aa r X i v : . [ m a t h . A T ] F e b EQUIVARIANT FORMALITY IN K -THEORY CHI-KWONG FOK
Abstract.
In this note we present an analogue of equivariant formality in K -theory andshow that it is equivalent to equivariant formality `a la Goresky-Kottwitz-MacPherson. Wealso apply this analogue to give alternative proofs of equivariant formality of conjugationaction on compact Lie groups, left translation action on generalized flag manifolds, andcompact Lie group actions with maximal rank isotropy subgroups.
Mathematics Subject Classification : 19L47; 55N15; 55N911.
Introduction
Equivariant formality, first defined in [GKM], is a special property of group actionson topological spaces which allows for easy computation of their equivariant cohomology.A G -action on a space X is said to be equivariantly formal if the Leray-Serre spectralsequence for the rational cohomology of the fiber bundle X ֒ → X × G EG → BG collapseson the E -page. The latter is also equivalent to H ∗ G ( X ; Q ) ∼ = H ∗ G (pt; Q ) ⊗ H ∗ ( X ; Q )as H ∗ G (pt; Q )-modules. There are various examples of interest which are known to beequivariantly formal, e.g. Hamiltonian group actions on compact symplectic manifolds andlinear algebraic torus actions on smooth complex projective varieties (cf. [GKM, Section1.2 and Theorem 14.1]).Though equivariant formality was first defined in terms of equivariant cohomology, insome situations working with analogous notions phrased in terms of other equivariant coho-mology theories may come in handy. The notion of equivariant formality in K -theory wasintroduced and explored by Harada and Landweber in [HL], where they instead used theterm ‘weak equivariant formality’ and exploited this notion to show equivariant formalityof Hamiltonian actions on compact symplectic manifolds. Definition 1.1 (cf. [HL, Def. 4.1]) . Let k be a commutative ring, G a compact Liegroup and X a G -space. We use K ∗ ( X ) (resp. K ∗ G ( X )) to denote the Z -graded complex(equivariant) K -theory of X , and K ∗ ( X ; k ) (resp. K ∗ G ( X ; k )) to denote K ∗ ( X ) ⊗ k (resp. K ∗ G ( X ) ⊗ k ). We denote the complex representation ring of G by R ( G ), and write R ( G ; k ) := R ( G ) ⊗ k , and I ( G ; k ) = I ( G ) ⊗ k , where I ( G ) is the augmentation ideal of R ( G ). Let f G : K ∗ G ( X ) → K ∗ ( X )be the forgetful map. A G -action on a space X is k -weakly equivariantly formal if f G induces an isomorphism K ∗ G ( X ; k ) ⊗ R ( G ; k ) k → K ∗ ( X ; k )We simply say the action is weakly equivariantly formal in the case k = Z . Date : October 3, 2018. Recall that one can use Z -grading in defining complex K -theory thanks to Bott periodicity. Harada and Landweber settled for weakly equivariant formality as in Definition 1.1as the K -theoretic analogue of equivariant formality, instead of the seemingly obviouscandidate K ∗ G ( X ) ∼ = K ∗ G (pt) ⊗ K ∗ ( X ), citing the lack of the Leray-Serre spectral sequencefor Atiyah-Segal’s equivariant K -theory. The term ‘weak’ is in reference to the conditionin Definition 1.1 being weaker than K ∗ G ( X ) ∼ = K ∗ G (pt) ⊗ K ∗ ( X ) because of the possiblepresence of torsion. We would like to define the following version of K -theoretic equivariantformality in exact analogy with another cohomological equivariant formality condition thatthe forgetful map H ∗ G ( X ) → H ∗ ( X ) be onto. Definition 1.2.
