Actions of Eilenberg-MacLane spaces on K-theory spectra and uniqueness of twisted K-theory

A CTIONS OF EILE NBER G-MA CLANE SP A CES ON K- T HEOR Y SPECT RA AND UNIQUENESS OF TWI ST ED K-THEOR Y BENJAMIN ANTIEAU ∗ , DA VID GEPNER, AND JOS ´ E MANUEL G ´ OMEZ Abstra ct. W e prov e t h e uniqueness of twisted K -theory in b oth t he real and complex cases using the computation of the K -t h eories of Eilen b erg-MacLane spaces due to Anderson and Ho d gkin. As an application of our method, we giv e some v anishing results for actions of Eilen b erg-MacLane spaces on K -theory spectra. 1. Introduction Twisted K -theory , due originally to Donov an and Karoubi [12], has b ecome an imp ortan t concept bridging the fields of analysis, geometry , top olo gy and string theory . It is the home of many top ological inv arian ts w hic h cannot b e seen by unt wisted K -theory in the same wa y that the fu ndamenta l class of a non-orien table m anifold must liv e in twisted cohomology . F or instance, an app ropriate t wisted K -theory is the receptor of a Th om-isomorphism for non-Spin c - v ector bun dles. Because of its imp ortance and its p lace in many differen t fields, there are a wide v ariet y of definitions app earing in t he literature. In addition to [1 2], there are the accoun ts of [4], [7], [8], [10], [13], [17], [21], [25] and [26] to name jus t a few. All of the th ese definitions exp loit differen t mo d els a v ailable f or K -theory and it is a natural question to determine the r elationship b et we en all p ossib le different approac hes. The goal of this article is to sho w that all r easonable definitions of t wisted (real or complex) K -theory essential ly agree. In our context w e tak e as a reasonable defin ition of (real or complex) K -theory as one arising from a map K ( Z , 3) → B GL 1 K or K ( Z / 2 , 2) → B GL 1 K O . In general if R is an A ∞ -sp ectrum w e can twist the generalized cohomology r epresen ted b y R o v er a space X . Let B GL 1 R denote the classifying s pace of th e space GL 1 R of homotop y un its in R . Given a map f : X → B GL 1 R w e h a v e an induced map of ∞ -cat egories f ∗ : Π ∞ X → Mo d R , where Π ∞ is f u ndamenta l ∞ - group oid of X and Mo d R is the ∞ -category of R -mo du les. W e refer the reader to [19] for an accoun t of ∞ -categories. This map factors thr ou gh the full subgroup oid of Mo d R spanned by the free rank one R -mo du le R . One can construct the f -t wisted R -sp ect rum R ( X ) f b y defining R ( X ) f = colim Π ∞ X f ∗ . This is an R -mo dule that can b e seen parametrized sp ectrum o ver X whose fib ers are rank one free R -mo d ules. One can th en defin e the f -t wisted R -cohomology group s of X b y taking sections of th is parametrized sp ectrum. These ideas are made pr ecise in [5] and are outlined b elo w in Section 3. W e are particularly in terested in th e case where R = K or R = K O , the sp ectra representing real or complex K -theory . Thus, from the viewp oin t of homotop y theory , there is only one Date : September 4, 2018. 2000 Mathematics Subje ct Classific ation. Primary 19L50, 55N15 . Key wor ds and phr ases. Twisted K -theory , units of ring sp ectra, top ological Brauer groups. ∗ The first author was supp orted in part by the NSF under Gran t R TG DMS 0838697. 1 2 B. ANTIEA U, D. GEPNER, AND J. M. G ´ OMEZ definition of t w isted K -theory: giv en a map f : X → B GL 1 K one p ro duces the f -t wisted K -theory sp ectrum K ( X ) f o v er X . Ho w ev er, in applications, one t ypically w an ts to associate t wists of K -theory arising from a geometrically accessible subspace of B GL 1 K . In the case of complex K -theory for example, w e ha v e an inclusion K ( Z , 3) → B GL 1 K and w e are in terested in twists of K -theory arising from m aps X → B GL 1 K th at factor through K ( Z , 3), at least up to homotopy . Su c h twists are classified by cohomology classes in H 3 ( X, Z ). F or in stance, the definition of Donov an and Karoubi asso ciates twisted K -theory sp ectra to the torsion classes in H 3 ( X, Z ), wh ile the definitions of Rosenber g and of A tiy ah and Segal define twisted K -theory for all classes of H 3 ( X, Z ). In these cases a map j : K ( Z , 3) → B GL 1 K is fixed , and the α -twiste d K -theory , where α ∈ H 3 ( X, Z ) is a cohomology class classifying a map f : X → K ( Z , 3), is the t wisted K -theory co rresp onding to the comp ositio n X f − → K ( Z , 3) j − → B GL 1 K. The ab o ve construction dep ends on the map j : K ( Z , 3) → B GL 1 K c h osen an d thus w e face the problem classifying maps K ( Z , 3) → B GL 1 K . Similarly , in th e real case, one twists b y elemen ts of H 2 ( X, Z / 2), and so desires a map K ( Z / 2) → B GL 1 K O . Our compu tations lead to the f ollo w in g theorem. Theorem 1.1. Ther e ar e natur al isomorphisms of gr oups [ K ( Z , 3) , B GL 1 K ] ∼ = [ K ( Z , 3) , K ( Z , 3)] ∼ = Z [ K ( Z / 2 , 2) , B GL 1 K O ] ∼ = [ K ( Z / 2 , 2) , K ( Z / 2 , 2)] ∼ = Z / 2 . Th us, an y t w o maps from K ( Z , 3) to B GL 1 K differ b y an endomorph ism of K ( Z , 3), u p to homotop y , and similarly an y t w o maps from K ( Z , 2) to B GL 1 K O differ b y an endomorphism of K ( Z / 2 , 2), up to homotop y . Next we outline our approac h. By [20], there is a d ecomp osition of infinite lo op spaces GL 1 K ≃ K ( Z / 2 , 0) × K ( Z , 2) × B S U ⊗ , where B S U ⊗ is th e infin ite lo op space classifying v ir tual complex v ecto r bund les of rank and determinan t one, equipp ed with the tensor pro du ct structure. Since