Equivariant K-theory and Resolution II: Non-Abelian actions
aa r X i v : . [ m a t h . K T ] D ec EQUIVARIANT K-THEORY AND RESOLUTIONII: NON-ABELIAN ACTIONS
PANAGIOTIS DIMAKIS AND RICHARD MELROSE
Abstract.
The smooth action of a compact Lie group on a compact manifoldcan be resolved to an iterated space, as made explicit by Pierre Albin and thesecond author. On the resolution the lifted action has fixed isotropy type cor-responding to the open stratum and also in an iterated sense, with connectingequivariant fibrations over the boundary hypersurfaces covering the resolutionsof the other strata. This structure descends to a resolution of the quotient asa stratified space. For an Abelian group action the equivariant K-theory canthen be described in terms of bundles over the bases ‘dressed’ by the represen-tations of the isotropy types with morphisms covering the connecting maps.A similar model is given here covering the non-Abelian case. Now the reducedobjects are torsion-twisted bundles over finite covers of the bases, correspond-ing to the projective action of the normalizers on the representations of theisotropy groups, again with morphisms over all the boundaries. This leadsto a closely related iterated deRham model for equivariant cohomology and,now with values in forms twisted by flat bundles of representation rings overthe bases, for delocalized equivariant cohomology. We show, as envisioned byBaum, Brylinksi and MacPherson, that the usual equivariant Chern character,mapping to equivariant cohomology, factors through a natural Chern charac-ter from equivariant K-theory to delocalized equivariant cohomology with thelatter giving an Atiyah-Hirzebruch isomorphism.
Contents
Introduction 11. Resolution and Lifting 62. Reduction 63. Single Isotropy type 84. Equivariant fibrations 135. Reduced models 176. Atiyah-Hirzebruch isomorphism 19References 20
Introduction
In this continuation of [6] the description, in terms of the normal resolution, ofthe equivariant K-theory of a compact manifold with an action by a compact Liegroup is extended to the general, non-Abelian, case. Doing so suggests a deRhamrealization of delocalized equivariant cohomology, in the sense of Baum, Brylinksiand MacPherson [2]. The natural Chern character(1) Ch : K G ( M ) −→ H even G, dl ( M ) yields an Atiyah-Hirzebruch isomorphism over R . The simplest cases of smooth G action can be analyzed without resolution. If G acts freely then the compact manifold is the total space of a principal bundle κ : M −→ Y, over the smooth orbit space Y = G \ M. The action on an equivariantbundle, V, over M gives descent data defining a bundle W over Y. Conversely, everyequivariant bundle is equivariantly isomorphic to the pull-back, V = κ ∗ W, of abundle over Y with G action gw x = w gx and equivariant bundle isomorphisms over M descend uniquely to bundle isomorphisms over Y. Passing to the Grothendieckgroups this induces a natural identification as rings(2) K G ( M ) = K ( Y ) . Thus the equivariant K-theory of M, as a ring, does not depend on the bundlestructure but only on the quotient. Note however that the representation ring of G acts on K G ( X ) by taking the tensor product with a representation viewed as anequivariant bundle; the composite is a ‘transfer map’(3) τ : R ( G ) −→ K ( Y )which does involve the geometry of the principal bundle.The second elementary case is that of a trivial action. Since this correspondsto the isotropy group below we denote the compact Lie group acting as H. Let H ′ be a complete set of irreducible unitary representations, with M ( π ) the Hermitianvector space on which π ∈ H ′ acts(4) H ′ ∋ π = ⇒ π : H −→ U( M ( π )) . Each fiber of an H -equivariant bundle over M may then be decomposed under theaction to a sum(5) V x = M π ∈ H ′ V x ( π ) , V ( π ) = W ( π ) ⊗ M ( π )where the W ( π ) form an unrestricted bundle with compact support on M × H ′ . An H -equivariant isomorphism of V induces a bundle isomorphism of each of the W ( π ) and the decomposition of representations under tensor product then resultsin the identification as rings(6) K H ( M ) = K ( M ) ⊗ Z R ( H ) . The more general case in which the G -action has a single stratum is intermediatebetween these two cases, where now all the isotropy groups are conjugate. Borelshowed that such a smooth action of a compact Lie group with fixed isotropy typecan be reduced to a principal action by the quotient Q = N ( H ) /H of the normalizerin G for a choice of isotropy group(7) Q X H (cid:31) (cid:127) / / $ $ ■■■■■■■■■■ X { { ✈✈✈✈✈✈✈✈✈✈ Y = X/G where X H is the submanifold where the isotropy group is H. As rings(8) K G ( X ) = K N ( X H ) . In this setting the equivariant K-theory was identified by Wassermann [10] interms of twisted K-theory. Our description of K-theory in the general case follows -THEORY AND RESOLUTION, II 3 from a related decomposition of equivariant vector bundles in terms of twistedbundles, or gerbe modules, rather than the C ∗ -theoretic approach in [10].Thus, suppose that H ⊂ N is a closed normal subgroup (the notation indicatingthat N is perhaps the normalizer of H in a larger group) acting, with fixed isotropygroup H, on a compact manifold (allowed to have corners) Z ; so Z is a principal Q = N/H bundle over a compact manifold with corners Y. The action of N on H by conjugation induces a locally trivial action of the quotient group Q = N/H on the irreducible representations, H ′ , so with finite orbits. This action lifts to anaction on Z × H ′ with the same isotropy group, H. Our ‘reduced’ description of N -equivariant bundles over Z utilizes the finite covers(9) Y ( π ) = ( Z × N π ) /N −→ Y given by replacing H ′ by the orbit N π ⊂ H ′ . Schur’s Lemma shows that the projective action of N on the representation spaceof π lifts to a projective action of N on the sum over the orbit of the representationsand so to a central extension(10) U(1) / / ˇ N ( π ) / / N ( π )which corresponds to Mackey’s obstruction. Since H lifts naturally into ˇ N ( π )this defines a central extension ˇ Q ( π ) of Q for each orbit. Making a choice ofrepresentatives π ∈ H ′ parameterizing the orbit space H ′ N = H ′ /N the space(11) Z ( π ) = ( Z × N π ) Q / / Y ( π )is a principal bundle for the action of Q. The decomposition (5) may be captured by considering the H -equivariant ho-momorphisms(12) W x ( π ) = { l : M ( π ) −→ V x ( π ); h ◦ l = l ◦ π ( h ) ∀ h ∈ H } . This has dimension, n, the multiplicity of π in V x . The rigidity of representationsof compact groups means that n is constant on components of X. Let W ( π ) denotethe collective bundle over Z ( π ) with fiber W x ( π ′ ) at ( x, π ′ ) ∈ ( X, N π ) . The action of N on V induces a b Q ( π )-equivariant action on W ( π ) by the re-verse extension to ˇ Q ( π ) . This makes the W ( π ) into twisted bundles over Y ( π ) orequivalently into gerbe modules for the lifting bundle gerbe defined by the cen-tral extension b Q ( π ) and the principal bundle Z ( π ) −→ Y ( π ) . Functorially this isassociated with the (torsion) Dixmier-Douady class α ( π ) ∈ H ( Y ( π ); Z ) which isthe transgression of the Mackey class. Denoting the category of twisted bundles B un( Y ( π ); α ( π )) (in which the morphisms depend on the trivialization of α ( π ))this gives an equivalence of categories(13) B un N ( Z ) −→ M π ∈ H ′ N B un( Y ( π ); α ( π ))where the sums on the right are finite. Wassermann’s ([10]) description of theequivariant K-theory for an action with fixed isotropy type(14) K G ( M ) ≃ M π ∈ H ′ N K ( Y ( π ); α ( π )) PANAGIOTIS DIMAKIS AND RICHARD MELROSE is a direct consequence. Note that this is a significantly functorial statement becausethe twisted K-groups on the right depend on the representations of the Dixmier-Douady class with different representations leading to isomorphisms correspondingto tensoring with line bundles.The localized representation ring, L ( H ) , consisting of the H -invariant polyno-mials on the Lie algebra of H (so L ( H ) is the real representation ring if H isconnected), is acted upon by the normalizer, N, of H and hence by the quotient Q. This action is locally trivial so the quotient(15) Z × L ( H ) /N = L H ( Y )is a flat bundle over Y modeled on L ( H ) . By an argument similar to that for freeactions (see for example [7], [8]), the Cartan model for the equivariant cohomologyof X can be reduced to the deRham cohomology of the complex of forms withvalues in L H (16) C ∞ ( Y Z ; Λ ∗ ⊗ L H ) . Replacing L ( H ) by the real representation ring generated by H ′ , R R ( H ) = ⊕ ( H ′ ⊗ R ) , the quotient of Z × R R ( H ) by the action of Q is again the total spaceof a flat coefficient bundle which we denote R R ( Y ) . Then delocalized equivariantcohomology, in the sense of Baum, Brylinski and MacPherson, corresponds to thedeRham cohomology of(17) C ∞ ( Y ; Λ ∗ ⊗ R R ( Y )) . As a consequence of the properties of torsion-twisted K-theory, in particular thatthe Chern character is a deRham class in the usual sense, the Chern character ap-plied to the twisted bundles W ( π ) over Y ( π ) gives an element of (17) and definesthe delocalized Chern character (1). The usual equivariant Chern character to equi-variant deRham cohomology corresponds to applying the twisted Chern characterto the (twisted) coefficient bundles defined by the elements of H ′ and so factorsthrough (1) by a surjective localization map.A general smooth action by a compact Lie group, G, on a compact manifold, M, is reduced to an iterated version of the case of a single isotropy type by resolution.The ‘normal resolution’ of the action (which may not be minimal as a resolution),see [1], is obtained by the radial blow up of the isotropy types, the strata, of theaction in any order compatible with the (reversed) partial inclusion order on theisotropy classes. The result is a compact manifold with corners with iterated blow-down map(18) β : X −→ M with interior the preimage of the open isotropy type. The resolutions of the closures, M a , of the components of the non-open isotropy types are in 1-1 correspondencewith the boundary hypersurfaces H a of X as the images of equivariant fibrations(19) φ a : H a −→ X a , β a : X a −→ M a with the action on each X a having the same conjugacy class of closed subgroups asthe action on the interior of M a . These fibrations give X an iterated structure, inparticular the fibrations are jointly compatible at each boundary face of X, forminga chain under fiber inclusion with respect to the isotropy partial order.As shown in [6] smooth equivariant bundles over M lift to smooth equivariantbundles over each of the X a , related by pull-back under the φ a . This functor to -THEORY AND RESOLUTION, II 5 iterated equivariant bundles over X projects to an equivalence on the Grothendieckgroups, i.e. the iterated equivariant bundles on X lead to the equivariant K-theoryof M. On each of the compact manifolds with corners, X a , including X = X , thedescription of equivariant bundles proceeds through the Borel reduction to corre-sponding manifolds Z a where the action has a fixed isotropy group. The quotientmanifolds Y a with quotient fibrations φ a form an iterated manifold giving a reso-lution of the stratified space M/G.
The main step in defining the iterated twistedbundles representing equivariant K-theory on the resolution of the quotient is todescribe the pull-back maps corresponding to the equivariant fibrations φ a . It is sig-nificant here that the existence of an equivariant fibration for a space, H a −→ X a with G -actions on both H a and X a having single isotropy type, imposes a partialtriviality condition on the action on H a . Changing notation, let φ : X −→ X be a fibration of smooth manifolds withcorners, equivariant for G -actions with fixed isotropy type, and with quotient fibra-tion ˜ φ : Y −→ Y . We may assume that Y is connected. First pass to the Borelreduction Z = X H of X for a choice of isotropy group H , with normalizer N . The preimage Z = φ − ( Z ) is a ‘reduction’ of X generalizing Borel’s reductionand made precise below. Choose a component of Z and an isotropy group H atsome point in it. The isotropy groups at all points of this component are necessarily H conjugate to H , not just conjugate under the N -action. Let Z ′ ⊂ Z be theunion of the components where all isotropy groups are H conjugate to H , this is areduction of Z , and hence X , with group action by N ′ ⊂ N the subgroup underwhich H is mapped to an H conjugate. The image, Z ′ = φ ( Z ′ ) is a reduction of Z with the same group, N ′ acting. Passing to the Borel reduction Z ′′ ⊂ Z ′ withits N ′ -action gives spaces with isotropy groups H ⊂ H and actions by groups N and N ′ connected by an equivariant fibration. As for Borel reduction underthese successive steps the G -equivariant bundles over X and X , respectively, areidentified with with N and N ′ -equivariant bundles over Z ′′ and Z ′ . This ‘double’ reduction of the G -action on the boundary hypersurfaces H a ⊂ X corresponds to a refinement of the decomposition of G -equivariant bundles intotwisted bundles and allows pull-back under φ for equivariant bundles to be iden-tified with an ‘augmented’ pull-back, φ , on the corresponding twisted bundles.Applied iteratively this gives the compatibility conditions on twisted bundles anda functorial reduction(20) B un G ( X ) ≡ B un red ( Y ∗ ) ⊂ M π ∈ H ′ N B un( Y ( π ); α ( π )) , ⊕ α W α ∈ B un red ( Y ∗ ) ⇐⇒ W α (cid:12)(cid:12) H β = φ α,β W β ∀ β > α identifying G -equivariant bundles over the original G -space M with iterated twistedbundles over the resolution of the quotient. Notice however that the pull-back mapshere involve additional information not found in the quotient itself, and the sameis true of the twistings.There is a corresponding description of equivariant cohomology in terms of it-erated forms on the resolved quotient with coefficients in the bundles formed fromthe localized representation rings of the isotropy groups. These are linked by aug-mented pull-back maps. Then delocalized equivariant cohomology is defined by PANAGIOTIS DIMAKIS AND RICHARD MELROSE analogy where the coefficient rings are the full real representation rings. This re-sults in an Atiyah-Hirzebruch isomorphism (1), as envisioned by Baum, Brylinskiand MacPherson. This is shown using the six term exact excision sequences corre-sponding to pruning the isotropy tree, essentially as in [6].The authors thank Victor Guillemin, Eckhard Meinrenken and David Vogan forhelpful discussions. 1.