We say that X is a rational K -theoretic equivariantly formal ( RKEF forshort) G -space if the forgetful map f G ⊗ Id Q : K ∗ G ( X ; Q ) → K ∗ ( X ; Q )is onto.Recall that K ( X ) (resp. K − ( X )) is the Grothendieck group of the commutativemonoid of isomorphism classes of (resp. reduced) complex vector bundles over X (resp.Σ X ) under Whitney sum, and K ∗ G ( X ) can be similarly defined using equivariant vectorbundles. The above condition then admits a natural interpretation in terms of vectorbundles: for every vector bundle V over X and its suspension Σ X , there are naturalnumbers p, q such that V ⊕ p ⊕ C q admits an equivariant G -structure.In this note, we will prove the following theorem, which asserts the equivalence of RKEFand equivariant formality in the classical sense. Theorem 1.3.
Let G be a compact and connected Lie group which acts on a finite CW-complex X . The following are equivalent.(1) X is a RKEF G -space.(2) X is an equivariantly formal G -space.(3) X is a Q -weakly equivariantly formal G -space. We will also give alternative proofs of equivariant formality of certain group actionswhich were proved in cohomological terms. These are conjugation action on compact Liegroups, left translation action on generalized flag manifolds, and compact Lie group actionswith maximal rank isotropy subgroups.We note that there is an analogue of Theorem 1.3 in the algebro-geometric setting ([Gr,Theorem 1.1]): it is also an assertion of surjectivity, but of the forgetful map from therational Grothendieck group of G -equivariant coherent sheaves on a G -scheme X to thecorresponding Grothendieck group for ordinary coherent sheaves, where G is a connectedreductive algebraic group. Theorem 1.3 confirms the expectation ([Gr, Introduction]) thatthe K -theoretic forgetful map is onto for equivariantly formal topological spaces.In the remainder of this note, the coefficient ring of any cohomology theory is always Q . Acknowledgment.
We would like to gratefully acknowledge the anonymous referee forthe critical comments on the early drafts of this paper and especially the suggestions forimproving Section 3.3. We would like to thank Ian Agol for answering a question relatedto the proof of Theorem 3.2.
QUIVARIANT FORMALITY IN K -THEORY 3 The proof
From now on, unless otherwise specified, X is a finite CW-complex equipped with anaction by a torus T or more generally a compact connected Lie group G . The following K -theoretic abelianization result enables us to prove K -theoretic results in this Section inthe T -equivariant case first and then generalize to the G -equivariant case. Theorem 2.1 (cf. [HLS, Theorem 4.9(ii)]) . Let T be a maximal torus of G and W theWeyl group. The map r ∗ : K ∗ G ( X ; Q ) → K ∗ T ( X ; Q ) restricting the G -action to the T -actionis an injective map onto K ∗ T ( X ; Q ) W . Here if w ∈ W and V is an equivariant T -vectorbundle, w takes V to the same underlying vector bundle with T -action twisted by w , andthis W -action on the set of isomorphism classes of equivariant T -vector bundles inducesthe W -action on K ∗ T ( X ) . Definition 2.2.
Let H ∗∗ G ( X ) be the completion of H ∗ G ( X ) as a H ∗ G (pt)-module at theaugmentation ideal J := H + G (pt) (cf. the paragraph preceeding [RK, Proposition 2.8]).The equivariant Chern character for a finite CW-complex with a G -action is the mapch G : K ∗ G ( X ; Q ) → H ∗∗ G ( X )which is defined by applying the Borel construction to the non-equivariant Chern character(cf. the discussion before [RK, Lemma 3.1]). Like the non-equivariant Chern character, ch G maps K G ( X ; Q ) to the even degree part of H ∗∗ G ( X ) and K − G ( X ; Q ) to the odd degree part.The image of ch G lies in H ∗∗ G ( X ) for the following reason which is borrowed from the proofof [RK, Lemma 3.1]: as X is a finite CW-complex, we can choose a , a , · · · , a m ∈ H ∗ G ( X )which generate H ∗ G ( X ) as a H ∗ G (pt)-module. Let a i · a j = m X k =1 f kij a k for f kij ∈ H ∗ G (pt), and c be c G ( L ) for some G -equivariant line bundle L such that c = m X i =1 g i a i for g i ∈ H ∗ G (pt). Soch G ( L ) = e c = 1 + X i g i a i + 12 X i,j,k g i g j f kij a k + 16 X i,j,k,l,p g i g j g l f kij f pkl a p + · · · . Write ch G ( L ) = 1+ P mi =1 p i a i , where p i are power series in g i and f kij . Identifying g i and f kij with W -invariant polynomials on t through the identification H ∗ G (pt) ∼ = H ∗ T (pt) W ∼ = S ( t ∗ ) W and using the estimate for p i given in the proof of [RK, Lemma 3.1], we have that p i arein H ∗∗ G (pt) and hence ch G ( L ) ∈ H ∗∗ G ( X ). The assertion ch G ( E ) ∈ H ∗∗ G ( X ) for generalequivariant G -vector bundle E follows from the splitting principle. Proposition 2.3.