this splitting resp ects the infinite lo op structures, it ma y b e delo op ed, so that we obtain the splitting B GL 1 K ≃ K ( Z / 2 , 1) × K ( Z , 3) × B B S U ⊗ . Let i : K ( Z , 3) → B GL 1 K b e th e canonical inclusion. W e v iew a reasonable d efinition of K -theory as one arising from a map j : K ( Z , 3) → B GL 1 K , thus we wish to compare i and j . Denote by bsu ⊗ the connectiv e sp ectrum such that Ω ∞ bsu ⊗ ≃ B S U ⊗ . The main result of this w ork sa ys in th e complex case that bsu 1 ⊗ ( K ( Z , 3)) = [ K ( Z , 3) , B B S U ⊗ ] = 0 . More generally , in Section 2 we p ro vide conditions on a fin itely generated ab elian group π and n that imply that bsu 1 ⊗ ( K ( π , n )) v anishes. Our calculations rely on a computation of the K -theory of Eilen b er g-MacLane spaces d u e to An derson-Ho dgkin [3]. In p articular, it follo ws that an y map K ( Z , 3) → B GL 1 K is homotopic to a in teger m ultiple of i . In practice, to fi gu r e out whic h in teger, it suffi ces to compute a d ifferen tial in the t wisted At iy ah-Hirzebruch sp ectral sequence, as is done in [8]. All constructions app ea ring in the literature d iffer by a u nit ± 1. In the real case, B GL 1 K O ≃ K ( Z / 2 , 1) × K ( Z / 2 , 2) × B B S O ⊗ , UNIQUENESS OF TWISTED K-THEOR Y 3 and w e sho w that bso 1 ⊗ ( K ( Z / 2 , 2)) = [ K ( Z / 2 , 2) , B B S O ⊗ ] = 0 . where B S O ⊗ is the infi nite lo op space classifying virtual real v ector bund les of rank and de- terminan t one, equip p ed with the tensor pro duct stru cture. Here, bso ⊗ denotes the asso ciated connectiv e sp ectrum . Therefore, an y m ap j : K ( Z / 2 , 2) → B GL 1 K O is either homotopically trivial or j is equiv alen t to the canonical inclusion, and therefore there is a unique non-trivial definition of twiste d real K -theory . This is prov en in a similar wa y to the complex case. One calls t wists associated to a map from X to B B S U ⊗ (resp. to B B S O ⊗ ) h igher t wists of K -theory on X . Th us our theorem amoun ts to sa ying that there are no higher twists of complex K -theory on K ( Z , 3) or of real K -theory on K ( Z / 2 , 2). In fact, in Pr op ositions 2.6, 2.7, and 2.8 we d etermine exactly when there are higher twists of K -theory on K ( π , n ) f or π finitely generated and n ≥ 2 (or n ≥ 3 if π is not torsion). The original r esult in this direction is due to the th ird named author [14] who sho w ed that there are no higher t wists for complex K - theory on the classifying spaces of compact Lie groups G . The resu lts in [14] imply in p articular that there are n o higher t wists of complex K -theory o ver K ( π , 1) w hen π is a finite group and also o v er K ( Z n , 2) for an y n ≥ 0 and th us our computations generalize these fact s. One might also b e in terested in twists of complex K -theory coming fr om r -torsion cla sses in H 3 ( X, Z ) for some fi xed in teger r as in [6]. W e s ho w that bsu 1 ⊗ ( K ( Z /r , 2)) = 0 so that the only t wists of K -theory b y r -torsion classes come fr om comp osin g the Bo c kstein map β : K ( Z /r, 2) → K ( Z , 3) with a map K ( Z , 3) → B GL 1 K . The actions of Eilen b erg-MacLane sp ectra app ear as f ollo ws . Give n a map K ( π , n ) → B GL 1 K, one ma y pass to the lev el of lo op sp aces and obtain an A ∞ -map K ( π , n − 1) → GL 1 K. F or example, by lo oking at the map on loop spaces asso cia ted to i : K ( Z , 3) → B GL 1 K w e obtain an A ∞ -map CP ∞ ≃ K ( Z , 2) → GL 1 K whic h classifies the action of CP ∞ on K giv en by tensoring with line bun dles. W e call an A ∞ map K ( π , n ) → GL 1 K an action of K ( π , n ) on the K -theory sp ectrum. If the map factors through B S U ⊗ , we call the action a higher action. As a corollary of out compu tations we obtain (Corollary 2.9) the classification of those K ( π , n ) with finitely generated ab elian group π and n ≥ 2 (or n ≥ 3 if π is not torsion) for whic h all higher actions are trivial. Here is the outline of the pap er. T h e tec hnical engine of the pap er is con tained in Section 2 where we compute th e generalized cohomolog y groups bsu 1 ⊗ and bso 1 ⊗ of Eilen b erg-MacLane spaces. After this, S ection 3 recalls the definition of t wisted K -theory via the method of [5 ], and we giv e there th e pr o of of the u niqueness theorem. Finally , in the app endix, w e giv e a nice geometric mo d el of K -theory in the spirit of At iy ah and Segal which h as the adv an tage that it is a structured ring sp ectrum and so ma y b e easily used to pro du ce a map K ( Z , 3) → B GL 1 K . Notation: W e will denote by k the sp ectrum represen ting connectiv e complex K -theory , by K th e sp ectrum r epresent ing complex K -theory , and by K O the s p ectrum rep r esen ting r eal K -theory . F or a prime p we w ill d enote b y Z p the ring of p -adic inte gers. Giv en a sp ectrum F and an ab elian group G w e can introduce G co efficients on F by considering the sp ectrum F G = F ∧ M G , w here M G is a Moore sp ectrum for the group G . Also, given an in teger n , w e denote the ( n − 1)-connected co ve r of F by F h n i . 4 B. ANTIEAU, D. GEPNER, AND J. M. G ´ OMEZ Ac kno w ledgmen ts: W e wo uld lik e to thank Ulric h Bunke for suggesting some ve ry useful remarks regarding this problem and for m aking v ery detailed commen ts on an early d raft. Also, w e thank Pe ter Bousfield for some comments on the K -theory of Eilen b er g-MacLane spaces. 