A new approach to twisted K-theory of compact Lie groups
aa r X i v : . [ m a t h . K T ] A ug A NEW APPROACH TO TWISTED K -THEORY OFCOMPACT LIE GROUPS JONATHAN ROSENBERG
Abstract.
This paper explores further the computation of thetwisted K -theory and K -homology of compact simple Lie groups,previously studied by Hopkins, Moore, Maldacena-Moore-Seiberg,Braun, and Douglas, with a focus on groups of rank 2. We give anew method of computation based on the Segal spectral sequencewhich seems to us appreciably simpler than the methods used pre-viously, at least in many key cases. Introduction
This paper is an outgrowth of the paper [32] by Mathai and the au-thor, where we started studying a new approach to the computationof the twisted K -theory of compact simple Lie groups. This problemwas first studied by physicists (e.g., [36, 31, 19, 9, 11, 12, 22]) becauseof interest in the WZW (Wess-Zumino-Witten) model, which appearsboth in conformal field theory and as a string theory whose underlyingspacetime manifold is a Lie group, usually compact and simple. Instring theories in general, D-brane charges are expected to take theirvalues in twisted K -theory of spacetime, so the study of WZW modelsled to the study of twisted K -theory of compact Lie groups. The cal-culation of twisted K -theory of Lie groups turned out to be sufficientlyinteresting so that it was eventually taken up by mathematicians (Hop-kins, unpublished, but quoted in [31], and Douglas [16]). Mathematics Subject Classification.
Primary 19L50. Secondary 81T30,57T10, 55T15, 55R20.
Key words and phrases. compact Lie group, twisted K-theory, D-brane, WZWmodel, Segal spectral sequence, Adams-Novikov spectral sequence, Hurewicz map.Partially supported by U.S. NSF grant number DMS-1607162. The authorwould also like to thank the Isaac Newton Institute for Mathematical Sciences,Cambridge, U.K., for support and hospitality during the Programme on OperatorAlgebras: subfactors and their applications, and the Hausdorff Institute for Mathe-matics, Bonn, Germany, for support and hospitality during the Trimester Programon K-Theory and Related Fields, both in 2017, when some work on this paperwas undertaken. This work was also supported by U.K. EPSRC grant numberEP/K032208/1.
Section 2 then revisits the topic of computing twisted K -theory K • ( G, h ), for arbitrary choices of the twisting h . This has been thesubject of an extensive literature, most notably [31, 36, 11, 16], andthe results are rather complicated and hard to understand. However,this is an important problem because of the connection, discoveredby physicists, between these twisted K -groups and fusion rings andrepresentations of loop groups. We therefore present in Section 3 aneasier way of computing these twisted K -groups for compact simplyconnected simple compact Lie groups of rank two. Theorems 8, 14, 15,and 12 recover all the known results for rank-2 groups using our directmethods. Section 4 goes on to discuss non-simply connected groups.While many of the results of this paper were previously known, to thebest of my knowledge, Theorems 6, 9, 16, and 17 are new.I would like to thank the referee for a careful reading of the manu-script, for noticing a mistake, and for making several very useful sug-gestions. 2. Review and Machinery
In this paper we will deal exclusively with complex periodic K -theory, which is 2-periodic by Bott periodicity. Given a topologicalspace X (which for present purposes we can take to be compact) anda principal bundle P over X with fibers the projective unitary group P U ( H ) = U ( H ) / T of an infinite-dimensional separable Hilbert space H , P defines a bundle of spectra over X with fibers the K -theoryspectrum, and from this one can construct twisted K -theory of X ina standard way (see for example [40, 41, 6, 27, 5, 28]—this is only asmall subset of the literature).Since P U ( H ) ≃ CP ∞ is a K ( Z ,
2) space, bundles P as above areclassified by classes h ∈ H ( X, Z ), and we will denote the twisted K -theory or twisted K -homology of X by K • ( X, h ) or K • ( X, h ), eventhough, strictly speaking, h only determines these groups up to non-canonical isomorphism. (The non-canonicity will not be important inanything we do.)Let G be a compact Lie group. In this section we restrict to thecase where G is simple, connected, and simply connected, which isthe most studied case. Since G is then 2-connected with π ( G ) ∼ = Z ,we have H ( G, Z ) ∼ = Z . There is in fact a canonical isomorphism of H ( G, Z ) with Z (i.e., a canonical choice of generator), due to the factthat ( X, Y, Z )
7→ h X, [ Y, Z ] i , h , i the Killing form, defines a canon-ical 3-form on the Lie algebra of G , and thus a preferred orientationon H ( G, R ). In what follows we will mostly consider the case of WISTED K -THEORY OF COMPACT LIE GROUPS 3 twistings h > H ( G, Z ) is identified with Z ). Changing thesign of h preserves the isomorphism types of K • ( X, h ) and K • ( X, h ),and when h = 0, Hodgkin [26] proved that K • ( G ) is an exterior algebraover Z with n generators, where n = rank G . Thus taking h ≥ G = SU(2) = Sp(1), the twisted K -theory K • ( G, h ) for h = 0was already computed in [40], with the result that it is 0 in even degreeand Z /h in odd degree. The following result was proved in [11, 16]: Theorem 1 ([16, Theorem 1.1]) . For G a simple, connected, and sim-ply connected compact Lie group, rank G = n , and for twisting h > , K • ( X, h ) ( even as a ring ) is the tensor product of an exterior algebraover Z on n − odd-degree generators with a finite cyclic group of order c ( G, h ) a divisor of h . As we will see, for the cases at least of SU( n + 1), Sp( n ), and G ,this is not particularly difficult, and the hard part is to compute thenumbers c ( G, h ).Incidentally, the distinction between K • ( X, h ) and K • ( X, h ) is notparticularly important here. Since these twisted K -groups are the ac-tual K -groups of a continuous trace C ∗ -algebra A over G (having h asDixmier-Douady class), K • ( X, h ) ∼ = K −• ( A ) and K −• ( X, h ) ∼ = K • ( A )are related by the universal coefficient theorem for type I C ∗ -algebras A [13], which says that there is a canonical exact sequence0 → Ext Z ( K • +1 ( A ) , Z ) → K • ( A ) → Hom Z ( K • ( A ) , Z ) → . Since A here has finitely generated K -theory and K • ( A ) is torsion, K • ( A ) has to be torsion, and so K • ( X, h ) and K • ( X, h ) agree exceptfor a degree shift. Thus, for SU(2) and h = 0, K • ( G, h ) is Z /h in even degree instead of odd degree, and in all other cases (again, with h = 0), K • ( G, h ) and K • ( G, h ) are actually non-canonically isomorphic.In [11], a simple form for the numbers c ( G, h ) was proposed, and wasproven modulo a conjecture about the commutative algebra of Verlinderings. (The conjecture is that Verlinde rings are the coordinate ringsof complete intersection affine varieties.) This conjecture is known forSU( n +1), Sp( n ), and G , but to the best of my knowledge it might stillbe open for the spin groups and the other exceptional groups (see, e.g.,[10, 17, 3] for partial results). Thus the following should be regarded asa definitive theorem for SU( n + 1), Sp( n ), and G , but a “conditionaltheorem” in the other cases. Theorem 2 ([11], but note comments above) . Assume the conjecturein the paragraph above, which is known at least in types A n , C n , and G .For G a simple, connected, and simply connected compact Lie group, JONATHAN ROSENBERG rank G = n , and for twisting h > , the order c ( G, h ) of the torsionin K • ( X, h ) and K • ( X, h ) is given by the formula c ( G, h ) = h gcd( h,y ( G )) ,where the number y ( G ) is given by the following table: G y ( G ) A n = SU( n + 1) lcm(1 , , · · · , n ) B n = Spin(2 n + 1) lcm(1 , , · · · , n − C n = Sp( n ) lcm(1 , , · · · , n, , , · · · , n − D n = Spin(2 n ) ( n >
3) lcm(1 , , · · · , n − G F E E E c ( G, h ) in [16, Theorem 1.2] for theclassical groups and [16, p. 797] for G , but they have a totally differentform; for example, c (SU( n + 1) , h ) = gcd (cid:0)(cid:0) h + ii (cid:1) − , ≤ i ≤ n (cid:1) and c (Sp( n ) , h ) = gcd X − h ≤ j ≤− (cid:0) j +2( i − i − (cid:1) , ≤ i ≤ n ! . Appendix C in [31] proved that the Douglas and Braun formulascoincide in the case of SU( n + 1). In Propositions 11 and 13, we willalso see that the Douglas and Braun formulas coincide in the case ofSp(2) and G .We now move on to the question of how to prove results like Theorem1 and Theorem 2 in an easier way. Computation of K • (SU( n + 1) , h )was discussed in [19, 31, 36] using methods motivated by physics, basedon a study of wrapping of branes in WZW theories. However, thosepapers don’t quite give a mathematically rigorous proof, except in thesimplest cases. More sophisticated methods for computing K • ( G, h )were used in [11, 16], but the techniques are decidedly not elemen-tary. [11] used the Hodgkin K¨unneth spectral sequence in equivari-ant K -theory together with the calculations of Freed-Hopkins-Teleman[20, 21] , while [16] used a Rothenberg-Steenrod spectral sequence and K -theory of loop spaces. So our purpose here is to give a more direct There is indirect physics input here since Freed-Hopkins-Teleman showed thatthe equivariant twisted K -theory is the same as the Verlinde ring of the associatedWZW model. WISTED K -THEORY OF COMPACT LIE GROUPS 5 approach. We will need the Segal spectral sequence (from [42, Propo-sition 5.2]), though for our purposes it is easiest to reformulate it inhomology instead of cohomology. Theorem 3.