Let G be a compact connected Lie group acting on a finite CW-complex X . Then the equivariant Chern characterch G : K ∗ G ( X ; Q ) → H ∗∗ G ( X ) is injective, and ch − G ( J ) = I ( G ; Q ) when X is a point. CHI-KWONG FOK
Proof.
By [AS, Theorem 2.1], K ∗ ( X × G EG ) ∼ = K ∗ G ( X × EG ) is the completion of K ∗ G ( X ) at I ( G ). The map ι : K ∗ G ( X ) → K ∗ ( X × G EG ) induced by the projection map X × EG → X is injective because the I ( G )-adic topology of the completion is Hausdorff if G is connected(cf. the Note immediately preceding [AH, Section 4.5]). It follows that the rationalizedmap ι ⊗ Q : K ∗ G ( X ; Q ) → K ∗ ( X × G EG ; Q ) is injective as well. On the other hand, let EG n be the Milnor join of n copies of G . Then X × G EG n is compact and the ordinaryChern character map ch n : K ∗ ( X × G EG n ; Q ) → H ∗ ( X × G EG n ) is an isomorphism. Notethat K ∗ ( X × G EG ; Q ) ∼ = lim ←− n K ∗ ( X × G EG n ; Q )(see [AS, Corollary 2.4, Proposition 4.1 and proof of Proposition 4.2]). It follows that themap ch : K ∗ ( X × G EG ; Q ) → H ∗∗ G ( X )is the inverse limit of the isomorphisms ch n and injective by the left-exactness of inverselimit. The map ch G is the composition of the two injective maps ι ⊗ Q and ch : K ∗ ( X × G EG ; Q ) → H ∗∗ G ( X ). Therefore ch G is injective. Next, consider the commutative diagram R ( G ; Q ) / / ch G (cid:15) (cid:15) K ∗ (pt; Q ) ch (cid:15) (cid:15) H ∗∗ G (pt) / / H ∗ (pt)where the two horizontal maps are forgetful maps. Since J is the kernel of the bottom mapand both ch G and ch are injective, ch − G ( J ) is the kernel of the top map, which is precisely I ( G ; Q ). (cid:3) Under the condition of weak equivariant formality, [HL, Proposition 4.2] asserts that thekernel of f is I ( G ) · K ∗ G ( X ). In fact, we also have Lemma 2.4.