2. Cohomology c omput a tions The goal of this section is to d etermine when the groups bsu 1 ⊗ ( K ( π , n )) and bso 1 ⊗ ( K ( π , n )) v anish. W e sh o w in d etail the computations for the complex case. T he real case is handled in a similar wa y and we only pr o vide the main p oint s lea ving the details to the reader. Lemma 2.1. Supp ose that π is a torsion ab elian gr oup. Then, ˜ K ∗ ( K ( π , n )) = 0 if n ≥ 2 . Supp ose that n ≥ 3 and that π is non-torsio n (not ne c essarily torsion-fr e e). Then, K 1 ( K ( π , n )) = 0 if and only if π ⊗ Z Q i s a 1 -dimensional Q - ve ctor sp ac e and n is o dd. Pr o of. Th e case of K ( π , n ) where π is torsion and n ≥ 2 is [28, Theorem 3]. No w, sup p ose that π is non-torsion. T h en, by [28 , T heorem 3], K 1 ( K ( π , n )) = K 1 ( K ( π ⊗ Z Q , n )) = M p + q =1 H p ( K ( π ⊗ Z Q , n ) , K q ( ∗ )) . Therefore, it suffices to p ro v e that if π is a free ab elian group and n ≥ 3, then K ( π ⊗ Z Q , n ) has in tegral cohomology concen trated in ev en d egrees if and only if π ⊗ Z Q is 1-dimensional and n is o dd. W e ma y as well assume that π = Z I for some non-empt y set I . Let { e i } i ∈ I b e a basis f or Z I . S in ce homology commute s w ith direct limits, by the computation of Cartan [11, Th ´ eor ` eme 1] of the in tegral homology of E ilen b erg-MacLane spaces, if n is o dd, then H ∗ ( K ( Z I , n ) , Q ) ∼ = Λ Q [ σ n e i ] , the r ational exterior algebra on symb ols σ n e i in degree n , and if n is ev en, then H ∗ ( K ( Z I , n ) , Q ) ∼ = Q [ σ n e i ] , a p olynomial algebra. S ince the reduced homology groups of K ( Q I , n ) are Q -v ector spaces, by unive rsal co efficien ts for homology , ˜ H ∗ ( K ( Q I , n ) , Z ) ∼ = ˜ H ∗ ( K ( Q I , n ) , Q ) . On the other hand, K ( Z I , n ) → K ( Q I , n ) is a rational homotopy equ iv alence (for instance, by [15, Corollary 7.6]), so ˜ H ∗ ( K ( Z I , n ) , Q ) ∼ = ˜ H ∗ ( K ( Q I , n ) , Q ) . Therefore, th e reduced integ ral h omology of K ( Q I , n ) is concen trated in o dd d egrees if and only if n is o dd and | I | = 1. But, since the reduced in tegral cohomology of K ( Q I , n ) consists of Q -v ector spaces, this imp lies th at the reduced integ ral co homology of K ( Q I , n ) is concen trated in ev en degrees if and only if n is o d d and | I | = 1, by the un iv ersal co efficient theorem.  Lemma 2.2. Supp ose that π is a torsion ab elian gr oup. Then, g K O ∗ ( K ( π , n )) = 0 if n ≥ 2 . Supp ose that n ≥ 3 and that π is non-torsio n. Then, g K O 1 ( K ( π , n )) = 0 UNIQUENESS OF TWISTED K-THEOR Y 5 if and only if π ⊗ Z Q i s at most 3 -dimensional as a Q - ve ctor sp ac e and n is o dd. Pr o of. If π is torsion, th en b y the previous lemma, e K ∗ ( K ( π , n )) = 0 so th at by [3, App en d ix], g K O ∗ ( K ( π , n )) = 0 . In general, K ( π , n ) → K ( π ⊗ Z Q , n ) induces an isomorph ism on K -homology b y [28], and th erefore also an isomorphism on K O - cohomology b y [23, Corollary 1.13]. Therefore, w e can assu me that π is a free ab elia n group. Then, it is evidently sufficient to pro ve the statemen t for π a fin itely generated free ab elian group. Ind eed if g K O 1 ( K ( τ , n )) 6 = 0 for τ of rank at least 4, then choosing a splitting π ∼ = τ ⊕ σ sho ws that g K O 1 ( K ( π , n )) 6 = 0 as well . Th us, let τ b e a fi nitely generated free ab elian group. W e sho w that g K O 1 ( K ( τ , n )) = 0 if and only if n ≥ 3 is o d d and the ran k of τ is at most 3. As w e are in the finitely generated case, by [3, App endix], g K O 1 ( K ( τ , n )) ∼ = M p + q =1 H p ( K ( τ ⊗ Z Q , n ) , K O q ( ∗ )) . Since the reduced cohomology of K ( τ ⊗ Z Q , n ) is a Q -vect or space, in the direct su m ab o ve, H p ( K ( τ ⊗ Z Q , n ) , K O q ( ∗ )) can only b e non-zero for q = 0 mo d 4. Therefore, g K O 1 ( K ( τ , n )) = 0 if and only if K ( τ ⊗ Z Q , n ) has no integral cohomology in degrees equal to 1 mo d 4. If n is ev en, then K ( τ ⊗ Z Q ) has in tegral cohomology in d egrees 1 mo d 4. Thus, g K O 1 ( K ( τ ⊗ Z Q , n )) 6 = 0. If n is o dd, supp osing that τ = Z m , the Cartan ca lculation we s aw in the pro of of the previous lemma sa ys that K ( τ ⊗ Z Q , n ) h as int egral cohomology in degrees n + 1 , 2 n + 1 , . . . , mn + 1 . If m ≤ 3, then these degrees are n + 1 , 2 n + 1 , 3 n + 1. Since n is o dd, none of th ese n umber s are equal to 1 mo d 4. If m ≥ 4, 4 n + 1 = 1 mo d 4. Th is completes th e pro of.  No w w e turn to the v anishing of bsu 1 ⊗ ( K ( π , n )). Let Σ 4 k ≃ K h 4 i denote the 3-c onnected co v er of K -theory . T his is a connectiv e sp ectrum with infi nite lo op space B S U ⊕ ≃ Ω ∞ Σ 4 k , though here the infin ite lo op sp ace structur e is additiv e and do es not agree with the multiplica tiv e one on B S U ⊗ . The main resu lt of [2, Corollary 1.4], h o w ev er, asserts that the infi nite lo op structures b ecome equiv alen t after lo calization or completion at any p rime p . This implies in particular that Σ 4 k ∧ M Z p ≃ bsu ⊗ ∧ M Z p for ev ery prime p . W e are going to use this fact to sho w the trivialit y of bsu 1 ⊗ ( K ( π , n )) for v arious π and n . F or this we n eed the follo wing lemma. Lemma 2.3. If (1) π is a finite ab elian g r oup and n ≥ 2 or (2) π is a finitely ge ner ate d ab elian gr oup with dim Q π ⊗ Z Q = 1 and n ≥ 3 is o dd, then k 5 Z p ( K ( π , n )) = 0 f or every prime p and k 5 ( K ( π , n )) = 0 . Pr o of. By Lemma 2.1 K 1 ( K ( π , n )) = 0 in cases (1) and (2). Note also that (1) ˜ H r ( K ( π , n ) , Z ) = 0 for 0 ≤ r ≤ 2 under the h yp otheses. 