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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.
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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
Ex-* ET<X)
370 M. F. ATIYAH
— (T3=ir
ON ^-THEORY AND REALITY 371
KR(X) (^) s KO(X).
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
374 M. F. ATIYAH
bundle over the real compact space X. Then
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
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).
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,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
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.
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
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
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 = 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
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
- — 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
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
| 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
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