Let F ι −→ E pr −→ B be a fiber bundle, say of CW complexes,and let h ∈ H ( E ) . Then there is a homological spectral sequence H p ( B, K q ( F, ι ∗ h )) ⇒ K • ( E, h ) . Proof.
In the absence of the twist, this is precisely the homology dualof the spectral sequence of [42, Proposition 5.2], in the case where thecohomology theory used is complex K -theory. If h = 0, E = B and F = pt, this reduces to the usual Atiyah-Hirzebruch spectral sequence(AHSS) for K -homology. Similarly, if E = B and F = pt, but h = 0,this is the AHSS for twisted K -homology. To get the general case, wefilter B by its skeleta. This induces a filtration of K • ( E, h ) for whichthis is the induced spectral sequence (by Segal’s proof). (cid:3)
Remark 4.
The spectral sequence of Theorem 3 will be strongly con-vergent if the ordinary homology of B is bounded. This will be thecase if B is weakly equivalent to a finite dimensional CW complex, andin particular covers all the cases considered in this paper.As a simple application of Theorem 3, we can immediately prove theeasiest part of Theorem 1. (However, this result is rather weak and wewill want to improve on it.) Theorem 5.
Let G be a simple, connected, and simply connected com-pact Lie group. For any twisting h > , K • ( X, h ) is a finite abeliangroup, and all elements have order a divisor of a power of h . In par-ticular, if h = 1 , then K • ( X, h ) vanishes identically, and if h = p r is aprime power, then K • ( X, h ) is a p -primary torsion group.Proof. First observe that G contains a subgroup H ∼ = SU(2) ∼ = Sp(1) ∼ =Spin(3) such that the inclusion H ֒ → G is an isomorphism on π j , j ≤ H → G → G/H . From Theorem 3, we get a spectralsequence converging to K • ( X, h ), with E p,q = H p ( G/H, K q (SU(2) , h )).But K q (SU(2) , h ) is non-zero only for q even, where it is Z /h . Since E is thus torsion with all elements of order dividing h , the same is trueof E ∞ . And even if there are nontrivial extensions involved in goingfrom E ∞ to K • ( X, h ), the result still follows.It still remains to verify the structural statement. For the classicalgroups, SU(2) sits in SU( n ), Spin(3) sits in Spin( n ), and Sp(1) sits inSp( n ) for all relevant values of n . The fact that these inclusions are JONATHAN ROSENBERG isomorphisms on π is standard, and follows from the classical fibrations SU( n ) → SU( n + 1) → S n +1 , Sp( n ) → Sp( n + 1) → S n +3 , Spin( n ) → Spin( n + 1) → S n , together with the facts that SU(2) and G are both 2-connected. In thecase of G , there is a fibration SU(2) → G → V , [8, Lemme 17.1]. Inthe case of F , there is a fibration Spin(9) → F → OP [7]. For the E -series we can use the fibration F → E → E /F along with whatwe know about F , then use the inclusions E ֒ → E ֒ → E . (cid:3) In order to apply Theorem 3 more precisely, in some cases we willneed an explicit description of some of the differentials. Thus the fol-lowing theorem is useful. It applies with basically the same proof toother exceptional homology theories, though we won’t need these here.
Theorem 6.
In the situation of Theorem , suppose that ι ∗ is anisomorphism ( or even just an injection ) on H ( so that the twistingclass on E can be identified with the restricted twisting class on F ) ,the differentials d , · · · , d r − leave E r, = H r ( B, K ( F, ι ∗ h )) unchanged,and one has a class x in this group which comes from a class α ∈ π r ( B ) under the composite π r ( B ) Hurewicz −−−−−→ H r ( B, K ( F, ι ∗ h )) . Then d r ( x ) ∈ E r ,r − , a quotient of K r − ( F, ι ∗ h ) , is the image of α under the composite π r ( B ) ∂ −→ π r − ( F ) Hurewicz −−−−−→ K r − ( F, ι ∗ h ) , where the first map is the boundary map in the long exact sequence ofthe fibration F ι −→ E pr −→ B . ( The Hurewicz map in twisted homology iseasy to understand as follows, at least if r ≥ : ι ∗ h defines a principal K ( Z , -bundle P ι ∗ h over F , and the pull-back of this bundle to the totalspace P ι ∗ h is trivial, so there is a natural map K • ( P ι ∗ h ) → K • ( F, ι ∗ h )[29, p. 536] ; the Hurewicz map is the composite π r − ( F ) ∼ = π r − ( P ι ∗ h ) ∧ −→ π r − ( P ι ∗ h ∧ K ) = K r − ( P ι ∗ h ) → K r − ( F, ι ∗ h ) , where π r − ( F ) ∼ = π r − ( P ι ∗ h ) if r > , by the long exact sequence of thefibration K ( Z , → P ι ∗ h → F . ) Proof.
Since the class x by assumption was not changed under theearlier differentials, and since the twisting comes entirely from the fiber,we can, without loss of generality, reduce to the case where B is a sphere S r and thus E = ( R r × F ) ∪ F , where R r × F corresponds to pr − of WISTED K -THEORY OF COMPACT LIE GROUPS 7 the open r -cell in the base. In this case the spectral sequence comesdirectly from the long exact sequence(1) · · · → K r ( F, ι ∗ h ) ι ∗ −→ K r ( E, h ) → K r ( E, F, h ) ∼ = K r ( E r F, h ) ∼ = K ( F, ι ∗ h ) ∂ −→ K r − ( F, ι ∗ h ) → · · · . Note here that H r ( B, K ( F, ι ∗ h )) can be identified with the term K ( F, ι ∗ h ) in (1). So the differential d r is the boundary map in (1),and we use commutativity of the diagram π r ( B ) ∂ / / Hurewicz (cid:15) (cid:15) π r − ( F ) Hurewicz (cid:15) (cid:15) H r ( B, K ( F, ι ∗ h )) ∂ / / K r − ( F, ι ∗ h ) , a consequence of naturality of the Hurewicz homomorphism. (cid:3) Another useful result for us will be the “universal coefficient theo-rem” of Khorami [29].
Theorem 7 (Khorami [29]) . Let X be a space ( say, a compact CW-complex ) , let h ∈ H ( X, h ) , and let P h be the associated principal bun-dle with structure group P U ( H ) ≃ CP ∞ . Then K • ( X, h ) ∼ = K • ( P h ) ⊗ R Z , where R = K ( CP ∞ ) is a ring under Pontrjagin product acting on K • ( P h ) via the principal CP ∞ -bundle structure on P h and on Z viathe ring homomorphism R → Z sending β j , j = 0 or , β j , j > . Here R is the free Z -module on generators β , β , · · · , where β j is dual to ( γ − j , γ the Hopf line bundle in K ( CP ∞ ) . In fact, Khorami mentions at the end of his paper that he suspectsthat his theorem can be used to recover Theorem 1, though he givesno details except in the case G = SU(2), where he points out thatfor P h as in Theorem 7, K • ( P h ) ∼ = R/ ( hβ ) and thus K • (SU(2) , h ) ∼ = R/ ( hβ ) ⊗ R Z ∼ = Z /h .3. Twisted K -theory of rank-two simple Lie groups The case of
SU(3) . To explain how we use these tools, we startwith the simplest nontrivial case, namely G = SU(3), which was firsttreated in [31, 36]. We recall the result: Theorem 8.
Let h be a positive integer, viewed as a twisting class for SU(3) . Then ( in both even and odd degree ) , K • (SU(3) , h ) ∼ = Z /h if h is odd, K • (SU(3) , h ) ∼ = Z / ( h/ if h is even. JONATHAN ROSENBERG
Proof.
We use the standard fibrationSU(2) = S ι −→ SU(3) pr −→ S . Here ι ∗ is an isomorphism on H , and we already know that K • (SU(2) , h )is Z /h in even degree, 0 in odd degree. So apply Theorem 3 and The-orem 6. The picture of the spectral sequence is given in Figure 1. To K • (SU(2) , h )4 • O O • • • • • / / b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ H • ( S )0 1 2 3 4 5 Figure 1.
