Twisted K-theory

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C*-algebras. Up to equivalence, the twisting corresponds to an element of H³(X; Z). We give a systematic account of the definition and basic properties of the...

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Дата:	2004
Автори:	Atiyah, M., Segal, G.
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Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2004
Назва видання:	Український математичний вісник
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Цитувати:	Twisted K-theory / M. Atiyah, G. Segal // Український математичний вісник. — 2004. — Т. 1, № 3. — С. 287-330. — Бібліогр.: 29 назв. — англ.

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spelling	irk-123456789-1246212017-10-01T03:02:57Z Twisted K-theory Atiyah, M. Segal, G. Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C*-algebras. Up to equivalence, the twisting corresponds to an element of H³(X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H³G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman. 2004 Article Twisted K-theory / M. Atiyah, G. Segal // Український математичний вісник. — 2004. — Т. 1, № 3. — С. 287-330. — Бібліогр.: 29 назв. — англ. 1810-3200 2000 MSC. 55-xx, 55N15, 55N91, 19Kxx. http://dspace.nbuv.gov.ua/handle/123456789/124621 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C*-algebras. Up to equivalence, the twisting corresponds to an element of H³(X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H³G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.
format	Article
author	Atiyah, M. Segal, G.
spellingShingle	Atiyah, M. Segal, G. Twisted K-theory Український математичний вісник
author_facet	Atiyah, M. Segal, G.
author_sort	Atiyah, M.
title	Twisted K-theory
title_short	Twisted K-theory
title_full	Twisted K-theory
title_fullStr	Twisted K-theory
title_full_unstemmed	Twisted K-theory
title_sort	twisted k-theory
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2004
url	http://dspace.nbuv.gov.ua/handle/123456789/124621
citation_txt	Twisted K-theory / M. Atiyah, G. Segal // Український математичний вісник. — 2004. — Т. 1, № 3. — С. 287-330. — Бібліогр.: 29 назв. — англ.
series	Український математичний вісник
work_keys_str_mv	AT atiyahm twistedktheory AT segalg twistedktheory
first_indexed	2025-07-09T01:44:17Z
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fulltext	Український математичний вiсник Том 1 (2004), № 3, 287 – 330 Twisted K-theory Michael Atiyah and Graeme Segal Abstract. Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C∗-algebras. Up to equivalence, the twisting corre- sponds to an element of H3(X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical cohomology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H3 G(X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman. 2000 MSC. 55-xx, 55N15, 55N91, 19Kxx. Key words and phrases. Algebraic topology, K-theory, equivariant cohomology, K-theory and operator algebras. 1. Introduction In classical cohomology theory the best known place where one en- counters twisted coefficients is the Poincaré duality theorem, which, for a compact oriented n-dimensional manifold X, relates to the pairing be- tween cohomology classes in complementary dimensions given by multi- plication followed by integration over X: Hp(X; Z) ×Hn−p(X : Z) → Hn(X; Z) → Z. If X is not orientable there is a local coefficient system ω on X whose fibre ωx at each point x is non-canonically Z, and the duality pairing is Hp(X; Z) ×Hn−p(X;ω) → Hn(X;ω) → Z. The difference between elements of Hn(X; Z) and of Hn(X;ω) is the difference between n-forms and densities. Received 18.03.2004 ISSN 1810 – 3200. c© Iнститут прикладної математики i механiки НАН України 288 Twisted K-theory In K-theory the Poincaré pairing involves twisting even when X is oriented. Let us, for simplicity, take X even dimensional and Rieman- nian. Then the analogue of the local system ω is the bundle C of finite dimensional algebras on X whose fibre Cx at x is the complex Clifford algebra of the cotangent space T ∗ x at x. Alongside the usual K-group K0(X) formed from the complex vector bundles on X there is the group K0 C(X) formed from C-modules, i.e. finite dimensional complex vector bundles E on X such that each fibre Ex has an action of the algebra Cx. On the sections of such a C-module E there is a Dirac operator DE = ΣγiDi (defined by choosing a connection in E; hereDi is covariant differentiation in the ith coordinate direction, and γi is Clifford multiplication by the dual covector). In fact the module E automatically has a decomposition E = E+ ⊕E−, and DE maps the space of sections Γ(E+) to Γ(E−), and vice versa. Each component D± E : Γ(E±) → Γ(E∓) is Fredholm, and associating to E the index of D+ E defines a homomor- phism K0 C(X) → Z which is the K-theory analogue of the integration map Hn(X;ω) → Z. Tensoring C-modules with ordinary vector bundles then defines the Poin- caré pairing K0(X) ×K0 C(X) → K0 C(X) → Z. (1.1) We can define a twisted K-group K0 A for any bundle A of finite di- mensional algebras on X. The interesting case is when each fibre Ax is a full complex matrix algebra: equivalence classes of such bundles A correspond, as we shall see, to the torsion elements in H3(X; Z). The class of the bundle C of Clifford algebras of an even-dimensional ori- entable real vector bundle E is the integral third Stiefel-Whitney class W3(E) ∈ H3(X; Z), the image of w2(E) ∈ H2(X; Z/2) by the Bockstein homomorphism. In this paper we shall consider a somewhat more general class of twistings parametrized by elements of H3(X; Z) which need not be of finite order. From one viewpoint the new twistings correspond to bundles of infinite dimensional algebras on X. In fact the bundle C of Clifford algebras on a manifold X is a mod 2 graded algebra, and the definition of K0 C should take the grading into Michael Atiyah and Graeme Segal 289 account. When this is done the pairing (1.1) expresses Poincaré duality even when X is not orientable. The existence of the twisted K-groups has been well-known to experts since the early days of K-theory (cf. Donovan-Karoubi [11], Rosenberg [22]), but, having until recently no apparent role in geometry, they at- tracted little attention. The rise of string theory has changed this. In string theory space-time is modelled by a new kind of mathematical struc- ture whose ”classical limit” is not just a Riemannian manifold, but rather one equipped with a so-called B-field [26]. A B-field β on a manifold X is precisely what is needed to define a twisted K-group K0 β(X), and the elements of this group represent geometric features of the stringy space- time. If the field β is realized by a bundle A of algebras on X then K0 β(X) is the K-theory of the non-commutative algebra of sections of A, and it is reasonable to think of the stringy space-time as the ”non-commutative space” — in the sense of Connes [8] — defined by this algebra. Many papers have appeared recently discussing twisted K-theory in relation to string theory, the most comprehensive probably being that of the Ade- laide school [6]. We refer to [19] for a physicist’s approach. A purely mathematical reason for being interested in twisted K- theory is the beautiful theorem proved recently by Freed, Hopkins, and Teleman which expresses the Verlinde ring of projective representations of the loop group LG of a compact Lie group G — a ring under the subtle operation of ”fusion” — as a twisted equivariant K-group of the space G. Here the twisting corresponds to the ”level”, or projective cocycle, of the representations being considered. In this paper we shall set out the basic facts about twisted K-theory simply but carefully. There are at least two ways of defining the groups, one in terms of families of Fredholm operators, and the other as the algebraic K-theory of a non-commutative algebra. We shall adopt the former, but shall sketch the latter too. The equivariant version of the theory is of considerable interest, but it has seemed clearest to present the non-equivariant theory first, using arguments designed to generalize, and only afterwards to explain the special features of the equivariant case. The plan of the paper is as follows. Section 2 discusses the main properties of bundles of infinite dimen- sional projective spaces, which are the ”local systems” which we shall use to define twisted K-theory. Section 3 gives the definition of the twisted K-theory of a space X equipped with a bundle P of projective spaces, first as the group of homo- topy classes of sections of a bundle on X whose fibre at x is the space of 290 Twisted K-theory Fredholm operators in the fibre Px of P , then as the algebraic K-theory of a C∗-algebra associated to X and P . The twistings by bundles of projective spaces are not the most general ones suggested by algebraic topology, and at the end of this section we mention the general case. Section 4 outlines the algebraic-topological properties of twisted K- theory. The relation of the twisted theory to classical cohomology will be discussed in a sequel to this paper. Section 5 describes some interesting examples of projective bundles and families of Fredholm operators in them, related to the ones occurring in the work of Freed, Hopkins, and Teleman [12]. In fact these are natu- rally equivariant examples. They have also been discussed by Mickelsson [18] (cf. also [7]). Section 6 turns to the equivariant theory, explaining the parts which are not just routine extensions of the non-equivariant discussion. Apart from that there are three technical appendices concerned with points of functional analysis with which we did not want to hold up the main text. The third is an equivariant version of Kuiper’s proof of the contractibility of the general linear group of Hilbert space with the norm topology. In a subsequent paper we shall discuss the relation of twistedK-theory to cohomology. We shall examine the effect of twisting on the Atiyah- Hirzebruch spectral sequence, on the Chern classes, and on the Chern character. We shall also see how twisting interacts with the operations in K-theory, such as the exterior powers and the Adams operations. 2. Bundles of projective spaces The ”local systems” which we shall use to define twisted K-theory are bundles of infinite dimensional complex projective spaces. This section treats their basic properties. We shall consider locally trivial bundles P → X whose fibres Px are of the form P(H), the projective space of a separable complex Hilbert space H which will usually, but not invariably, be infinite dimensional (we shall at least require that it has dimension ≥ 1, so that P(H) is non-empty). We shall assume that our base-spaces X are metrizable, though this could easily be avoided by working in the category of compactly generated spaces. The projective-Hilbert structure of the fibres is supposed to be given. This means that P is a fibre bundle whose structural group is the projective unitary group PU(H) with the compact-open topology.