Let X be a finite CW-complex which is acted on by a compact connected Liegroup G equivariantly formally. Then the kernel of the forgetful map f G ⊗ Id Q : K ∗ G ( X ; Q ) → K ∗ ( X ; Q ) is I ( G ; Q ) · K ∗ G ( X ; Q ) .Proof. In the following diagram, K ∗ G ( X ; Q ) f G ⊗ Id Q / / ch G (cid:15) (cid:15) K ∗ ( X ; Q ) ch (cid:15) (cid:15) H ∗∗ G ( X ) e g G ⊗ Id Q / / H ∗ ( X )(2.1)where e g G ⊗ Id Q is the forgetful map, H ∗∗ G ( X ) is the completion of H ∗ G ( X ) at the augmenta-tion ideal J of H ∗ G (pt). Since X is an equivariantly formal G -space, H ∗ G ( X ) is isomorphicto H ∗ G (pt) ⊗ H ∗ ( X ) as a H ∗ G (pt)-module, and the forgetful map g G ⊗ Id Q : H ∗ G ( X ) → H ∗ ( X )has J · H ∗ G ( X ) as the kernel. Since H ∗ G ( X ) is a finitely generated module over the Noetherianring H ∗ G (pt), a simple result on completions (cf. [Ma, Theorem 55]) implies that H ∗∗ G ( X ) ∼ = QUIVARIANT FORMALITY IN K -THEORY 5 H ∗ G ( X ) ⊗ H ∗ G (pt) H ∗∗ G (pt). So the kernel of e g G ⊗ Id Q is J · H ∗∗ G ( X ). By Proposition 2.3, thepreimage ch − G ( J ) is I ( G ; Q ) and ch G is injective. It follows that the kernel of f G ⊗ Id Q isch − G ( J · H ∗∗ G ( X )) = I ( G ; Q ) · K ∗ G ( X ; Q ). (cid:3) Proof of Theorem 1.3, (1) ⇐⇒ (2) . We first deal with the T -equivariant case, where T isa maximal torus of G . We claim that, if X is an equivariantly formal T -space, we have thefollowing string of (in)equalities.dim Q K ∗ ( X T ; Q ) = rank R ( T ; Q ) K ∗ T ( X ; Q ) ≤ dim K ∗ T ( X ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) ≤ dim K ∗ ( X ; Q ) . Applying Segal’s localization theorem to the case of torus group actions (cf. [Se, Proposi-tion 4.1]), we have that the restriction map K ∗ T ( X ; Q ) → K ∗ T ( X T ; Q ) becomes an isomor-phism after localizing at the zero prime ideal, i.e. to the field of fraction of R ( T ; Q ). Sorank R ( T ; Q ) K ∗ T ( X ; Q ) = rank R ( T ; Q ) K ∗ T ( X T ; Q ). By [Se, Proposition 2.2], K ∗ T ( X T ; Q ) is iso-morphic to R ( T ; Q ) ⊗ K ∗ ( X T ; Q ), whose rank over R ( T ; Q ) equals dim Q K ∗ ( X T ; Q ). Thefirst equality then follows. Next, by [Se, Proposition 5.4] and the discussion thereafter, wehave that K ∗ T ( X ; Q ) is a finite R ( T ; Q )-module. After localizing K ∗ T ( X ; Q ) at I ( T ; Q ) andreduction modulo the same ideal, we have that K ∗ T ( X ; Q ) I ( T ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) I ( T ; Q ) is a finite dimensional Q -vector space. We let n be the dimension of this vector space,and x , · · · , x n ∈ K ∗ T ( X ; Q ) I ( T ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) I ( T ; Q ) be its basis. Finite genera-tion of K ∗ T ( X ; Q ) as a module over the Noetherian ring R ( T ; Q ) enables us to invokeNakayama lemma, and have that there exist lifts b x , · · · , b x n ∈ K ∗ T ( X ; Q ) I ( T ; Q ) that gener-ate K ∗ T ( X ; Q ) I ( T ; Q ) as a R ( T ; Q ) I ( T ; Q ) -module. It follows, after further localization to thefield of fraction of R ( T ; Q ), that b x , · · · , b x n span K ∗ T ( X ; Q ) (0) as a R ( T ; Q ) (0) -vector space,and thatdim R ( T ; Q ) (0) K ∗ T ( X ; Q ) (0) ≤ dim Q K ∗ T ( X ; Q ) I ( T ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) I ( T ; Q ) = n. Noting the isomorphism K ∗ T ( X ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) ∼ = K ∗ T ( X ; Q ) I ( T ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) I ( T ; Q ) ,we arrive at the first inequality. Finally, the last inequality follows from Lemma 2.4.If X is an equivariantly formal T -space, then dim H ∗ ( X ) = dim H ∗ ( X T ) (see [Hs, p. 46]).The Chern character isomorphism implies that dim K ∗ ( X T ; Q ) = dim K ∗ ( X ; Q ) which, to-gether with the (in)equalities in the above claim, yields dim K ∗ T ( X ; Q ) /I ( T ; Q ) · K ∗ T ( X ; Q ) =dim K ∗ ( X ; Q ) or, equivalently, that X is RKEF.Assume on the other hand that X is RKEF. Consider the commutative diagram (2.1).Since f T ⊗ Id Q is onto and ch is an isomorphism, e g T ⊗ Id Q is onto. By [Ma, Theorem55], we have that H ∗∗ T ( X ) ∼ = H ∗ T ( X ) ⊗ H ∗ T (pt) H ∗∗ T (pt). Applying e g T ⊗ Id Q gives H ∗ ( X ) =Im( e g T ⊗ Id Q ) = Im( g T ⊗ Id Q ) ⊗ Q Q = Im( g T ⊗ Id Q ). Hence X is T -equivariantly formal.With the equivalence of equivariant formality and RKEF for T -action we have justproved and the fact that, if T is a maximal torus of G which is compact and connected, T -equivariant formality is equivalent to G -equivariant formality (cf. [GR, Proposition 2.4]),it suffices to show that f T ⊗ Id Q is onto if and only if f G ⊗ Id Q is onto in order to establishthe equivalence of equivariant formality and RKEF for G -action. One direction is easy: if f G ⊗ Id Q is onto, so is f T ⊗ Id Q because f G ⊗ Id Q = ( f T ⊗ Id Q ) ◦ r ∗ . Conversely, supposethat f T ⊗ Id Q is onto. Then any x ∈ K ∗ ( X ; Q ) admits a lift e x ∈ K ∗ T ( X ; Q ). Note that for CHI-KWONG FOK any w ∈ W , ( f T ⊗ Id Q )( w · e x ) = x . It follows that the average x := 1 | W | X w ∈ W w · e x is also a lift of x . Moreover, by Theorem 2.1, x ∈ r ∗ K G ( X ; Q ). So ( r ∗ ) − ( x ) ∈ K ∗ G ( X ; Q )is a lift of x and f G ⊗ Id Q is onto as well. (cid:3) Proof of Theorem 1.3, (1) ⇐⇒ (3) . That Q -weakly equivariant formality implies RKEFis immediate (cf. [HL, Definition 4.1]). On the other hand, if X is a RKEF G -space, thenby Theorem 1.3, (1) = ⇒ (2), X is an equivariantly formal G -space. The map K ∗ G ( X ; Q ) ⊗ R ( G ; Q ) Q → K ∗ ( X ; Q ) α ⊗ z f G ( α ) z is injective by Lemma 2.4 and surjective by RKEF. Hence X is a Q -weakly equivariantlyformal G -space. This completes the proof. (cid:3) Some applications
In this Section, we shall demonstrate the utility of Theorem 1.3 by giving alternativeproofs of some previous results.3.1.
Conjugation action on compact Lie groups.