The Segal SS for twisted K -homology ofSU(3). Heavy dots indicate copies of Z /h . The diag-onal arrow shows the differential d .compute the differential d , we use Theorem 6, along with the exacthomotopy sequence π (SU(2)) → π (SU(3)) → π ( S ) ∂ −→ π (SU(2)) → π (SU(3)) . Here it is classical that π (SU(2)) ∼ = π (SU(2)) ∼ = Z /
2, and π (SU(3)) ∼ = Z , π (SU(3)) = 0 by [35]. Thus the boundary map ∂ in this sequencehas kernel of index 2. Now we need to understand the Hurewicz maps π ( S ) → H ( S , K (SU(2) , h )) ,π (SU(2)) → K (SU(2) , h ) ∼ = K (SU(2) , h ) . The generator of π ( S ) is suspended from the generator of π ( S ),just as H ( S , K (SU(2) , h )) ∼ = K (SU(2) , h ) via suspension, so thegenerator 1 of π ( S ) goes to the generator 1 of the cyclic group Z /h .To finish the proof, we need the following Theorem 9. Thus we see that d in the Segal spectral sequence has kernel of order 2 if h is even andtrivial kernel if h is odd, and the result follows. (cid:3) Theorem 9.
Let h ∈ Z , h = 0 . Then the Hurewicz map π ( S ) → K ( S , h ) ∼ = Z /h is non-zero if and only if h is even. WISTED K -THEORY OF COMPACT LIE GROUPS 9 Proof.
Since π ( S ) ∼ = Z /
2, obviously the Hurewicz map is 0 if h is odd,since then there is no 2-torsion in K ( S , h ). So assume h is even. TheHurewicz map in twisted K -homology is a bit more mysterious thanthe usual Hurewicz map in K -homology, but we can apply Theorem7 to help clarify things. Let P h be the principal CP ∞ -bundle over S classified by the non-zero integer h ∈ Z ∼ = H ( S , Z ). The Serre spec-tral sequence for the fibration CP ∞ → P h → S has only two columns,so in cohomology the only differential is d , which sends the generator u of H ( CP ∞ ) to h times the usual generator y of H ( S ). Since d is aderivation, d ( u n ) = nhu n − y , and the homology differential is similar,but just points in the opposite direction. Hence H n ( P h ) ∼ = Z / ( nh ) for n ≥
1, and H odd ( P h ) vanishes. Thus the AHSS for P h collapses, andKhorami computed that K ( P h ) ∼ = R/ ( hβ ) as an R -module, where R = K ( CP ∞ ) (with multiplication defined by Pontrjagin product),and K ( S , h ) ∼ = Z /h is gotten from this by tensoring with Z (viewedas an R -module under β β j j > S h −→ S of degree h pulls the CP ∞ -bundle P over S back to the CP ∞ -bundle P h over S . So we get a pull-back square(2) CP ∞ / / = (cid:15) (cid:15) P h / / f h (cid:15) (cid:15) S h (cid:15) (cid:15) CP ∞ / / P / / S , and the map f h : P h → P induces a map of R -modules R/ ( hβ ) → R/ ( β ) on K -homology. Comparison of the Serre spectral sequencesalso shows that ( f h ) ∗ is surjective on integral homology. From the longexact homotopy sequences associated to the two rows, we also see that π j ( P h ) ∼ = Z /h for j = 2 and ∼ = π j ( S ) for j ≥
4, and that the map f h : P h → P induces multiplication by h on π j for j ≥ P is the homotopy fiber of the canonical map S → K ( Z ,
3) inducing an isomorphism on π , and thus P → S → K ( Z , S . Thus P is 3-connectedand π j ( P ) ∼ = π j ( S ) for j ≥
4. So by the Hurewicz theorem, theHurewicz map π ( P ) → H ( P ) ∼ = Z / P (cid:15) (cid:15) i h ~ ~ P h / / S h / / K ( Z , , where the downward solid arrow is the bundle projection of the CP ∞ -bundle P over S . From the exact homotopy sequence[ P , P h ] → [ P , S ] → [ P , K ( Z , , we see that we get a lifting i h : P → P h , which is the first stage of thePostnikov fibration P i h −→ P h → K ( Z /h,
2) for P h . Unlike the map f h in the other direction, i h is not a map of CP ∞ -bundles.Putting everything together, we see that the Hurewicz map π j ( S ) → K j ( S , h ) (for j ≥ π j ( S ) ∼ = π j ( P ) ( i h ) ∗ −−→ ∼ = π j ( P h ) → K j ( P h ) = R/ ( hβ ) → Z /h. Now K even ( P ) and K even ( P h ) have skeletal filtrations F = Z ⊂ F ⊂ F ⊂ · · · , where F j is generated (additively) by the images of β , · · · , β j ,and since the AHSS for K -homology of P collapses, we have maps F j → H j identifying F j /F j − with H j . The image of π under theHurewicz map must lie in F (just on dimensional grounds). Thus sincethe Hurewicz map π ( P ) → H ( P ) is an isomorphism, the Hurewiczmap in K -homology for P maps π ( P ) ∼ = Z / β , of order 2 in R/ ( β ), that maps onto H ( P ). (Onecan compute that 2 β = β − β lies in the ideal generated by β .)The Hurewicz map in ordinary homology π ( P h ) → H ( P h ) can beidentified with the edge homomorphism ( i h ) ∗ : H ( P ) → H ( P h ) as-sociated to the Serre spectral sequence for P i h −→ P h → K ( Z /h, K ( Z /h,
2) is a bit complicated, but we only need its 2-primary partin low degree. When h = 2, Serre showed that H • ( K ( Z /h, , F )is a polynomial ring on generators ι, Sq ι, Sq Sq ι , · · · , where ι isthe canonical generator in degree 2 [24, p. 500]. Thus the F -Bettinumbers of K ( Z / ,
2) are 1 , , , , , , · · · . In particular we can seefrom this that rank H ( K ( Z / , Z ) = 1. The complete calculation of H ( K ( Z /h, Z ) may be found in [18, Theorem 21.1] and in [14, 15](which even computes the integral homology in arbitrary degree, atleast in principle), and it turns out that H ( K ( Z /h, Z ) ∼ = Γ ( Z /h ),where Γ is the functor defined in [45], and for h even, this is a cyclicgroup of order 2 h , while H ( K ( Z /h, Z ) ∼ = Z / H ( P h ; Z ) ∼ = Z / (2 h ). Thus the red arrow in Figure 2has to be an isomorphism and the edge homomorphism ( i h ) ∗ : H ( P ) → H ( P h ), which is the Hurewicz map, vanishes. One way of thinkingabout this is that we can view the Hurewicz map as being about theembedding of an S in P h via the generator η of π ( P h ). This spheredoesn’t bound a disk (if it did, the homotopy class of η would be WISTED K -THEORY OF COMPACT LIE GROUPS 11 H • ( P )4 • O O • • • • • • • • / / d b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ H • ( K ( Z /h, Figure 2.
The Serre SS for the first Postnikov fibrationof P h .trivial), but it does bound a homology chain, and even an orientedmanifold (in this low dimension, oriented bordism is almost the sameas homology). The question for us now is: does it bound a Spin c man-ifold? This determines the Hurewicz map in K -homology since (whenwe localize everything at 2) the low-degree summand of M Spin c is ku (connective K -theory) ([38, §
8] and [43, Ch. XI]) and the K -homologyclass of this 4-sphere, which is the image of the Hurewicz map, comesfrom its ku -homology class.So let’s reconsider Figure 2 redone in ku -homology, which is Figure3. Since ku • ( P h ) is all concentrated in even degree, everything in odddegree must cancel. We have f ku ( P ) = 0, f ku ( P ) ∼ = H ( P ) ∼ = Z / f ku ( P h ) ∼ = H ( P h ) ∼ = Z /h , and f ku ( P h ) is an extension of H ( P h ) ∼ = Z / (2 h ) by H ( P h ) ∼ = Z /h . The Hurewicz map π ( P h ) → K even ( P h ) nowcomes from the edge homomorphism f ku ( P ) → f ku ( P h ), and so therelevant question is which of the two arrows ( d and d ) starting at theposition (5 ,
0) in Figure 3 is non-zero. ku • ( P )4 • O O • • • • • • • • d d d ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ • • • • / / d b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ d d d ❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏❏ H • ( K ( Z /h, Figure 3.