* The An account of the compact-open topology can be found in Appendix 1. Michael Atiyah and Graeme Segal 291 significance of this topology is that a map X → PU(H) is the same thing as a bundle isomorphism X × P(H) → X × P(H). In fact, essentially by the Banach-Steinhaus theorem, the same is true if PU(H) has the slightly coarser topology of pointwise convergence, which is called the ”strong operator topology” by functional analysts. Let us stress that we do not always want to assume that the structural group of our bundles is PU(H) with the norm topology, i.e. that there is a preferred class of local trivializations between which the transition functions are norm-continuous, for doing so would exclude most naturally arising bundles. For example, if Y → X is a smooth fibre bundle with compact fibres Yx then the Hilbert space bundle E on X whose fibre Ex is the space of L2 half-densities on Yx does not admit U(H) with the norm topology as structure group, for the same reason that if H = L2(G) is the regular representation of a group G the action map G → U(H) is not norm-continuous, even if G is compact. Nevertheless, it follows from Proposition 2.1(ii) below that for many purposes we lose nothing by working with norm-continuous projective bundles, and it is simpler to do so. When we have a bundle P → X of projective spaces we can construct another bundle End(P ) onX whose fibre at x is the vector space End(Hx) of endomorphisms of a Hilbert space Hx such that Px = P(Hx). For, although Hx is not determined canonically by the projective space Px, if we make another choice H̃x with P(H̃x) = P(Hx) then End(H̃x) is canonically isomorphic to End(Hx), and it makes sense to define End(Px) = End(Hx) = End(H̃x). This observation will play a basic role for us, and we shall use several variants of it, replacing End(Hx) by, for example, the subspaces of com- pact, Fredholm, Hilbert-Schmidt, or unitary operators in End(Hx). We must beware, however, that if the structural group of P does not have the norm topology we must use the compact-open topology on the fibres of End(P ), Fred(P ), or U(P ). In the case of the compact or Hilbert-Schmidt operators there is no problem of this kind, for, as is proved in Appendix 1, the group U(H) with the compact-open topology acts continuously on the Banach space K(H) of compact operators and the Hilbert space H∗ ⊗H of Hilbert-Schmidt operators. Each bundle P → X of projective spaces has a class ηP ∈ H3(X; Z) defined as follows. Locally P arises from a bundle of Hilbert spaces 292 Twisted K-theory on X, so we can choose an open covering {Xα} of X and isomorphisms P \|Xα ∼= P(Eα), where Eα is a Hilbert space bundle on Xα. If the covering {Xα} is chosen sufficiently fine the transition functions between these ”charts” can be realized by isomorphims gαβ : Eα\|Xαβ → Eβ\|Xαβ , where Xαβ = Xα∩Xβ , which are projectively coherent, so that over each triple intersection Xαβγ = Xα ∩Xβ ∩Xγ the composite gγαgβγgαβ is multiplication by a circle-valued function fαβγ : Xαβγ → T. These functions {fαβγ} constitute a cocycle defining an element η̃P of the Čech cohomology group H2(X; sh(T)), where sh(T) denotes the sheaf of con- tinuous T-valued functions on X. Using the exact sequence 0 → sh(Z) → sh(R) → sh(T) → 0 we can define ηP as the image of η̃P under the coboundary homomorphism H2(X; sh(T)) → H3(X; sh(Z)) = H3(X; Z) (which is an isomorphism because H i(X; sh(R)) = 0 for i > 0 by the existence of partitions of unity). Before stating the main result of this section let us notice that bun- dles of projective Hilbert spaces can be tensored: the fibre (P1 ⊗ P2)x is the Hilbert space tensor product P1,x ⊗ P2,x, i.e. the projective space of the Hilbert space of Hilbert-Schmidt operators E∗ 1,x → E2,x, where E∗ i,x is the dual space of Ei,x, and P(Ei,x) ∼= Pi,x. Furthermore, for any bundle P there is a dual projective bundle P ∗ whose points are the closed hyperplanes in P , and P ∗ ⊗ P comes from a vector bundle. In fact P ∗⊗P = P(E), where E is the bundle of Hilbert-Schmidt endomor- phisms of P . (This is a first application of the observation above that the vector space End(H) is functorially associated to the projective space P(H), even though H itself is not.) Proposition 2.1. (i) We have ηP = 0 if and only if the bundle P of projective spaces comes from a vector bundle E on X. This is a slight oversimplification. Most spaces of interest posses arbitrarily fine open coverings {Xα} such that the intersections Xαβ are contractible, and then the maps gαβ can be lifted to vector bundle isomorphisms, e.g. by fixing the phase of some matrix element (which is continuous in the compact-open topology). But in general we must use the standard technology of sheaf theory, which takes a limit over coverings rather than using a particular covering. Michael Atiyah and Graeme Segal 293 (ii) Each element of H3(X; Z) arises from a bundle P , even from one whose structure group is PU(H) with the norm topology. (iii) If the fibres of P are infinite dimensional and separable then P is determined up to isomorphism by ηP . (iv) If P has finite dimensional fibres P(Cn) then nηP = 0. (v) Every torsion element of H3(X; Z) arises from a finite dimensional bundle P , though a class of order n need not arise from a bundle with fibre P(Cn). (vi) If P0 → P is a tame embedding of projective bundles, in the sense explained below, then ηP0 = ηP . In particular, if P has a continuous section then ηP = 0, and if P is a fixed projective space then ηP = ηP⊗P. (vii) We have ηP1⊗P2 = ηP1 + ηP2. (viii) We have ηP ∗ = −ηP . In (vi) above, a tame embedding means one which is locally isomor- phic (on X) to the inclusion of X × P(H0) in X × P(H), where H0 is a closed subspace of H. A typical example of a non-tame embedding is the following. Let H be the standard Hilbert space L2(0, 1). Then in the trivial bundle X ×H on the closed interval X = [0, 1 2 ] the subbundle whose fibre at x is L2(x, 1) is not tame. Proposition 2.1, whose proof is given below, tells us that the group of isomorphism classes of projective bundles (with infinite dimensional separable fibres) under the tensor product is precisely H3(X; Z). We also need to know about the automorphism groups of these bundles. An automorphism α : P → P defines a complex line bundle Lα on X: the non-zero elements of the fibre of Lα at x are the linear isomorphisms Ex → Ex which induce α\|Px, where Px = P(Ex). (We have already pointed out that the choice of Ex is irrelevant.) Proposition 2.2. For a projective bundle P with infinite dimensional separable fibres the assignment α 7→ Lα identifies the group of connected components of the automorphism group of P with the group H2(X; Z) of isomorphism classes of complex line bundles on X. The proof will be given presently. Proof of Proposition 2.1. (i) This is immediate because the vanishing of the Čech cohomology class η̃P ∈ H2(X; sh(T)) defined by transition 294 Twisted K-theory functions {gαβ} is precisely the condition that the gαβ can be multiplied by functions λαβ : Xαβ → T to make them exactly coherent. (ii) Because the unitary group U(H) of an infinite dimensional Hilbert space is contractible — with either the norm topology, or the compact- open topology (see Appendix 2) — the projective group PU(H) has the homotopy type of an Eilenberg-Maclane space K(Z, 2), and its classifying space BPU(H) is accordingly a K(Z, 3). Thus any element of H3(X; Z) corresponds to a map f : X → BPU(H), and hence to the bundle of projective spaces pulled back by f from the universal bundle on BPU(H). (iii) Any bundle can be pulled back from the universal bundle, and homotopic maps pull back isomorphic bundles. (iv) The commutative diagram of exact sequences µn −→ SUn −→ PUn ↓ ↓ ↓ T −→ Un −→ PUn, where µn is the group of nth roots of unity, and the right-hand vertical map is the identity, shows that the invariant η̃P ∈ H2(X; sh(T)), when P has structural group PUn, comes from H2(X; sh(µn)), and hence has order dividing n. (v) (The following argument is due to Serre, see [14].) If l divides m — say m = lr — we have an inclusion PUl → PUm given by tensoring with Cr. By Bott periodicity the homotopy groups πi(BPUl) for i < 2l − 1 are given by π2(BPUl) = Z/l πi(BPUl) = Z for i even and > 1 πi(BPUl) = 0 for i odd. The inclusion PUl → PUm induces multiplication by r = m/l on all homotopy groups, so we have π2(BPU∞) = Q/Z πi(BPU∞) = Q for i even and > 1 πi(BPU∞) = 0 for i odd. Thus BPU∞ can be constructed from the Eilenberg-Maclane space K(Q/Z, 2) by successively forming fibrations over it with fibres K(Q, 2j). A fibration with fibre K(Q, 2j) on a base-space Y is determined by an element of H2j+1(Y ; Q). Now K(Q/Z, 2) has the rational cohomology of Michael Atiyah and Graeme Segal 295 a point, while the other Eilenberg-Maclane spaces involved have rational cohomology only in even dimensions. So BPU∞ ≃ K(Q/Z, 2) ×K(Q, 4) ×K(Q, 6) × . . . . This means that every element η′ of H2(X; Q/Z) can be realized by a BPU∞-bundle P whose invariant ηP is the image of η′ in H3(X; Z). But from the Bockstein sequence for 0 → Z → Q → Q/Z → 0 the torsion elements of H3(X; Z) are precisely the image of H2(X; Q/Z). There is, however, no reason to expect that when nη = 0 we can represent the class of η by a bundle with fibre P(Cn). We have seen, for example, that the class of the bundle Cliff(E) of Clifford algebras of a 2k-dimensional real vector bundle E — or, equivalently, of the projective bundle of spinors of E — is W3(E) ∈ H3(X; Z), which is of order 2, while the projective bundle of spinors has dimension 2k − 1, and its class need not be represented by a bundle of lower dimension. To have a concrete counterexample we can reason as follows. The invariant of a bundle with fibre P(C2) is given by a map BPU2 → K(Z/2, 2) → K(Z, 3). If every invariant of order 2 came from a PU2-bundle then the map K(Z/2, 2) → K(Z, 3) would factorize K(Z/2, 2) → BPU2 → K(Z, 3), and taking loops would give K(Z/2, 1) → PU2 → K(Z, 2), which is impossible because the Bockstein map K(Z/2, 1) → K(Z, 2) (i.e. RP∞ →֒ CP∞) clearly does not factorize through a finite dimensional space. (vi) This follows by the argument of case (iv) from the diagram T −→ U(H0) −→ PU(H0) ↑∼= ↑ ↑ T −→ U(H,H0) −→ PU(H,H0) ↓∼= ↓ ↓ T −→ U(H) −→ PU(H) where U(H,H0) = {u ∈ U(H) : u(H0) = H0}. 296 Twisted K-theory (vii) Here we consider T × T −→ U(H1) × U(H2) −→ PU(H1) × PU(H) ↓ ↓ ↓ T −→ U(H1 ⊗H2) −→ PU(H1 ⊗H2) where the left-hand vertical map is composition in T. (viii) This follows from (vii). Proof of Proposition 2.2. An automorphism of P is a section of a bundle on X whose fibre is PU(H). This bundle, however, comes from one with fibre U(H), and so it is trivial. The group of automorphisms can therefore be identified with the maps from X to PU(H), which is an Eilenberg-Maclane space K(Z, 2). Remark 2.1. In fact the natural objects that can be used to twist K- theory are not simply bundles P → X of projective spaces, but rather are bundles of projective spaces in which a unitary involution is given in each fibre Px. An involution in a projective space P expresses it as the join of two disjoint closed projective subspaces P+ and P− which, despite the notation, are not ordered. We shall always assume that P+ x and P− x fit together locally to form tame subbundles of P . Thus the involution defines a double covering of X, and hence a class ξP ∈ H1(X; Z/2). Let Proj±(X) denote the group of isomorphism classes of infinite di- mensional projective Hilbert space bundles with involution on X, under the operation of graded tensor product. Proposition 2.3. As sets we have Proj(±)(X) ∼= H1(X; Z/2) ×H3(X; Z) canonically, but the tensor product of bundles induces the product (ξ1, η1).