Resolution and Lifting
We refer to the corresponding sections of [6] for a description of the normalresolution of a compact manifold with a smooth action by a compact Lie groupand the categorical lifting of G -bundles. If the initial manifold has corners thenwe demand that the action be ‘boundary free’ in the sense that if one connectedboundary hypersurface is mapped to another by an element of the group then theydo not intersect. If this is not the case then it can be arranged by replacing themanifold by its total boundary blowup. This boundary-free property is preservedunder normal resolution and so applies throughout.2. Reduction
Borel, see [4], showed how to reduce the geometry of the action of a compactLie group with a fixed isotropy type to the special case of a fixed, i.e. normal,isotropy group. The geometry is thereby reduced to that of a principal bundlefor the quotient of the normalizer by the chosen isotropy group. We need a moregeneral version of this construction which we term ‘reduction’ below.Let X be a compact manifold (meaning possibly with corners always) with asmooth action by a compact Lie group, G, with unique isotropy type. This is afibre bundle X −→ Y = X/G with fibre modeled on
G/H for a choice of isotropygroup, H. By a reduction of X we mean a smooth submanifold Z ⊂ X which meetsevery orbit, so defines a subfibration over Y, is stabilized by a closed subgroup K in the strong sense that(2.1) g ∈ G, x ∈ Z then gx ∈ Z iff g ∈ K and on which K acts with fixed isotropy type. In particular Borel’s reduction isthe case where Z = X H is the set of points where the isotropy group is a givenelement of the conjugacy class. Equivariant K-theory behaves in the same way inthis more general setting. Proposition 1. If X is a compact manifold with a smooth action, with uniqueisotropy type, by a compact Lie group, G, and Z ⊂ X is a reduction of it, withstabilizer K, as defined above, then restriction gives an equivalence of categories (2.2) B un G ( X ) → B un K ( Z ) . Proof. If V is a G -equivariant bundle over X then its restriction to Z is K -equivariant. This defines a map (2.2) which is clearly functorial with respect toequivariant morphisms. Conversely if U is a K -equivariant bundle over Z then itlifts to G × Z under projection to the second factor. The left G -action induces atrivial G -action on the lifted bundle. The diagonal K -action in which k maps ( g, z )to ( gk − , kz ) is free, commutes with the G -action and is covered by the action of K on U. By (2.1) the quotient space given by the K -action is X and the K -action onthe bundle gives descent data defining a G -equivariant bundle. That these bundles -THEORY AND RESOLUTION, II 7 and operations are smooth follows from the local product decompositions impliedby the uniqueness of the isotropy types.Clearly restriction is a left equivalence for this extension. To see it is a rightequivalence set U = V (cid:12)(cid:12) Z for a G -equivariant bundle V on X. Then the action on V gives a bundle map of fibre isomorphisms(2.3) { g } × V z −→ V gz which is equivariant for the G actions and commutes with the K action { g }× V z −→{ gk − } × V kz and so extends to an equivariant isomorphism of V and the extensionof V (cid:12)(cid:12) Z . (cid:3) Borel reduction corresponds to a choice among the isotropy groups H ⊂ G. Thenormalizer N ( H ) = N G ( H ) acts on X H with the subgroup H acting trivially andthe quotient group Q = N G ( H ) /H acts freely. Thus X H is a principal Q -bundleover the orbit space Y = Q \ X H (2.4) Q X H (cid:15) (cid:15) Y which is independent of the choice of H up to equivalence.However, the choice of a particular isotropy group is not natural, in general,and at points below we need to allow changes of this choice. The closed subgroupsconjugate to H in G may be identified with the homogeneous space G/N ( H ) overwhich the space X fibres with the fibre above H g = gHg − being a principal Q ( H g ) = gQg − -bundle. In this more equivariant picture(2.5) Q ( H g ) X (cid:15) (cid:15) ι / / G/N ( H g ) z z ✉✉✉✉✉✉✉✉✉ Y the group G still acts as a fibre-preserving map.It is convenient to consider another setting closely related to reduction. Thus,suppose that a compact Lie group N acts on a compact manifold with corners Z with a fixed, hence normal, isotropy group H. A smooth submanifold Z ′ ⊂ Z willbe called a restriction of the action if it is a subbundle of Z as a principal N/H bundle with the subgroup N ′ ⊂ N acting fibre-transversally on Z ′ with isotropygroup H ′ = H ∩ N ′ . Clearly there is then a restriction functor(2.6) B un N ( Z ) −→ B un N ′ ( Z ′ ) . This functor arises in the connecting maps in the reduced picture of equivariantK-theory below.Borel’s model for the equivariant cohomology of a G -space X, for a compact Liegroup G, is to take a classifying bundle for G, so a contractible space EG on which G acts freely. Then the product action on X × EG is free. Writing the quotientspace as X × G EG (2.7) H G ( X ) = H ( X × G EG )where we are interested in the case of real coefficients. PANAGIOTIS DIMAKIS AND RICHARD MELROSE
Proposition 2.
Under a reduction of a smooth action of a compact Lie group, asdefined above (2.8) H N ( Z ) = H G ( X ) Proof.
Consider the product G action on G × Z × EG which commutes with the N action recovering X from G × Z. This has G action has the transversal { e }× Z × EG so the quotient may be identified with Z × EG.
Projected to this transversal theaction of N on G × Z × EG is identified with the product action on Z × EG withquotient Z × N EG.
On the other hand taking the quotient by the N action firstthis is also identified with the quotient X × G EG.
