Cliff(i^-fl+1)-modu]e then a extends over SP'Q+1. Since the projection

t This notation diverges from that of (4) [§ 1] where (for q = 0) this involution is called a and 'bar' is reserved for an anti-automorphism.

382 M. F. ATIYAH

After these preliminaries we can now proceed to the proof of Lemma 3.9. What we have to show is that under the map

, x 4 :

the element of KR^^S*-^0 ) defined by M lifts to the element of KR-^S'-^0 ) denned by N. To do this it is clearly sufficient to exhibit a commutative diagram of real isomorphisms

S<°xRBxN

SWxWxN^SWx&xM (4.1)

where v is compatible with fit (i.e. v(a, x,y,n) = (a, x --iay, m) for some m) and the vertical arrows are given by the module structures (i.e. (a, x, y, n) H>- (a, x, y, (x, y)n). Consider now the algebra Cnff(ii^4 '^0 ) = C 4. The even part CJ is isomorphic to H © H [(4) Table 1]. Moreover its centre is generated by 1 and w = ejejege^, the two projections being A(l±w)- To be quite specific let us define the embedding £ : H -

2

_ 1+w

- — 2~ ei«4- Then we can define an embedding T?: S(U) -> Spin(4) c T 4 by ?() £()+$•(!—">)> where F 4 is the Clifford group [(4) 3.1] and S(H) denotes the quaternions of norm 1. It can now be verified that the composite homomorphism S(H) -»• Spin(4) -> <SO(4) defines the natural action of <S(H) on R = H given by left multiplica- tion.f I n other words (^1) ?(*)!/ (^7) ?( (^5) )~ (^1) = y (« G <S(H), J/ G R). (4.2) If we give <S(H) the anti-podal involution then 17 is not compatible with involutions, since the involution on the even part _C_ is trivial. f We identify 1, t, j , k with the standard base Bj, «,, «,, et in that order.

ON X - T H E O R Y AND R E A L I T Y 383 Regarding CUff(iJ*-°) as embedded in Clifff-R^4 -^4 ) in t h e natural way we now define the required m a p v b y

v{8,x,y,n) = (s,x+isy,Tj(s)n). From the definition of w it follows that

and so rj(— s)fi = ^(— a){—wn) = 77(5)71 = rj(s)n, showing that v is a real map. Equation (4.2) implies that 7](8)(x,y)n = (x+isy)r)(8)n, showing that v is compatible with the module structures. Thus we have established the existence of the diagram (4.1) and this completes the proof of Lemma 3.9. The definitions of M(p, q) and A(p, q) given were the natural ones from our present point of view. However, it may be worth pointing out what they correspond to in more concrete or classical terms. To see this we observe that if M is a real (7(.Rp'^8 )-module we can define a new action [ ] of It?™ on M by (^) r n. • J (^) [x,y}m = xm+iym.

Then [x,yfm = {-|N!+||«/||s}m. Moreover for the involutions we have [x, y]m = xm--iym = xm+iym (since y = —y) = [z,y]m. Thus MR is now a real module in the usual sense for the Clifford algebra CPA of the quadratic form

It is easy to see that we can reverse the process. Thus M(p, q) can equally well be defined as the Qrothendieck group of real graded Cpa-modules. From this it is not difficult to compute the groups AiPfg) on the lines of (4) [§ 4,5] and to see that they depend only on p—q (mod 8) [cf. also (8)]. Using the result of (4) [11.4] one can then deduce that a: A(p,q) -»- -/LRP-«(point) is always an isomorphism. The details are left to the reader. We should perhaps point out at this stage that our double index notation was suggested by the work of Karoubi (8).

ON i f - T H E O R Y AND R E A L I T Y 385

defines in the usual way an element

[cr(P)]eKR(B(X),S(X)) '

where B(X), the unit ball bundle of S(X), has the associated real

structure, f

The kernel and cokernel of a real elliptic operator have natural real

structures. Thus the index is naturally an element of ^iJ(point). Of

course since r D.... p ,. ,.

iT.R(point) -> Z(point)

is an isomorphism there is no immediate advantage in denning this

apparently refined real index. However, the situation alters if we con-

sider instead a family of real elliptic operators with parameter or base

space 7. In this case a real index can be defined as an element of KR(Y)

and KR{Y) -+ K(T)

is not in general injective.

All these matters admit a natural extension to real elliptic complexes

(9). Of particular interest is the Dolbeault complex on a real algebraic

manifold. This is a real elliptic complex because the holomorphic map

T : X ->• X maps the Dolbeault complex of £ into the Dolbeault complex

of X. If X is such that the sheaf cohomology groups HQ(X, 0) = 0 for

q > 1, H°(X, 0) ^ C, the index, or Euler characteristic, of the Dolbeault

complex is 1. Based on this fact one can prove the following result:

PROPOSITION. Let f: X -+- Y be a fibering by real algebraic manifolds, where the fibre F is such that H«(F, 0) = 0 (q ^ 1, H°(F, <P) s C), then there is a homomorphism /„ : KR(X) -+ KR(Y) which is a left inverse of

The proof cannot be given here but we observe that a special case is given

by taking X = Y X F where F is a (compact) homogeneous space of a real

algebraic linear group. For example we can take F to be a complex

quadric, as required to prove (3.13). We can also take F = S0{2n)\U{n),

or S0(2n)/Tn, the flag manifold of S0(2n). These spaces can be used to

establish the splitting principle for orthogonal bundles. It is then

significant to observe that the real space

{S0(2n)IU(n)}xR°-in

| All thia extends of course to integral (or pseudo-differential) operators.

386 ON ^ - T H E O R Y AND R E A L I T Y

has the structure of a real vector bundle. A point of S0(2n)IU(n)

defines a complex structure of i?** and conjugate points give conjugate

structures. For n = 2 this is essentially^ what we used in § 3 to deduce

the orthogonal periodicity from Theorem 2.1.

t In (3.1) we used the 3-sphere S*-°. We could just as well have used the 2- sphere S^3 -^0_._ This coincides with SO(4)/U(2).

R E F E R E N C E S

  1. D. W. Anderson (Thesis: not yet published). 2. M. F. Atiyah, Lectures on JT-theory (mimeographed notes, Harvard 1965).
  2. M. F. Atiyah and R. Bott, 'On the periodicity theorem for complex vector bundles', Ada Math. 112 (1964) 229-47.
  3. M. F. Atiyah, R. Bott, and A. Shapiro, 'Clifford modules', Topology 3 (1964) 3-38.
  4. M. F. Atiyah and F. Hirzebruch, 'Vector bundles and homogeneous spaces', Proc. Symposium in Pure Math. Vol. 3, American Mathematical Society (1961).
  5. M. F. Atiyah and Q. B. Segal, 'EquivariantX-theory' (Lecture notes, Oxford 1965).
  6. P. S. Green, 'A cohomology theory based upon self-conjugacies of complex vector bundles, Bull. Amer. Math. Soc. 70 (1964) 522.
  7. M. Karoubi (Thesis: not yet published).
  8. R. Palais, "The Atiyah-Singer index theorem \Annale. of Math. Study 57 (1965).

The Mathematical Institute

Oxford University