The Segal SS in ku for the first Postnikovfibration of P h .To answer this question we can consider the map P h → K ( Z /h, ku • ( K ( Z /h, e K • ( K ( Z /h, ku • ( K ( Z /h, d : H → H is non-zero, since the dual differential for computing ku -cohomology, d = Sq : H ( K ( Z /h, Z ) → H ( K ( Z /h, Z )is non-zero, and thus the blue arrow in Figure 3 is non-trivial. Thisimplies that the map f ku ( P ) → f ku ( P h ) is non-trivial, and the imagemust go to (cid:0) h (cid:1) β in e K even ( P h ). So under the map K ( P h ) → K ( S , h ),it maps to h in Z /h , which is non-trivial. This gives the desired result. (cid:3) The Braun-Douglas Theorem for
Sp(2) ∼ = Spin(5) . We beginby showing that the Douglas and Braun formulas coincide in the caseof Sp(2) ∼ = Spin(5). Here it is convenient to use the following standarddefinition. Definition 10.
Fix a prime p , and for x a positive integer, let ν p ( x )be the number of times that p divides x . In other words, ν p ( x ) is WISTED K -THEORY OF COMPACT LIE GROUPS 13 defined by the property that x = p ν p ( x ) x ′ , where gcd( x ′ , p ) = 1. Thus x = Q p prime p ν p ( x ) . Proposition 11.
Let h be a positive integer, let h ′ = h gcd( h, lcm(1 , , h gcd( h, , which is the Braun formula for c ( G , h ) , and let h ′′ = gcd (cid:0) h, (cid:0) h (cid:1) + (cid:0) h (cid:1)(cid:1) , which is Douglas’ formula for c ( G , h ) in [16, Theorem 1.2] . Then h ′ = h ′′ .Proof. It will suffice to show that ν p ( h ′ ) = ν p ( h ′′ ) for all primes p .Clearly ν p ( h ′ ) = ν p ( h ′′ ) = 0 if p does not divide h . So assume p divides h , and we’ll consider in turn the cases of p = 2 ,
3, and p >
3. If p = 2and h is even, then ν ( h ′ ) = ν ( h ) −
1, while 2 (cid:0) h (cid:1) = h ( h − h − (cid:0) h (cid:1) = h ( h −
1) has the same divisibility by 2 as h .Thus ν ( h ′′ ) = ν ( h ) − ν ( h ′ ). If p = 3 and h ≡ ν ( h ′ ) = ν ( h ) −
1, while ν (cid:0)(cid:0) h (cid:1)(cid:1) = ν ( h ) and ν (cid:0) (cid:0) h (cid:1)(cid:1) = ν (cid:18) h ( h − h − (cid:19) = ν ( h ) − . Thus ν ( h ′′ ) = ν ( h ) −
1. Finally, if p > p divides h , then ν p ( h ) = ν p (cid:0)(cid:0) h (cid:1)(cid:1) = ν p (cid:0) (cid:0) h (cid:1)(cid:1) and so ν p ( h ′ ) = ν p ( h ′′ ) = ν p ( h ). (cid:3) Theorem 12 (Braun-Douglas Theorem for Sp(2)) . Let h be a positiveinteger and let G = Sp(2) ∼ = Spin(5) . Then in any degree, K • ( G, h ) iscyclic of order h gcd( h, .Proof. We argue as in Theorem 8, using the usual fibration Sp(1) → G → S , where Sp(1) ∼ = SU(2) ∼ = S . The inclusion of Sp(1) into Sp(2)induces an isomorphism on H , and the groups K j (Sp(1) , h ) are cyclicof order h for j even, zero for j odd. We just need to compute thedifferential in the Segal spectral sequence d : H ( S , K (Sp(1) , h )) → K (Sp(1) , h ) . As explained in Theorem 1, this differential is related to the boundarymap ∂ in the long exact homotopy sequence π (Sp(1)) → π (Sp(2)) → π ( S ) ∂ −→ π (Sp(1)) → π (Sp(2)) . Here π ( S ) ∼ = Z / π ( S ) ∼ = Z / π (Sp(2)) ∼ = Z , and π (Sp(2)) =0. (See for example [24, p. 339] for the homotopy groups of S and[35] for the homotopy groups of Sp(2).) From this exact sequence, ∂ is surjective onto π ( S ) ∼ = Z /
12. Away from the primes 2 and 3, ∂ vanishes and so does the differential in the spectral sequence for K • ( G, h ). So we only need to analyze what happens at the primes2 and 3. This involves understanding the Hurewicz homomorphism π ( S ) → K ( S , h ) or π ( P h ) → K ( P h ), where P h is the principal CP ∞ -bundle over SU(2) associated to the twist h as in the proof ofTheorem 9.We can proceed as we did there. If h is divisible by neither 2 nor 3,then obviously the Hurewicz homomorphism is zero. If h is divisible by3 and we localize at 3, then everything is largely as in the proof of The-orem 9, and we keep the notation used there. The fiber P of the Post-nikov fibration P h −→ P h → K ( Z /h,
2) is 3-locally 5-connected, so bythe mod- C Hurewicz Theorem (with C the Serre class of prime-to-3 tor-sion groups), the 3-torsion subgroup Z / π ( S ) ∼ = π ( P ) ∼ = π ( P h )maps isomorphically to H ( P ) ∼ = Z /
3. We have H ( P h ) ∼ = Z / (3 h )and H ( K ( Z /h, ∼ = Z / (3 h ) by [18, Theorem 21.1], so the differential d : H ( K ( Z /h, → H ( P ) in the analogue of Figure 2 must be non-zero and the Hurewicz homomorphism in ordinary homology, which canbe identified with the map ( i h ) ∗ : H ( P ) → H ( P h ), vanishes. To studythe corresponding map in ku -homology, we compare the diagram analo-gous to Figure 3 with the AHSS for ku • ( K ( Z /h, H • ( K ( Z /h, F p )has generators ι , β r ι , P β r ι , etc. ( β r the r th-power Bockstein, 3 r thebiggest power of 3 dividing h ) and the first nontrivial differential in theAHSS for computing K • ( K ( Z / r , H • ( K ( Z / r , Z ) is (upto a non-zero constant) d = β r P : β r ι β r P β r ι . This is dualto a non-zero differential d with target H ( K ( Z / r , Z ), and so theHurewicz map ku ( P ) → ku ( P h ) will be non-zero, just as in the proofof Theorem 9.The hardest step is the 2-local calculation in the case where h is even,which involves the 2-local part of the Hurewicz map π ( P h ) → ku ( P h )for h even. We defer this calculation to Theorem 16. (cid:3) The Braun-Douglas Theorem for G . The following resultand its proof are partially modeled on Appendix C in [31], and provesthat the Douglas and Braun formulas coincide in the case of G . Proposition 13.
Let h be a positive integer, let h ′ = h gcd( h, , and let h ′′ = gcd (cid:18) h, (cid:0) h +22 (cid:1) − , ( h + 1)( h + 2)(2 h + 3)(3 h + 4)(3 h + 5)120 − (cid:19) . WISTED K -THEORY OF COMPACT LIE GROUPS 15 Note that h ′ is Braun’s formula for c ( G , h ) and h ′′ is Douglas’ formulafor c ( G , h ) . Then h ′ = h ′′ .Proof. We again use Definition 10. It will suffice to show that ν p ( h ′ ) = ν p ( h ′′ ) for all primes p .First consider p = 2. If h is odd, then ν ( h ) = ν ( h ′ ) = ν ( h ′′ ) = 0.If ν ( h ) = 1, then since ν (gcd( h, ν ( h ′ ) = 0. Consider h ′′ .We have h ≡ h + 2 ≡ (cid:0) h +22 (cid:1) is even, hence (cid:0) h +22 (cid:1) − ν ( h ′′ ) = 0 = ν ( h ′ ) in this case. If ν ( h ) ≥ ν (60) = 2, ν ( h ′ ) = ν ( h ) −
2. But (cid:0) h +22 (cid:1) − ( h +2)( h +1) − = h ( h +3)2 . Thus if ν ( h ) ≥ ν (gcd( h, (cid:0) h +22 (cid:1) − ν ( h ) −
1. On theother hand( h + 1)( h + 2)(2 h + 3)(3 h + 4)(3 h + 5)120 −
1= 18 h + 135 h + 400 h + 585 h + 422 h + 120 − h (18 h + 135 h + 400 h + 585 h + 422)120 . The denominator is 2 ·
15 and since h ≡ ≡ ν of the numerator is ν ( h ) + 1. Thus ν of this fraction is ν ( h ) + 1 − ν ( h ) − ν ( h ′ ). So again ν ( h ′ ) = ν ( h ′′ ).Next, consider p = 3. If ν ( h ) = 0, then clearly ν ( h ′ ) = ν ( h ′′ ) = 0.If ν ( h ) ≥
1, then ν (gcd( h, ν ( h ′ ) = ν ( h ) −
1. Onthe other hand, ν (cid:16) h ( h +3)2 (cid:17) > ν ( h ), so taking the gcd with h ( h +3)2 doesn’t change ν ( h ). With regard to h (18 h +135 h +400 h +585 h +422)120 , if h is divisible by 3, then ν of the numerator is the same as for h (since422 ≡ ν (120) = 1, so ν of the fraction, as well as ν ( h ′′ ), is ν ( h ) −
1, which agrees with ν ( h ′ ).Consider now p = 5. If ν ( h ) = 0, then clearly ν ( h ′ ) = ν ( h ′′ ) = 0.If ν ( h ) ≥
1, then ν (gcd( h, ν ( h ′ ) = ν ( h ) −
1. For h divisible by 5, h + 3 ≡ ν (cid:16) h ( h +3)2 (cid:17) = ν ( h ), and takingthe gcd with h ( h +3)2 doesn’t change ν ( h ). Again, for h divisible by 5,18 h + 135 h + 400 h + 585 h + 422 ≡ ν (120) = 1, so ν of the big fraction, as well as ν ( h ′′ ), is ν ( h ) −
1, which agrees with ν ( h ′ ).Finally, suppose p ≥
7. Then 2, 60, and 120 are all relatively primeto p . If ν p ( h ) = 0, then clearly ν p ( h ′ ) = ν p ( h ′′ ) = 0. If ν p ( h ) ≥
1, then ν p (gcd( h, ν p ( h ′ ) = ν p ( h ). On the other hand, if ν p ( h ) ≥ then ν p (cid:18) h ( h + 3)2 (cid:19) = ν p ( h ) , and ν p (cid:18) h (18 h + 135 h + 400 h + 585 h + 422)120 (cid:19) ≥ ν p ( h ) , so ν p ( h ′′ ) = ν p ( h ) = ν p ( h ′ ). This concludes the proof. (cid:3) We now want to give an elementary but rigorous proof of the Braun-Douglas Theorem for G . We start with analysis of the odd torsion. Forconvenience in what follows, if x is a positive integer, let x odd denotethe maximal odd factor of x . Of course, x odd = Q p prime ≥ p ν p ( x ) . Theorem 14.