(ξ2, η2) = (ξ1 + ξ2, η1 + η2 + β(ξ1ξ2)) on the cohomology classes, where ξ1ξ2 ∈ H2(X; Z/2) is the cup-product, and β : H2(X; Z/2) → H3(X; Z) is the Bockstein homomorphism. Proof. In other words, we have an exact sequence 0 → H3(X; Z) → Proj(±)(X) → H1(X; Z/2) → 0 Michael Atiyah and Graeme Segal 297 which is split (because every element of the middle group has order 2), but not canonically split. The Bockstein cocycle describing the extension expresses the extent to which the forgetful functor from projective spaces with involution to projective spaces does not respect the tensor product. The proof of Proposition 2.3 is very simple. We can think of elements of H1(X; Z/2) as real line bundles on X, and can define a map H1(X; Z/2) → Proj(±)(X) by taking a line bundle L to P(SL⊗H), where SL is an irreducible graded module for the bundle of Clifford algebras C(L), and H is a fixed Hilbert space. Now P(SL1 ⊗H) ⊗ P(SL2 ⊗H) ∼= P(SL1 ⊗ SL2 ⊗H⊗H) ∼= P(SL1⊕L2 ⊗H), where everything is understood in the graded sense. But W3(L1 ⊕ L2) = β(w2(L1 ⊕ L2)) = β(w1(L1)w1(L2)), which is the assertion of Proposition 2.3. For simplicity, in the rest of this paper we shall not pursue this gener- alization, but for the most part will keep to the twistings corresponding to elements of H3. The other extreme, when the twisting is given by an element of H1(X; Z/2) alone, is a special case of the version of K-theory developed by Atiyah and Hopkins [2]. 3. The definition It is well known (see [1] Appendix) that the space Fred(H) of Fred- holm operators in an infinite dimensional Hilbert space H, with the norm topology, is a representing space for K-theory, i.e. that K0(X) ∼= [X; Fred(H)] for any space X, where [ ; ] denotes the set of homotopy classes of continuous maps. The basic observation for twisting K-theory is that when P is a bundle on X with fibre P(H) there is an associated bundle Fred(P ) with fibre Fred(H), and we can define K0 P (X) as the set of homotopy classes of sections of Fred(P ). If the bundle P admits the projective unitary group PU(H)norm with the norm topology as its structure group this is straightforward, as 298 Twisted K-theory PU(H)norm acts on Fred(H) by conjugation. But, as we have explained, we want to avoid that assumption. (To be quite clear, for any given projective bundle P we could, by Proposition 2.1(ii), choose a reduction of the structure group to PU(H)norm, but we could not then expect a natural family of Fredholm operators in P to define a continuous section of Fred(P ).) We can, of course, in any case construct a bundle whose fibre is Fred(H)c.o. with the compact-open topology, but Fred(H)c.o. does not represent K-theory: it is a contractible space (see Appendix 2), and the index is not a continuous function on it. We can deal with this problem in various ways. The simplest is to replace Fred(H) by another representing space for K-theory on which PU(H)c.o. does act continuously. One such space is the restricted Grass- mannian Grres(H) described in Chap. 7 of [21]. In practical applications of the theory, however, K-theory elements are more commonly repre- sented by families of Fredholm operators — often elliptic differential op- erators — than by maps into Grassmannians. We therefore stay with Fredholm operators, and we can do this by defining a modified space of operators, bearing in mind that a continuously varying Fredholm opera- tor usually has a natural continuously varying parametrix. An operator A : H → H is Fredholm if and only if it is invertible modulo compact operators, i.e. if there exists a ”parametrix” B : H → H such that AB−1 and BA − 1 are compact. Let us therefore consider the set Fred′(H) of pairs (A,B) of Fredholm operators related in this way. Ignoring topology for the moment, notice that the projection (A,B) 7→ A makes Fred′(H) a bundle of affine spaces over Fred(H) whose fibres are isomorphic to the vector space K of compact operators. We shall give Fred′(H) the topology induced by the embedding (A,B) 7→ (A,B,AB − 1, BA− 1) in B × B × K × K, where B is the bounded operators in H with the compact-open topology and K is the compact operators with the norm topology. A proof of the following proposition is implicit in [25], where a more general situation is treated. But for clarity we have included a direct proof of Proposition 3.1(i) in Appendix 2, while Proposition 3.1(ii) is proved in Appendix 1. Proposition 3.1. (i) Fred′(H) is a representing space for K-theory. (ii) The group PGL(H) with the compact-open topology acts continu- ously on Fred′(H) by conjugation. Michael Atiyah and Graeme Segal 299 If P → X is an infinite dimensional bundle of projective spaces Propo- sition 3.1 allows us to define the associated bundle Fred′(P ), and we can define K0 P (X) as the group of homotopy classes of its actions. To deal with the multiplicativity properties ofK-theory, however, it is convenient, following [3], to introduce the mod 2 graded space Ĥ = H⊕H = H⊗C2 and to replace Fred′(H) by Fred′′(Ĥ), the bundle whose fibres are the pairs (Â, B̂) of self-adjoint degree 1 operators in Ĥ such that ÂB̂ and B̂Â differ from the identity by compact operators. The space Fred′′(Ĥ) is, of course, homeomorphic to Fred′(H), but it allows us to use a slightly larger class of twistings. For if H = H+⊕H− has a mod 2 grading we can give Ĥ = H⊗C2 the usual tensor product grading. As the space Fred′′(Ĥ) of self-adjoint degree 1 operators in Ĥ does not change if the grading of H is reversed, the bundle Fred′′(P̂ ) associated to a projective bundle P with involution is well-defined. It will be technically more convenient, however, to modify the fibre Fred′′(Ĥ) still further, without changing its homotopy type. Let us recall that for any bounded operator A there is a unique positive self-adjoint operator \|A\| such that \|A\|2 = A∗A. If now Â = ( 0 A A∗ 0 ) and B̂ = ( 0 B∗ B 0 ) are self-adjoint degree 1 Fredholm operators which are inverse modulo compact operators then Ã = ( 0 \|B\|A A∗\|B\| 0 ) is another operator of the same type, but with the property that Ã2 differs from the identity by a compact operator. It can be connected to Â in Fred′′(H) by the path {Ãt}t∈[0,1] where Ãt = ( 0 \|B\|tA A∗\|B\|t 0 ) . Definition 3.1. If Ĥ is a mod 2 graded Hilbert space, let Fred(0)(Ĥ) denote the space of self-adjoint degree 1 Fredholm operators Ã in Ĥ such that Ã2 differs from the identity by a compact operator, with the topology coming from its embedding Ã 7→ (Ã, Ã2 − 1) in B × K. Of course Fred(0)(Ĥ) is a representing space for K-theory, and when- ever we have a projective Hilbert bundle P with involution we can define an associated bundle Fred(0)(P ). 300 Twisted K-theory Definition 3.2. For a projective Hilbert bundle P with involution, we write K0 P (X) for the space of homotopy classes of sections of Fred(0)(P̂ ), where P̂ = P ⊗ P(Ĥ), where Ĥ is a fixed standard mod 2 graded Hilbert space such that both Ĥ+ and Ĥ− are infinite dimensional. Addition inK0 P (X) is defined by the operation of fibrewise direct sum, so that the sum of two elements naturally lies in K0 P⊗P(C2)(X), which is canonically isomorphic to K0 P (X) (see below). Of course in Fred′(H) we can define the sum ”internally” simply by composition of operators, but nothing real is gained by that as one needs to pass to H⊕H to see that composition is homotopy-commutative. Remark 3.1. If P admits a norm-topology structure then Definition 3.2 agrees with the ”naive” definition in terms of sections of Fred(P )norm, for the map of bundles Fred(P )norm → Fred(0)(P̂ ) is a fibre-homotopy equivalence (see [10]). Remark 3.2. The group K0 P (X) is functorially associated to the pair (X,P ), and an isomorphism θ : P → P ′ of projective bundles induces an isomorphism θ∗ : K0 P (X) → K0 P ′(X). In particular the group Aut(P ) ∼= H2(X; Z) acts naturally on K0 P (X). The choice of a definite bundle P representing a class inH3(X; Z) is analogous to the choice of a base-point x0 in defining the homotopy group πi(X,x0), when a path γ from x0 to x1 induces γ∗ : πi(X,x0) ∼=→ πi(X,x1), and π1(X,x0) acts on πi(X,x0). If we give only the class of P inH3(X; Z) then the twisted K-group is defined only up to the action of H2(X; Z). Note, however, that to identify K0 P⊗P(C2) with K0 P above we have only to choose an isomorphism between Ĥ ⊗ C2 and Ĥ, and the space of these isomorphisms is contractible. Remark 3.3. The standard proof that Fred(H) is a representing space for K-theory (see Appendix 2 or the appendix to[1]) proceeds by showing that a family of Fredholm operators parametrized by a space X can be deformed to a family for which the kernels and cokernels of the oper- ators have locally constant dimension. These finite dimensional spaces then form vector bundles on X, and their difference is the element of K0(X) corresponding to the family. In the twisted case, however, such a deformation is never possible if the class of the bundle P in H3(X; Z) is not of finite order, for if it were possible then the kernels would define a Michael Atiyah and Graeme Segal 301 finite dimensional sub-projective-bundle P0 of P , and by Proposition 2.1 (iv) and (vi) the class [P ] = [P0] would have finite order. Remark 3.4. Another peculiarity of twisted K-theory when the class [P ] is of infinite order is that the index map K0 P (X) → Z is zero. In other words, any section of Fred(P ) takes values in the index zero com- ponent of the fibre Fred(H). This follows easily from the cohomology spectral sequence of the fibration Fred(P ) on X, a topic which will be examined in our subsequent paper. In particular we shall show that, for the component Fredk(P ) formed by the index k components of the fibres, we have d3(c1) = k[P ] where c1 ∈ H2(Fred(H); Z) is the universal first Chern class. The spec- tral sequence gives rise to an exact sequence H2(Fred(H); Z) d3→ H3(X; Z) π∗ → H3(Fredk(P ); Z), where π is the projection of the fibre bundle. Thus π∗ ◦ d3 = 0, and hence π∗(k[P ]) = 0. If a section of Fredk(P ) exists then π∗ is injective, and hence k[P ] = 0. Since [P ] is assumed not to have finite order this implies that k = 0, as asserted. 