Let G be a compact connected Liegroup with conjugation action by itself. It is well-known that this action is equivariantlyformal. See, for example, [GS, Sect. 11.9, Item 6]) for a sketch of proof for the case G = U ( n ), and [J] for an explicit construction of equivariant extensions of the generatorsof H ∗ ( G ). We will show equivariant formality of conjugation action by proving that G isa RKEF G -space. By [Ho, II, Theorem 2.1], K ∗ ( G ; Q ) ∼ = ^ ∗ Q ( R ⊗ Q ) , where R is the image of the map δ : R ( G ) → K − ( G )which sends ρ ∈ R ( G ) to the following complex of vector bundles −→ G × R × V −→ G × R × V −→ g, t, v ) (cid:26) ( g, t, − tρ ( g ) v ) , if t ≥ , ( g, t, v ) , if t ≤ . For any ρ , δ ( ρ ) admits an equivariant lift in K ∗ G ( G ) because G × R × V can be equippedwith the G -action given by g · ( g, t, v ) = ( g gg − , t, ρ ( g ) v ) , with respect to which the middle map of the above complex of vector bundles is G -equivariant. Thus f G ⊗ Id Q : K ∗ G ( G ; Q ) → K ∗ ( G ; Q ) is onto, i.e., G is a RKEF G -space. The map δ , which was defined in [BZ] and corrected in [F], is the same as the map β defined in [Ho]. QUIVARIANT FORMALITY IN K -THEORY 7 Left translation action on
G/K where rank G = rank K . Let G be a com-pact connected Lie group and K a connected Lie subgroup of the same rank. The lefttranslation action on G/K by G is well-known to be equivariantly formal, which can beproved by noting that G/K satisfies the sufficient condition for equivariant formality thatits odd cohomology vanish (cf. [GHV, Chapter XI, Theorem VII]). Alternatively, by therationalized version of [Sn, Theorem 4.2] and the remark following it, K ∗ ( G/K ; Q ) ∼ = R ( K ; Q ) ⊗ R ( G ; Q ) Q ∼ = R ( K ; Q ) /r ∗ I ( G ; Q ) , where r ∗ : R ( G ; Q ) → R ( K ; Q ) is the restriction map. The forgetful map f G ⊗ Id Q : K ∗ G ( G/K ; Q ) ∼ = R ( K ; Q ) → K ∗ ( G/K ; Q ) is simply the projection map and hence surjective(in fact the forgetful map sends any representation ρ ∈ R ( K ) to the K -theory class of thehomogeneous vector bundle G × K V ρ , where V ρ is the underlying complex vector space for ρ ). Thus G/K is a RKEF G -space, and equivalently an equivariantly formal G -space. Remark 3.1.
In the more general case where equality of ranks of G and K is not assumed,a representation theoretic characterization of equivariant formality of the left translationaction of K on G/K is given by virtue of RKEF in [CF].3.3.
Actions with connected maximal rank isotropy subgroups.
In this section wewill prove the following equivariant formality result.
Theorem 3.2.
Let G be a compact connected Lie group and X a finite G -CW complex.Suppose that the G -action on X has maximal rank connected isotropy subgroups. Then X is an equivariantly formal G -space. Remark 3.3.
In fact, Theorem 3.2 follows from [GR, Corollary 3.5], where connectednessof isotropy subgroups is not assumed. Though the space under consideration in [GR,Corollary 3.5] is the subset of a compact G -manifold consisting of those points with maximalrank isotropy subgroups, its proof does not make use of this assumption and can be easilyadapted to the more general case of G -CW complexes. Indeed the proof hinges on theobservation that for any compact space X with maximal rank isotropy subgroups and amaximal torus T , the map G × N G ( T ) X T → X given by [ g, x ] gx is onto and that the fibersof the map are acyclic. This enables one to assert the isomorphism H ∗ G ( X ) ∼ = H ∗ N G ( T ) ( X T ).The latter, by abelianization, is H ∗ T ( X T ) W , which in turn by a commutative algebra result([GR, Lemma 2.7]) is a free module over H ∗ T (pt) W ∼ = H ∗ G (pt). Hence X is an equivariantlyformal G -space. Remark 3.4. If G in addition satisfies the condition that π ( G ) be torsion-free, then K ∗ G ( X ; Q ) is a free R ( G ; Q )-module with rank dim Q K ∗ ( X T ; Q ) ([AG, Theorem 1.1]).We would like to give a different proof of this result by using Theorem 1.3 and inductionon the dimension of X . We shall point out that the group actions considered in Sections3.1 and 3.2 are examples of group actions we discuss in this section. However, equivariantformality of left translation actions on generalized flag manifolds as in Section 3.2 is usedin the proof. Lemma 3.5.