Let h be a positive integer, viewed as a twisting classon G . Then K • ( G , h ) is a finite torsion group in all degrees. Its oddtorsion ( in any degree ) is cyclic of order c ( G , h ) odd = h odd / gcd( h odd , . Proof.
We use the fibration [8, Lemme 17.1]SU(2) → G → V , , and get from Theorem 3 a spectral sequence E p,q = H p ( V , , K q (SU(2) , h )) ⇒ K • ( G , h ) . (The restriction map H ( G ) → H (SU(2)) is an isomorphism.) Here K q (SU(2) , h ) ∼ = Z /h for q even and is 0 for q even. Since E is torsion,so is K • ( G , h ). The Stiefel manifold V , is 11-dimensional and hasonly one nontrivial homology group below the top dimension, namelya Z / § V , becomes homotopy equivalent to S by the Hurewicz Theoremmodulo the Serre class of 2-primary torsion groups. Thus from thepoint of view of odd torsion, we are in the situation of Theorem 6 witha unique differential d . We have the long exact homotopy sequence π ( G ) → π ( V , ) → π (SU(2)) → π ( G ) , and π (SU(2)) ∼ = Z / π ( G ) = 0 [34]. Thus the boundary map π ( V , ) → π (SU(2)) has kernel of order 15, and the theorem followsfrom Theorem 6, exactly as in the proof of Theorem 8.Let’s first deal with the 5-primary torsion. If gcd( h,
5) = 1, then K • ( G , h ) can’t have 5-primary torsion, by Theorem 5. So assume h is divisible by 5 and localize everything at 5. Once again, let’s use thenotation of Theorem 9. The first 5-primary torsion in the homotopygroups and homology groups of P occurs in degree 10. So the Hurewiczmap π ( S ) ∼ = π ( P ) → H ( P ) ∼ = Z / WISTED K -THEORY OF COMPACT LIE GROUPS 17 is the Hurewicz map to f ku ( P ). Just as in the proof of Theorem 9,we need to show that the map f ku ( P ) → f ku ( P h ) is injective onthe 5-torsion. And again, we do this by comparing the Segal spectralsequence H p ( K ( Z /h, , ku q ( P )) ⇒ ku • ( P h )with the AHSS for computing ku • ( K ( Z /h, (cid:3) Theorem 15.
Let h be a positive integer, viewed as a twisting classon G . Then K • ( G , h ) is a finite torsion group in all degrees. Its -primary torsion ( in any degree ) is cyclic of order c ( G , h ) -primary = 2 max(0 ,ν ( h ) − . In other words, ν ( c ( G , h )) = ( , ν ( h ) ≤ ,ν ( h ) − , ν ( h ) > . Proof.
First suppose that ν ( h ) ≤
1. This time we use the fibrationSU(3) → G → S , coming from the action of G on the unit sphere of the imaginary octoni-ans. We can apply Theorem 8 together with Theorems 3 and 6. The in-clusion SU(3) ֒ → G induces an isomorphism on H , and K • (SU(3) , h )has no 2-torsion, and that proves the theorem in this case. Note thatin the case ν ( h ) = 1, we see that the picture for the Segal spectralsequence attached to SU(2) → G → V , has to look like Figure 4, with the red arrows isomorphisms, so thateverything cancels out.If ν ( h ) ≥
2, then in Figure 4, the red d arrows will still be non-zerofor the same reasons as before , but this time the copies of Z / ν ( h ) arereduced to Z / ν ( h ) − at the E stage. At this point the dots in the H and H columns have disappeared and the spectral sequence nowlooks like one for a fibration with S as the base and with Z / ν ( h ) − The map E , → E , is determined by Theorem 6 and the calculation of theHurewicz map in twisted K -homology from Theorem 9. The other map E , → E , is linked to this one by the module action of K • ( G ), which is an exterioralgebra over Z , on K • ( G , h ). K • (SU(2) , h )4 • O O · · • · · • · b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ · • / / b b ❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊❊ H • ( V , )0 1 2 3 4 5 6 7 8 9 10 11 Figure 4.
The Segal SS for 2-primary twisted K -homology of G when ν ( h ) ≥
1. Heavy dots indicatecopies of Z / ν ( h ) . Light dots indicate copies of Z / d .in even degrees in the twisted K -homology of the fiber. There is stillthe differential d : E ,j → E ,j +10 to be reckoned with. We claim that this differential has image isomor-phic to Z /
2, which will give the desired result.This comes about in the following way. If G really fit into a fibration S → G → S and we were looking at the associated 2-local Segalspectral sequence for computing K • ( G , h ), then d would vanish asa consequence of Theorem 6, since π ( S ) has no 2-torsion. But inour situation, the groups Z / ν ( h ) − in E , k and in E , k +10 are reallydifferent. The former arose as kernels of d , a map Z / ν ( h ) → Z / cokernels of a map Z / → Z / ν ( h ) (see Figure 4again). By [26], K • ( G ) ∼ = V ( x , x ), and exterior algebra on two oddgenerators, and the module action of x ∈ K • ( G ) on K • ( G , h ) and onthe spectral sequence sets up an isomorphism E , k → E , k which hasto pass to an isomorphism K k ( G , h ) → K k +11 ( G , h ). This can onlyhappen if E , k and E , k are each quotients of an index-2 subgroupof K k ( S , ν ( h ) ) ∼ = Z / ν ( h ) by a subgroup of order 2, and we end upwith a 2-torsion subgroup of order 2 ν ( h ) − .An alternative way to prove the theorem when ν ( h ) ≥ → G → S tocompute the order of KU • ( G , h ), along with the previous argumentwith the other spectral sequence to deduce that the group in each WISTED K -THEORY OF COMPACT LIE GROUPS 19 degree is cyclic. For example, suppose h = 4 and consider the dif-ferential H ( S , K (SU(3) , h )) → H ( S , K (SU(3) , h )). Both groupshere are cyclic of order 2, and the generator of the domain is the im-age of the Hurewicz map from π ( S ). Since π (SU(3)) ∼ = Z [35] and π ( G ) = 0 [34], the boundary map ∂ : π ( S ) → π (SU(3)) is anisomorphism and we just need to determine what happens to the gen-erator of π (SU(3)) under the Hurewicz map to K (SU(3) , h ). But theHurewicz map π (SU(3)) → H (SU(3)) has image which is of index2 [44, (4.3)], so the image of π (SU(3)) in K (SU(3) , h ) correspondsexactly to the kernel of d : H ( S , K (SU(2) , → H ( S , K (SU(2) , K • (SU(3) ,
4) from the fibration S → SU(3) → S , and this is the entire group K (SU(3) , h ) ∼ = Z /
2. Thus inthe Segal spectral sequence, the differential H ( S , K (SU(3) , h )) → H ( S , K (SU(3) , h ))is an isomorphism. The differential with the other parity can alsobe seen to be an isomorphism, and in this way one can show that K • ( G ,
4) = 0. (cid:3)
Analysis of Hurewicz maps.