3.1. Algebraic K-theory We shall now explain how the twisted K-theory of a compact space can be defined as the algebraic K-theory of a Banach algebra, just as the usual group K0(X) is the algebraic K-theory of the algebra C(X) of continuous complex-valued functions on X. We shall content ourselves with the basic case of twisting by a projective bundle, ignoring bundles with involution. A bundle P of projective spaces on X gives us a bundle End(P ) of algebras, and we might guess that K0 P (X) is the algebraic K-theory of the algebra Γ End(P ) of sections of End(P ). This is wrong, however — even ignoring the problem of topology we encountered in defining Fred(H) — unless P is finite dimensional. If X is an infinite dimensional Hilbert space then H ∼= H⊕H, so End(H) ∼= Hom(H⊕H;H) ∼= End(H) ⊕ End(H) as left-modules over End(H), and so the algebraic K-theory of End(H) is trivially zero. Instead of End(H) we need the Banach algebra K = Endcpt(H) of compact operators in H, with the norm topology, which is 302 Twisted K-theory an algebra without a 1. The K-theory of such a non-unital algebra K is defined by K0(K) = ker : K0(K̂) → K0(C), where K̂ = C ⊕K is the algebra obtained by adjoining a unit to K. The unital algebra K̂ has two obvious finitely generated projective modules: K̂ itself, and also H. In fact (see [15]) K0(K̂) ∼= Z ⊕ Z with these two generators, and K0(K) ∼= Z with generator H. (Notice that C ⊗K̂ H = 0, so H maps to zero in K0(C).) With this in mind, we associate to the projective space bundle P the bundle KP of non-unital algebras whose fibre at x is Endcpt(Px). This makes sense because U(H)c.o. acts continuously on K (see Appendix 1). Definition 3.3. The group K0 P (X) is canonically isomorphic to the al- gebraic K-theory of the Banach algebra Γ(KP ) of sections of KP . Proof. There does not seem to be an obvious map between the two groups, so we shall proceed indirectly, using Bott periodicity ([29],[8], [15]) for the Banach algebra Γ̂ formed by adjoining a unit to Γ = Γ(KP ). For Γ̂, periodicity asserts that K0(Γ̂) ∼= π2(BGL(Γ̂)) ∼= π1(GL(Γ̂)), where GL(Γ̂) = ⋃ GLn(Γ̂) is the infinite general linear group. We readily deduce K0(Γ) ∼= lim n π1(GLn(Γ)), where GLn(Γ) denotes the group of invertible n×n matrices of the form 1 + A, where A has entries in Γ. Now GLn(Γ) is the group of sections of the bundle on X associated to P with fibre GLn(K). Furthermore GL1(K) is isomorphic to GLn(K), and the inclusion GL1(K) → GLn(K) is a homotopy equivalence. Finally, GL1(K) is known [20] to have the homotopy type of the infinite unitary group lim → Un, so that its loop- space is Z ×BU . Putting everything together we find that K0(Γ) is the homotopy classes of sections of the bundle associated to P with fibre Z ×BU , and this is precisely K0 P (X). Michael Atiyah and Graeme Segal 303 Remark 3.5. The fact that elements of K0 P (X) cannot be represented by families of Fredholm operators with kernels and cokernels of locally constant dimension corresponds to the existence of two kinds of projec- tive module for K̂ — ”big” modules like K̂ and ”small” modules like H. Elements of K0(K) can be described using only ”small” modules, but, when we have a twisted family, elements of K0(ΓKP ) cannot. 3.2. More general twistings From the point of view of generalized cohomology theories the twist- ings of K-theory which we consider are not the most general possible. A cohomology theory h∗ is represented by a spectrum hq(X) ∼= [X;hq], where [ , ] denotes homotopy classes of maps, and {hq} is a sequence of spaces with base-point equipped with homotopy equivalences hq → Ωhq+1. (Here Ω denotes the based loop-space.) Any theory possesses a topological group Gh of automorphisms which is well-defined up to homo- topy. (In principle an automorphism is a sequence of maps Tq : hq → hq which commute with the structural maps; but the details of the theory of spectra need great care.) In any event, the homotopy groups of Gh are unproblematic: πi(Gh) is the group of transformations of cohomology theories h∗ → h∗ which lower degree by i. Thus if h∗ is classical cohomol- ogy with integer coefficients Gh is (up to homotopy) the discrete group {±1} of units of Z, for there are no degree-lowering operations. On the other hand, if h∗ is complex K-theory then Gh is much larger. Whenever we have a principal Gh-bundle P on X we can form the associated bundle of spectra, and can define twisted cohomology groups h∗P (X). But for a multiplicative theory h∗ — such as K-theory — it may be natural to restrict to module-like twistings, i.e. those such that h∗P (X) is a module over h∗(X). These correspond to a subgroup Gmod h of Gh of Gh with π0(Gmod h ) = h0(point)× πi(Gmod h ) = h−i(point) for i > 0. It is twistings of this kind with which we are concerned here. We can think of Gmod K as the ”group” Fred±1 of Fredholm operators of index ±1 under tensor product: it fits into an exact sequence Fred1 → Fred±1 → (±1). The group Fred1 is a product Fred1 ≃ P∞ C × SFred1, 304 Twisted K-theory where SFred1 is the fibre of the determinant map Fred1 ∼= BU → BT ∼= P∞ C , and the twistings of this paper are those coming from (±1)×P∞ C . We do not know any equally geometrical approach to the more general ones. 4. Basic properties of twisted K-theory In this section we could without any loss use the norm topology on the spaces of Fredholm operators. One advantage of using the mod 2 graded version Fred(0)(P̂ ) of the bundle of Fredholm operators associated to a projective bundle P is that it gives us at once a multiplication K0 P (X) ×K0 P ′(X) → K0 P⊗P ′(X) (4.1) coming from the map (A,A′) 7→ A⊗ 1 + 1 ⊗A′ defined on the spaces of degree 1 self-adjoint Fredholm operators. (The operator B = A⊗1+1⊗A′ is Fredholm, for B2 is the positive self-adjoint operator A2 ⊗ 1 + 1 ⊗ (A′)2, as A ⊗ 1 and 1 ⊗ A′ anticommute by the usual conventions of graded algebra. If we use the compact-open topology we need to observe that B2 nevertheless varies continuously in the norm topology, so that λf(λB2)B is a parametrix for B for sufficiently large λ, where f : R → R is a smooth function such that f(t) = t−1 for t ≥ 1. We thank J.-L. Tu for pointing out a mistake at this point in an earlier version of this paper.) In particular, each group K0 P (X) is a module over the untwisted group K0(X): this action extends the action of the Picard group Aut(P ) = H2(X; Z), which is a multiplicative subgroup of K0(X). The bilinearity, associativity, and commutativity of the multiplications (4.1) are proved just as for untwisted K-theory. The next task is to define groups Ki P (X) for all i ∈ Z, and to check that they form a cohomology theory on the category of spaces equipped with a projective bundle. The bundle Fred(0)(P̂ ) has a base-point in each fibre, represented by a chosen fibrewise identification P̂+ x ∼= P̂− x . We can therefore form the fibre- wise iterated loop-space Ωn X Fred(0)(P̂ ), whose fibre at x is ΩnFred(0)(P̂x). The homotopy-classes of sections of this bundle will be denoted K−n P (X). Just as in ordinary K-theory these groups are periodic in n with period 2, and we can use this periodicity to define them for all n ∈ Z. We have Michael Atiyah and Graeme Segal 305 only to be careful to use a proof of periodicity which works fibrewise, i.e. we need a homotopy equivalence Fred(0)(H) → Ω2Fred(0)(H) which is equivariant with respect to U(H)c.o.. The easiest choice is the method of [3]. For any n we consider the complexified Clifford algebra Cn of the vector space Rn with its usual inner product. This is a mod 2 graded algebra, for which we choose an irreducible graded module Sn. Then Sn ⊗H is also a graded module for Cn, and we define Fred(n)(H) as the subspace of Fred(0)(Sn ⊗H) consisting operators which commute with the action of Cn, in the graded sense. In [3] there is defined an explicit homotopy equivalence Fred(n)(Sn ⊗H) → ΩnFred(0)(Sn ⊗H) ∼= ΩnFred(0)(H). (4.2) On the other hand, when n is even, say n = 2m, the algebra Cn is simply the full matrix algebra of endomorphisms of the vector space Sn ∼= C2m , and so tensoring with Sn is an isomorphism Fred(0)(H) → Fred(n)(Sn ⊗H). (4.3) The maps (4.2) and (4.3) are completely natural in H, and make sense fibrewise in Fred(0)(P ). To be a cohomology theory on spaces with a projective bundle means that K∗ P must be homotopy-invariant and must possess the Mayer-Vieto- ris property that if X is the union of two subsets X1 and X2 whose interiors cover X, and P is a projective bundle on X, there is an exact sequence . . . d−→ Ki P (X) → Ki P1 (X1)⊕Ki P2 (X2) → Ki P12 (X12) d−→ Ki+1 P (X) → . . . where X12 = X1 ∩ X2, and P1, P2, P12 are the restrictions of P to X1, X2, X12. The proof of this is completely standard, and we shall say no more about it than that the definition of the boundary map d, when i = −1, is as follows. One chooses ϕ : X → [0, 1] such that ϕ\|X1 = 0 and ϕ\|X2 = 1. Then if s is a section of ΩXFred(0)(P ) defined over X12 we define the section ds of Fred(0)(P ) to be the base-point outside X12, and at x ∈ X12 to be the evaluation of the loop s(x) at time ϕ(x). 4.1. The spectral sequence Once we have a cohomology theory we automatically have a spectral sequence defined for any space X with a projective bundle P , relating K∗ P (X) to classical cohomology. More precisely, 306 Twisted K-theory Proposition 4.1. There is a spectral sequence whose abutment is K∗ P (X) with Epq2 = Hp(X;Kq(point)). The coefficients here are twisted by the class ξP of P in H1(X; Z/2). The spectral sequence is constructed exactly as in the untwisted case, e.g. by the method of [23]. We shall discuss this further in the sequel to this paper, where we shall determine the first non-zero differential d3, and shall use the spectral sequence to describe K∗ P (X) ⊗ Q. 5. Examples An important source of projective spaces which do not have canon- ically defined underlying vector spaces is the fermionic Fock space con- struction, due originally to Dirac. If H is a Hilbert space with an or- thonormal basis {en}n∈Z one can consider the Hilbert space F(H) span- ned by an orthonormal basis consisting of the formal symbols en1 ∧ en2 ∧ en3 ∧ . . . where n1 > n2 > n3 > . . . and nk+1 = nk−1 for all large k. We can think of F(H) as a ”renormalized” version of the exterior algebra of H. The important thing for our purposes is that the projective space PF(H) of F(H) depends only weakly on the choice of the orthonormal basis {en}. Because F(H) ∼= Λ(H+) ⊗ Λ(H̄−) it clearly depends only on the decomposition H = H+ ⊕H−, where H+ is spanned by {en}n≥0; but, less obviously, it depends only on the polar- ization of H, i.e. on the class of the decomposition in a sense explained in [21] Chap. 7. The case of interest here is when H = HE is the space of sections of a smooth complex vector bundle E on an oriented circle S. If we choose a parametrization θ : S → R/2πZ and a trivialization E ∼= S × Cm then the class of the splitting for which H+ is spanned by vke inθ for n ≥ 0, where {vk} is the basis of Cm, is independent of both the parametrization θ and the trivialization, so that the projective space PE = PF(HE) depends only on E. We can apply this as follows. For each element u of the unitary group Um let Eu be the vector bundle on S1 = R/2πZ with holonomy u. (In other words, Eu is obtained from R × Cn by identifying (x+ 2π, ξ) with Michael Atiyah and Graeme Segal 307 (x, uξ).) Then the spaces PEu form a projective bundle on the group Um. We shall denote this bundle again by PE : we hope the notation will not prove confusing. The bundle PE on Um is equivariant with respect to the action of Um on itself by conjugation: an element g ∈ Um defines an isomorphism Eu → Egug −1 , and hence an isomorphism PEu → P Egug−1 . We shall return to this aspect of the bundle in Section 6. We can also regard PE as a projective bundle with involution, for multiplication by {±1} on H induces a projective action of the group {±1} on F(H). Proposition 5.1. The class of the projective bundle PE on Um is a generator of H3(Um; Z) ∼= Z, and as a bundle with involution its class is the non-zero element of H1(Um; Z/2) ∼= Z/2. Before justifying this assertion we shall mention a similar example, which is actually the one used by Freed, Hopkins, and Teleman. For a finite dimensional complex vector space W with an inner product the pro- jective space of the exterior algebra Λ(W ) is independent of the complex structure on W , as it is canonically isomorphic to the projective space of the spin module ∆(V ) of the real vector space V underlying W . Another way of saying this is that if we start with an even-dimensional real vec- tor space V then there is a canonical factorization of complex projective spaces P(Λ(VC)) ∼= P(∆(V )) ⊗ P(∆(V )), (5.1) where VC is the complexification of V . There is an infinite dimensional