6 B. ANTIEAU, D. GEPNER, AND J. M. G ´ OMEZ Let K ( π , n ) b e end ow ed with a CW-complex structure with m -skele ton F m , a finite CW- complex. Note that ˜ H r ( F m , Z ) = 0 for 0 ≤ r ≤ 2 and m large enough. Then, b y [24], there are exact sequences 0 → lim 1 m →∞ K 4 ( F m ) → K 5 ( K ( π , n )) → lim m →∞ K 5 ( F m ) → 0 , 0 → lim 1 m →∞ k 4 ( F m ) → k 5 ( K ( π , n )) → lim m →∞ k 5 ( F m ) → 0 . W e will p ro v e separately that lim 1 m →∞ k 4 ( F m ) = 0 and lim m →∞ k 5 ( F m ) = 0. Let’s sh ow firs t th at lim 1 m →∞ k 4 ( F m ) = 0. Since K 5 ( K ( π , n )) = 0 w e ha v e lim 1 m →∞ K 4 ( F m ) = 0 and it is easy to see that this imp lies that lim 1 m →∞ ˜ K 4 ( F m ) = 0. Fix m large enough and consider the Atiy ah-Hirzebru ch s p ectral sequences compu ting K ∗ ( F m ) and k ∗ ( F m ) E r,s 2 = H r ( F m , k s ( ∗ )) = ⇒ k r + s ( F m ) , (2) ˜ E r,s 2 = H r ( F m , K s ( ∗ )) = ⇒ K r + s ( F m ) . (3) Both of these sp ectral sequences con ve rge strongly as F m is a finite C W-complex. W e hav e a map of s p ectra k → K indu cing an isomorph ism on homotopy groups in non-negativ e degrees. This provides a map of sp ectral sequences f r,s ∗ : E r,s ∗ → ˜ E r,s ∗ suc h th at f r,s 2 is an isomorphism whenev er s ≤ 0. Moreo ver, since ˜ H r ( F m , Z ) = 0 for 0 ≤ r ≤ 2 w e ha v e that f r,s 2 is an isomorphism wheneve r r + s = 4 and r > 0. Also note that there are no differen tials that kill elemen ts in total degree 4 in the case of K that fail to do so in the case of k . T his is b ecause the only p ossible such differen tials must ha v e sour ce ˜ E 1 , 2 ∗ but this is tr ivial as ˜ E 1 , 2 2 = H 1 ( F m , Z ) = 0. This pro v es that f r,s ∗ induces an isomorphism f r,s ∞ : E r,s ∞ → ˜ E r,s ∞ whenev er r + s = 4 and r > 0. Also f 0 , 4 ∞ : E 0 , 4 ∞ = 0 → ˜ E 0 , 4 ∞ ∼ = Z since ˜ E 0 , 4 2 = H 0 ( F m , Z ) ∼ = Z and any differentia l with source ˜ E 0 , 4 ∗ ∼ = Z is trivial as one sees by comparing ˜ E r,s ∗ with the Atiy ah-Hirzebruch sp ectral sequence computing K ( ∗ ). This in turn prov es that the map of sp ectra k → K ind uces a short exact sequence 0 → k 4 ( F m ) → K 4 ( F m ) → Z → 0 . Note that in fact k 4 ( F m ) ⊂ ˜ K 4 ( F m ). W e conclude that the map k → K induces an isomorphism k 4 ( F m ) ∼ = ˜ K 4 ( F m ). Sin ce lim 1 m →∞ ˜ K 4 ( F m ) = 0 we conclud e that lim 1 m →∞ k 4 ( F m ) = 0 . Let’s pro v e no w that lim m →∞ k 5 ( F m ) = 0. T o p ro v e this compare agai n the sp ectral sequences (2) E r,s ∗ and ˜ E r,s ∗ for F m . In total degree 5 th e map f r,s ∗ is such that f r,s 2 is an isomorphism whenever s ≤ 0. A similar argum en t as b efore shows also in this case there are no differen tials that kill elemen ts in total degree 5 in the case of K that fail to do so in the case of k . Th erefore f r,s ∞ : E r,s ∗ → ˜ E r,s ∗ is an isomorp hism whenever r + s = 5 and s ≤ 0. Also note that E r,s ∞ = 0 whenev er r + s = 5 and s > 0. These facts sh o w that the map of sp ect ra k → K ind u ces an injectiv e map k 5 ( F m ) → K 5 ( F m ) for m large enough. Ind eed, a map of filtered ab elian groups with finite decreasing filtrations is injectiv e if the map on eac h slice is injectiv e by an iterated UNIQUENESS OF TWISTED K-THEOR Y 7 use of th e snak e lemma. Giv en the commutat iv e diagram (4) k 5 ( F m +1 ) i ∗ − − − − → k 5 ( F m )   y   y K 5 ( F m +1 ) i ∗ − − − − → K 5 ( F m ) . and the fact that lim m →∞ K 5 ( F m ) = 0, it f ollo ws that lim m →∞ k 5 ( F m ) = 0 since lim is left-exact. The fact that k 5 Z p ( K ( π , n )) = 0 is prov ed in the same wa y once we kno w th at K 5 Z p ( K ( π , n )) = 0. T o see this note that K Z p = holim k →∞ K Z / ( p k ) , with stru cture m aps coming from the maps Z / ( p k +1 ) → Z / ( p k ). Because of this, we hav e a short exact sequence (5) 0 → lim 1 k →∞ K 4 Z / ( p k ) ( K ( π , n )) → K 5 Z p ( K ( π , n )) → lim k →∞ K 5 Z / ( p k ) ( K ( π , n )) → 0 . On th e one hand, by [1, Prop osition 6.6] we hav e a sh ort exact sequence (6) 0 → K 5 ( K ( π , n )) ⊗ Z Z / ( p k ) → K 5 Z / ( p k ) ( K ( π , n )) → T or Z 1 ( K 6 ( K ( π , n )) , Z / ( p k )) → 0 . Under the giv en hyp othesis K 5 ( K ( π , n )) = 0 by Lemma 2.1. By [3, Theorem I] w e h a v e ˜ K ∗ ( K ( π , n )) = 0 when π is as in (1) and b y [3, Theorem II ] w e h a v e that ˜ K ∗ ( K ( π , n )) = H ∗∗ ( K ( π ⊗ Z Q , n ) , Z ) = 0 wh en π satisfies (2). In particular ˜ K 6 ( K ( π , n )) is vect or space o ver Q in this case. In either case it follo ws that T or Z 1 ( K 6 ( K ( π , n )) , Z / ( p k )) = 0 and we conclude that K 5 Z / ( p k ) ( K ( π , n )) = 0. This pr ov es that the righ t hand side in the short exact sequ ence (5) v anishes. W e are left to pro ve that lim 1 k →∞ K 4 Z / ( p k ) ( K ( π , n )) = 0 . T o show th is we u se the exact s equence (7) 0 → K 4 ( K ( π , n )) ⊗ Z / ( p k ) → K 4 Z / ( p k ) ( K ( π , n )) → T or Z 1 ( K 5 ( K ( π , n )) , Z / ( p k )) → 0 . Since K 5 ( K ( π , n )) = 0, w e conclude from (7) that K 4 Z / ( p k ) ( K ( π , n )) = K 4 ( K ( π , n )) ⊗ Z Z / ( p k ) . F rom here we can see that the maps K 4 Z / ( p k +1 ) ( K ( π , n )) → K 4 Z / ( p k ) ( K ( π , n )) are su rjectiv e and th us the lim 1 term in the sh ort exact sequence (5 ) v anishes. This pro v es that K 5 Z p ( K ( π , n )) = 0.  A similar computation can b e done in the real case. C onsider K O h 2 i , the 1-connected co v er of K O . Th en K O h 2 i is a connectiv e sp ect rum with Ω ∞ K O h 2 i ≃ B S O ⊕ . By [2, Corollary 1.4] it follo ws that K O h 2 i ∧ M Z p ≃ bso ⊗ ∧ M Z p for ev ery p r ime p . Lemma 2.4. If (1) π is a finite ab elian g r oup and n ≥ 2 or (2) π is a finitely ge ner ate d ab elian gr oup with 1 ≤ d im Q π ⊗ Z Q ≤ 3 and n ≥ 3 is o dd, then K O h 2 i 1 Z p ( K ( π , n )) = 0 f or every prime p and K O h 2 i 1 ( K ( π , n )) = 0 . 