To complete all the theoremsfrom Section 3, we need the following technical result.
Theorem 16.
Let h be a positive integer and let P h be the prin-cipal CP ∞ -bundle over S classified by a map of degree h from S to K ( Z , , as in the proof of Theorem 9. Then the Hurewicz maps π j ( S ) ∼ = π j ( P h ) → ku j ( P h ) are injective on the (1) 2 -torsion Z / when j = 4 and h is even ( this case was in The-orem 9) , (2) 2 -primary torsion Z / when j = 6 and h is divisible by — if h ≡ , the map is non-zero ( these cases were neededfor Theorem 12) , (3) 3 -torsion Z / when j = 10 and divides h ( this case was neededfor Theorem 14) , (4) 5 -torsion Z / when j = 10 and divides h ( this case was alsoneeded for Theorem 14) .In all cases, the image maps injectively under the quotient map K ( P h ) → K ( S , h ) . Before starting the proof, let’s explain a bit about strategy. Wehave lumped all of these results together and included cases (1) and(4) (even though we already did those by another method) to illustrate a common method of attack using either the Adams-Novikov SpectralSequence (ANSS) or the classical Adams Spectral Sequence (ASS). Forthis we localize at the appropriate prime (2, 3, or 5). We will usethe fact that the unstable Hurewicz homomorphism π j ( P h ) → ku j ( P h )factors through the stable Hurewicz homomorphism π sj ( P h ) → ku j ( P h ),and the latter can be computed by comparing π s • and ku • with the helpof the ANSS or the ASS. Proof.
We localize at the appropriate prime. It suffices to look at theassociated Brown-Peterson homology BP • , since M U • splits as a wedgeof shifted copies of BP and K • can be recovered from M U • by theConner-Floyd isomorphism M U • ( P h ) ⊗ MU • K • ∼ = K • ( P h ). The ANSShas the form ([39, 23] or [1, Part III, § sBP • BP (Σ t BP • , g BP • ( P h )) ⇒ { S t − s , P h } , and the edge homomorphism { S t , P h } → Hom BP • BP (Σ t BP • , g BP • ( P h ))is the stable Hurewicz map in BP -homology. One can compare thiswith study of the classical Adams spectral sequence (ASS)Ext s A ∗ (Σ t F p , e H • ( P h ; F p )) ⇒ { S t − s , P h } , for which the edge homomorphism { S t , P h } → Hom A ∗ (Σ t F p , e H • ( P h ; F p ))is the stable Hurewicz map in ordinary mod- p homology. (We havedeliberately ignored a few localization and completion issues whichdon’t cause problems in our case. Here A ∗ is the dual Steenrod algebraat the prime p .) Proof of case (4).
Let’s start with case (4), taking p = 5, starting withordinary homology. We have H • ( P h ; F ) ∼ = F [ u ] ⊗ V ( y ), where u isin degree 2 and y is in degree 5. The Bockstein β ν ( h ) sends u to y ,and P j ( u j ) = u j . In particular, there is no non-zero A -module map H • ( P h ; F ) → Σ F , since any such map would send u to 0 and P u tosomething non-zero, and thus the Hurewicz map in F p -homology (whichwould be dual to this map in cohomology) has to be 0. Of course, wecould also observe this from the fact that H ( P h ; Z ) ∼ = Z / (5 h ), whichhas 5-primary subgroup cyclic of order 5 ν ( h )+1 ≥
25, and since the5-primary subgroup of π ( P h ) ∼ = π ( S ) ∼ = Z /
15 is cyclic of order 5,the image of the Hurewicz map has to reduce mod 5 to 0. But in factthe integral
Hurewicz map in degree 10 vanishes, for any ν ( h ) ≥ WISTED K -THEORY OF COMPACT LIE GROUPS 21 the fibration P → P h → K ( Z /h,
2) as in Figure 2. In the range ofdimensions we’re interested in, H • ( K ( Z / ν ( h ) , F ) agrees with F [ ι , β ν ( h ) P β ν ( h ) ι ] ⊗ ^ ( β ν ( h ) ι , P β ν ( h ) ι ) . The generators here have degrees 2 , , ,
11, respectively. Via the cal-culation of the Bockstein spectral sequence in [33, Theorem 10.4], H ( K ( Z / ν ( h ) , Z ) ∼ = H ( K ( Z / ν ( h ) , Z ) ∼ = ( Z / ν ( h ) )( ι ) ( β ν ( h ) ι ) ⊕ ( Z / P β ν ( h ) ι . The k -invariant of the 5-local Postnikov approximation K ( Z / , → X → K ( Z / ν ( h ) ,
2) to P h can be identified with the image under d of the canonical generator of H ( K ( Z / , F ) in the Serre spec-tral sequence for this Postnikov approximant, and has to be non-zero, since otherwise the 5-primary torsion in H ( P h ; Z ) would be Z / ⊕ H ( K ( Z / ν ( h ) , ∼ = ( Z / ⊕ ( Z / ν ( h ) ), not Z / ν ( h )+1 , so the k -invariant can be seen to be a non-zero multiple of P ( β ν ( h ) ι ), andthe Hurewicz map (which corresponds to the image of H ( K ( Z / , BP • are v in degree 2( p −
1) = 8, v in degree 2( p −
1) = 48, etc. Since theseare all in even degree and the homology of P h is also all in even degree,the AHSS for BP • collapses and BP odd ( P h ) vanishes identically. Sincewe’ve already seen that the Hurewicz map π ( P h ) → H ( P h ) vanishes,the image of the Hurewicz map π ( P h ) → BP • ( P h ) has to map to 0in E ∞ , ∼ = H ( P h ) and thus has to lie in E ∞ , ∼ = ( Z / ν ( h ) ) v (here theindexing of E ∞ corresponds to the AHSS for BP • ). Note that BP • ( P h )does not necessarily split as a direct sum of BP • BP -comodules corre-sponding to the summands of E ∞ for the AHSS, but it has a filtrationfor which this is the associated graded BP • BP -comodule.Note that BP • / ( hj ) = BP • /p ν p ( h )+ ν p ( j ) , so we get a spectral sequenceconverging to the E -term of the ANSS for which E is a sum of copiesof Ext s,t − jBP • BP ( BP • , BP • /p ν p ( h )+ ν p ( j ) ) , j ≤ t. Under the map BP • ( P h ) → K ( S , h ), 1 and v map to 1 and the othergenerators v j , j ≥
2, map to 0. So we are particularly interested inwhat happens in low topological degree ( t − s = 10 for case (4) of thetheorem, other values no larger than this for the other cases) and withregard to v .For our case at hand with p = 5, where ν ( h ) = 1 for simplicity, adiagram of the Ext groups may be found in [39, Figure 4.4.16]. In lowdegrees [39, Theorem 4.4.15], Ext • , • BP • BP ( BP • , BP • /p ) is a polynomial algebra on v (which has bidegree s = 0, t = 8) tensored with anexterior algebra on h , (which has bidegree s = 1, t = 8). We see thatnot very much can contribute, except forExt BP • BP (Σ BP • , Σ BP • / ∼ = F v corresponding to E ∞ , in the AHSS (which is where we expected theHurewicz homomorphism to land). This can’t be killed by a differential,so the Hurewicz map is non-zero, and since v
1, this maps to anelement of order 5 in K ( S , h ). Proof of case (3).
The other cases of the theorem are treated in asimilar fashion. Let’s next deal with the other odd torsion case, (3),with p = 3 and again degree 10. We’ll take ν ( h ) = 1 (again forsimplicity—when ν ( h ) is larger, things are similar but the bookkeepingis more complicated). Again, the Hurewicz map π ( P h ) → H ( P h ; F )vanishes since if there were a map f : S → P h which were non-zeroon homology with F coefficients, the dual map on cohomology wouldsend the generator u ∈ H ( P h ; F ) to 0, and thus would have to kill u , which is the generator in degree 10. So once again we look at theANSS to study the Hurewicz map in BP homology. This time, v is indegree 2 · (3 −
1) = 4, v in degree 2 · (3 −
1) = 16, etc., so the Hurewiczmap in BP homology will have target in a BP • BP -subcomodule M of BP • ( P h ) which is an extension0 → Σ BP • / → M → Σ BP • / → , where the subobject comes from H ( P h ) ∼ = Z / H ( P h ) ∼ = Z /
9. This extension is nontrivial since in cohomology, P is nonzero from H ( P h ; F ) to H ( P h ; F ). We get a long exact se-quence of Ext groups (all over BP • BP , which we omit for conciseness):0 → Ext , ( BP • , Σ BP • / → Ext , ( BP • , M ) → Ext , ( BP • , Σ BP • / → Ext , ( BP • , Σ BP • / → · · · . Here v gives a non-vanishing contribution to Ext , ( BP • , M ) whichcan’t be killed under any differential of the ANSS. The upshot of all ofthis is that the Hurewicz map in BP -homology is non-zero π ( P h ) → BP ( P h ), and that under the map to K ( S , h ), this goes to non-zero3-torsion. Proof of case (1).