Thus it follows that there is anatural identification (2.8) for any reduction of a G action. (cid:3) Note that for a reduction the image gZ of Z under any element g ∈ G is areduction with group action by gN g − and (2.8) extends to provide natural iden-tifications between the various H gNg − ( gZ ) . From the naturality of these constructions the G -equivariant Chern characterof a G bundle over X can be realized as the N -equivariant Chern character forthe restriction of the bundle to a reduction Z and this is natural with respect torestriction in the Cartan model for equivariant cohomology.3. Single Isotropy type
The resolution of the manifold with group action reduces it to a tree of compactmanifolds with corners on each of which the group acts with all isotropy groupsconjugate; the resolved pieces are linked by equivariant fibrations over the bound-ary hypersurfaces. Borel’s reduction replaces each component manifold X by thesubmanifold Z = X H where the isotropy group, H, is fixed with the group replacedby the normalizer, i.e. H ⊂ N is a normal subgroup. Then Z is the total space of aprincipal Q = N/H -bundle; by treating the components separately we can assumethat the base, Y, is connected. We proceed to show how equivariant bundles arereduced to twisted bundles over finite covers of the base.Let H ′ be the category of irreducible unitary representations of H with M thetautological ‘bundle’ over H ′ with fibre the representation space. The conjugationaction of N on H induces an action of N on H ′ with finite orbits, o ( π ) = N π.
Thenover Z × o ( π ) consider the trivial bundle obtained by pulling back the tautoligicalbundle from H ′ , (3.1) M ′ −→ Z ( π ) = Z × o ( π ) , so with fibre at { z } × { π ′ } the representation space M ( π ′ ) of π ′ ∈ o ( π ) . Each n ∈ N conjugates the representation π ′ to nπ ′ and by Schur’s Lemma the unitaryconjugation is determined up to a multiple of the identity. This determines a centralextension(3.2) U(1) −→ ˇ N ( π ) −→ N. Thus ˇ N ( π ) acts equivariantly on M ′ covering its action through N on Z ( π ) . Wewrite ˇ N ( π ) since it is the ‘opposite’ extension, b N ( π ) , given by the dual of the circlebundle (3.2), which appears in the decomposition of equivariant bundles.The action by N on Z ( π ) has isotropy group H which naturally includes in ˇ N ( π )since it acts consistently on each M ( π ) . Thus Z ( π ) is the total space of a principal -THEORY AND RESOLUTION, II 9 bundle with base Y ( π ) and structure group Q = N/H whereas M ′ is a bundle over Z ( π ) with an equivariant action of ˇ N ( π ) . We now turn to the decomposition of an N -equivariant bundle, V, over Z onwhich N acts with fixed isotropy group H. The pointwise action of H induces afinite decomposition into subbundles(3.3) V = M π ∈ H ′ V ( π )where the action of H on V ( π ) is conjugate to the action on C n ⊗ M ( π ) for some n. This can be formalized by considering the bundle over Z × H ′ with fibre at ( z, π ′ )the space of H -equivariant homomorphisms(3.4) W ( z, π ′ ) = { l : M ( π ′ ) −→ V z ; l ◦ π ′ ( h ) = h ◦ l } = hom H ( M ( π ′ ) , V z , ) . The rigidity of representations implies that these are smooth bundles, although thedimension of W ( z, π ′ ) may vary between components of Z. Proposition 3.
The fibres (3.4) define a smooth bundle bundle W over Z × H ′ which carries an equivariant action of b Q ( π ) = b N ( π ) /H on the restriction W ( π ) toeach orbit Z ( π ) and is such that summing over the fibres (3.5) V = ( π Z ) ∗ ( W ⊗ M ′ ) as N -equivariant bundles over Z. This defines an equivalence of categories (3.6) B un N ( Z ) −→ M π ∈ H ′ /N B un b Q ( π ) ( Z ( π )) . Proof.
The given equivariant bundle V decomposes as in (3.3) and so lifts to abundle V ′ over Z × H ′ where the fibre at ( z, π ) is the image of W ( z, π ) . Then(3.7) W = hom H ( M ′ , V ′ ) = V ′ ⊗ H ( M ′ ) ∗ is the bundle of H -equivariant homomorphisms over Z × H ′ . Since the ˇ N ( π ) actionon M ′ induces an b N ( π ) action on the dual, W has (over each orbit Z ( π )) anaction of b N ( π ) on the second factor and of N on the first. Combined this gives anequivariant action of b N ( π ) in which H acts trivially. Thus W carries an equivariantaction of b Q ( π ) . Then V ′ is recovered as the bundle W ⊗ M ′ over Z × H ′ in which theextensions in the b N ( π ) and ˇ N ( π ) actions cancel to give an N -equivariant bundlewhich pushes forward, i.e. sums over the finite number of representations present,to give V. This relationship is an equivalence of categories. (cid:3) If W was actually Q -equivariant its restriction to Z ( π ) would descend to a bundleover the base, Y ( π ) , for the action of Q. Since the action is only projective itbecomes a twisted bundle over Y ( π ) . We treat these in the context of gerbe modules which we now recall. The central extension of Q ( π ) induces a lifting bundle gerbe, in the sense of Murray [9], over Y ( π ) :(3.8) C ( π ) (cid:15) (cid:15) / / ❴❴❴ b Q ( π ) (cid:15) (cid:15) Q ( π ) Z (cid:15) (cid:15) Z [2] o o o o | | ①①①①①①①①① / / Q ( π ) Y ( π )where C ( π ) is the pull-back of the circle giving the central extension.This gerbe is classified by its (torsion) Dixmier-Douady class α ( π ) ∈ H ( Y ( π ); Z )which in this case is the transgression of the Mackey class of the extension. Avector bundle W ( π ) over Z ( π ) with a b Q ( π )-action covering the Q ( π )-action is agerbe module in the sense of [5] since the induced isomorphisms over each point( p, q ) ∈ Z [2] (3.9) γ p,q : W p ⊗ C ( π ) p,q −→ W q are consistent with the gerbe structure. Thus: Proposition 4. A G -equivariant vector bundle on X, for an action with uniqueisotropy type, induces a gerbe module over Y ( π ) for each of the lifting bundle gerbes (3.8) , trivial outside a finite subset of H ′ /N and conversely; G -equivariant bundleisomorphisms correspond uniquely to b Q ( π ) -equivariant bundle isomorphisms, andhence gerbe module isomorphisms resulting in an equivalence of categories (3.10) B un G ( X ) = M π ∈ H ′ /N B un( Y ( π ); α ( π )) . Thus, each G -equivariant bundle over X induces gerbe modules over a finitecollection of the lifting gerbes (3.8) with π lying in different orbits for the actionof N on H ′ . Different choices of base point in the orbits induce gerbe moduleisomorphisms.
Theorem 5 (A. Wassermann) . For the action of a compact group G on a compactmanifold X with unique isotropy type, the equivariant K-theory can be identifiedwith the collective twisted K-theory with coefficients in the representation ring ofthe isotropy group (3.11) K G ( X ) = K ( Y ( π ); α ( π )) ⊗ H ′ /N R ( H ) Proof.