8 B. ANTIEAU, D. GEPNER, AND J. M. G ´ OMEZ Pr o of. Let { F m } m ≥ 0 b e th e skeleto n fi ltration of CW-complex stru cture on K ( π , n ) in su c h a w a y that F m is a finite CW-complex. Note that in these cases we also ha ve f or large m ˜ H r ( F m , Z ) = 0 f or 0 ≤ r ≤ 2 . Also, g K O ∗ ( K ( π , n )) = 0 when π is finite ab elian and n ≥ 2, and g K O 1 ( K ( π , n )) = 0 for π finitely generated and n ≥ 3 o dd, as prov ed ab o ve. W e argue in a sim ilar wa y as in the previous lemma. W e can compare the A tiyah-Hi rzebru ch sp ectral sequ en ces computing K O h 2 i ∗ ( F m ) and K O ∗ ( F m ). By d oing so w e pro ve that lim 1 m →∞ K O h 2 i 0 ( F m ) = 0 and lim m →∞ K O h 2 i 1 ( F m ) = 0 . The lemma follo ws using the lim 1 exact sequence in K O h 2 i 1 asso ciated to the filtration { F m } m ≥ 0 . The argument for p -completed K O -theory of K ( π , n ) f ollo ws the same lines as the complex case using the fact that K O h 2 i Z p = holim k →∞ K O h 2 i Z / ( p k ) .  Definition 2.5. An in v erse system of groups { G n } , i.e., a diagram of the form · · · → G n +1 → G n → · · · → G 2 → G 1 , is said to satisfy the Mittag-Leffler condition if f or ev ery i w e can fi nd a j > i such that for ev ery k > j , im( G k → G i ) = im( G j → G i ) . It is we ll kn o wn [27 , Prop osition 3.5.7] that if { G n } satisfies the Mittag-Leffler condition th en lim 1 k →∞ G k = 0 . On th e other hand, if eac h G k is a coun table group and lim 1 m →∞ G m = 0, then b y [22, Theorem 2] w e ha v e that the s ystem { G n } m ust satisfy the Mittag-Leffler condition. Using the previous prop ositio ns we can show th at bsu 1 ⊗ ( K ( π , n )) v anishes for certain v ales of n and some ab elia n groups π . This computation is central for our treatmen t on t wisted K -theory . Prop osition 2.6. If (1) π is a finite ab elian gr oup and n ≥ 2 or (2) π i s a finitely gener ate d ab elian gr oup with dim Q π ⊗ Z Q = 1 and n ≥ 3 is o dd, then we ha ve bsu 1 ⊗ ( K ( π , n )) = 0 . In p articular, ther e ar e no higher twists of c omplex K -the ory on K ( π , n ) in e ither c ase. Pr o of. As ab o v e, since π is fi nitely ge nerated, we can give K ( π , n ) a CW-complex structure in suc h a w a y that the m -sk eleton F m is a finite CW-complex. The filtration { F m } m ≥ 0 induces a short exact sequence 0 → lim 1 m →∞ bsu 0 ⊗ ( F m ) → bsu 1 ⊗ ( K ( π , n )) → lim m →∞ bsu 1 ⊗ ( F m ) → 0 . Belo w we prov e that the sequence of groups { bsu 0 ⊗ ( F m ) } m ≥ 0 satisfies the Mittag-Le ffler condition. Th us the lim 1 in the previous sequence v anishes yielding (8) bsu 1 ⊗ ( K ( π , n )) = lim m →∞ bsu 1 ⊗ ( F m ) . UNIQUENESS OF TWISTED K-THEOR Y 9 W e are going to show that lim m →∞ ( bsu 1 ⊗ ( F m ) ⊗ Z p ) = 0 for all pr ime num b ers p . Because eac h bsu 1 ⊗ ( F m ) is finitely generated group th en by [14, Lemma 7] lim m →∞ bsu 1 ⊗ ( F m ) = 0 and thus bsu 1 ⊗ ( K ( π , n )) = 0 by (8). Consider the sh ort exact sequ ence (9) 0 → lim 1 m →∞ ( bsu ⊗ ∧ M Z p ) 0 ( F m ) → ( bsu ⊗ ∧ M Z p ) 1 ( K ( π , n )) → lim m →∞ ( bsu ⊗ ∧ M Z p ) 1 ( F m ) → 0 . By [2, Corollary 1.4] we ha v e Σ 4 k ∧ M Z p ≃ bsu ⊗ ∧ M Z p . Th is together w ith Lemma 2.3 giv es ( bsu ⊗ ∧ M Z p ) 1 ( K ( π , n )) ∼ = k 5 Z p ( K ( π , n )) = 0 . W e conclude th at th e m id dle term in (9) v anishes and h ence we s ee th at lim m →∞ ( bsu ⊗ ∧ M Z p ) 1 ( F m ) = 0 . By [1, Prop osition I I I.6.6] there is a short exact sequence 0 → bsu 1 ⊗ ( F m ) ⊗ Z p → ( bsu ⊗ ∧ M Z p ) 1 ( F m ) → T or 1 Z ( bsu 2 ⊗ ( F m ) , Z p ) → 0 . The T or term in this sequence v anish es as Z p is flat as a Z -mo dule. Therefore ( bsu ⊗ ∧ M Z p ) 1 ( F m ) = bsu 1 ⊗ ( F m ) ⊗ Z Z p and this in turn sho ws that for ev ery prime p lim m →∞ ( bsu 1 ⊗ ( F m ) ⊗ Z Z p ) = 0 . W e are left to pr o v e that the sys tem of groups B m := bsu 0 ⊗ ( F m ) satisfies the Mittag-Leffler condition. F or ev ery m ≥ 0 let A m := k 4 ( F m ). W e saw in th e pro of of Lemma 2.3 that lim 1 m →∞ A m = 0 . Also since F m is a finite CW-complex, w e ha ve that A m and B m are fin itely generated ab elian groups for ev ery m ≥ 0, in particular they are coun table. Th erefore the system { A m } m ≥ 0 satisfies the Mittag- Leffler condition. On the other h an d , as p oin ted out ab o ve Σ 4 k ∧ M Z p ≃ bsu ⊗ ∧ M Z p , th us for ev ery m ≥ 0 and an y prime num b er p (10) A m ⊗ Z Z p = k 4 Z p ( F m ) ≃ → ( bsu ⊗ ∧ M Z p ) 0 ( F m ) = B m ⊗ Z p . The outer equalities follo w by [1, Pr op osition I I I.6.6] since the T or terms also v anish here. T his yields a comm utativ e diagram in wh ich the v ertical arro ws are isomorp hisms → A m ⊗ Z Z p · · · → A 2 ⊗ Z Z p → A 1 ⊗ Z Z p ↓ ↓ ↓ → B m ⊗ Z Z p · · · → B 2 ⊗ Z Z p → B 1 ⊗ Z Z p . Using th is diagram and the fact that { A m } m ≥ 0 satisfies th e Mittag-Leffler pr op erty it can b e seen that the system { B m } m ≥ 0 also satisfies the Mittag-Leffler condition u s ing an argument similar to that in [14, Theorem 5].  The previous p rop osition has the follo win g r eal analogue that can b e prov ed in the same w a y using the fact that K O h 2 i ∧ M Z p ≃ bso ⊗ ∧ M Z p for ev ery prime p . 