Let G be a compact connected Lie group acting on a finite CW-complex X equivariantly formally. Let V and V be vector bundles on X which are isomorphicnonequivariantly. Then there exist positive integers a and b such that V ⊕ a ⊕ C b and V ⊕ a ⊕ C b can be made equivariant G -vector bundles which are isomorphic equivariantly. CHI-KWONG FOK
Proof.
By Theorem 1.3 and the discussion preceding it, there exists p and q such that T := V ⊕ p ⊕ C q and T := V ⊕ p ⊕ C q admit equivariant structures. Let e T and e T denotethe corresponding equivariant G -vector bundles. They then define the equivariant K -theoryclass [ e T ] − [ e T ] ∈ K ∗ G ( X ; Q ) which lies in the kernel of the forgetful map f G ⊗ Id Q . ByLemma 2.4, there exist a positive integer m , representations ρ i and ρ i of G with the samedimension, and equivariant G -vector bundles A i and B i such that m ([ e T ] − [ e T ]) = X i ([ ρ i ] − [ ρ i ]) · ([ A i ] − [ B i ])Here, for ρ ∈ R ( G ) with V ρ being the complex vector space underlying the representa-tion, ρ means the vector bundle X × V ρ with the diagonal G -action. By the definitionof Grothendieck construction, there exists an equivariant G -vector bundle C such that wehave the following G -vector bundle isomorphism. e T ⊕ m ⊕ M i ( ρ i ⊗ A i ⊕ ρ i ⊗ B i ) ⊕ C ∼ = e T ⊕ m ⊕ M i ( ρ i ⊗ A i ⊕ ρ i ⊗ B i ) ⊕ C. By [Se, Proposition 2.4], there exists an equivariant G -vector bundle D such that L i ( ρ i ⊗ A i ⊕ ρ i ⊗ B i ) ⊕ C ⊕ D ∼ = ρ for some ρ ∈ R ( G ). Taking the direct sum of both sides with D and forgetting the equivariant structures, we have V ⊕ pm ⊕ C qm +dim ρ ∼ = V ⊕ pm ⊕ C qm +dim ρ . Taking a = pm and b = qm + dim ρ finishes the proof. (cid:3) Proof of Theorem 3.2.
Consider the n -skeleton X n . It is obtained by gluing the equivariantcells G/K i × D n for 1 ≤ i ≤ k and K i compact, connected and of maximal rank, tothe ( n − X n − through some G -equivariant attaching maps. For convenienceof exposition and without loss of generality we will consider the case of attaching oneequivariant cell G/K × D n . Let f : G/K × ∂ D n → X n − be the equivariant attaching map and F : G/K × D n → X n be the inclusion of the equivariant cell into X n . We also let V be any given vector bundleover X n . To prove Proposition 3.2, it suffices, by Theorem 1.3 and the discussion afterDefinition 1.2, to show that, for some p and q , V ⊕ p ⊕ C q admits an equivariant structure,assuming by induction hypothesis that V := V | X n − satisfies the condition that V ⊕ p ⊕ C q admits an equivariant structure for some p and q .Note that V can be obtained by gluing V → X n − and W → G/K i × D n , where W := F ∗ V , through the clutching maps, i.e. vector bundle homomorphism h : W | G/K × ∂ D n → V which covers the map f and send fiber to fiber isomorphically. By the discussion in Section3.2 and the contractibility of D n , there exist r and s such that W ⊕ r ⊕ C s is isomorphicto a certain homogeneous vector bundle which is obviously G -equivariant. If we take QUIVARIANT FORMALITY IN K -THEORY 9 p = LCM( p , r ) and q = max { q , s } then both V ⊕ p ⊕ C q and W ⊕ p ⊕ C q admit equivariantstructures. Consider the clutching map j : W ⊕ p | G/K × ∂ D n ⊕ C q → V ⊕ p ⊕ C q built from h for the vector bundles W ⊕ p ⊕ C q and V ⊕ p ⊕ C q . The vector bundle V ⊕ p ⊕ C q admits an equivariant structure if j is homotopy equivalent to another clutching map whichis G -equivariant. Now we define the map α : W ⊕ p | G/K × ∂ D n ⊕ C q → f ∗ V ⊕ p ⊕ C q such that j is the composition of α and the natural map f ∗ V ⊕ p ⊕ C q ∼ = f ∗ ( V ⊕ p ⊕ C q ) → V ⊕ p ⊕ C q ( x, v ) v, where f ( x ) = π ( v ) , x ∈ G/K × ∂ D n , v ∈ V ⊕ p ⊕ C q . The latter map is obviously G -equivariant. If the map α is homotopy equivalent to a G -equivariant map (and hence so is the clutching map j ), then V ⊕ p ⊕ C q , which is obtainedby gluing V ⊕ p ⊕ C q and W ⊕ p ⊕ C q through the clutching map, admits the G -equivariantstructure inherited from those of V ⊕ p ⊕ C q and W ⊕ p ⊕ C q . In fact it suffices to show thefollowing Claim 3.6.