Now let’s consider cases (1) and (2), which involvethe prime p = 2. First consider case (1), which is relatively easy;we want to compute the Hurewicz map in BP in degree 4 for P h , h even, using the ANSS. This time the generators are v in degree 2, WISTED K -THEORY OF COMPACT LIE GROUPS 23 v in degree 6, etc., and Ext BP • BP (Σ BP • , g BP • ( P h )) potentially hascontributions fromExt , ( BP • , BP • / ν ( h ) ) and Ext , ( BP • , BP • / ν ( h )+1 ) . Since the Hurewicz map vanishes in ordinary homology, the composite π ( P h ) → BP ( P h ) → H ( P h ) (the last map being the edge homomor-phism of the AHSS) has to vanish, so we are only interested in thefirst term. Say that ν ( h ) = 1; then the picture of Ext s,t ( BP • , BP • / v ∈ Ext , ; this is a permanent cycle as one can seefrom the picture, so the Hurewicz map is non-zero. And v reduces to1 in K ( S , h ). Proof of case (2).
Finally we have the case (2) in topological degree6. First take ν ( h ) = 1; then H • ( P h ; F ) = F [ u ] ⊗ V Sq u , with thepolynomial generator u in degree 2. The ordinary Hurewicz map hasto vanish, since there is no non-zero ring homomorphism H • ( P h ; F ) → H • ( S , F ). So candidates for the BP Hurewicz map have to live in ina BP • BP -subcomodule M of BP • ( P h ) which is an extension0 → Σ BP • / → M → Σ BP • / → , where the subobject comes from H ( P h ) ∼ = Z / H ( P h ) ∼ = Z /
4. Once again the contribution of v ∈ Ext , to Ext , ( BP • , BP • ( M )) is a permanent cycle mapping nontrivially to K ( S , h ). An alternate method.
Before we deal with higher p -primary torsion,we should mention another approach to our theorem using the clas-sical ASS, which is discussed in this context in [1, Part III, § p = 2 andcases (1) and (2) of the theorem. Following Adams’ notation, let B be the subalgebra of the mod-2 Steenrod algebra A generated by Sq and Q = Sq Sq + Sq Sq . This is an exterior algebra on genera-tors of degrees 1 and 3, so it has total dimension 4. By a change-of-rings argument, Adams [1, Part III, Proposition 16.1] proves that theASS for f ku • ( X ) has E term which simplifies to Ext s,t B ∗ ( F , e H • ( X ; F )).We can study the Hurewicz map π s • ( X ) → ku • ( X ) by comparing thisASS with the one with E terms Ext s,t A ∗ ( F , e H • ( X ; F )) converging to π s • ( X ). The natural map Ext s,t A ∗ → Ext s,t B ∗ comes from the forgetfulfunctor from A ∗ -comodules to B ∗ -comodules. The advantage of thisapproach, applied to X = the suspension spectrum of P h , is that weknow H • ( P h ; F ) quite explicitly as a module over A (and in particular over B ). Indeed, if h is even, in the Serre spectral sequence for comput-ing H • ( P h ; F ) from CP ∞ → P h → S , the only differential d vanishes,and so H • ( P h ; F ) = F [ u ] ⊗ V ( y ), where u is in degree 2 and y is indegree 3. Since H ( P h ; Z ) = 0 and H ( P h ; Z ) ∼ = Z /h , if ν ( h ) = 1,Sq u = y , whereas if ν ( h ) >
1, Sq u = 0 and β ν ( h ) ( u ) = y . In bothcases we have Sq u = u , Sq j y = 0 for j ≥
1. (The last identityfollows from the fact that y is pulled back from H ( S ; F ), on which A acts trivially.) The rest of the action of the Steenrod algebra canbe determined from the Cartan relations. For the sake of definiteness,let’s take ν ( h ) = 1. Note that the inclusion CP ∞ ֒ → P h induces anisomorphism of F [ u ] ⊂ H • ( P h ; F ) onto H • ( CP ∞ ; F ). So if A even isthe subalgebra of A generated by the Sq j , j ≥
1, and a ∈ A even , then a ( u ) must be a linear combination of primitive elements of F [ u ], i.e., ofthe elements u j , j ≥
0, by the same argument found in [2, pp. 19–21].Most of the work in computing the Ext and stable homotopy groupswas done by Liulevicius [30] and Mosher [37]. Let M be the left A -module e H • ( CP ∞ ; F ), and let N be the left A -module e H • ( Y ; F ), where Y is the result of attaching a 3-cell to CP ∞ via a map S h −→ CP ⊂ CP ∞ of degree h . Y can be indentified with a subcomplex of P h and thecofiber of the inclusion Y → P h can be identified with Σ CP ∞ . So wehave exact sequences of A -modules(3) (a) 0 → Σ F → N → M → , (b) 0 → Σ M → e H • ( P h ; F ) → N → . These extensions are nontrivial since we have the relations Sq ( u j +1 ) = u j y , and so there are classes v ∈ Ext , A ( M, F ), w ∈ Ext , A ( N, M ),associated to (3) (a) and (b), respectively.In low dimensions Ext s,t A ( M, F ) was computed in [30], and there isonly one Adams differential in this range. There is a unique nonzeroelement in Ext , A ( M, F ), so that is v , and the connecting map in thelong exact sequence coming from (3)(a) is Yoneda product with v by[39, Theorem 2.3.4]. From knowledge of Ext s,t A ( F , F ) [39, Theorem3.2.11] and of Ext s,t A ( M, F ) [30, Proposition II.3] in low dimensionsalong with the long exact sequence, we get the diagram of the longexact sequence for Ext s,t A ( N, F ) shown in Figure 5. Here Ext s,t A ( F , F )is depicted at the left, Ext s,t A ( M, F ) at the right, and red dots indicateelements paired under the connecting map (i.e., under product with v ).From this picture we can read off the stable homotopy groups of Y , since there are no Adams differentials in the dimension range we’reinterested in. So for example π s ( Y ) ∼ = Z ⊕ Z /
2, with the Z coming from WISTED K -THEORY OF COMPACT LIE GROUPS 25 t − s t − ss s Figure 5.
The groups Ext s,t A ( F , F ) (left) andExt s,t A ( M, F ) (right, following Liulevicius) in low dimen-sions. Red dots indicate elements which cancel underthe connecting map (green arrows) in the long exactsequence for Ext s,t A ( N, F ) (for ν ( h ) = 1). CP ∞ (the t − s = 4 column on the right in Figure 5) and the Z / S (the dot at t − s = 1, s = 1 on the left in Figure 5—rememberthat we shift up in dimension by 3). Similarly, π s ( Y ) ∼ = Z ⊕ Z /
4, withthe Z coming from CP ∞ (the t − s = 6 column on the right in Figure5) and the Z / S (the dots at t − s = 3, s = 2 and 3 onthe left in Figure 5).To compute Ext s,t A ( e H • ( P h ; F ) , F ), we need one more exact sequencecoming from (3) (b). Since all the reduced homology of P h is torsion, sois π s • ( P h ), and the connecting map Ext s,t A ( M, F ) → Ext s +1 ,t +3 A ( N, F )kills off the Z summands. The Hurewicz maps we are interested income from the composites π j ( S ) → π sj ( Y ) → π sj ( P h ) with j = 4 and j = 6, so they come from the Z / π s ( Y ) and the Z / π s ( Y )coming from the dots on the left in Figure 5 in bidegrees ( s = 1 , t = 2),resp., ( s = 2 , t = 5) and ( s = 3 , t = 6). In both the cases j = 4 and 6,the torsion summand in π sj ( Y ) cannot be killed by π sj +1 (Σ CP ∞ ).Next let’s compute the B -module structure on H • ( X ), needed forthe right side of Figure 6. When ν ( h ) >
1, Sq vanishes identically on H • ( P h ; F ), and so the B -module structure is trivial andExt s,t B ∗ ( F , e H • ( X ; F )) ∼ = e H t ( P h ; F ) ⊗ Ext s, ∗B ∗ ( F , F ) . When ν ( h ) = 1, then Sq ( u j ) = ju j − y and Sq ( u j y ) = 0, while byan induction using the Cartan formula, we have Sq ( u j ) = u j +1 for j odd, 0 for j even, and Sq ( u j y ) = u j +1 y for j odd, 0 for j even. Thus Q ( u j ) = (Sq Sq + Sq Sq )( u j ) = Sq ( ju j +1 ) + Sq ( ju j − y ) = 0 t − s t − ss s Figure 6.