10 B. ANTIEAU, D. GEPNER, AND J. M. G ´ OMEZ Prop osition 2.7. If (1) π is a finite ab elian gr oup and n ≥ 2 or (2) π i s a finitely gener ate d ab elian gr oup with 1 ≤ dim Q π ⊗ Z Q ≤ 3 and n ≥ 3 is o dd, then we ha ve bso 1 ⊗ ( K ( π , n )) = 0 . In p articular, ther e ar e no higher twists of r e al K -the ory on K ( π , n ) in either c ase. On th e other hand, Prop osition 2.6 is sharp as w e sh o w next. A real analogue can b e pro v ed in a similar w a y . Prop osition 2.8. If (1) π is a non-torsion (not ne c essarily torsion-fr e e) finitely g e ner ate d ab elian gr oup and n > 3 is even or (2) π is a finitely g e ner ate d ab elian gr oup with dim Q π ⊗ Z Q > 1 and n ≥ 3 is o dd, then bsu 1 ⊗ ( K ( π , n )) 6 = 0 . Pr o of. By Lemma 2.1 w e kno w that K 5 ( K ( π , n )) 6 = 0 in these case s. Let’s sho w first that k 5 ( K ( π , n )) 6 = 0. Assume b y con tradiction that k 5 ( K ( π , n )) = 0. As b efore giv e K ( π , n ) a structure of a CW-complex suc h that F k , the k -sk eleton of K ( π , n ), is a finite C W-complex. Since w e are assuming that k 5 ( K ( π , n )) = 0 we h a v e that lim 1 k →∞ k 4 ( F k ) = 0 and lim k →∞ k 5 ( F k ) = 0. By comparing the A tiy ah-Hirzebruc h sp ectral sequences computing K ∗ ( F k ) and k ∗ ( F k ) as in Lemma 2.3 we can see th at lim 1 k →∞ K 4 ( F k ) = 0 and lim k →∞ K 5 ( F k ) = 0. This in tur n p ro v es that K 5 ( K ( π , n )) = 0 whic h is a con tradiction. Let’s sho w no w that bsu 1 ⊗ ( K ( π , n )) 6 = 0. Reasoning b y con tradiction again assu me that bsu 1 ⊗ ( K ( π , n )) = 0. The sh ort exact sequence 0 → lim 1 k →∞ bsu 0 ⊗ ( F k ) → bsu 1 ⊗ ( K ( π , n )) → lim k →∞ bsu 1 ⊗ ( F k ) → 0 sho ws that lim k →∞ bsu 1 ⊗ ( F k ) = 0 and lim 1 k →∞ bsu 0 ⊗ ( F k ) = 0. Sin ce F k is a fin ite CW-complex w e ha v e that B k := bsu 0 ⊗ ( F k ) is finitely ge nerated for ev er y k ≥ 0. In particular we conclude that the system { B k } k ≥ 0 satisfies the Mittag-Leffler condition. Let A k = k 4 ( F k ). As in the p ro of of the previous prop osition we hav e a commutativ e diagram in w h ic h the ve rtical arro ws are isomorphisms → A n ⊗ Z Z p · · · → A 2 ⊗ Z Z p → A 1 ⊗ Z Z p ↓ ↓ ↓ → B n ⊗ Z Z p · · · → B 2 ⊗ Z Z p → B 1 ⊗ Z Z p . This diagram, the fact that { B k } k ≥ 0 satisfies the Mittag-Leffler prop ert y and an argument similar to the one pro vided in [14, Theorem 5] pro v e that { A k } k ≥ 0 also satisfies the Mitt ag- Leffler prop erty and in particular lim 1 k →∞ A k = lim 1 k →∞ k 4 ( F k ) = 0 . On the other hand, b y [3, Theorem I I I] w e ha v e l im k →∞ K 5 ( F k ) = 0. Comp aring th e A tiy ah- Hirzebruc h sp ectral sequ ences computing K ∗ ( F k ) and k ∗ ( F k ) we can see that K 5 ( F k ) ∼ = k 5 ( F k ), in particular we obtain lim k →∞ k 5 ( F k ) = 0. Finally , the short exact sequence 0 → lim 1 k →∞ k 4 ( F k ) → k 5 ( K ( π , n )) → lim k →∞ k 5 ( F k ) → 0 sho ws that k 5 ( K ( π , n )) = 0 wh ic h is a con tradiction.  Next w e consider actions of Eilenberg-MacLane spaces on the K -theory sp ectrum. W e call an A ∞ -map K ( π , n − 1) → GL 1 K an action of K ( π , n − 1) on K . Giv en giv en an A ∞ -action of K ( π , n − 1) on K , it can b e de-lo op ed to obtain a map K ( π , n ) → B GL 1 K . Conv ersely , UNIQUENESS OF TWISTED K-THEOR Y 11 giv en a map K ( π , n ) → B GL 1 K , we obtain an A ∞ -map K ( π , n − 1) → GL 1 K b y passing to the lev el of lo op space. In fact actions of K ( π , n − 1) on the K -theory sp ectrum are in one to one corresp onden ce with maps K ( π , n ) → B GL 1 K . W e call an action of K ( π , n − 1) on K a higher action if the corresp onding map K ( π , n ) → B GL 1 K factors thr ough B B S U ⊗ . Th e ab ov e w ork can b e rephrased as follo ws. Corollary 2.9. L et π b e a finitely gener ate d ab elian gr oup and n ≥ 2 an inte ger. (2) Ther e ar e no higher actions of K ( π , n ) on K if and only if π is torsion or n is even and dim π ⊗ Z Q = 1 . (3) Ther e ar e no higher actions of K ( π , n ) on K O if and only if π is torsion or n is even and 1 ≤ d im π ⊗ Z Q ≤ 3 . Corollary 2.10. L et π b e a finite ab e lian gr oup. Then, ther e ar e no higher actions of K ( π , 1) on either K or K O . Corollary 2.10 w as obtained by Gomez [1 4]. 3. Unique ness of twisted K -th eor y In this section we us e the computations of bsu ⊗ and bso ⊗ -cohomolog y in the pr evious section to establish a uniqueness s tatement for definitions of t wisted K -theory for b oth the real and complex cases. Let R denote an A ∞ -ring sp ectrum. W e can twist the generalized cohomology r epresen ted b y R o v er a space X . Let GL 1 R the space of homotopy units of R . Th is space is defined as the homotop y pullback in the diagram GL 1 R − − − − → Ω ∞ R   y   y ( π 0 Ω R ) × − − − − → π 0 Ω ∞ R. The space GL 1 R is a group-lik e A ∞ -space and t wists of the theory R o v er a space X are classified b y homotopy classes of m aps X → B GL 1 R . Sp ecifically , giv en a map f : X → B GL 1 R , we obtain (b y asso ciating to a sp ace X its fun damen tal ∞ -group oid Π ∞ X , as in [19 ]) an indu ced map of ∞ -categories Π ∞ X f − → Π ∞ B GL 1 R ≃ B Aut R ( R ) i − → Mo d R . Here Aut R ( R ) denotes th e group-like A ∞ -space of automorphisms of R in Mo d R , B Aut R ( R ) its delo oping, and B Au t R ( R ) → Mo d R the inclusion of the full subgroup oid of Mod R spanned b y the f ree rank one R -mo d ule R . The R -mo dule sp ectrum R ( X ) f , or the f -t wisted R -theory sp ectrum of X , is the resulting “Thom sp ectrum” R ( X ) f = colim Π ∞ X i ◦ f , the colimit in Mo d R of the comp osite map i ◦ f : Π ∞ X → Mod R . The colimit exists since Mo d R admits colimits indexed by an arb itrary small ∞ -categ ory . See [19] for an accoun t of the ∞ -categ orical theory of col imits. The f -t wisted R -cohomology groups of X are R n ( X ) f := π 0 F R ( R ( X ) f , Σ n R ) , where F R ( R ( X ) f , Σ n R ) is the fu nction sp ectrum of R -mo du le maps R ( X ) f → Σ n R . 