analogue of this phenomenon, explained in Chapter 12 of [21]. If H is a real Hilbert space a complex polarization of H will mean a preferred class of complex structures — equivalently, a class of decompositions HC = H+ ⊕ H− with H+ and H− complex conjugate. If H has a complex polarization then we can define a projective spin module P(∆(H)), and PF(HC) ∼= P(Λ(H+) ⊗ Λ(H̄−)) ∼= P(∆(H)) ⊗ P(∆(H)). (5.2) Before applying this to bundles on the circle we need a little more discussion. The first point is that the isomorphisms (5.1) and (5.2) are functorial in the category of projective spaces with involution. This is im- portant because an orientation-reversing automorphism of V interchanges the components of ∆(V ). Next, if we have an odd-dimensional real vector space V we define ∆(V ) = ∆(V ⊕R), but we must think of it as having an additional action of the Clifford algebra C1 on one generator (commuting in the graded sense with the action of the Clifford algebra C(V ) which ∆(V ) possesses in all cases). For odd dimensional V the isomorphism 308 Twisted K-theory (5.1) is replaced by P(Λ(VC)) ⊗ P(S2) ∼= P(∆(V )) ⊗ P(∆(V )), as projective spaces with involution, where, on the left, the space S2 ∼= C2 is the irreducible module for the Clifford algebra C2 ∼= C1 ⊗C1. There is exactly the same distinction between ”odd” and ”even” dimensionality for polarized real Hilbert spaces H, according as H or H⊕R has a preferred class of complex structures. Now let us consider the real Hilbert space HE of sections of a smooth real vector bundle E on the circle S1. The Fourier decomposition gives either HE or HE ⊕ R a class of complex structures: in fact HE is ”even- dimensional” if E is even-dimensional and orientable, or if E is odd- dimensional and non-orientable, and HE is ”odd-dimensional” otherwise. We shall write P spin E for the projective Hilbert space P∆(HE). As be- fore, we can consider the family of m-dimensional real bundles Eu on S1 parametrized by elements u of the orthogonal group Om. The corre- sponding projective spaces P spin Eu form a bundle P spin E on Om. Proposition 5.2. The class of the bundle P spin E — with its involution — on Om is (ε, η) ∈ H1(Om : Z/2) ⊕ H3(Om; Z), where ε restricts to the non-trivial element, and η to a generator, on each connected component of Om. To prove Propositions 5.1 and 5.2, let us take a slightly different point of view on the preceding constructions. If G is a compact connected Lie group, let LG denote the group of smooth loops S1 = R/2πZ → G, and let PG be the space of smooth maps f : R → G such that θ 7→ f(θ + 2π)f(θ)−1 is constant. Then LG acts freely on PG by right multiplication, and the map PG → G given by f 7→ f(2π)f(0)−1 makes PG a principal LG-bundle over G. Thus for any projective representation P of LG we have an associated projective bundle PG ×LG P on G — in fact a G-equivariant bundle, when G acts on itself by conjugation, in view of the action of G on PG by left multiplication. The invariant of PG×LG P in H3(G; Z) is clearly represented by the composite G→ BLG→ BPU(H) ≃ K(Z, 3), where the first map is the classifying map for PG and the second is induced by the representation LG → PU(H). This implies that the transgression H3(G; Z) → H2(LG; Z) takes the invariant to the class of the circle bundle on LG which is the central extension defined by P. The bundle PE on Um which we described above is obtained from PUm by Michael Atiyah and Graeme Segal 309 what is called the basic representation of LUm. (To see this, think of an element of PUm over u ∈ Um as defining a trivialization of the bundle Eu.) Because the maps H3(Um; Z) → H3(SUm; Z) → H2(LSUm; Z) ∼= Z are isomorphisms, we need only ask which central extension of LSUm acts on the basic representation, and we know from [21] that we get a generator of H2(LSUm; Z). The other part of Proposition 5.1, concerning the class in H1(Um; Z/2), is much easier, as all we need to know is that an element of LUm of winding number 1 acts on the Fock space F(L2(S1; Cn)) by an operator which raises degree by 1. Proposition 5.2 follows easily from Proposition 5.1. First, one may as well assume m is even. Then the bundle P spin E on O2k restricts to PE on Uk, while the maps H1(SO2k; Z/2) → H1(Uk; Z/2) and H3(SO2k; Z) → H3(Uk; Z) are isomorphisms; this deals with the invariants on the identity compo- nent of O2k. The other component can be treated by embedding Uk−1 in it by adding a fixed non-orientable bundle and using the multiplicativity of the Fock space construction. Let us now describe some families of Fredholm operators in the pro- jective bundles we have just constructed. In the representation theory of a loop group LG one usually studies projective representations H which are of positive energy and finite type. This means that the circle T of rotations of the loops acts unitarily on H, compatibly with its action on LG, and decomposes H into finite dimensional eigenspaces H = ⊕ n≥0 Hn, where T acts on Hn by the character eiθ 7→ einθ. (One calls Hn the part of ”energy” n.) The infinitesimal generator L0 of the circle action is an unbounded positive self-adjoint operator in H. When we consider the family P×LGP(H) on G the group R acts on P by translation, compatibly with the action of T = R/2πZ on LG and P(H). So R acts fibrewise on the bundle. If we identify the fibre Pg at g ∈ G with P(H) by choosing f ∈ P such that f(θ + 2π)f(θ)−1 = g then the infinitesimal generator L (g) 0 of the R-action on Pg is clearly given by L (g) 0 = L0 + f−1f ′, 310 Twisted K-theory where f−1f ′, which is periodic, is regarded as an element of the Lie algebra of LG. In fact we can choose f to be a 1-parameter subgroup of G generated by an element ξ ∈ g = Lie(G) such that exp(2πξ) = g, and then L (g) 0 = L0 + ξ. As ξ commutes with L0 it acts separately in each energy level Hn. In fact we know from [21](9.3.7) that if Vλ is an irreducible representation of G with highest weight λ contained in Hn then ‖λ‖2 ≤ an+ b, where a and b are constants depending on the representation H. On the other hand the eigenvalues of ξ in Vλ are bounded by ‖λ‖‖ξ‖, so the eigenvalues of ξ in Hn grow only like n1/2 as n → ∞. This shows that for any g ∈ G the operator L (g) 0 decomposes the Hilbert space Hg underlying the projective space Pg into the orthogonal direct sum of a sequence of finite- dimensional eigenspaces Hg,λ corresponding to a sequence of eigenvalues {λ}, depending on g and tending to ∞. In particular, the zero-eigenspace of L (g) 0 is always finite-dimensional. The family {L(g) 0 }, being positive, is not directly interesting in K- theory. It is analogous to the family of Laplace operators on the fibres of a bundle of compact manifolds, and we need something analogous to the family of Dirac operators. For a positive energy representation H of a loop group LG Freed, Hopkins, and Teleman consider the projective bundle P(H)G = PG×LGP(H) on G which we have already described. Its fibre Pg = P(Hg) at g ∈ G is a representation of the twisted loop group LgG whose Lie algebra Lgg is the space of sections of the real vector bundle Eg on S1 with fibre g and holonomy g. They tensor P(H)G with the spinor bundle P spin E . There is then a Dirac-type operator DH = {Dg} acting fibrewise in P(H)G ⊗ P spin E , defined for ξ ⊗ ψ ∈ Hg ⊗ ∆(Lgg) by Dg(ξ ⊗ ψ) = ∑ eiξ ⊗ e∗iψ, where {ei} is a basis of Lgg ∗, and {e∗i } is the dual basis of Lgg∗, regarded as elements of the Clifford algebra C(Lgg∗). (If ξ and ψ are in L (g) 0 - eigenspaces, and we choose the basis {ei} to consist of L (g) 0 -eigenvectors in L(g)gC, then the sum on the right is finite.) The operator Dg is, of course, an unbounded operator, but of a very tractable kind. It is defined on the dense subspace which is the algebraic direct sum of the finite-dimensional eigenspaces of L (g) 0 , and its square is a scalar multiple of L (g) 0 . It therefore decomposes as the sum of finite-dimensional operators acting in the L (g) 0 -eigenspaces. We can obtain a family {Ag} of bounded Michael Atiyah and Graeme Segal 311 Fredholm operators from the family {Dg} by defining Ag = (D2 g + 1)−1/2Dg. The family {Ag} defines an element of the twisted K-theory of G — in fact of the G-equivariant twistedK-theory — for each projective represen- tation H of the loop group LG. This is the map which Freed-Hopkins- Teleman prove to be an isomorphism. (If G is odd-dimensional, so is, as we have seen, the polarized Lie algebra Lg, and then the additional C1-action on P spin E gives us an odd-dimensional K-theory class.) 6. The equivariant case When a compact group G acts on a space X we can define equivariant K-theory K∗ G(X). If X is compact then K0 G(X) is the Grothendieck group of G-vector-bundles on X. If X is not compact, however, then one normally defines K0 G(X) as the equivariant homotopy classes of G- maps from X to a suitable representing G-space K0 G. Just as in the non-equivariant case, the space K0 G can be chosen in quite a variety of ways. If HG is what we shall call a stable G-Hilbert-space, i.e. a Hilbert space representation of G in which each irreducible representation of G occurs with infinite multiplicity (or, equivalently, one such that HG ∼= HG⊗L2(G)), then any G-vector-bundle on a compact base-space X can be embedded as a G-subbundle of X × HG, and so can be pulled back from the Grassmannian Gr(HG) of all finite dimensional vector subspaces of HG. Stabilizing in a familiar way gives us a natural candidate for K0 G. (A convenient choice of the stabilization is the restricted Grassmannian Grres(HG) mentioned in Section 3.) The space Fred(HG) of Fredholm operators in HG, with the norm topology, might seem another natural choice for K0 G, but unfortunately the action of G on Fred(HG) is very far from continuous. This can be dealt with in two ways. One is to replace FredHG) by the G-continuous subspace FredG−cts(HG) = {A ∈ Fred(HG) : g 7→ gAg−1 is continuous}, which is closed in Fred(HG), and is a representing space for K0 G, as is proved in Appendix 3. The other way is to pass to the more sophisticated space Fred0)(HG) introduced in Section 3. To twist equivariantK-theory we need a bundle P of projective spaces on which G acts, mapping Px to Pgx by a projective isomorphism. We shall call P stable if P ∼= P⊗L2(G). As before, we must decide whether or 312 Twisted K-theory not to require that the structural group of P is U(H) with the norm topol- ogy. Either way, we must be more careful than in the non-equivariant case. If P has structural group U(H)norm when the G-action is ignored it is impossible for G to act continuously on the associated principal bundle of P (unless G acts almost freely on X). Instead, we must require that (i) each point x ∈ X with isotropy group Gx has a Gx-invariant neighbourhood Ux such that there is an isomorphism of bundles with Gx-action P \|Ux ∼= Ux × P(Hx) for some projective space P(Hx) with Gx-action, and (ii) the transitions between these trivializations are given by maps Ux ∩ Uy → Isom(Hx;Hy) which are continuous in the norm topology. When P satisfies these conditions we can associate to it the bundle Fred(P ), defined without using the G-action of P , and with the norm topology in each fibre. Although the natural action of G on Fred(P ) is not continuous, it makes sense to define K0 G,P (X) as the group of homotopy classes of G-equivariant continuous sections of Fred(P ). As in the non-equivariant case, however, we prefer to avoid the norm topology. For any locally trivial projective bundle P with G-action the group G acts continuously on the associated bundle Fred(0)(P ). Even using Fred(0)(P ), however, it seems essential to require the bundle P to satisfy condition (i): otherwise we do not, for example, see how to show that Fred(0)(P ) is equivariantly trivial when P = P(E) comes from a stable equivariant bundle E of Hilbert spaces on X (cf. the action of G = (±1) on E = [0, 1] × L2([0, 1]) given by (−1).(x, φ) = (x, εxφ), where εx(y) = 1 when y ≤ x = −1 when y > x.) If condition (i) holds then we can trivialize E over a compact base X by constructing a G-equivariant section of the bundle on X with fibre Isom(HG;Ex) at x. This can be done by induction on the number of sets in a covering of X by G-invariant open sets of the form G.Si, where Si is a Gxi -invariant ”slice” (see [5]Chap.7, and [24] page 144) at a point xi ∈ X, and E\|Si is Gxi -equivariantly trivial. Michael Atiyah and Graeme Segal 313 Definition 6.1. For stable projective bundles P which satisfy condition (i) above we define K0 G,P (X) as the group of homotopy classes of equiv- ariant sections of Fred(0)(P ). The passage from twisted K-theory to the equivariant twisted the- ory is now quite unproblematical, at any rate for those accustomed to ordinary equivariant K-theory [24]. There seems no point in spelling it out. The most interesting thing to discuss is the classification of stable G-projective-bundles P , i.e. the analogue of Proposition 2.1 and Propo- sition 2.2. A G-projective bundle has an invariant ηP in the equivariant cohomology group H3 G(X; Z). This group can be defined by means of the ”Borel construction”, i.e. the functor which takes a G-space X to XG = (X × EG)/G, where EG is a fixed contractible space on which G acts freely. Definition 6.2. H∗ G(X; Z) = H∗(XG; Z). In particular, H∗ G(point; Z) = H∗(BG; Z), where BG is the classifying space EG/G. Let us write PicG(X) for the group of isomorphism classes of com- plex G-line-bundles on X (or, equivalently, of principal T-bundles with G-action), and ProjG(X) for the group of stable G-projective-bundles satisfying condition (i). Applying the Borel construction to line bundles and projective bundles gives us homomorphisms PicG(X) → Pic(XG) ∼= H2 G(X; Z) ProjG(X) → Proj(XG) ∼= H3 G(X; Z), which we shall show are bijective. Remark 6.1. A mod 2 graded projective bundle, in the sense of Sec- tion 2, is a projective bundle with Z/2-action on a base X with trivial Z/2-action. If G = Z/2 acts trivially on X then H∗ G(X) = H∗(X × RP∞) ∼= H∗(X;H∗(RP∞)), so that H3 G(X; Z) ∼= H1(X; Z/2) ⊕H3(X; Z). This agrees set-theoretically with Proposition 2.3, but the tensor product of G-spaces is not the same as the graded tensor product. Proposition 6.1. (i) H2 G(point; Z) ∼= Hom(G; T) 314 Twisted K-theory (ii) H3 G(point; Z) ∼= Ext(G; T), the group of central extensions 1 → T → G̃→ G→ 1. (iii) PicG(X) ∼= H2 G(X; Z) (iv) ProjG(X) ∼= H3 G(X; Z), and this remains true if we replace the left- hand side by the group of stable G-projective bundles with norm- topology structural groups. Of course the assertions (i) and (ii) here follow from (iii) and (iv), but they are easier to prove, and seem worth making explicit. Because PG = P(HG) is a classifying space for G-line-bundles when HG is an ample G-Hilbert-space, (iii) is simply the fact that PG represents the functor H2 G( ; Z), which can be proved quite easily in a variety of ways. The method we follow is chosen for its wider applications. Before giving the proof of Proposition 6.1, let us review the bundles of Fock spaces on a group G which were described in Section 5. These bundles are G-equivariant when G acts on the base-space by conjugation. They satisfy the equivariant local triviality conditiion (i) because the principal fibration PG → G has the corresponding property. They are not the most general possible equivariant bundles, as the action of the isotropy group on each fibre extends (non-canonically) to an action of G. They do not, however, have a natural norm-continuous structure, for the natural identifications of the fibre Pg at g with P1 differ among themselves by the action of elements of LG on P1, and so the natural transition maps between local trivializations will factorize through LG, which sits as a discrete subspace in U(H)norm. These equivariant projective bundles are determined by their classes in H3 G(Gconj; Z). The Borel construction EG×G Gconj is simply the free loop space LBG, which for connected G is the same as BLG. In the connected case this is most clearly seen by writing EG×G Gconj = EG×G (P/LG) = (EG×G P)/LG ≃ BLG, as G\P can be identified with the affine space of connections in the trivial G-bundle on the circle, so that EG×GP is a contractible space on which LG acts freely. From this point of view it is clear that the class of the bundle on Gconj coming from a projective representation H of LG is simply the topological class of the bundle BT → BL̃G→ BLG Michael Atiyah and Graeme Segal 315 with fibre BT ≃ P∞ C , where L̃G is the central extension of LG which acts on H. If G is connected and semisimple, the Serre spectral sequence for H∗ G gives us an exact sequence 0 → Ext(G; T) → H3 G(G; Z) → H3(G; Z), where the inclusion of Ext(G; T) is split by restriction to 1 ∈ G. Thus the class of an equivariant projective bundle — or of a representation of LG — is determined by its non-equivariant class together with its class as a projective representation of G, and the examples of Section 5 show us that when G = SOm any class in H3(G; Z) can arise. When G = Um, on the other hand, the spectral sequence gives us an exact sequence 0 → H2(BUm;H1(Um; Z)) → H3 Um (Um; Z) → H3(Um; Z) → 0. When m = 1 this tells us that H3 U1 (U1; Z) ∼= Z, the invariant being the flow of the grading of a Z-graded projective bundle around the base circle. When m > 1, we have H3 Um (Um; Z) ∼= Z ⊕ Z by the map H3 Um (Um; Z) → H3 U1 (U1; Z) ⊕H3(Um; Z). To prove Proposition 6.1 it is helpful to introduce groups H∗ G(X;A) defined for any topological abelian group A. These are the hypercoho- mology groups of a simplicial space X whose ”realization” is the space XG (see [23]). Whenever a group G acts on a space X we have a topological category whose space of objects is X and whose space of morphisms from x0 to x1 is {g ∈ G : gx0 = x1}. (Thus the complete space of morphisms is G×X.) A topological category can be regarded as a simplicial space X whose space Xp of p-simplexes is the space of composable p-tuples of morphisms in the category: in our case Xp = Gp ×X. For any simplicial space X and any topological abelian group A we can define the hypercohomology H∗(X ; sh(A)) with coefficients in the sheaf of continuous A-valued functions. It is the cohomology of a double complex C .., where, for each p ≥ 0, the cochain complex Cp. calculates H∗(Xp; sh(A)). Definition 6.3. H∗ G(X;A) = H∗(X ; sh(A)). 316 Twisted K-theory If A is discrete, the hypercohomology is just a way of calculating the cohomology of the realization XG of X , so the new definition of H∗ G(X;A) agrees with the old one. In any case, the groups H∗ G(X;A) are the abutment of a spectral sequence with Epq1 = Hq(Gp ×X; sh(A)). Lemma 6.1. If G is a compact group, then Hp+1 G (X; Z) ∼= Hp G(X; T) for any p > 0. Proof. Because of the exact sequence 0 → sh(Z) → sh(R) → sh(T) → 0 it is enough to show thatHp G(X ; R) = 0 for p > 0. As Epq 1 = 0 for q > 0 in the specctral sequence when A = R, we see that H∗ G(X; R) is simply the cohomology of the cochain complex of continuous real-valued functions on the simplicial space X , which is easily recognized as the complex of continuous Eilenberg-Maclane cochains of the group G with values in the topological vector space Map(X; R) of continuous real-valued functions on X. This complex is well-known to be acyclic in degrees > 0 when G is compact. (It is the G-invariant part of the contractible complex of so-called ”homogeneous cochains”, and taking the G-invariants is an exact functor, simply because cochains can be averaged over G.) Proof of Proposition 6.1. (i) When X is a point we have Eoq 1 = 0 in the spectral sequence for H∗ G, and we have already pointed out that Epo2 = Hp c.c.(G;A) is the cohomology of G defined by continuous Eilenberg- Maclane cochains. So H1 G(point;A) ∼= E10 2 ∼= H1 c.c.(G;A) ∼= Hom(G,A) for any topological abelian group A. (ii) In this case the spectral sequence gives us an exact sequence 0 → E20 2 → H2 G(point; T) → E11 2 → E30 2 , i.e. 0 → H2 c.c.(G; T) → H2 G(point; T) → Pic(G)prim → H3 c.c.(G; T), Michael Atiyah and Graeme Segal 317 for E11 1 = H ′(G; sh(T)) = Pic(G), and E11 2 is the subgroup of primitive elements, i.e. of circle bundles G̃ on G such that m∗G̃ ∼= pr∗1G̃⊗ pr∗2G̃, where pr1, pr2,m : G × G → G are the obvious maps. Equivalently, Pic(G)prim consists of circle bundles G̃ on G equipped with bundle maps m̃ : G̃ × G̃ → G̃ covering the multiplication in G. It is easy to see that the composite Ext(G; T) → H2 G(point; T) → Pic(G) (6.1) takes an extension to its class as a circle bundle. On the other hand H2 c.c.(G; T) is plainly the group of extensions T → G̃→ G which as circle bundles admit a continuous section, so its image in Ext(G; T) is precisely the kernel of (6.1). It remains only to show that the image of Ext(G; T) in Pic(G)prim is the kernel of Pic(G)prim → H3 c.c.(G; T). This map, however, associates to a bundle G̃ with a bundle map m̃ as above precisely the obstruction to changing m̃ by a bundle map G×G→ T to make it an associative product on G̃. (iii) The spectral sequence gives 0 → E10 2 → H1 G(X; T) → E01 2 → E20 2 . Now E01 2 = Pic(X), and E01 2 is the subgroup of circle bundles S → X which admit a bundle map m̃ : G × S → S covering the G-action on X. As before, m̃ can be made into a G-action on S if and only if an obstruction in H2 cc(G; Map(X; T)) vanishes. Finally, the kernel of PicG(X) → Pic(X) is the group of G-actions on X × T, and this is just E10 2 = H1 cc(G; Map(X; T)). (iv) This is the essential statement for us, and is distinctly harder to prove than the other three. If we knew a priori that the functor X 7→ ProjG(X) was representable by a G-space the argument would be much simpler; but we do not see a simple proof of representability. Instead we shall prove by the preceding methods that the map ProjG(X) → H3 G(X; Z) is injective, and then we shall construct a G-space P with a natural G-projective-bundle on it, and shall show that the composite map [X;P]G → ProjG(X) → H3 G(X; Z) is an isomorphism. 318 Twisted K-theory To prove the injectivity of ProjG(X) → H2 G(X; T) ∼= H3 G(X; Z) we consider the filtration ProjG(X) ⊇ Proj(1) ⊇ Proj(0), where Proj(1) consists of the stable projective bundles which are triv- ial when the G-action is forgotten, i.e. those that can be described by cocycles α : G×X → PU(H) such that α(g2, g, x)α(g1, x) = α(g2g1, x), and Proj(0) consists of those such that α lifts to α : G×X → U(H) such that α(g2, g1x)α(g1, x) = c(g2, g1, x)α(g2g1, x) (6.2) for some c : G×G×X → T. We shall compare the filtration of ProjG(X) with the filtration H2 G(X; T) = H(2) ⊃ H(1) ⊃ H(0) defined by the spectral sequence. By definition H(1) is the kernel of H2 G(X; T) → E02 1 = H2(X; sh(T)) = Proj(X), and the composite ProjG(X) → H2 G(X; T) → Proj(X) is clearly the map which forgets the G-action. Thus ProjG(X)/Proj(1) maps injectively to H2 G(X; T)/H(1) ∼= E02 ∞ →֒ E02 1 = Proj(X). Now let us consider the map Proj(1) → H(1). The subgroup H(0) is the kernel of H(1) → E11 2 , while E11 1 = Pic(G×X). We readily check that an element of Proj(1) defined by the cocycle α : G×X → PU(H) Michael Atiyah and Graeme Segal 319 maps to the element of Pic(G × X) which is the pull-back of the circle bundle U(H) → PU(H), and can conclude that α maps to zero in E11 2 if and only if it defines an element of Proj(0). Thus Proj(1)/Proj(0) injects into H(1)/H(0) = E11 ∞ = ker : E11 2 → E30 2 . Finally, assigning to an element α of Proj(0) the class in E20 2 = H2 c.c.(G; Map(X; T)) of the cocycle c occurring in (6.7), we see that if [c] = 0 then the projective bundle comes from a G-Hilbert-bundle, which is necessarily trivial, as we have already explained. So Proj(0) injects into H(0) = E20 2 . We now turn to the construction of the potential universal G-space P mentioned above. We shall begin with a few general remarks about G-equivariant homotopy theory when G is a compact group. If Y is a G-space we can consider the space Y H of H-fixed-points for any subgroup H of G. This is a space with an action of WH = NH/H, where NH is the normalizer of H in G. To give the space Y H clearly determines [X;Y ]G when X is a G-space of the form X = (G/H) ×X0, where G acts trivially on X0; and to give Y H together with its WH -action determines [X;Y ]G whenever X is isotypical of type H (i.e. all isotropy groups in X are conjugate toH), for then [X;Y ]G is the homotopy classes of sections of a bundle on X/G with fibre Y H associated to the principal WH -bundle XH → X/G. To give an element of ProjG(X) on an H-isotypical G-space X is the same as to give a stable NH -equivariant bundle on XH . Because isomorphism classes of stable H-Hilbert-spaces correspond to elements of Ext(H; T), these bundles are classified by WH -equivariant maps from XH to PH = ∐ H∈Ext(H;T) BPU(H)H , where we represent an element of Ext(H; T) by the essentially unique Hilbert space H with a stable projective representation of H inducing the extension. The group PU(H)H is disconnected, its group of components being Hom(H; T), but each connected component has the homotopy type of BT ∼= P∞ C . As the classifying space functor B commutes with taking H-invariants, the space PH , being a space ofH-fixed-points, has a natural action of WH . We shall give each group PU(H) the norm topology: there is then a natural projective bundle on PH with fibres P(H) which satisfies both conditions (i) and (ii) from the beginning of this section. There is now a standard procedure — unappealingly abstract — for cobbling together a G-space P so that for each subgroup H of G we have 320 Twisted K-theory PH ≃ PH . We introduce the topological category O of G-orbits (i.e. transitive G-spaces) and G-maps. Any G-space Y gives a contravariant functor from O to spaces by S 7→ MapG(S;Y ). If S = G/H, then MapG(G/H;Y ) ∼= Y H . Conversely, suppose that F is a contravariant functor from O to spaces. Let OF denote the topological category whose objects are triples (S, s, y), where S is an orbit, s ∈ S, and y ∈ F (S). A morphism (S0, s0, y0) → (S1, s1, y1) is a map θ : S0 → S1 in O such that θ(s0) = s1 and θ∗(y1) = y0. The group G acts on the category OF by g.(S, s, y) = (S, gs, y), and so the ”realization” \|OF \| (in the sense of [23]) is a G-space, and the fixed-point set \|OF \|H plainly contains F (G/H). If each space F (S) is an ANR then \|OF \| is a G-ANR. Proposition 6.2. The inclusion F (G/H) → \|OF \|H is a homotopy- equivalence. We shall omit the proof, which is quite elementary. We apply it to the functor F defined by F (G/H) = PH . There is no trouble in seeing that P = \|OF \| carries a tautological G-projective-bundle, so that we have a G-map P → Map(EG;BPU(H)) (6.3) into the space which represents the functor X 7→ H3 G(X; Z). To see that (6.3) induces an isomorphism [X;P]G → H3 G(X; Z) it is enough (by the result of [16]) to check the cases X = (G/H) × Si, when Si is an i-sphere; but this reduces to the isomorphism πi(PH)) ∼= H3−i(BH; Z) which we have already pointed out. Appendix 1. The compact-open topology The compact-open topology on the space Map(X;Y ) of continuous maps from a space X to a metric space Y is the topology of uniform convergence on all compact subsets of X. (In fact there is no need for Y Michael Atiyah and Graeme Segal 321 to be metrizable, for the compact-open topology can also be defined as the coarsest topology for which the subsets FC,U = {f : X → Y such that f(C) ⊂ U} are open whenever C is compact in X and U open in Y .) With this topology it is clear that a map Z → Map(X;Y ) is continuous if and only if the adjoint map Z ×X → Y is continuous on all subsets of the form Z×C, where C is compact in X. If Z and X are metrizable this is simply saying that Z ×X → Y is continuous. On the space Hom(H0;H1) of continuous linear maps between two Hilbert spaces the compact-open topology is only very slightly finer than the topology of pointwise convergence, which is called ”the strong oper- ator topology” by functional analysts. The Banach-Steinhaus theorem tells us that exactly the same subsets are compact in these two topologies; and on compact subsets the topologies must of course coincide. In partic- ular, if Z is a metrizable space the continuous maps Z → Hom(H0;H1) are the same for both topologies. For a Hilbert space H the groups GL(H) and U(H) are subsets of End(H), but when we speak of the compact-open topology on these groups we mean their subspace topology not in End(H) but in End(H)× End(H), in which they are embedded by g 7→ (g, g−1). The reason is that on the subset G of invertible elements of End(H) the map G→ End(H) given by inversion is not continuous. (For example, let gn be the diagonal transformation of the standard Hilbert space l2 of sequences defined by (gnξ)k = ξk if k 6= n, = n−1ξn if k = n. then gnξ → ξ as n→ ∞ for every ξ ∈ l2. But if ξ ∈ l2 is the vector with ξk = k−1 then ‖g−1 n ξ − ξ‖ → 1 as n→ ∞, so g−1 n ξ 6→ ξ.) Even when we define the compact-open topol- ogy so as to make inversion continuous, however, neither GL(H) and U(H) are quite topological groups, for the multiplication map is contin- uous only on compact subsets. One can say that they are ”groups in the category of compactly generated spaces”. (See [27]. Functional analysts use the word hypocontinuous for bilinear maps which are continuous on *Strictly, the Banach-Steinhaus theorem ([28] Thm 33.1, [4] chap.III §3,thm 2), which holds whenever H0 is Fréchet and H1 is locally convex, asserts that a set of maps which is compact for the topology of pointwise convergence is equicontinuous. But it is easy to see ([28] 32.5) that on equicontinuous subsets the compact-open and pointwise topologies coincide. 322 Twisted K-theory compact subsets: the tensor product of distributions is a well-known ex- ample.) In any case, for any metrizable space Z the space of continuous maps into GL(H) or U(H) forms a group, and that is quite enough for our purposes. We should also point out that the involution End(H) → End(H) given by A 7→ A∗ is not continuous for the compact-open topology. For example let An = e0 ⊗ e∗n be the operator of rank 1 in l2 which takes ξ = (ξk) to Anξ = (ξn, 0, 0, 0, . . .). Clearly An → 0 pointwise as n → ∞. But A∗ n = en ⊗ e∗0 takes the unit basis vector e0 = (1 0 0 0 . . .) to the unit vector en, and so A∗ ne0 6→ 0. The most important positive result for our purposes is Proposition A1.1. The group U(H) with the compact-open topology acts continuously by conjugation on the Banach space K(H) of compact operators in H, and also on the Hilbert space H∗ ⊗H of Hilbert-Schmidt operators. Proof. (i) We must show that for each unitary operator u0, each compact operator k0, and each ε > 0, we can find a compact subset C of H, and a δ > 0 such that if ‖k − k0‖ < δ and ‖u(ξ) − u0(ξ)‖ < δ for all ξ ∈ C then ‖uku−1 − u0k0u −1 0 ‖ < ε. Now ‖uku−1 − u0k0u −1 0 ‖ ≤ ‖uku−1 − uk0u −1‖ + ‖uk0u −1 − u0k0u −1‖ + ‖u0k0u −1 − u0k0u −1 0 ‖ = ‖k − k0‖ + ‖(u− u0)k0‖ + ‖k0(u ∗ − u∗0)‖ = ‖k − k0‖ + ‖(u− u0)k0‖ + ‖(u− u0)k ∗ 0‖, where in the last line we have used ‖A∗‖ = ‖A‖. Because k0 and k∗0 are both compact operators we can find a compact subset C of H which con- tains k0ξ and k∗0ξ for all unit vectors ξ, and we get the desired inequality by taking δ = ε/3. (ii) If k and k0 are Hilbert-Schmidt operators, the preceding cal- culation remains true if the operator norms ‖ ‖ are replaced by the Hilbert-Schmidt norm ‖ ‖HS , given by ‖A‖2 HS = ∑ ‖Aen‖2, Michael Atiyah and Graeme Segal 323 where {en} is an orthonormal basis of H. It is therefore enough to show that for any Hilbert-Schmidt k0 we have ‖(u− u0)k0‖HS < ε if u− u0 is small in the compact-open topology. But as ‖u− u0‖ < 2 we have ∑ n>N ‖(u− u0)k0en‖2 ≤ 4 ∑ n>N ‖k0en‖2, which is < ε/2 for suitable N , and we can make ‖(u− u0)k0en‖ small for all n ≤ N . That essentially completes our discussion of the compact-open topol- ogy, but we shall briefly mention a few other points. Because a compact subset of End(H) is equicontinuous, it is bounded in the operator norm (even though the example of the sequence {e0⊗e∗n} above shows that the norm is not itself a continuous function). This implies that A 7→ A∗A is continuous on compact sets, though A 7→ A∗ is not. Polynomial maps A 7→ p(A) are also continuous on compact sets, and hence — as a continuous function on the spectrum can be uniformly approximated by polynomials — so is the retraction map A 7→ (A∗A)t used on the space of Fredholm operators in Section 3. From the point of view of homotopy theory the one really bad feature of the compact-open topology is that the subspaces GL(H) and Fred(H) are neither open nor closed in the vector space End(H), and so are not ANRs. In other words, if X0 is a closed subspace of a space X then a continuous mapX0 → GL(H) need not be extendable to a neighbourhood of X0 in X. Appendix 2. Fredholm operators Proposition A2.1. For a separable Hilbert space H the spaces GL(H), U(H), and Fred(H) are contractible in the compact-open topology, by a homotopy h = {ht} : X × [0, 1] → X which is continuous on compact subsets. Proof. A single map h : End(H)×[0, 1] → End(H) will deal with the three cases simultaneously: it will have the property that ht(g −1) = (ht(g)) −1, 324 Twisted K-theory which is needed in view of the definition of the compact-open topology on GL(H) and U(H) which was explained in Appendix 1. The essential point is that we can identify H with the standard Hilbert space L2([0, 1]) of complex-valued functions on the unit inter- val, and that then the projection operator Pt which projects on to the first factor in L2([0, 1]) = L2([0, t]) ⊕ L2([t, 1]) depends continuously on t ∈ [0, 1] on the compact-open topology. (For it is obviously continuous in the topology of pointwise convergence.) Let us factorize Pt as itRt, where Rt : L2([0, 1]) → L2([0, t]) is the restriction and it is the inclusion of L2([0, t]) in L2([0, 1]), and when 0 < t ≤ 1 let us write Qt : L2([0, t]) → L2([0, 1]) for the isometric isomorphism given by (Qtf)(x) = t1/2f(tx). Then we define ht : End(H) → End(H) by ht(A) = itQ −1 t AQtRt + (1 − Pt) when t ∈ (0, 1], and h0(a) = 1. Because ‖QtRtξ‖ = ‖Ptξ‖ is continuous in t and → 0 as t→ 0, while ‖itQ−1 t A‖ = ‖A‖, the homotopy ht from h1 = (identity) to h0 = (constant) is continuous as claimed, and it preserves the subsets GL(H), U(H), and