This is a direct consequence of Proposition 4, by passing to the correspondingGrothendieck groups, and the characterization of twisted K-theory in terms of gerbemodules by Bouwknegt, Carey, Varghese, Murray and Stevenson in [5]. (cid:3)
Although (3.10) gives a complete description of equivariant bundles in this case,the twisted bundles can be further reduced, in the sense of §
2, and this is relevantto the discussion of pull-back maps below.In the action of N on H ′ for a normal subgroup H, consider the stabilizer N ( π )and its normalizer Γ( π ) . The submanifold(3.12) Z ′ ( π ) = Z × Γ( π ) π ⊂ Z ( π ) = Z × N π -THEORY AND RESOLUTION, II 11 is a reduction of Z ( π ) for the action of b Q ( π ) which is reduced to the action of b Q ′ ( π ) = b Γ( π ) /H where b Γ( π ) is the central extension of Γ( π ) induced by b N ( π ) . As noted in the Introduction, and proved already by Cartan, the equivariantcohomology for a free action is the cohomology of the base. This can be seendirectly in terms of the Borel model and there is a corresponding argument for theCartan model using a connection, see the book of Guillemin and Sternberg, [7].Namely if α is a connection on X as a principal G -bundle over Y then the tracefunctional on g ⊗ g ∗ defines a linear map(3.13) C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) ∋ u −→ Au = Tr g ( α ∧ u ) ∈ C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) . Here S ( g ∗ ) is realized as the totally symmetric polynomials with the trace acting inthe first variable. This lowers the polynomial order by one. Acting on G -invariantforms which are homogeneous of degree p as polynomials on g (3.14) d G ( Au ) − Ad G + pu = pB p u where B p is given in terms of contraction with the curvature 2-form of α and is ofdegree p − B p to the terms of highestpositive degree, eventually stabilizes on each form, replacing an equivariant formby a basic one as discussed by Meinrenken [8].The same approach can be used to study the Cartan cohomology in the case ofan action with single isotropy type. It follows from the relationship between theBorel and Cartan models, see [7, Chap 4], that under Borel reduction the restrictionmap(3.15) C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) G −→ C ∞ ( X H ; Λ ∗ ⊗ S ( n ∗ )) N induces an homotopy equivalence of complexes realizing the isomorphism of H ∗ G ( X )and H ∗ N ( X H ) where N = N ( H ) is the normalizer.To see directly that the restriction, both on the manifold and the Lie algebra,gives an isomorphism the contraction operator A in (3.13) given in terms of aconnection form, can be replaced by contraction with a ‘partial connection’. Forconvenience, give the group an invariant metric. By such a partial connectionform we mean a G -equivariant 1-form on X, with values in g , which restrictedto the tangent space to the orbit at each point restricts to the identity on theorthocomplement of the Lie algebra of the normalizer of the isotropy group atthat point. For the homogeneous space G/H such a form can be constructed byorthogonal projection of the Maurier-Cartan form and in general by using a G -invariant metric on X to project the orthogonal subspace to the Lie algebra ofthe normalizer in the tangent space to the orbits. Application of the analogue of(3.13) lowers the polynomial order pointwise in this subspace and gives a homotopyequivalence for corresponding to (3.15).In the reduced case written as the action by N on Z with fixed isotropy group H the conjugation action of N on the Lie algebra h induces a locally trivial equivariantaction on Z × ( S ∗ ( h ∗ ) H ) defining a flat bundle S H over the base. Proposition 6. If N acts on a compact manifold Z with unique isotropy group H and base Y, a choice of invariant inner product on N and a connection on Z as the total space of a principal Q = N/H bundle defines a homotopy equivalenceintertwining the equivariant deRham differential on Z and the deRham differential on Y with coefficients (3.16) B ∞ : ( C ∞ ( Z ; Λ ∗ ⊗ S ( n ∗ )) N −→ C ∞ ( Y ; Λ ∗ ⊗ S H ) . Proof.
The given connection form α ∈ C ∞ ( Z ; Λ ⊗ q ) for Z as a principal Q bun-dle can be interpreted as a form with values in n using the embedding of q ⊂ n as the space orthogonal to h . Then the identity (3.14) allow a successive lower-ing of the polynomial order in q of Cartan forms on Z so identifying the equi-variant cohomology with the deRham cohomology of basic Q -invariant sections of C ∞ ( Z ; Λ ∗ ⊗ ( S ∗ ( h ∗ ) H )) . The action of Q is locally trivial and defines the descentto the cohomology of Y with coefficients in S H . (cid:3) For the extension to a general action below it is important that (3.16) definesan explicit homotopy equivalence.In this reduced case of an action with single isotropy group, an equivariantbundle V decomposes as a b Q ( π )-equivariant bundle W ( π ) over each Z ( π ) . theChern character can be see directly in terms of Proposition 6. Namely, given anequivariant connection, the b Q ( π )-equivariant Chern character of W ( π ) is a Q ( π )-equivariant form on Z ( π ) . The character map given by the trace defines a sectionof the trivial polynomial bundle(3.17) tr : H ′ −→ ( S ( h ∗ )) H over Z ( π ) which is invariant under the action of Q projected from the action ofˇ Q ( π ) on M ′ . The form and section combine to give a form over the quotient, Y ( π )with values in the flat descended bundle and this further pushes forward, summingover the fibres of each Y ( π ) , and over H ′ /N, to give the closed form(3.18) Ch( V ) ∈ C ∞ ( Y ; Λ even ⊗ S H ) . This descends to the equivariant Chern character in H N ( Z ) = H G ( X ) . The delocalized equivariant cohomology in this case of a single isotropy group isobtained by replacing the character ring by the real representation ring R ( H ) . As avector space this is freely generated by H ′ . Thus there is a linear subspace o ( π ) ⊗ R associated with each orbit of the action of Q on H ′ which can be viewed as thepush forward under the map from o ( π ) to a point. The action of Q on Z × o ( π )therefore generates a flat real line bundle over Y ( π ) which pushes forward to Y todefine a flat bundle. The formal sum over H ′ /N of these spaces is the flat bundle R H ( Y ) . Definition . The delocalized equivariant cohomology, H ∗ G, dl ( X ) , of a manifold withsmooth action, with unique isotropy type, by a compact Lie group is the deRhamcohomology of the base with coefficients in R H ( Y ) . Note that the forms appearing here, in C ∞ ( X/G ; Λ ∗ ⊗R H ( Y )) , are each requiredto have coefficients in a fixed finite span of the subspaces corresponding to the orbitsof the normalizer of an isotropy group on its representation ring. This space of formsis functorial under change of isotropy group as in (2.5). Theorem 7 (Following [2]) . For the action of a compact Lie group G on a com-pact manifold X with unique isotropy type, the equivariant Chern character factors -THEORY AND RESOLUTION, II 13 through a natural ‘delocalized Chern character’ as in (1)(3.19) K G ( M ) Ch dl / / Ch % % ▲▲▲▲▲▲▲▲▲▲ H even G, dl ( M ) loc x x qqqqqqqqqq H even G ( M ) with the top map an isomorphism after tensoring with R . So this is an Atiyah-Hirzebruch isomorphism.