12 B. ANTIEAU, D. GEPNER, AND J. M. G ´ OMEZ W e can use this metho d of t w istin g in the particular cases of t wisted (real or complex) K - theory . In the complex case th e decomposition GL 1 K ≃ K ( Z / 2 , 0) × K ( Z , 2) × B S U ⊗ is compatible with the evident A ∞ -structures so it delo ops to a decomp osition B GL 1 K ≃ K ( Z / 2 , 1) × K ( Z , 3) × B B S U ⊗ . Therefore, t wists of complex K -theory ov er X are classified by h omotop y classes of maps X → B GL 1 K ≃ K ( Z / 2 , 1) × K ( Z , 3) × B B S U ⊗ . The t wisted cohomology group s K n ( X ) f dep end only on the homotop y class of f , through non-canonical isomorphisms, and thus by the decomp ositio n B GL 1 K ≃ K ( Z / 2 , 1) × K ( Z , 3) × B B S U ⊗ w e ha v e t wists of complex K -theory asso cia ted to elemen ts in H 1 ( X, Z / 2) × H 3 ( X, Z ) × bsu 1 ⊗ ( X ) . In the geometric applications ho w ev er, one sp ecializes to twists of K -theory asso ciated to maps f representing a cohomology class in H 3 ( X, Z ). More p recisely , let i : K ( Z , 3) − → B GL 1 K. b e the in clusion map and supp ose that f : X → K ( Z , 3) is a map representing a cohomology class in H 3 ( X, Z ). Then the f -t wisted K -theory sp ectrum of X is defined as K ( X ) i ◦ f . Differen t constructions or mo dels of t wisted K -theory asso ciated to cohomology classes in H 3 ( X, Z ) can b e constru cted by sp ecifying a map j : K ( Z , 3) − → B GL 1 K. Giv en suc h a map we can defin e the t wisted K -groups as ab o ve. Th us a particular mo del or definition of t wisted (complex) K -theory asso ciated to cohomology classes in H 3 ( X, Z ) amounts to pro ducing a particular map K ( Z , 3) − → B GL 1 K . W e discuss here ho w the construction giv en in [8] by Atiy ah and Segal fits in to this framew ork. Let H b e a fixed infinite dimensional separable Hilb ert space. The space of F redholm op erato rs F r ed ( H ) with the n orm top ology is then a classifying space for complex K -theory . The space of unitary op erators U ( H ) acts by conjugation on F r ed ( H ). This ind u ces an action of th e pro jectiv e unitary group P U ( H ) on F r ed ( H ). Th e space P U ( H ) is a K ( Z , 2) and give n a map f : X → B P U ( H ) ≃ K ( Z , 3) there is an associated p rincipal P U ( H )-bundle P → X . W e can then f orm the b undle ξ := P × P U ( H ) F r ed ( H ) → X and A tiy ah and Segal define K 0 ( X ) f ,AS := π 0 Γ( ξ → X ) , the group of homotop y classes of sections of ξ → X . In the app end ix, w e giv e d etails on ho w to us e the symmetric sp ectrum mo d el of K -theory du e to J oac h im [16] to obtain a map K ( Z , 3) → B GL 1 K which is very muc h in the spirit of Atiy ah and S egal. The case of twisted real K -theory can b e handled in the same w a y . As in the complex case, we are interested in th ose twists asso ciated to a map f representing a cohomology class H 2 ( X ; Z / 2). Th us giv en a m ap f : X → K ( Z / 2 , 2) represent ing a cohomology class in H 2 ( X ; Z / 2) we can define th e f -t wisted real K -theory as K O ( X ) i ◦ f , where i : K ( Z / 2 , 2) → B GL 1 K O UNIQUENESS OF TWISTED K-THEOR Y 13 is the inclusion map. Therefore construction of t wisted complex K -theory asso cia ted to in tegral cohomolog y classes in H 3 ( X, Z ) amounts to a p oin ted map j : K ( Z , 3) → B GL 1 K. Similarly , a construction of t wisted real K -theory asso ciated to cohomol ogy classes in H 2 ( X, Z / 2) amoun ts to a p ointe d map j : K ( Z / 2 , 2) → B GL 1 K O . On the other h and, a construction for r -torsion inte gral classes in the complex case is determined b y a p ointed map j r : K ( Z /r , 2) → B GL 1 K. A construction of all integ ral classes yields one for the r -torsion ones by comp osition with the Bo c kstein β : K ( Z /r , 2) → K ( Z , 3). Our main theorem sa ys there are no h igher t wists of complex K -theory on K ( Z , 3) or K ( Z /r , 2). I n the real case w e establish the nonexistence of higher t wists of real K -theory on K ( Z / 2 , 2). Let p : B GL 1 K → K ( Z , 3) and q : B GL 1 K O → K ( Z , 2) b e th e pro jection maps. Theorem 3.1. The map p induc es isomorp hisms σ : [ K ( Z , 3) , B GL 1 K ] → [ K ( Z , 3) , K ( Z , 3)] ∼ = Z σ r : [ K ( Z /r , 2) , B GL 1 K ] → [ K ( Z /r , 2) , K ( Z , 3)] ∼ = Z /r . The map q induc es an isomorp hism τ : [ K ( Z / 2 , 2) , B GL 1 K O ] → [ K ( Z / 2 , 2) , K ( Z / 2 , 2)] ∼ = Z / 2 . Pr o of. Th e inclusion of the K ( Z , 3) comp onen t into B GL 1 K giv es surjectivit y . So, it suffices to p r o v e that σ is in jectiv e. In other words, we wish to sh o w that if σ ( j ) = 0, then j is n ull-homotopic. But, if σ ( j ) = 0, then the map K ( Z , 3) → K ( Z / 2 , 1) × K ( Z , 3) × B B S U ⊗ is homotopically trivial on the first t w o comp onen ts. By Prop osition 2.6, it is also trivial on the third comp onent. The statemen t for σ r is similar. Th e statemen t for τ is p ro v ed in th e same w a y by using Prop ositio n 2.7.  The p r evious theorem sho ws th at in particular, an y d efinition of complex twisted K -theory arising through a map K ( Z , 3) → B GL 1 K agrees, up to m ultiplication of an inte ger, with the definition giv en ab ov e. F or a give n defi nition this inte ger can b e obtained b y determining the differential d 3 in the A tiy ah-Hirzebruc h sp ectral sequence computing the t wisted equiv ariant K -groups. Equiv alen tly , we can determine th is in teger by computing th e t wisted K -groups on the sphere S 3 for a generator α ∈ H 3 ( S 3 , Z ) ∼ = Z . A s im ilar situatio n occur s f or twisted K -theory asso cia ted to r -to rsion integral classes. F or the case of t w isted real K -theory the situation simplifies. In this case, b y Theorem 3.1 w e ha v e [ K ( Z / 2 , 2) , B GL 1 K O ] ∼ = Z / 2. In particular, an y t w o non-trivial defi n itions of t wisted r eal K -theory arising through a m ap K ( Z / 2 , 2) → B GL 1 K O m ust coincide. 