There exist some positive integers l and m such that the map α ⊕ m ⊕ Id C l : ( W ⊕ p | G/K × ∂ D n ⊕ C q ) ⊕ m ⊕ C l → ( f ∗ V ⊕ p ⊕ C q ) ⊕ m ⊕ C l is homotopy equivalent to a G -equivariant map. The claim will imply that V ⊕ pm ⊕ C qm + l admits an equivariant structure by the aboveclutching argument. We may then replace p and q with pm and qm + l respectively.We shall prove the above claim. Note that α is a vector bundle isomorphism as it coversthe identity map on G/K × ∂ D n and send fiber to fiber isomorphically. Bearing in mindthat G/K is an equivariantly formal G -space (cf. Section 3.2) and so is ∂ D n due to thetrivial G -action, G/K × ∂ D n is an equivariant formal G -space because it is a product ofequivariant formal G -spaces. By Lemma 3.5, there exist positive integers a and b andequivariant G -vector bundle isomorphism β : ( f ∗ V ⊕ p ⊕ C q ) ⊕ a ⊕ C b → ( W ⊕ p | G/K × ∂ D n ⊕ C q ) ⊕ a ⊕ C b . The composition γ := β ◦ ( α ⊕ a ⊕ Id C b ) then is a vector bundle automorphism of U := W ⊕ pa | G/K × ∂ D n ⊕ C qa + b . Let Y be the vector bundle U × [0 , / (( u, ∼ ( γ ( u ) , G/K × ∂ D n × S , which is an equivariantly formal G -space by the above argument. ByTheorem 1.3, G/K × ∂ D n × S is RKEF. It follows that for some positive integers c and d , Y ⊕ c ⊕ C d can be made an equivariant G -vector bundle, and thus γ ⊕ c ⊕ Id C d is homotopyequivalent to some G -equivariant clutching map δ : U ⊕ c ⊕ Id C d → U ⊕ c ⊕ Id C d . It followsthat α ⊕ ac ⊕ Id C bc + d = (( β ) − ) ⊕ c ⊕ Id C d ) ◦ ( γ ⊕ c ⊕ Id C d ) is homotopy equivalent to theequivariant G -vector bundle isomorphism (( β − ) ⊕ c ⊕ Id C d ) ◦ δ . Now taking m = ac and l = bc + d finishes the proof of the claim.We have shown that, by induction on the dimension of X , for any given vector bundle V → X , V ⊕ p ⊕ C q admits an equivariant structure for some p and q . The same is true for the suspension Σ X because it is also a G -CW complex with maximal rank connectedisotropy subgroups. It follows that the G -action on X is equivariantly formal by Theorem1.3. (cid:3) References [AG] A. Adem, J. M. G´omez,
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National Center for Theoretical Sciences, Mathematics Division, National Taiwan Univer-sity, Taipei 10617, TaiwanSchool of Mathematical Sciences, the University of Adelaide, Adelaide, SA 5005, AustraliaE-mail : [email protected] URL ::