Proof.
As in the discussion of the standard equivariant Chern character above, anequivariant K-class is represented by a formal difference V ⊖ C N where the trivialbundle contributes a constant form. The decomposition (5) yields the bundle W ( π )over each of the principal Q bundles Z ( π ) . A b Q -invariant connection on W ( π ) yieldsa Q -invariant form on Z ( π ) . Tensored with the canonical section on the trivial R -bundle over H ′ with its Q action this descends to a form with values in the flatline bundle over the quotient Y ( π ) and then pushes forward to a closed elementof C ∞ ( Y ; Λ even ⊗ R H . ) The deRham class of this form is well-defined, giving themap Ch dl in (3.19). The trace map on representations induces the localization mapcompleting the commutative diagram.For torsion-twisted K-theory there is an Atiyah-Hirzebruch isomorphism(3.20) Ch : K ( Y ; α ) ⊗ R −→ H even ( Y ; R ) . The crucial injectivity of this map follows from the multiplicative property oftwisted K-theory(3.21) K ( Y ; α ) × K ( Y ; β ) −→ K ( Y ; α + β )(given consistent trivializations of the torsion classes). At the level of bundles (3.20)corresponds to the fact that if nα = 0 then the n -fold tensor product of a twistedbundle is untwisted and division by n is an isomorphism over R . Applying (3.20), at the level of bundles over the Y ( pi ) , to (3.10) shows the in-jectivity of the delocalized Chern character over R . Since the surjectivity of theusual equivariant Chern character is known the delocalized Atiyah-Hirzebruch iso-morphism follows. (cid:3) Equivariant fibrations
Consider two compact manifolds X i , i = 0 , , with smooth G actions with fixedisotropy type where X is the total space of a G -equivariant fibre bundle over X (4.1) G Id (cid:15) (cid:15) X φ (cid:15) (cid:15) / / Y φ (cid:15) (cid:15) G X / / Y with Y connected.In the absence of the connectedness condition on Y (which becomes significant inthe stability of subgroups) we proceed component by component.First choose an isotropy group H for the lower action and pass to the Borelreduction Z = X H ⊂ X with its N = N ( H ) action. Next consider the preimage(4.2) Z = φ − ( X H ) ⊂ X . As the preimage of a smooth manifold under a fibration this is smooth and meetseach orbit of the G -action on X . Furthermore the pointwise stabilizer is exactly N since if x ∈ φ − ( X H ) and gx ∈ φ − ( X H ) then φ ( x ) ∈ X H and gφ ( x ) = φ ( gx ) ∈ X H so g ∈ N and conversely. The isotropy groups at points of Z remainunchanged since they are necessarily subgroups of H ⊂ N . Thus Z is a reductionof the action on X . We have thereby reduced the equivariant fibration to the special case where thelower action has unique isotropy group H (4.3) N (cid:15) (cid:15) Z φ (cid:15) (cid:15) / / Y φ (cid:15) (cid:15) N Z / / Y with normal isotropy group H . Now, consider the components of Z . Since Y is connected each of these con-nected submanifolds meets every orbit in Z and hence projects onto Y . Choose apoint in Z at which the isotropy group is H . The rigidity of subgroups, alreadyutilized above, shows that within the component containing this point all isotropygroups fall in the H -conjugacy class of H as a subgroup of H . Indeed, near thechosen point, the isotropy groups are contained in H and are conjugate to H byan element of N near the identity and hence by the product n h with n in thenormalizer of H and h ∈ H . Thus they are H -conjugate to H , in fact by anelement in the component of the identity in H . This therefore remains true overthe component of Z . Let Z ′ be the union of the images of this component underthe action of H ; all the isotropy groups on Z ′ are H -conjugate to H and all suchgroups occur in each orbit. The stabilizer of Z ′ is the subgroup N ′ ⊂ N (of finiteindex) fixing the H -isotropy class of H . This is a reduction of the group actionon Z and hence of that on X . The image φ ( Z ′ ) = Z ′ is a reduction of the action on Z to the subgroup N ′ ; eachof the components of Z ′ projects to a smooth submanifold and for two componentsthe images are either equal or disjoint. Thus Z ′ ⊂ Z is fixed by the conditionthat at every point in the preimage in Z the isotropy group is H -conjugate to H . Thus we arrive at the reduced equivariant fibration(4.4) N ′ (cid:15) (cid:15) Z ′ φ (cid:15) (cid:15) / / Y φ (cid:15) (cid:15) with isotropy groups H -conjugate to H N ′ Z ′ / / Y with normal isotropy group H . From this we deduce
Proposition 8.
For an equivariant fibration (4.1) with Y connected there arechoices of isotropy groups H ⊂ H for the two actions and a (‘ double ’) reduction, Z ′′ , of X with fixed isotropy group H and group action of N = N G ( H ) ∩ N G ( H ) and Z ′ of X with isotropy groups H and group action by N ′ = N H giving anequivariant fibration (4.5) N (cid:127) _ (cid:15) (cid:15) Z ′′ φ (cid:15) (cid:15) / / Y φ (cid:15) (cid:15) with normal isotropy group H N ′ Z ′ / / Y with normal isotropy group H . -THEORY AND RESOLUTION, II 15 Proof.