14 B. ANTIEAU, D. GEPNER, AND J. M. G ´ OMEZ Corollary 3.2. L et j : K ( Z , 3) → B GL 1 K b e a p ointe d map. Then, Ω j : K ( Z , 2) → GL 1 K sends a line bund le L on X to the e quivalenc e K ( X ) ∼ → K ( X ) give n by tensoring with L ⊗ σ ( j ) . Similarly, if j : K ( Z / 2 , 2) → B GL 1 K O is a p ointe d map, then Ω j is the auto-e quivalenc e of K O g iven by tensoring with the τ ( j ) -th p ower of r e al line bund les. Pr o of. Th is is true b y construction for the morphism K ( Z , 3) → B GL 1 K , whic h h as σ ( j ) = 1, constructed in [4]. Thus, it is true for all other definitions.  4. App endix: a geometric model of twiste d K -theor y W e outline h ow the symmetric sp ectrum mo d el of K -theory (or K O -theory) du e to Joac him [16] ma y b e used to twist K -theory in a fashion th at is n ice fr om b oth the geometric and homotopical p ersp ectiv es. The original mo del f or twisted K -theory sp ectra is due to Ati ya h and Segal [7], but it is not easy to s ee directly ho w it fits in to the h omotopical framework of twists in the sense of [5]. On th e other hand, in [5] and [4] it is h ard to see the concrete analysis and geometry w hic h w ere the original foun dation f or t wisted K -theory . J oac h im’s sp ectrum pro vides a v anta ge wh ere b oth views may b e app r eciated. In h is pap er, Joac h im w orks with r eal p eriod ic K -theory , but no alterations except replacing 8 b y 2 at v arious places are required to app ly the same arguments to complex p eriod ic K -theory . F or sim p licit y , w e presen t the complex case. Let H b e a fi xed infin ite dimensional separable Hilb ert space, and let H ∗ = H 0 ⊕ H 1 b e the Z / 2 -graded Hilb ert space w ith H 0 = H 1 = H . Th e sp ace U ( H ) is th e group of all u nitary op erators on H , equipp ed w ith the norm top ology; it is a contract ible space by Kuip er’s theorem. The quotien t of U ( H ) by the sub group U (1) of diagonal op erators is P U ( H ) ≃ K ( Z , 3). If F : H → H is an op erato r and P ∈ P U ( H ), then P − 1 F P is another op erator. Let F 1 ( H ∗ ) b e the space of self-adjoin t o dd F redholm op erato rs on H ∗ . Thus, any elemen t of F of F 1 ( H ∗ ) can b e represented by a matrix  0 ˜ F ˜ F ∗ 0  , where ˜ F is a F r edholm op erato r on H . Th e group P U ( H ) acts co nti nuously on F 1 ( H ∗ ) by P ·  0 ˜ F ˜ F ∗ 0  =  0 P − 1 ˜ F P P − 1 ˜ F ∗ P 0  , where we p urp osely confu se P with any op erato r in U ( H ) representing P . If one is only intereste d in t wisted K -groups, then this is already enough setup to do so. If P is a principal P U ( H ) norm -bundle on X , Atiy ah and Segal define K 0 P ( X ) as the group of homotop y classes of sections of the asso ciated bu ndle P × P U ( H ) F 1 ( H ∗ ) → X with fib er F 1 ( H ∗ ). Ho we v er, w e are of course intereste d in an entire sp ectrum. T o describ e t wisted K -theory sp ectrum of J oac h im, we r ecall the Clifford alg ebra C l ( n ), the complex Clifford algebra of C n equipp ed with the quadratic form q (( z 1 , . . . , z n )) = − n X i =1 z 2 i . UNIQUENESS OF TWISTED K-THEOR Y 15 There are ca nonical isomorp h isms C l ( p ) b ⊗ C l ( q ) → C l ( p + q ) for p, q ≥ 1, where b ⊗ denotes the Z / 2-graded tensor pro duct. F or detail s on Clifford algebras, consult [18]. If J ∗ is a Z / 2-graded Hilb ert space mo dule for C l ( n ), w e let F 1 C l ( n ) ( J ∗ ) den ote the sp ace of o dd self-a djoint F redholm op erators on J ∗ whic h are C l ( n )-mo dule morphisms wh en n is ev en or the complement of the t wo contract ible comp onents of this space identi fied in [9] w hen n is o dd. Let H ( n ) ∗ = ( C l (1) b ⊗H ∗ ) ˆ ⊗ n . Then, H ( n ) ∗ is naturally a graded C l ( n )-mo d ule. Joac him shows that the multiplicati on maps µ p,q : F 1 C l ( p ) ( H ( p ) ∗ ) × F 1 C l ( q ) ( H ( q ) ∗ ) → F 1 C l ( p + q ) ( H ( p + q ) ∗ ) , ( F , G ) 7→ F ⋆ G = F b ⊗ I d + I d b ⊗ G are con tinuous. Th e maps µ p,q are Σ p × Σ q -equiv arian t, wh ere Σ n acts n atur ally on F 1 C l ( n ) ( H ( n ) ∗ ). T o create based m aps, let K n = F 1 C l ( n ) ( H ( n )) + , top ologized as in [16, Section 3] so that F 1 C l ( n ) ( H ( n )) → K n is a homotop y equiv alence. T hen, the µ p,q induce contin uou s maps K p ∧ K q → K p + q for p, q ≥ 1. Let P U ( H ) act on F 1 C l ( n ) ( H ( n ) ∗ ) in the n atural w a y , through the diagonal action of U ( H ) on H b ⊗ n ∗ . Th is extends to a con tin uous action of P U ( H ) on K n , where P U ( H ) fixes the basep oint. Prop osition 4.1. The natur al P U ( H ) actions on the sp ac es of Jo achim’s sp e c trum K make K into a P U ( H ) -sp e ctrum in the sense that the actions ar e c omp atible with the multiplic ation maps and the symmetric gr oup actions . Pr o of. Th is is cle ar as P U ( H ) acts d iagonally on H b ⊗ n ∗ .  Corollary 4.2. The c onjugation action of P U ( H ) on Jo achim’s sp e ctrum K determines an A ∞ -map P U ( H ) → GL 1 K which delo ops to a map K ( Z , 3) ≃ B P U ( H ) → B GL 1 K . Referen ces [1] J. F. 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