Fred(H). Proposition A2.2. The space Fred′(H) of Proposition 3.1 is a repre- senting space for K-theory, i. e. for every compact space X we have a natural bijection [X; Fred′(H)] → K0(X). Michael Atiyah and Graeme Segal 325 The proof, which follows closely the corresponding argument in the Appendix of [1], will be presented as a sequence of lemmas in which we shall denote a map X → Fred′(H) by (A,B) = ({Ax}, {Bx})x∈X , where each Ax is a Fredholm operator in H with parametrix Bx, and AxBx− 1 and BxAx− 1 depend continuously on x in the norm topology. Lemma A2.1. If Ax is surjective (resp. injective) when x = x0 then it is surjective (resp. injective) for all x in a neighbourhood of x0. Proof. Suppose that Ax0 is surjective. Because the Fredholm operator Ax0Bx0 is of the form 1+(compact) it has index 0, and so we can find a finite rank operator F such that Ax0(Bx0 + F ) is surjective, and hence an isomorphism. As Ax(Bx+F ) depends continuously on x in the norm topology, and invertible operators form an open set in the norm topology, we find that Ax(Bx+F ) is invertible for x near x0, and so Ax is surjective there. A similar argument applies when Ax0 is injective. Lemma A2.2. Suppose that Ax is surjective for all x ∈ X. Then the spaces Ex = ker(Ax) form a finite dimensional vector bundle on X. Proof. Given x0 ∈ X, let H0 = E⊥ x0 , and let i0 : H0 → H be the inclusion. Then Ax ◦ i0 is bijective when x = x0, and hence for all x near x0 by the preceding lemma. Considering the map of short exact sequences H0 i0−→ H −→ Ex0 Axi0 ↓ Ax ↓ ↓ H −→ H −→ 0 we conclude that orthogonal projection defines an isomorphism Ex → Ex0 for all x near x0. Lemma A2.3. There is a subspace H1 of finite codimension in H such that p ◦ Ax is surjective for all x ∈ X, where p is orthogonal projection H → H1. Proof. By lemma A2.1 we can achieve this for x in a neighbourhood of a chosen point of X. But X can be covered by a finite number of such neighbourhoods, and we can take the intersection of the corresponding subspaces H1. 326 Twisted K-theory Proof of Proposition A2.2. To each Fredholm family (A, b) we can now associate the element χA,B = [{ker(p ◦Ax)}] − [X × ker(p)] of K0(X), where p is as in the preceding lemma. The only choice made was of H1, but replacing H1 by a smaller subspace adds the same trivial bundle to both ker(p ◦ A) and X× ker(p), so the K-theory class χA,B, for a homotopy gives us an element of K0(X × [0, 1]) ∼= K0(X). Finally, we must show that if χA,B = 0 then (A,B) is homotopic to a constant map. But if χA,B = 0 we can assume (by making H1 smaller) that the bundle {ker(p◦Ax)} is trivial, and isomorphic to X× ker(p). We can then add a finite rank family {Fx} to {Ax} so that Ãx = Ax + Fx is an isomorphism for all x; and (Ã, B) is still a map into Fred′(H), and is homotopic to (A,B). Because GL(H) is contractible in the compact-open topology, we can deform (Ã, B) to (1, Ã−1B), where Ã−1B is of the form 1+ (compact), and then we can deform this family linearly to (1,1). Appendix 3. Equivariant contractibility of the unitary group of Hilbert space in the norm topology The results in this appendix are not, strictly speaking, needed in the paper, except to show that for a projective bundle with norm-continuous structure the two possible definitions of twisted equivariant K-theory coincide. We have included them partly for their intrinsic interest, and partly to correct a number of misstatements by the second author and others which have often been repeated in the literature. Let H be a stable G-Hilbert-space, and U(H) the unitary group with the norm topology. We have pointed out that the G-action on H does not induce a continuous action of G on U(H). The G-continuous elements UG−cts(H) = {u ∈ U(H) : g 7→ gug−1 is continuous} do, however, form a closed subgroup of U(H), in fact a sub-Banach-Lie-group. It is the inter- section of U(H) with the closed linear subspace EndG−cts(H) of End(H). To get a feeling for this subspace, notice that if H = L2(G) then multipli- cation by an L∞ function f on G is a G-continuous operator if and only if f is continuous. If G is the circle group T then a T-action on H defines a grading H = ⊕Hk, and any continuous linear map A : H → H can be represented by a block matrix (Akl), where Akl : Hl → Hk. Roughly, A is G-continuous if \|\|Akl\|\| → 0 sufficiently fast as \|k − l\| → ∞ Proposition A3.1. The group UG−cts(H) is equivariantly contractible. Michael Atiyah and Graeme Segal 327 Corollary A3.1. The space FredG−cts(H) of G-continuous Fredholm operators in H, with the norm topology, is a representing space for K0 G. The corollary follows from the proposition by exactly the same argu- ment used in the non-equivariant case in Appendix 2, and we shall say no more about it. One can think of the results in the following way. Although G does not act continuously on U(H) or Fred(H) it does make sense to say that a continuous map from a G-space X to these spaces is G-equivariant. Then A3.1 says that any two G-maps X → U(H) are homotopic, while A3.2 says that K0 G(X) is the set of homotopy classes of G-maps X → Fred(H). In this sense the misstatements referred to are innocuous. Proof of Proposition A3.1. Because U = UG−cts(H) is a G-ANR (see [JS]) it is enough to show that any G-map f : X → U from a compact G-space X can be deformed to the constant map at the identity. By a well-known ”Eilenberg swindle” argument it is enough to show that f can be deformed into the subgroup of elements of the form ( u 0 0 1 ) with respect to an orthogonal decomposition H = H1 ⊕ H2 of H into stable G-Hilbert-spaces. (For there is a canonical path from u ⊕ u−1 to the identity, and hence from u⊕ 1 = u⊕ (1 ⊕ 1) ⊕ (1 ⊕ 1) ⊕ . . . to u⊕ (u−1 ⊕ u) ⊕ (u−1 ⊕ u) ⊕ . . . = (u⊕ u−1) ⊕ (u⊕ u−1) ⊕ . . . , and hence to the identity.) It is also enough if we perform the deformation in the larger group GL = GLG−ctr(H), for GL can be equivariantly retracted to U by the usual polar decomposition. The essential step in Kuiper’s proof is the Lemma A3.1. For any ε > 0 there is an orthogonal decomposition H = H1 ⊕H2 ⊕H3 into stable G-Hilbert-spaces wuch that f(x)(H1) is ε-orthogonal to H3 for every x ∈ X. (We say that subspaces P and Q are ε-orthogonal if \|〈p, q〉\| < ε‖p‖‖q‖ for all p ∈ P and q ∈ Q.) 328 Twisted K-theory Granting the lemma, the proof of Proposition A3.1 is as follows. For each x ∈ X we have an ε-orthogonal decomposition H = f(x)H1 ⊕Hx ⊕H3, (A3.1) where Hx = H ⊖ (f(x)H1 ⊕ H3), and the projections on to each sum- mand depend continuously on x (in the norm topology). Choose a fixed isomorphism T : H1 → H2. Then the nearly unitary transformation ϕx of H which, in terms of the decomposition (A3.1), takes f(x)ξ ⊕ η ⊕ Tζ to −f(x)ζ ⊕ η ⊕ Tξ belongs to GL, and is connected to the identity by the path obtained by conjugating the unitary rotation from ξ ⊕ η ⊕ ζ to (−ζ) ⊕ η ⊕ ξ in H1 ⊕ H2 ⊕ H1 by f(x) ⊕ 1 ⊕ T . This path depends continuously on x. The original map f is therefore G-homotopic in GL to f1, where f1(x) = ϕ−1 x f(x). Now f1(x)\|H1 is simply the fixed map T : H1 → H3 ⊂ H, so we can perform a rotation interchanging H1 and H3 to deform f1 to a map f2 such that f2(x)\|H1 is the identity for all x ∈ X. Proof of Lemma A3.1. Thinking of f : X → UG−ctr(H) as a map into the Banach space End(H) we can find, because X is compact, a map f̃ arbitrarily close to f such that f̃(X) is contained in a finite dimensional subspace V of End(H). In fact, because vectors ξ ∈ H with finite di- mensional G-orbits are dense in H (cf. [9] p. 93), we can suppose V is a G-subspace of H, and, by averaging over G, that f̃ is a G-map, the image is automatically in GL. Now suppose that we have found three orthogonal finite dimensional G-subspaces P1, P2, P3 of H such that P1 ∼= P3 and α(P1) ⊂ P1⊕P2 for all α ∈ V . Let Q1 be an arbitrary irreducible G-subspace of H orthogonal to P1 ⊕P2 ⊕P3. We can clearly find two other finite dimensional subspaces Q2 and Q3, orthogonal both to each other and to P1 ⊕ P2 ⊕ P3 ⊕Q1 so that Q1 ∼= Q3 and α(Q1) ⊂ P1 ⊕Q1 ⊕P2 ⊕Q3 for all α ∈ V . Now define P (1) i = Pi ⊕Qi for i = 1, 2, 3. We have α(P (1) 1 ) ⊂ P (1) 1 ⊕ P (1) 2 and P (1) 1 ∼= P (1) 3 . Michael Atiyah and Graeme Segal 329 Repeating the process we find increasing sequences of subspaces Pi ⊂ P (1) i ⊂ P (2) i ⊂ . . . such that P (k) 1 , P (k) 2 , P (k) 3 are mutually orthogonal for all k, while α(P (k) 1 ) ⊂ P (k) 1 ⊕ P (k) 2 and P (k) 1 ∼= P (k) 2 . Finally we define H1 as the closure of the union of the subspaces P (k) 1 for k = 1, 2, . . ., and H3 as the closure of the union of the P (k) 3 . Then H2 is defined so that H = H1 ⊕H2 ⊕H3. It is obvious that we can make the choices so that all three subspaces Hi are stable. We have now finished, for f̃(x)(H1) is orthogonal to H3 for all x ∈ X, and so f(x)(H1) is ε-orthogonal to H3 as ‖f(x)− f̃(x)‖ < ε. References [1] M. F. 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Student Texts 32, Cambridge Univ. Press, 1995. [10] A. Dold, Partitions of unity in the theory of fibrations // Ann. of Math., 78 (1963), 223–255. [11] P. Donovan, M. Karoubi, Graded Braner groups and K-theory with local coeffi- cients // Publ. Math. I. H. E. S. Paris, 38 (1970), 5–25. [12] D. Freed, Twisted K-theory and loop groups. Proc. Int. Congress of Mathemati- cians Vol. III, Beijing, 2002, 419–430. [13] I. B. Frenkel, H. Garland, G. J. Zuckerman, Semi-infinite cohomology and string theory // Proc. Nat. Acad. Sci. U. S. A., 83 (1986), 8442–8446. 330 Twisted K-theory [14] A. Grothendieck, Le groupe de Brauer. Séminaire Bourbaki. Vol. 9, exp. 290, Soc. Math. France 1995. [15] N. Higson, J. Roe, Analytic K-homology. Oxford Univ. Press, 2000. [16] I. M. James, G. B. Segal On equivariant homotopy type // Topology, 17 (1978), 267–272. [17] N. H. Kuiper, The homotopy type of the unitary group of Hilbert space // Topo- logy, 3 (1965), 19–30. [18] J. Mickelsson, Twisted K-theory invariants. Preprint AT/0401130 [19] G. Moore, K-theory from a physical perspective. Topology, Geometry, and Quan- tum field theory. Ed. U. Tillmann. Cambridge Univ. Press, 2004. [20] R. S. Palais, On the homotopy of certain groups of operators // Topology, 3 (1965), 271–279. [21] A. N. Pressley, G. B. Segal, Loop groups. Oxford Univ. Press, 1986. [22] J. Rosenberg, Continuous-trace algebras from the bundle-theoretic point of view // J. Austral. Math. Soc., A47 (1989), 368-381. [23] G. B. Segal, Classifying spaces and spectral sequences // Publ. Math. I. H. E. S., Paris, 34 (1968), 105–112. [24] G. B. Segal, Equivariant K-theory // Publ. Math. I. H. E. S., Paris, 34 (1968), 129-151. [25] G. B. Segal, Fredholm complexes // Quarterly J. Math. Oxford, 21 (1970), 385– 402. [26] G. B. Segal, Topological structures in string theory // Phil. Trans. R. Soc. Lon- don, A359 (2001), 1389–1398. [27] N. E. Steenrod, A convenient category of topological spaces Michigan Math. J., 14 (1967), 133–152. [28] F. Trèves, Topological vector spaces, distributions and kernels. Academic Press, 1967. [29] R. Wood, Banach algebras and Bott periodicity. Topology, 4 (1965/6), 371–389. Contact information Michael Atiyah School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Kings Buildings, Mayfield Road, Edinburgh, EH9 3JZ, United Kingdom E-Mail: M.Atiyah@ed.ac.uk Graeme Segal All Souls College, Oxford OX1 4AL United Kingdom E-Mail: segalg@maths.ox.ac.uk, graeme.segal@all-souls.oxford.ac.uk

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