Having arrived at (4.4) we make the Borel reduction of the upper actioncorresponding to the isotropy group H and with action by N , the normalizer of H in N ′ . The fibres of φ above points of Z ′ have actions by H and isotropygroups, for the N ′ action on Z ′ , H conjugate to H . Thus, each fibre meets Z ′′ which fibres over Z ′ . The fact that the N ′ -conjugates of H are also H -conjugatesmeans that for each n ∈ N ′ there exists h ∈ H such that(4.6) n H n − = h H h − = ⇒ h − n H ( h − n ) − = H = ⇒ N ′ = H N . Since H is normal in N ′ , N ′ = N H and H acts trivially on Z ′ . (cid:3) Note that these reductions occur precisely because the principal bundles corre-sponding to the G actions are partially trivialized by the existence of the equivariantfibration. Since Z ′′ is a reduction of the G action on X and Z ′ is a reduction for X there are, by Proposition 1, equivalences of categories and hence a pull-backfunctor φ (4.7) B un G ( X ) / / B un N ( Z ′′ ) / / L π ∈ H ′ /N B un b Q ( π ) ( Y ( π )) B un G ( X ) / / φ ∗ O O B un N ′ ( Z ′ ) φ O O / / L σ ∈ H ′ /N ′ B un b Q ′ ( σ ) ( Y ( σ )) φ O O which we proceed to describe more explicitly.The pull-back operation on the right in (4.7) is the composite of pull-back for N ′ -equivariant bundles and restriction to the Borel reduction to N -equivariantbundles. It follows that it can also written as the composite of restriction, from N ′ -equivariant to N -equivariant bundles on Z ′ followed by pull-back of N -equivariantbundles. The functoriality of reduction to twisted bundles shows that the secondstep is just pull-back of twisted bundles so we concentrate on the first step whichcan be understood in terms of ‘branching maps’.For a compact Lie group and closed subgroup H ⊂ H consider the bundle overthe product of the sets of unitary irreducibles(4.8) τ / / H ′ × H ′ with fibre at ( σ, π ) the space hom H ( π, σ ) of H -equivariant linear maps from therepresentation space M ( π ) of π to the representation space M ( σ ) of σ. The sup-port of this bundle is proper as a relation.For simplicity we will denote N ′ in the discussion by N . Thus suppose as abovethat H ⊂ N is a normal subgroup of a compact Lie group, H ⊂ H , and N ⊂ N are closed subgroups with H normal in N and N = H · N . As a subgroup N ⊂ N acquires a central extension ˇ N ( σ ) ⊂ ˇ N ( σ ) from the central extension of N associated to the orbit of σ ∈ H ′ . The tautological bundle M ′ over H ′ thereforehas an action of ˇ N ( σ ) when restricted to N σ. In consequence the bundle τ hasan action of b N ( π ) × ˇ N ( σ ) over N σ × N π. Proposition 9. If X is a compact manifold with smooth action by N with fixedisotropy group H and H , N are as above then for each pair ( σ, π ) ∈ H ′ × H there is a ‘branching map’ (4.9) τ : B un( Y ( σ ); α ( σ )) −→ B un( Y ( π ); α ( π )) from b N ( σ ) -equivariant bundles over ( X × N σ ) /N to b N -equivariant bundles over ( X × N π ) /N given by tensor product with τ pulled back to X . Corollary 10.
The pull-back map on the right in (4.7) is the composite (4.10) φ = φ ∗ ◦ τ where φ ∗ is the pull-back of bundle gerbe modules covering the pull-back of principalbundles. For the Cartan model for equivariant cohomology, as noted in (3.15), reductioncorresponds to an homotopy equivalence to the complex of equivariant forms onthe reduction. Thus under the multiple reductions on X , (4.11) ( C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) G −→ ( C ∞ ( Z ′′ ; Λ ∗ ⊗ S ( n ∗ )) N where n is the Lie algebra of N which normalizes the isotropy group H . Thequotient group Q = N /H acts freely on Z ′′ and (3.16) gives the further homotopyequivalence(4.12) ( C ∞ ( Z ′′ ; Λ ∗ ⊗ S ( n ∗ )) N −→ C ∞ ( Y ; Λ ∗ ⊗ S H )to the deRham complex with coefficients in the flat bundle S H over Y . Similarly in the base the successive reductions and ‘twisted reduction’ give ho-motopy equivalences(4.13) ( C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) G −→ ( C ∞ ( Z ′ ; Λ ∗ ⊗ S ( n ∗ )) N ′ −→ C ∞ ( Y ; Λ ∗ ⊗ S H )to the deRham complex with coefficients in S H which has typical fibre S ( n ∗ ) . Since Q = N ′ /H = N /H , the actions on S ( n ∗ ) and n ∗ are consistent with restriction,induced by the inclusion of N ֒ → N ′ and hence there is a natural pull-back andrestriction map covering φ :(4.14) ( S H ) φ ( y ) −→ ( S H ) y , y ∈ Y and hence an augmented, smooth, pull-back map on ‘deRham’ sections(4.15) φ : C ∞ ( Y ; Λ ∗ ⊗ S H ) −→ C ∞ ( Y ; Λ ∗ ⊗ S H ) . Proposition 11.
Pull-back and iterated reduction gives a commutative diagram (4.16) ( C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) G / / C ∞ ( Y ; Λ ∗ ⊗ S H )( C ∞ ( X ; Λ ∗ ⊗ S ( g ∗ )) Gφ ∗ O O / / C ∞ ( Y ; Λ ∗ ⊗ S H ) φ O O where the horizontal maps are homotopy equivalences. Pull-back for the forms corresponding to delocalized equivariant cohomologybehaves similarly. The quotient bundle Q acts on both H ′ and H ′ . As in the caseof a general equivariant bundle the tautological bundle M ′ of H -representationsover Z ′ × H ′ decomposes into a bundle over the product with H ′ (4.17) M ′ −→ Z ′ × H ′ × H ′ with proper support, such that M ′ ⊗ M ′ pushes forward off H ′ to M ′ over Z ′ × H ′ . Restricting to a Q -orbit o ( σ ) ⊂ H ′ the support of M ′ is restricted to the finitenumber of orbits o ( π ) ⊂ H ′ of H -subrepresentations of σ. The bundle M ′ over Z ′ × o ( σ ) has an action of the corresponding central extension ˇ Q ( σ ) and following -THEORY AND RESOLUTION, II 17 the argument above M ′ has an action by the central extension ˇ Q ( π, σ ) which hasthe property(4.18) ˇ Q ( π, σ ) ⊗ ˇ Q ( σ ) = ˇ Q ( π )with the tensor product referring to the circle bundles giving the central extensions.5. Reduced models
Each of the three equivariant cohomology theories, K-theory, delocalized (bydefinition) and Cartan, has a model over the iterated space resolving the quotientby the group action and at this level the behavior of the Chern character can beseen rather directly.Not surprisingly, the most complicated of these is equivariant K-theory. Each ofthe strata, X α , of the normal resolution, has Borel reduction to a principal bundlewith total space Z α and base Y α for the action of Q α = N ( H α ) /H α where H α isa choice of isotropy group. We need to choose the isotropy groups consistently todefine the pull-back maps so we proceed categorically, allowing all possible choicesof isotropy group. Once a choice is made the spaces(5.1) Y α ( π ) = Z α ( π ) /N ( H α ) , π ∈ H ′ α , X α ( π ) = X α × o ( π ) , o ( π ) = N ( H α ) π are defined for each orbit of N ( H α ) acting on the category of irreducible unitaryrepresentations H ′ α . Then the reduced version of an equivariant bundle consists of afinite collection of twisted bundles for the Dixmier-Douady classes, a α ( π ) , of Z α ( π )over the corresponding Y α ( π ) . As discussed above, these are bundles gerbes overthe X α ( π ) as the total space of a lifting bundle gerbe given by the free action of Q α ( π ) and the central extension b Q α ( π ) arising from the action of N ( H α ) on o ( π ) . Thus for each stratum we have a category of reduced equivariant bundles(5.2) B un α = M H ′ α /N ( H α ) B un( Y α ( π ) , a α ( π )) . This behaves naturally under change of choices.For each β < α there is a boundary hypersurface H β,α of X β with fibration φ β,α : H β,α −→ X α . Proposition 4.9 describes the corresponding augmented pull-back map(5.3) φ β,α : B un α −→ B un β (cid:12)(cid:12) H β,α . Then the ‘chain space’ for K-theory consists of W α ∈ B un α for each α such thatfor all α < β (5.4) W α (cid:12)(cid:12) H β,α ≃ φ β,α W β where the implied isomorphisms must compose under iteration. The collectionof reduced bundles is then a category and the reduced K-theory K red ( Y ∗ ) is thecorresponding Grothendieck group; of course, despite the notation, this depends onmore information than just the quotient spaces. Proposition 12.
The equivariant K-theory of X is naturally isomorphic to thereduced K-theory K red ( Y ∗ ) of the resolution of the quotient.Proof. This follows by application of the reduction results above. (cid:3)
For equivariant cohomology there is a closely related ‘reduced’ Cartan model.In the same setting as above, let C ( H α ) be the space of H α -invariant polynomialson the dual of the Lie algebra of H α . The normalizer N ( H σ ) acts on this space byconjugation, with the action descending to Q α and locally trivial. Lifting C ( H α )to a trivial bundle over X α this action defines a flat bundle C α over the base Y α ofthe G action. Then the reduced Cartan model consists of forms(5.5) u α ∈ C ∞ ( Y α ; Λ ∗ ⊗ C α )over each Y α with values in C α but connected by (reduced) pull-back maps as in(4.16),(5.6) u α (cid:12)(cid:12) H β,α = φ ∗ β,α u β . Again by repeated application if the reduction results above we arrive at
Proposition 13.
The equivariant cohomology of X is naturally isomorphic to thereduced Cartan cohomology which is the deRham cohomology of forms (5.5) , (5.6) . Finally we define delocalized equivariant cohomology by replacing the Cartanspace C ( H α ) by the real representation ring of H α , R ( H α ) = I ( H α ) ⊗ R , to whichit is isomorphic in the connected case. The action of N ( H α ) by conjugation on H α induces an equivariant action on I ( H α ) and hence on R ( H α ) and this defines a flatbundle R α over Y α . Then the admissible ‘delocalized’ forms are(5.7) v α ∈ C ∞ ( Y α ; Λ ∗ ⊗ R α )connected by the pull-back maps augmented by branching(5.8) v α (cid:12)(cid:12) H β,α = φ ∗ β,α v β . The delocalized cohomology H ∗ G, dl ( X ) is defined to be the deRham cohomology ofthis complex of forms. In this sense there is no reduction theorem but a definition.It is very natural to expect the possibility a sheaf-theoretic formulation directlyover the original space.The surjective character maps R ( H ) −→ C ( H ) for the isotropy groups inducesurjective ‘localization’ maps from the forms (5.7), (5.8) to the Cartan forms (5.5),(5.6) through relaxation of the local coefficients from R α to C α . This thereforedescends to a surjective map(5.9) loc : H ∗ G, dl ( X ) −→ H ∗ G ( X ) . To define the equivariant Chern character as a homomorphism giving a commu-tative triangle(5.10) K G ( X ) Ch G, dl / / Ch G % % ❑❑❑❑❑❑❑❑❑❑ H even G, dl ( X ) loc x x qqqqqqqqqq H even G ( X )we start with an appropriate notion of ‘reduced connection’ on the bundles (5.3),(5.4).Each bundle W α can be equipped with a connection which is invariant underthe action of b N α . The Chern character is then an even closed form on X α which isinvariant under the action of b N α , since the center acts trivially on forms thus is atrue form on the base Y α . The same is true for the ˇ N α action on M ′ as a bundle -THEORY AND RESOLUTION, II 19 over X α and the sum over the fibres of the product of these closed forms is anelement of the space (5.7). The collection of these forms satisfies (5.8) provided theconnections are chosen consistently. This defines the Chern character (5.10) withcommutativity following from the lifting of the standard definition of the equivariantChern character, see for example [3].6. Atiyah-Hirzebruch isomorphism
Theorem 14.
For any smooth action by a compact Lie group on a compact man-ifold the delocalized equivariant Chern character defines an isomorphism (6.1) Ch G, dl : K G ( X ) ⊗ R −→ H ∗ G, dl ( X ) . The proof follows the same lines as in [6]. The delocalized equivariant cohomol-ogy is naturally Z -graded (not Z -graded because of the Chern character factorsin the augmented pull-back maps). Odd K-theory is defined in the usual way bysuspension, i.e. as the null space of the restriction morphism(6.2) K G, red ( S × Y ∗ ) −→ K G, red ( { } × Y ∗ ) . Consider the ‘pruning’ of the isotropy tree. Thus let A be an ordered collectionof the indices, so(6.3) β < α, β ∈ P = ⇒ α ∈ P. In each cohomology theory the notion of triviality on the X α for α ∈ P is well-defined giving relative versions(6.4) K ∗ G, red ( Y ∗ ; P ) , H ∗ G, dl ( Y ∗ ; P ) and H ∗ G ( Y ∗ ; P ) . In all three cohomology theories there is an excision sequence corresponding totwo ordered sets P and P ′ = P ∪ { γ } . Namely for (reduced) equivariant K-theory(6.5) K G, red ( Y ∗ ; P ′ ) / / K G, red ( Y ∗ ; P ) / / K G, red ( Y γ ; P ) (cid:15) (cid:15) K G, red ( Y γ ; P ) O O K G, red ( Y ∗ ; P ) o o K G, red ( Y ∗ ; P ′ ) . o o The delocalized Chern character gives an exact complex(6.6) K G, red ( Y ∗ ; P ′ ) / / Ch (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ K G, red ( Y ∗ ; P ) / / Ch (cid:15) (cid:15) K G, red ( Y γ ; P ) Ch (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ (cid:15) (cid:15) H ( Y ∗ ; P ′ ) / / H ( Y ∗ ; P ) / / H ( Y γ ; P ) (cid:15) (cid:15) H ( Y γ ; P ) O O H ( Y ∗ ; P ) o o H ( Y ∗ ; P ′ ) . o o K G, red ( Y γ ; P ) O O Ch ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ K G, red ( Y ∗ ; P ) o o Ch O O K G, red ( Y ∗ ; P ′ ) . Ch _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ o o This complex remains exact when tensored with R . We proceed by inductionover P starting from the case that P contains all elements except the minimalone corresponding to the open isotropy type. The top right and lower left Cherncharacter maps are isomorphism as discussed above and by induction the other twocorner Chern maps are isomorphisms. The fives lemma shows the central maps tobe isomorphism, so the induction continues. The case that P = ∅ is the Atiyah-Hirzebruch-Baum-Brylinski-MacPherson isomorphism. References [1] Pierre Albin and Richard Melrose,
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