Comparing the 2-local Adams spectral se-quences for computing π s • ( P h ) and ku • ( P h ) for h =2 k , k odd. Red dots indicate the contribution fromExt s, t C ∗ ( F , F ). Some other contributions on the rightare omitted.in all cases, and similarly Q ( u j y ) = 0 in all cases. Let C = V ( Q )be the subalgebra of B generated by Q . Then we’ve seen that for j odd, u j and u j − y span a B -module M j on which Sq acts cyclicallyand C acts trivially. So this module is B ⊗ C F and again by change ofrings, Ext s, j + t B ∗ ( F , M j ) ∼ = Ext s, j + t C ∗ ( F , F ). However, when j is even, u j and u j − y each span a trivial one-dimensional B -module. Thus, for ν ( h ) = 1,Ext s,t B ∗ ( F , e H • ( P h ; F )) ∼ = M j odd Ext s, j + t C ∗ ( F , F ) ⊕ M j even Ext s, j + t B ∗ ( F , F ) ⊕ M j even Ext s, j +1+ t B ∗ ( F , F ) . Note that a simple calculation gives Ext s,t C ∗ ( F , F ) ∼ = F for all s ≥ t = 3 s (0 for other values of t ) and Ext s,t B ∗ ( F , F ) is a sum ofcopies of F , one for each s , s ≥ s = s + s , t = 3 s + s (theformulas for t come from the fact that Sq raises topological degree by1 and Q raises topological degree by 3). Increasing s corresponds tomultiplying by h ∈ Ext , . (This is also all in [39, Theorem 3.1.16].)Thus in case (1) with ν ( h ) = 1, we get in the E of the ASS for e ku • ( P h )copies of F in bidegrees( s, t ) = ( s, s ) , ( s + s , s + s ) , ( s + s , s + s ) , etc . These are shown on the right side of Figure 6. Note that the termscoming from homology in degrees 2 j and 2 j + 1 correspond to the WISTED K -THEORY OF COMPACT LIE GROUPS 27 image of Z β j ⊂ K ( P h ) (in Khorami’s notation in [29]). Since β j maps to 0 in K ( S , h ) for j ≥
2, we are really only interested in theterms with j = 1, which are indicated by red dots in Figure 6. Thenontriviality of the green arrows in Figure 6 (which is easy to checkpurely algebraically) immediately gives another proof of cases (1) and(2) when ν ( h ) = 1. Proof of cases with ν ( h ) > . Finally, we consider cases (1) and (2)when ν ( h ) >
1, say for definiteness ν ( h ) = 2. Then the A -moduleextensions in equation (3) now split, and Figure 6 is modified as follows.On the left-hand side, since π ( X ) = Z / t − s = 2 , H ( P h ) ∼ = Z /
4, and the B -submodule of H • ( X ; F ) generated by u and y is now trivial. That changes the picture on the right as shownin Figure 7. The differentials on the right are determined by the factsthat H ( P h ) = Z /h and that the AHSS for ku collapses at E . Thispicture proves the remaining cases of the theorem with p = 2. Cases(3) and (4), with p = 3 or 5, can also be handled by the same methodsas cases (1) and (2). The picture analogous to Figure 6 for p = 3, ν ( h ) = 1, and case (3) appears as Figure 8. (At an odd prime p , C becomes the exterior algebra on Q , which is of degree 2 p − (cid:3) t − s t − ss s Figure 7.
Comparing the 2-local Adams spectral se-quences for computing π s • ( P h ) and ku • ( P h ) for h =4 k , k odd. Red dots indicate the contribution fromExt s, t B ∗ ( F , F ) ⊕ Ext s, t B ∗ ( F , F ). Other contributionson the right are omitted. Blue arrows show d . t − s t − ss s h Π h Figure 8.
Comparing the 3-local Adams spectral se-quences for computing π s • ( P h ) and ku • ( P h ) for h = 3 k ,gcd(3 , k ) = 1. Red dots indicate the contribution fromExt s, t C ∗ ( F , F ). Other contributions on the right areomitted.4. The nonsimply connected cases
Similar techniques can also be used to compute twisted K -theoryfor the non-simply connected simple rank-2 groups. There are twoof these, PSU(3) with fundamental group Z / ∼ = SO(5)with fundamental group Z /
2. The case of PSU(3) was studied in [32,Theorem 19 and Remark 20], so we consider here the case of PSp(2).Note first of all that the covering map Sp(2) π −→ PSp(2) induces anisomorphism on H by [32, Theorem 1], and that PSp(2) fits into afibration(4) S = Sp(1) → PSp(2) → RP , which replaces the fibration Sp(1) → Sp(2) → S used in the proof ofTheorem 12. We have transfer and push-forward maps π ∗ : K • (PSp(2) , h ) → K • (Sp(2) , h ) and π ∗ : K • (Sp(2) , h ) → K • (PSp(2) , h ) , and π ∗ ◦ π ∗ is multiplication by 2. Since K • (Sp(2) , h ) is cyclic in botheven and odd degree, this implies that when we localize at an oddprime p , K • (PSp(2) , h ) ( p ) ∼ = K • (Sp(2) , h ) ( p ) . If p = 3, this is a cyclicgroup of order 3 max(0 ,ν ( h ) − , and if p ≥
5, this is a cyclic group oforder p ν p ( h ) . The only issue is therefore what happens with 2-primarytorsion. Recall from Theorem 12 that K • (Sp(2) , h ) (2) is a cyclic groupof order 2 max(0 ,ν ( h ) − . We have by Theorem 3 from (4) a Segal spectral WISTED K -THEORY OF COMPACT LIE GROUPS 29 sequence(5) H p ( RP , K q ( S , h )) ⇒ K • (PSp(2) , h ) . If h is odd, this gives 0 after localizing at 2. So assume that h = 2 k with k odd. After localizing at 2, the left side of (5) becomes H p ( RP , F ) for q even, 0 for q odd. The transfer argument shows that multiplicationby 2 on K • (PSp(2) , h ) (2) factors through K • (Sp(2) , h ) (2) = 0, so all2-primary torsion is of order 2.Now if h is even, it is 0 mod 2, so we have natural maps K ( S , h ) reduce mod 2 −−−−−−−→ K ( S , h ; F ) ∼ = K ( S ; F ) , and K • (PSp(2) , h ) reduce mod 2 −−−−−−−→ K • (PSp(2) , h ; F ) ∼ = K • (PSp(2); F ) , the first of which is an isomorphism. So we get a map of spectralsequences(6) H p ( RP , K q ( S , h )) (2) + (cid:127) _ (cid:15) (cid:15) K • (PSp(2) , h ) (2) (cid:15) (cid:15) H p ( RP , K q ( S ; F )) + K • (PSp(2); F ) . The K -theory of PSp(2) ∼ = SO(5) was computed in [25, Satz 5.15];as an abelian group it is Z ⊕ Z / H p ( RP , K q ( S )) ⇒ K • (PSp(2)), whichhas a Z / E in bidegrees (2 j − , k ), j = 1 , ,
3, there is room foronly one differential. In fact, from the description of the torsion in K • in [25], the torsion in K is generated by the pull-back of the gener-ator of e K ( RP ), and the generator of the torsion in K is generatedby the product of this class with an odd generator λ of a torsion-free exterior algebra, which is precisely the canonical representationPSp(2) ∼ = SO(5) → U (5) viewed as a class in K . So this determinesthe differentials in the Segal spectral sequence in K -cohomology; theremust be differentials killing off H ( RP , K ( S )) and H ( RP , K ( S )).From the universal coefficient theorem, K • (PSp(2); F ) ∼ = F in botheven and odd degree. (We get a group of rank 3 from reducing the in-tegral K -homology mod 2, and pick up another F from the Tor term.)If we compare the bottom spectral sequence in (6) with the one for in-tegral K -homology and with the one for twisted K -homology of SO(4)(in which there are no differentials at all) we see that the only non-zerodifferentials are d : E p +2 ,q → E p,q +1 with p = 4 or 5. Now go back tothe commuting diagram (6). There cannot be a non-zero differentialin the upper spectral sequence, since it would imply existence of a for-bidden differential in the lower sequence. So the spectral sequence for K • (PSp(2) , h ) (2) collapses, and since all torsion is of order 2, we con-clude that the 2-primary torsion in K • (PSp(2) , h ) (2) is ( Z / in botheven and odd degree. Putting everything together, we see that we haveproved the following: Theorem 17.
Suppose that h is either odd or mod . Then K • (PSp(2) , h ) is finite, and is the same in both even and odd degree. The odd tor-sion in K • (PSp(2) , h ) is cyclic of order h odd / gcd( h, . The -primarytorsion vanishes if h is odd, and if h is mod , it is ( Z / in eachdegree. Cases where h is divisible by a higher power of 2 can be handledsimilarly, though the results are more complicated. References
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E-mail address , Jonathan Rosenberg:, Jonathan Rosenberg: