Torsion in the full orbifold K-theory of abelian symplectic quotients

TORSION IN THE FULL ORBIFOLD K -THEOR Y OF ABELIAN SYMPLE CTIC QUOTIENTS REBECCA GOLDIN, MEGUMI HARADA, AND T ARA S. HOL M A B S T R A C T . Let ( M , ω , Φ) be a Hamiltonian T -space and let H ⊆ T be a clos ed Lie subtorus. Under some technical hypotheses on the moment map Φ , we prove that there is no additive torsio n in the integral full o rbifold K -theory of the orbifold symplectic quotient [ M / /H ] . Our main technical tool is an extension to the case of moment map le vel sets the wel l -known result that many components of the moment map of a Hamiltonian T -space M are Mo r se-Bott functions on M . As first applications, we conclude that a large class of symplectic tor ic or bifo lds, as well as certain S 1 -quotients of GKM spaces, have integral full orbifold K -theory that is free o f additive torsion. Finally , we i ntroduce the notion of semilocally Delzant which allows us to formulate sufficient conditions under which the hypotheses o f the main theorem hold. W e illustrate o ur results using low-rank coadjoint orbits of type A and B . C O N T E N T S 1. Introduction 1 2. A local normal form and Mo r s e-Bott the ory on level sets of moment maps 3 3. The pr oof and a corollary o f the main theorem 6 4. Symplectic to ric o r bifolds 8 5. GKM spaces 10 6. Semilocally Delza nt spaces 13 References 16 1. I N T R O D U C T I O N The main purpo se of this manuscript is t o show that the in tegral full orbifold K -theory of se v- eral classe s of orbifolds X arising as abelian symplectic quo t ients are free of additive t o rsion. An important subclass of symplectic qu otients to which our results app ly are orbifold toric varietie s , of which weigh ted projective spaces are themselves a sp e cial case. Orbifold toric va rieties ar e global quotients of a manifold by a torus action, and are there- fore a natu r al st arting point for a study o f o rbifolds. Many conjectures on orbifolds and o rb- ifold invariants in active areas of r es ear ch (algebraic geometry , equivariant topology , the t heory of mirr or symmetry , t o name a few) h ave been first tested in the real m of orbifold t oric varieties. Date : November 3, 2018. 2000 Mathematics Sub ject Classification. Primary: 19L47; Secondary: 53D20. The fir st author was partially suppor ted by NSF-DMS Grant #0606869 and by a Geo rge Mason University Provost’s Seed Grant. The second author was par tially suppor ted by an NSERC Dis covery Grant, an NSERC University F aculty A ward, and an Ontario Ministry o f Research and Innovation Early Researcher A ward. The third author was partially supported by NSF-DM S Gr ant #0835507 and and by a President’s Co uncil of Cornell W omen Affinito-Stewart Grant. 1 2 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM More specifically , there has been historically [2 – 4, 29, 35] and also qu ite recently a burs t of inter- est in weighted projective s paces (and their integral inva riants); for more r ecent work , s e e for instance [7, 9, 12, 17, 26, 38]. W e note t h at the application of our main result to orbifold toric varieties is in the s pirit of the previous work in t he study of (both or dinary and orbifold) top o logical inva riants of weighted projective sp aces [3 , 1 5, 29]. Moreover , recent work of Hua [27] uses algebro-geometric methods t o show that Gr othend ieck groups of a lar g e class of toric Deligne-Mumford s t acks are fr ee of addi- tive torsion. This part of o ur results (Theo r em 1.2 below) can be viewed as a full orbifold K-t h e ory analogue of his results in the t o pological categ o ry , proved via sy mplectic ge ometric meth o ds. However , the scope of our results is more gen e ral. While the above-mentioned work all d eal with certain case s of orbifold to r ic varieties, the te chniques in this manuscript, w hich build upon the symplectic and equivariant Mo rse theoretic methods d eveloped in [15], allow us to prove th at the full orbifold K -theory is fr ee of additive torsion in more gene ral set tings. In particular , we discuss non-toric e xamples in t h e later sections. Let T denote a compact connected abelian Lie group, i.e., a torus. Supp ose ( M , ω , Φ ) is a Hamil- tonian T -space, with moment map Φ : M → t ∗ . Furthermore, let β : H ֒ → T be a closed Lie subgroup (i.e. a subtorus). Let Φ H : M → h ∗ be the induced H -moment ma p obtained as the composition of Φ : M → t ∗ with the linear projection 1 β ∗ : t ∗ → h ∗ . Supp ose η ∈ h ∗ is a regular value of Φ H , and denote by Z := Φ − 1 H ( η ) ⊆ M the corresponding level set. Since η is a regular value, Z is a smooth submanifold of M , and H acts locally fr eely on Z . Let (1.1) X := [ M / / η H ] = [ Z/H ] denote the quotient s t ack ass ociated to the locally free H -a ction on Z. This is an orbifold, also referr ed to as a Deligne-Mumford stack in the diff erentiable category . Let ξ ∈ t and reca ll that Φ ξ := h Φ , ξ i : M → R deno t es the corresponding component of the momen t map. The full orbifold K -theory K orb ( X ) over Q was introduced by Jarvis, Kaufman and Kimura in [28]. In the case that X is formed as an abelian quotient of a manifold Z by a locally free action of a to rus T , the authors of this manuscript and Kimura gave an integ ral lift K orb ( X ) in te rms of the inertial K -the ory N K T ( Z ) in [15]. T his satisfie s K orb ( X ) ⊗ Q ∼ = K orb ( X ) as rings [8]. Specifica lly , the full o rbifold K -theory may be d e scribed additively as a mod ule o ver K T (pt) by K orb ( X ) = M t ∈ T K T ( Z t ) . This dif fers fr om previous de finitions of “orbifold K -th e ory ,” e.g. that of Ade m and Ruan [1]. W e refer the r eade r to the introduction of [15] for a more detailed discussion of other notions of orbifold K -theory in th e literature. In t his manuscript, for G a compact Lie group and Y a G -space, we let K G ( Y ) = K 0 G ( Y ) deno t e t he Atiyah-Segal topolog ical G -equivariant K -theo ry [36]. This is built fr om G -equivariant ve cto r bundles whe n Y is a compact G -space, and G -equivaria nt maps [ Y , F red( H G )] G if Y is noncompact (he re H G is a Hilbert s pace t hat cont ains infinitely many copies of ever y irreducible representation of G , s ee e.g. [6]). W e may now s tate our main theorem about the structur e of K orb ( X ) . Theorem 1.1. Let ( M , ω , Φ) be a H amiltonian T -space, β : H ֒ → T a connected subtorus with induced moment map Φ H := β ∗ ◦ Φ : M → h ∗ . S u ppose η ∈ h ∗ is a reg ular value of Φ H , let Z := Φ − 1 H ( η ) denote its level set, and X := [ Z/H ] the associat ed qu otient orbifold stack. Su ppose ther e exists ξ ∈ t such that the following conditi ons hold: 1 By s light abuse of notation we use β to also denote the linear map h → t obtained as the derivative of the inclusion map β : H ֒ → T . TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 3 (1) H ⊆ exp( tξ ) , the closur e of the one-par ameter subgr oup generated by ξ in T ; (2) f := Φ ξ | Z is pr oper and bounded below; (3) for each t ∈ H , π 0 (Crit( f | Z t )) is finite; (4) for each t ∈ H and each connected component C of C rit( f | Z t ) , (a) K 0 H ( C ) contains n o additive torsion, and (b) K 1 H ( C ) = 0 . Then K orb ( X ) contains no additive torsion. A direct consequence is th at when the components of the cri tical set are isolated H -orbits, K orb ( X ) contains no additive torsion (se e Cor ollary 3.2). W e use this corolla ry to prove the fol- lowing. Theorem 1.2. L et X be a sympl ectic toric orbifold obtained as a symplectic quotient of a linear H -action on a complex affine space , wher e H is a conn ected compact torus. T hen K orb ( X ) is free of additive torsion. As mentione d above, t h is corollary is similar in spirit t o Kawasaki’s result that the integral cohomology o f (the underlying topological s paces of) weighted projectiv e spaces are fr ee of addi- tive torsion. Ka wasaki showed in [29] that the integral cohomolog y groups of (th e coarse moduli space of) a weighte d projective s pace agr ee with those of a smooth pr ojective space, but the ring structure diff ers , with structure constants that depen d on the weights. Hence we expect t h at the richness o f the data in K orb ( X ) for toric orbifolds is also contained not in additive tors ion but rather in the multiplicative structure constants of t he ring . W e leave this for future work. The main the orem may be applied in situations othe r than that of the Delzant construction of orbifold toric varieties; the cont e nt of the last two se ctions of this manuscript is an exploration o f other s ituations in equivariant symp lectic geometry in which the hypothe ses also hold. First, we observe in Se ction 5 that the hypothese s above on t he relevant connected components C hold for S 1 -symplectic quotients of Hamiltonian T -spaces which are GKM, under a technical cond ition on the choice of subgroup S 1 ⊆ T . Spaces with T -action which satisfy the so-called “GKM cond i- tions,” introduced in the influential work of Goresky-Kottw itz-Macpherso n [16], are exte n s ively studied in equivariant algebraic g eometry , s ymplectic g eometry , and g eometric representation theory , and encompass a wide array o f examples. W e use a corollary of the main the orem t o prove Theorem 5.1, which state s that t h e full orbifold K -the ory o f the quotient of a GKM sp ace by certain cir cle s ubgroups is free of add itive to rsion. Second ly , we explore in Se ction 6 how the es sential properties of th e Delzant construction (which allow us to prove The orem 1.2) may in fact be placed in a more gene ral framework of pheno mena in t orus-equivariant symplectic ge ometry which may be informall y described as ‘taking p lace within a T -equivariant Darboux neighborhood of an iso- lated T -fixed p o int.’ The precise statements ar e g iven in detail in Section 6 , where we introduce the notion of a closed H -invariant s ubset of a H amiltonian T -sp ace being semilocally Delzant (with respect t o H ), and make some initial r emarks on situations in which this n o tion applies. One class of spaces to which our de finitions apply ar e the ge neralized flag varieties G/B and G/P , which may be covered by Darboux neighborhoods given by t he W ey l translates of the open Bruhat cell. Furthermore, the natural T -action o n generalized flag varieties (that of the maximal torus T in G ) is also well-known to be GKM. In both Sections 5 and 6, we illustrate o u r results using examples of this ty pe. 2. A L O C A L N O R M A L F O R M A N D M O R S E - B O T T T H E O R Y O N L E V E L S E T S O F M O M E N T M A P S W e begin with our main t echnical lemma (Lemma 2.2) regarding t h e Mo rse-Bott theory of mo- ment maps in equivaria nt symplectic geo me try . T he techniques used t o prove t his result are fairly 4 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM standard in the field, but we have no t seen this particular formulation in the literatur e. It is well- known that compone nts o f moment maps Φ ξ = h Φ , ξ i : M → R ar e Morse-Bott functions o n a Hamiltonian T -s p ace M , for any ξ ∈ t . In addition, t hese compon e nts induce Mo rse-Bott func- tions on smooth symplectic quotients M / / η H , w here H is a closed Lie subgroup of T , and η is a regular value of the H -m oment map Φ H [34]. What seems her et ofore unnoticed 2 is that a component Φ ξ of the T -moment map, r est ricted t o the level set Φ − 1 H ( η ) itse lf, is also a Morse-Bot t function w hen H is contained in the closure of the subgroup ge nerated by ξ . Th is may be d educed fr om the following local nor mal form result of Hilgert, Nee b, and Plank [25, Lemmata 2.1 and 2.2], which builds on work of Guillemin and Sternberg [19, Chapter II]. Note that g eneric ξ satisfy th is condition. Proposition 2.1 (Hilgert, Neeb, Plank) . L et ( M , ω , Φ ) be a Hamiltonian T -space with moment map Φ : M → t ∗ . Let p ∈ M . Then ther e exists a T -invariant neighbor hood U ⊆ M of the orbit T · p ⊆ M , a subtorus T 1 ⊆ T and a symplecti c vector space V such that: 1. Ther e is a decompo sition T = T 0 × T 1 , wher e T 0 = S tab ( p ) 0 is the connected component of the identity in the stabilizer gr oup of p in T . 2. Ther e is a T -equivariant sympl ectic open covering from an open subset U ′ of T 1 × t ∗ 1 × V onto U , wher e the T -action on T 1 × t ∗ 1 × V is given by ( T 0 × T 1 ) × ( T 1 × t ∗ 1 × V ) → ( T 1 × t ∗ 1 × V ) (( t 0 , t 1 ) , ( g , γ , v )) 7→ ( t 1 · g , γ , ρ ( t 0 ) v ) , (2.1) wher e ρ : T 0 → S p ( V ) is a linear symplectic re pres entation. 3. Ther e exists a complex str ucture I on V such that h v, w i := ω V ( I v , w ) defines a positi ve defi- nite scalar pr oduct on V . Let V = L α V α be the d ecomposition of V into i sotypic compo nents corr esponding to weights α ∈ t ∗ 0 . W ith r espect to these local coord inates, the moment map Φ ′ on U ′ ⊆ T 1 × t ∗ 1 × V is given by Φ ′ : U ′ ⊆ T 1 × t ∗ 1 × V → t ∗ ∼ = t ∗ 0 ⊕ t ∗ 1 ( g , η , v ) 7→ Φ ′ (1 , 0 , 0) +  1 2 X || v α || 2 α, η  . (2.2) For any ξ ∈ t , let T ξ := exp( tξ ) den o te the closure of the one-parameter s ubgroup gene rated by ξ ∈ t . Using the no t ation se t in the Introduction, we now have the following. Lemma 2.2. Let ( M , ω , Φ) be a Hamilto nian T -space, and H ⊆ T a subtorus. Let Z := Φ − 1 H ( η ) be a level set of the moment map for the H action at a regular value. The function f := Φ ξ | Z : Z → R is a Morse-Bott function on Z f or every ξ ∈ t such that H ⊆ T ξ . Pro of. W e show th at for any point p ∈ Z su ch that d f p = 0 , 1. the connected compo nent of Cr it( f ) containing p is a submanif old, where C rit( f ) is the critical se t o f f , and 2. the Hes s ian of f at p is no n-degenerate in t h e d ir ections normal to the connected compo- nent of Crit( f ) containing p . 2 However , a result o f this nature appears to be implicit in the work of Lerman and T olman on the classification of orbifold toric varieties [33] , and even earlie r in wor k of Mars den and W ei nstein [34] and Atiyah [5]. TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 5 Since the conditions to be checked are purely local, we may ar gue separately for each point p in the critica l set Crit( f ) . For the purposes of t his ar gument, we may assume without loss of generality t hat the T - equivariant sy mplectic open cover U ′ → U of Pr opos ition 2.1 is in fact a T -equ ivariant sym- plectomorphism. The only part of t his claim requiring justification is the relationship, in gen- eral, betwe en the moment maps Φ 1 and Φ 2 associated to Hamiltonian T -s p aces ( M 1 , ω 1 , Φ 1 ) and ( M 2 , ω 2 , Φ 2 ) wher e ther e exists a T -equivariant symplectic o pen cove r π : M 1 → M 2 . Sinc e by assumption π ∗ 1 ω 2 = ω 1 and π ∗ ( ξ ♯ M 1 ) = ξ ♯ M 2 for all ξ ∈ t where ξ ♯ M i denote s the infinitesmal vector fields generated by ξ on the M i , it follows immediately fr om Hamilton’s equations that π ∗ Φ 2 may be chosen as a moment map Φ 1 for the T -action on M 1 . In particular , since π is an o pen covering, the local ar gument for Φ 1 in a small enoug h neighborhood of a point p in M 1 translates directly to an analogous argument in M 2 for Φ 2 . Therefore we henceforth assume that (2.1) and (2.2) locally repr esent a ne ighborhood of p , and Φ near p ∈ Z , respectively . W e continue with a characterization of t he critical points Crit( f ) ⊆ Z . Recall T ξ := exp( tξ ) . L et Stab T ξ ( p ) d enote the st abili zer group in T ξ of p and co dim( H , T ξ ) t h e codimens ion o f the subgroup H in T ξ . Sup pose p ∈ Z. W e claim that p ∈ C rit( f ) if and only if dim(Stab T ξ ( p )) = co dim( H , T ξ )) . Note that p ∈ Z immediately implies d im S tab T ξ ( p ) ≤ co d im ( H , T ξ )) , s ince H acts locally freely on Z . By definition, a point p ∈ Z is critica l fo r f if and o nly if d f p ( v ) = h d Φ p ( v ) , ξ i = ω p ( ξ ♯ p , v ) = 0 , ∀ v ∈ T p Z, where T p Z denot e s the tangent sp ace at p to Z . Note also that the tangent s p ace T p Z = T p Φ − 1 H ( η ) = ( T p ( H · p )) ω p ⊆ T p M . Thus p ∈ Z is critical for f if and only if ξ ♯ p ∈ (( T p ( H · p )) ω p ) ω p = T p ( H · p ) . Since ξ generates T ξ , it follows that p ∈ Crit( f ) if and only if (2.3) T p ( T ξ · p ) ⊆ T p ( H · p ) , Hence dim Stab T ξ ( p ) ≥ cod im( H , T ξ ) . Thus p ∈ Z is critical for f if and only if dim Stab T ξ ( p ) = co dim( H , T ξ ) . The above argument shows that for any ξ ∈ t with H ⊆ T ξ , the critical set Cr it( f ) is precisely the union of sets of the form Z ( T ′ ) for s ubtori T ′ of T ξ such t hat dim( T ′ ) = cod im( H , T ξ ) , where Z ( T ′ ) := { p ∈ Z : Stab T ξ ( p ) = T ′ } consists o f the points whose stabiliz er group in T ξ is precisely T ′ . Since H acts locally fr eely o n Z , a subtorus T ′ of T ξ as above has maxima l dimension among s ubtori of T ξ with nonempty Z ( T ′ ) . Now let p ∈ C rit( f ) . Consider local coordinates near p as in (2.1), with Φ near p described by (2.2). W rite ξ = ξ 0 + ξ 1 for ξ 0 ∈ t 0 , ξ 1 ∈ t 1 . W e first d etermine the intersection of C rit( f ) with this coordinate chart, in terms of these local coo rdinates. From the description of t he T = T 0 × T 1 - action in (2.1), and from the fact observed above that p is in C rit( f ) pr ecisely wh e n its stabilizer subgroup is o f maximal possible dimension, it follows t h at Cr it( f ) is the se t of points of the form { ( g, γ , v ) : v ∈ V 0 } where V 0 is the subspace of V on which T 0 acts trivially . In particular , Crit( f ) is a submanifol d of Z near p . Finally , we show t h at the He ssian of f near p is nondege n e rate on those tangent directions in T p Z corr esp onding to tang e nt vectors of the form { (0 , 0 , P α 6 =0 v α ) : v α ∈ V α , α 6 = 0 } in the chosen 6 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM local coordinates. Recall that for t ang e nt vectors v , w ∈ T p Z, the He ssian Hess( f ) p ( v , w ) is com- puted by L ˜ v L ˜ w ( f ) w h e re ˜ v , ˜ w are arbitrary extensions of v , w to vector fields in a neighborhood of p in Z (a nd L X denote s a Lie d erivative along a vector field X ). In the local coordina tes of Propo- sition 2.1, any t wo vecto rs o f t he form v = (0 , 0 , P α 6 =0 v α ) , w = (0 , 0 , P α 6 =0 w α ) may be ex t ended to a neighborhood as the constant vector fie ld ˜ v ≡ (0 , 0 , P α 6 =0 v α ) , ˜ w ≡ (0 , 0 , P α 6 =0 w α ) . W e then observe that the des cription of Φ in (2.2) implies that for such a ˜ w , L ˜ w ( f ) = d f ( ˜ w ) = d (Φ ξ 0 | Z )( ˜ w ) , since ˜ w contains no component in t ∗ 1 . It t hen suffices t o show that the Hessian of the t ∗ 0 -component of Φ is nond egenerate in the dir ections ⊕ α 6 =0 V α . From the local normal form of Φ in (2.2) , this is just a standard quadratic moment map for a linear symplectic action of a torus o n a sy mplectic vector space, so this non-degener acy is classical (see e.g. [5]).  3. T H E P R O O F A N D A C O R O L L A R Y O F T H E M A I N T H E O R E M W e now pr ove the main theo r em. The argument u ses equivariant Morse t heory of the moment map, most of which is standard (see , for example, [24, 31, 32, 37]). The no vel featur e here involves the use o f a compo nent of t h e mome n t map on a level se t of a mome nt map for a p artial torus action. W e use the same notation as in t he introduction. Proof of T h eorem 1.1. W e first note that since the st atement of the theorem involves only the ad- ditive structur e of K orb , we need only r ecall the definition (a nd computation) of K orb ( X ) as an additive group. In [15] (cf. also [8]), the integ r al full orbifold K -theory of orbifolds X arising as abelian s ymplectic quotients (by a torus H ) is described via an isomorphism [15, R emark 2.5] K orb ( X ) ∼ = N K H ( Z ) := M t ∈ H K H  Z t  where the middle term is the H -equivariant inte gral inertial K -theory of the manifold Z := (Φ H ) − 1 ( η ) , defined additively as the dir ect sum above. W e n o w sho w that the right-hand side is torsion free. Note that Z t = (Φ H | M t ) − 1 ( η ) , so it is itself a level se t for the H -moment map on M t for each t ∈ T . Suppo se ξ ∈ t satisfies the hy pothese s o f t he theo rem, and let f = Φ ξ | Z . Since f is proper and bounded be low , then clearly f | Z t is also pr ope r and bounded be low . It is now immediate that ξ satisfie s conditions (1)–(4) for th e H amiltonian T -space M t . Thus without loss of ge nerality , we need o nly check that K H ( Z ) is torsion-free; all other cases follow similarly . By Lemma 2.2, f is a Morse -Bott function. Denote the connected components of Crit( f ) by { C j } ℓ j = 1 , where ℓ is finite by condition (3) and assu me without loss of gene rality that f ( C i ) < f ( C j ) if i < j . B e cause f is bounded be low and proper , all component s are close d and compact, and there exists a minimal compone n t , which we d enote C 0 . Assume Z is nonempty . W e build the equivariant K -theo ry of Z inductively by s tudying t h e critical sets, beg inning with the base case. B y assumption, K 0 H ( C 0 ) has no additive t o rsion and K 1 H ( C 0 ) = 0 . F or s mall e nough ε > 0 , conside r the submanifolds Z + j = f − 1 (( −∞ , f ( C j ) + ε )) , Z − j = f − 1 (( −∞ , f ( C j ) − ε )) , TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 7 where ε is chos en so that C j is the only critical compo nent con t ained in Z + j \ Z − j . Using th e 2 - periodicity of (e q u ivaria nt) K -the ory , the re is a periodic long ex act sequ e nce (3.1) K 0 H ( Z + j ) / / K 0 H ( Z − j ) & & ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ K 0 H ( Z + j , Z − j ) 8 8 q q q q q q q q q q q K 1 H ( Z + j , Z − j ) x x q q q q q q q q q q q K 1 H ( Z − j ) f f ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ K 1 H ( Z + j ) o o in equivariant K -the ory for the pair ( Z + j , Z − j ) . Choo s e an H -invariant metric on Z , and iden- tify K ∗ H ( Z + j , Z − j ) w ith K ∗ H ( D ( ν − j ) , S ( ν − j )) , where D ( ν − j ) , S ( ν − j ) are the disc and sphere bundles, respectively , o f the neg ative normal bundle to C j with respect to f . Th e e quivariant Thom iso- morphism also s ays th at K ∗ H ( D ( ν − j ) , S ( ν − j )) ∼ = K ∗ H ( C j ) . The re is n o d egree shift since t he (real) dimension of the negative no rmal bundle is even (as can be seen from Proposition 2.1) and K - theory is 2 -period ic. By assumption, K 1 H ( C j ) = 0 , and by t h e indu ctive assumption we have K 1 H ( Z − j ) = 0 . Hence we may immediately conclude fr om (3.1) that K 1 H ( Z + j ) = 0 and that there is a short e xact se q u ence (3.2) 0 → K 0 H ( Z + j , Z − j ) → K 0 H ( Z + j ) → K 0 H ( Z − j ) → 0 . By induction, K 0 H ( Z − j ) has no additive torsion, and by assumption, K 0 H ( Z + j , Z − j ) ∼ = K 0 H ( D ( ν − j ) , S ( ν − j )) ∼ = K 0 H ( C j ) does not either . W e conclude that K 0 H ( Z + j ) is also free of add itive torsion. H e nce by induction we conclude that K 0 H ( Z + ℓ ) is free o f additive to rsion. Since C ℓ is t he maximal critical component, there ar e no higher critical sets , so the negative gradien t flow with respect to f yields an H -equivariant deformation retraction from Z to Z + ℓ . Hence K H ( Z ) ∼ = K H ( Z + ℓ ) , and in particular we may con- clude t hat K 0 H ( Z ) is free of additive torsion, as de sired.  Remark 3.1. In the course of the proof, we have also sho wn that K 1 H ( Z t ) = 0 for all t ∈ H . In the inductive arguments given in S e ctions 4 and 5, we will need this additional fact to obtain Theorems 4.1 and 5.1 . W e now turn to the first application of Theorem 1.1, the case when the critical set consists of isolated H -or bits . Corollary 3.2. Let X = [ Z/H ] be an orbifold constructed as in (1.1) . A s above, suppose that ther e exist s ξ ∈ t such that • H ⊆ T ξ , • f := Φ ξ | Z is pr oper and bounded below , and • for every t ∈ H , C r it ( f | Z t ) consists of finitely many isolated H -orbits. Then K orb ( X ) contains no additive torsion. Fu rthermor e, K 1 H ( Z t ) = 0 for all t ∈ H . Pro of. It suffic es to check that the hypothese s o f Theo rem 1.1 are satisfied, and it is evident t h at the only assumpt ion needing comment is (4). Since e ach conne cted compo nent C is an isolated H -orbit, and by assumpt ion H acts locally fr ee ly on Z , we have K 0 H ( C ) ∼ = K 0 H ( H · p ) ∼ = K 0 H ( H / Γ) , 8 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM where p ∈ C and Γ is the finite stabilizer subgroup Stab T ( p ) in H . The H -equivariant K -theo ry of a homogeneous sp ace is the repr esent ation ring of the stabiliz er o f the iden t ity coset , K 0 H ( H / Γ) ∼ = K 0 Γ (pt) ∼ = R (Γ) , which has no additive to rsion. Mor eover , K 1 H ( H / Γ) ∼ = K 1 Γ (pt) = 0 . H ence, assumpt ions (4a) and (4b) hold, and we may apply the Main Theorem. The result follows.  This corollary provides the starting point for inductive arguments which show that the integral full orbifold K -the ory of an abeli an symplectic quotien t is torsion free. Remark 3.3. I t follows immediately from this proof that t he integ ral full orbifold K -the ory K orb ( X ) of an orbifold X = [ Z /H ] satisfying the hypothese s of the Main Theorem is add itively t he dir ect sum of representation rings R (Γ) for those s ubgroups Γ of H appe aring as stabilizer gr oup s in the level set of the moment map Z = Φ − 1 H ( η ) . It would be interesting to compare t his description via repr esent ation rings to the comput ation given in [15 ] in te rms of the Kirwan surjectivity theo rem in full or bifold K -theory . 4. S Y M P L E C T I C T O R I C O R B I F O L D S W e now provide a first applica tion of the Main Theo rem and its corollary , namely: for a lar ge class of toric orbifolds, the integral full orbifold K -th e ory contains no additive torsion. In the case of weighted p r ojective spaces similar results we re obtained by Kawasaki in ordinary integral cohomology in the 1970s [29 ], the n in ordinary K -theo r y (us ing results of [29]) by Al A mrani in [3]. Mo re recently , Zheng Hua [27] has independently s hown us ing algebro-geometric meth o ds that, when the gene ric point is stacky , the Gr othend ieck group K 0 ( X Σ ) of a smoot h complet e toric Deligne-Mumford stack is a fr ee Z -module. Here, Σ is a stacky fan as defined in [10] and K 0 is the al ge braic K -the ory defined via coherent sheaves. Since it i s str aightfo r w ar d to see fr om the definition (given below) of symplectic t oric o rbifolds X that the t wisted s ectors arising in the computation of th e full orbifold K -theory K orb ( X ) ar e th e mselves stacks which ar e symplectic toric orbifolds, the substantive statement (which is t h e topological K -theory analogue of Hua’s r es ult) is that each twist ed sector individually has K -theo ry free of add itive t orsion. Hence T heorem 4.1 should be viewed as a straightforward integral full orbifold K -theo ry analogue, in t h e topological category and for symp lectic toric orbifolds X , of Hua’s result [27]. Ho wever , our methods of proof, which us e the e quivariant Morse t heory of symplectic g eometry develop ed in Sections 2 and 3, are significantly dif ferent fr om those of [27]. W e first est ablish notation for both the Delzant construction of toric varieties and the stateme nt of the theorem. In the smooth case, this construction may be found in [13] (for an accessible account, se e [11]). Th is construction is ge neralized to the orbifold case in [33]. Let T n = ( S 1 ) n be the standar d compact n -torus, acting in the stand ard linear fashion on C n (via t he e mbedding of T n into U ( n, C ) as diagonal matrices with unit complex entries). This is a H amiltonian T n -action on C n with respect to the standard K ¨ a hler structure on C n . Let Φ : C n → ( t n ) ∗ denote a moment map for this action. Fo r a connecte d cl os ed subtorus β : H ֒ → T n , let Φ H := β ◦ Φ : C n → h ∗ denote the induced moment map. For a r eg ular value η ∈ h ∗ of Φ H , let Z := Φ − 1 H ( η ) be its level set. By regularity of η , H acts locally freely on Z . The symplectic toric o rbifold s p ecified by β : H ֒ → T n and η is then defined by X := C n / / η H = [ Z/H ] . TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 9 The procedur e just recounted is often called the Delzant construction of the toric orbifold X , al- though hist orically it was the underlying t opological space of X that w as studied , not th e associ- ated stack 3 . Symplectic toric orbifolds were classified in [33]; we conside r only those obtained by a quotient by a connecte d subto rus H . W e w ill call an e leme nt ξ ∈ t of t he Lie algebra generic if its associated 1 -param et e r subgroup exp ( tξ ) in T is dense : in the notation of Section 2, T ξ = T . Note that if there exist s any ξ ∈ t such that Φ ξ | Z is proper and bounded below , then there also exists a generic ξ ∈ t satisfying t he s ame conditions. Theorem 4.1 . Let X = C n / / η H be a symplect ic toric orbifo ld, where β : H ֒ → T a connected closed subtorus of T and η ∈ h ∗ a r egular value. Let Z = Φ − 1 H ( η ) denote the η -level set of Φ H . Then K orb ( X ) has no addit ive torsion. Furthermor e, K 1 H ( Z t ) = 0 for all t ∈ H . Pro of. Since the original T -action o n C n is a standard linear action by diagonal matrices, for any t ∈ H , the fixed point s et ( C n ) t is a coordinate subspace, i.e. ( C n ) t ∼ = C m ⊂ C n , dete rmined by the values of the T -weights on each coordinate line { (0 , 0 , . . . , z j , 0 , . . . , 0) } ⊆ C n . It is immediate that ( C n ) t is a linear symplectic su bspace of C n and that the restriction Φ H | ( C n ) t : ( C n ) t → h ∗ is a moment map for this action. Thus Z t is equal to  Φ H | ( C n ) t  − 1 ( η ) , the level set of a mome n t map for a H - action on a possibly-smaller -dimensional vector s pace. Choose a gene ric ξ ∈ t s uch that Φ ξ | Z is proper and bounde d below . Such a ξ exists because there ar e such component s for Φ : C n → t ∗ , and Z is a T -invariant closed s ubset of C n . Let f = Φ ξ | Z . In order to apply Corollary 3.2, we must check that for all t ∈ H , the critical set Crit( f | Z t ) consists of finitely many isolated H - orbits. W e first observe that since ( C n ) t ∼ = C m is itse lf a s ymplectic linear space equipped w ith a linear T -action, it suffices to prove this statement for the special case t = id ; the other cases follow simila rly . Let C be a connected component in Crit( f ) and p ∈ C . Since f is T -invariant, H · p ⊂ C . Since C is compact and connecte d , it suffices to s h o w t hat C consists o f one o rbit locally . R ecall from the proof of Lemma 2.2 that p ∈ Crit( f ) exactly if dim(Stab T ( p )) = co dim( H ) = n − k . Thus d im(Stab T ( p )) = n − k exactly if p = ( z 1 , z 2 , . . . , z n ) ∈ C n has precisely n − k coordinates e qual to 0 , i.e. p lies in a coordinate subspace of C n isomorphic to C k . No te that H acts on C n preserving this C k , and t he r eg ularity ass umption on η implies th at the restriction of Φ H to C k is Q -linearly isomorphic to the standard moment map for t he standard H -action on C k (up to a translation by a constant in h ∗ ). In particular , this implies that the condition p ∈ Z := Φ − 1 H ( η ) for a regular value η uniquely det ermines the non-zero norms of the coo r dinates k z i 1 k 2 , k z i 2 k 2 , . . . , k z i k k 2 . Therefore, the only nearby points p ′ ∈ Z with dim(Stab T ( p ′ )) = n − k are those in the H orbit of p . W e conclude that each conn e cted compo nent C of Cr it( f ) is a single H -orbit. Moreover , there are only finitely many critical components because there ar e only finitely many k -dimensional coordina te subspaces o f C n . The same argument for each ( C n ) t and an application of Corolla ry 3.2 completes the pr oof.  3 Indeed, the underlyi ng topological space Z/H corresponding to the s tack X is often als o called the symplectic quotient o f C n by H at the value η . In the current literature, there is an unfortunate ambiguity: the “symplectic quotient” may refer to the s tack or the underly ing topolog ical s pace. 10 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM 5. G K M S P A C E S Let ( M , ω , Φ) be a compact Hamiltonian T -space. Suppos e in add ition that the T -fixed points ar e isolated, and that t he s et of points with codimension- 1 stabilizer M 1 := { x ∈ M | codim(Stab( x )) = 1 } has real dimension dim( M 1 ) ≤ 2 . Wh e n these conditions are satisfied, we say that M i s a GKM space and that the T -action on M is GKM . 4 It is also well-known in the theory of GKM spaces (in the cont ext o f the study o f H amiltonian T -actions) that the se cond itions imply t hat t h e equiv ariant 1 -skelet on o f M , i.e. the closure M 1 = M 1 ∪ M T , is a union of symplectic 2 -sphe r es S 2 . Moreover , each such 2 -sphe re is itself a Hamiltonian T -space; t he T -action on S 2 is given by a nontrivial character T → S 1 (equivalently , a no nzer o weight α ∈ t ∗ Z ) where th e S 1 acts on S 2 by rotation. Here the weight α is obtained fr om the linear T -isot ropy data at either of t h e two T -fixed points in S 2 . (For details see e. g . the expository article [39].) Hamiltonian T -sp aces ( M , ω , Φ ) (or algebraic varieties equipped w ith algebraic torus actions) for which the T -action is GKM have been e xtensively studied in modern equivariant symplectic and algebraic g e ometry , primarily due to t he link p rovi ded by GKM theo ry betwee n T -e q u ivaria nt topology and t he combinatorics o f what is called the mo m ent graph (or GKM graph ) of M . Many natural e xamples arise in the realm of g eometric repr es entation theo ry and Schubert calculus, including generalized flag varieties G/B and G/P of Kac-Moody gr oup s G (where B is a Borel subgroup and, more g enerally , P a parabolic subgroup). Hence the o r bifold invariants o f the orbifold symplectic quotien t s of GKM spaces is a natural ar ea of study . If the T -action is GKM, then for a lar ge class of circle s ubgroups of T , the ass ociated orbifold symplectic quo t ients M / / η S 1 have n o additive t orsion in full integral orbifold K -theory , as w e now see. Theorem 5.1. Su ppose that ( M , ω , T , Φ) is a compact Hamiltonian T -space, and suppose further that the T -action is GKM. Su ppose that β : S 1 ֒ → T is a cir cle subgroup in T such that M S 1 = M T , and let Φ S 1 := β ∗ ◦ Φ : M → Lie( S 1 ) ∗ denote the induced moment map. Let η ∈ Lie( S 1 ) be a reg ular value of Φ S 1 , and X = M / / η S 1 the orbifo ld symplectic quotient. Then K orb ( X ) is free of additive torsion. Pro of. W e show th at the hypot heses of Corol lary 3.2 hold. Le t Z := Φ − 1 S 1 ( η ) , choose ξ ∈ t such that its 1 -parameter subgroup in T is dense in T , and let f := Φ ξ | Z . Properness of f is immediate since M is compact. He nce it suffices t o s how th at the critical sets C rit( f ) and Crit( f | Z t ) are isolated S 1 -orbits. Obser ve that when M is a GKM space, M t is also a GKM space for any t ∈ S 1 . Hence it suffices to ar gue only for t he case of Crit( f ) ; the ot hers follow similarl y . By the ar gument given in the pr oof of Lemma 2.2, Cr it( f ) cons ists pr ecisely of those points p ∈ Z satisfying co dim(S tab T ( p )) = 1 . In other words, Crit( f ) = Z ∩ M 1 . The closure M 1 consists of a union o f 2 -sphe res, and the T -action on each S 2 is s pecified by a non -zero weight α ∈ t ∗ Z obtained fr om the T -isotropy decompos ition at o ne of the t wo fixed points of the S 2 . By assumption on the cir cle subgroup S 1 , the kernel of t h e character φ α : T → S 1 specified by α do es not contain S 1 . Therefor e, S 1 acts nontrivially on each S 2 ⊆ M 1 , implying Φ S 1 | S 2 is nontrivial, and Φ − 1 S 1 ( η ) ∩ S 2 consists of a single S 1 -orbit. (Not e that Z does not contain any 0 -dimensional orbits of S 1 since, by assumption on regularity of η , S 1 acts locall y freely on Z .) Thus the hypo theses of Coroll ary 3.2 are satisfied , so K orb ( X ) is additively torsion-free.  4 There are many variants on the de finition of GKM actions (see e.g. [20–23]). In particular , in les s restr ictive versions, the T -space M need not be compact nor symplectic, nor even finite-dimensional. TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 11 Remark 5 .2. W e r estrict to the case of compact symplectic manifolds in this section for sake of br evity . H owever , the arguments given above could be altered to prove analogous r es ults in less- restrictive contexts of GKM the o ry (see e.g. [23], [22]). Remark 5.3. It may be an interesting e xerci se to g e neralize Theorem 5.1 to s ymplectic quotient s of GKM spaces by higher dimensional tori. One appr oach wo uld be t o cons ide r quotients o f a k -independent GKM space (cf. [20]) by a ( k − 1) -dimensional torus. W e now ill ust rate us e of T heorem 5.1 for so me coadjoint orbits of low-rank Lie type. W e will analyze examples derived from the natural G -action on coadjoint or bits of G , but we must be car eful to avoid the possibility of non-ef fective actions (so the s ymplectic quot ien t is an ef fective orbifold). Therefore, in Examples 5 .4 , 5.5, and 6.4, we use the quotient group P G := G/ Z ( G ) where Z ( G ) d enotes the (finite) center of G ; by slight abuse of notation, we also notate by T the image of t he usual maximal torus und er the quotient G → P G . Example 5.4. Le t M = O λ ∼ = F ℓ ag s ( C 3 ) be a full coadjoint orbit of the Lie group P S U (3 , C ) with maximal t orus T given by the standard diagonal s ubgroup. Here λ ∈ t ∗ ⊆ su (3) ∗ and O λ is the λ -orbit of P S U (3) with respect to the usual coadjoint action. Equip M = O λ with the Kostant- Kirillov-Souriau form ω λ and let Φ : O λ → t ∗ be the T -moment map obtained by compo s ing the projection π : su (3 , C ) ∗ → t ∗ with the incl usion O λ ֒ → su (3 , C ) ∗ . It is w ell-known that the T -action on M is GKM, and that t h e equivariant 1 -skeleto n o f O λ maps u nder Φ to the GKM graph pictured in gr ey in Figure 5.1. P S f r a g r e p l a c e m e n t s ξ β ∗ Lie ( S 1 ) ∗ F I G U R E 5 . 1 . In grey , we indicate the image of t he equivariant 1 -skelet on of M . The T -fixed points correspond to the six (corner) vertices of the graph. The blac k line interse cting the polytope repr esent s the mo me nt image of the level s et Z o f an S 1 -moment map Φ S 1 . There ar e 5 critical compo nents C i in Crit( f ) , corresponding to t he 5 thick black dots (the images of the C i under Φ ). 12 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM For a choice of β : S 1 ֒ → T such that O S 1 λ = O T λ , the level s e t Z of the S 1 -moment map Φ S 1 = β ∗ ◦ Φ is schematical ly indicated in Figure 5.1 by the thick black line; the (images under Φ of t he) components of Crit( f ) for a ge neric choice of f = Φ ξ | Z ar e indicated by the th ick black dots. The standard maximal-torus T -action on a coadjoint o rbit of a compact conne cted Lie group G is GKM; hence we may apply Theo r em 5.1. Fr om Figure 5.1 we see that, additively , N K S 1 ( Z ) = K orb ([ Z/S 1 ]) is a dir ect s u m o f representation rings R (Γ i,t ) , one for each critical compon e nt C i,t in Crit( f | Z t ) , as t ranges in S 1 . In fact, only finitely many t ∈ S 1 will contribute nontrivial sum- mands. Here the subgroup Γ i,t of S 1 is the finite s tabiliz er group of a point p in C i,t , wh ich in t urn may be compu t ed in a straightforward manner by analyzing the intersection of the chosen S 1 with each of the stabilizer subgroups appearing in the T -orbit stratification of O λ (cf. [18, A ppendix B]). Example 5.5. Now we consider the Lie t ype B 2 . Here we find it convenient to work with the complex form P S O (5 , C ) . W e recal l that the maximal torus T o f t ype B 2 is 2 -dimensional and th e roots are given as in Figur e 5.2. W e conside r a coadjoint orbit M = O λ , which may be identified with the homogene ous space S O (5 , C ) /P α 1 where P α 1 is the parabolic subgroup correspond ing to the positive s imple root { α 1 } . Mor e specifically , we may t ake O λ to be t he coadjo int orbit through the element λ ∈ t ∼ = t ∗ indicated in Figure 5.2 . The image of the equivariant 1 -skeleto n for t he Hamiltonian T -action on O λ ∼ = P S O (5 , C ) /P α 1 is depicted in Figure 5.3. P S f r a g r e p l a c e m e n t s α 1 α 2 + 2 α 1 α 2 + α 1 α 2 λ F I G U R E 5 . 2 . The type B 2 root sy stem. The dotted line is the hy perplane d istinguishing the positive from th e negative roots. The el- ement λ lying on th e line spanned by α 1 + α 2 specifies t he coadjo int orbit O λ . P S f r a g r e p l a c e m e n t s ξ β ∗ Lie ( S 1 ) ∗ F I G U R E 5 . 3 . In gr ey , we indicate the image of the equivariant 1 -skeleton of M = O λ ∼ = S O (5 , C ) /P α 1 . The T -fixed points are the 4 outer vertices. The blac k line intersect- ing the p olytope represents the mome nt im- age o f the level set Z of an S 1 -moment map Φ S 1 . There are 3 critical component s C i in Crit( f ) , correspond ing to the 3 thick black dots (the images of the C i under Φ ). Given S 1 ⊂ T with M S 1 = M T and corresponding moment ma p Φ S 1 , the level set Φ − 1 S 1 ( η ) indicated (unde r its i mage under Φ ) in the fig ure e vidently lies entir ely within an op e n Bruhat TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 13 cell of M . This Bruhat cell may be modelled on a single linear T -repr ese n t ation with T -weights − α 2 , − α 1 − α 2 , − 2 α 1 − α 2 , which renders the explicit computation o f the r elevant finite stabilizer subgroups Γ i,t ⊆ S 1 particularly straightforward. This o bservation motivates the discus s ion in the next s ection. 6. S E M I L O C A L L Y D E L Z A N T S P A C E S W e have alr eady seen in Sections 4 and 5 t hat the hyp otheses o f Corollary 3.2 are satisfied in several situations familiar in equivariant sy mp lectic g e ometry . W e will now see that the method s of proof used th u s far i n this manuscript allow us to make i nductive use of the Ma in Theorem to c over mor e cases of orbifold sy mp lectic quotients. S pecifically , we observe that the proof of Theorem 4.1 sh o ws that th e H - equ ivariant K -theory of the level s et Z arising from a Delzant construction has the properties that K 0 H ( Z ) is additive-torsion-free and K 1 H ( Z ) = 0 . Ther efore, for ( M , ω , Φ) a Hamiltonian T -space and β : H ֒ → T a connected subtorus, if it can be shown that each o f t he conne cted c ompo n e nts of the critic al s ets appearing in Theorem 1.1 can be H - equivariantly identified with a level set of a Delzant con s truction, then the hypothe s es (3a) and (3b) o f Theo rem 1.1 would be satisfied , thus allowing us to apply the Main T heorem to a wider class of s ymplectic quot ien t s. T o this end , we make the following definition. Definition 6.1. Le t ( M , ω , Φ H ) be a Hamiltonian H -space with momen t map Φ H : M → h ∗ . W e will say that an H -invariant s ubset C ⊂ M is semilocally Delzant with respect to H if the follow- ing conditions are s atisfied: (1) Ther e e x ist s a 2 n -dimensional H -invariant symplectic submanifold N ⊆ M , an H -invariant open ne ighborhood U ⊆ N of C , and a H -equivariant symplectomorp hism ψ : U → V ⊆ C n for an op en H -invaria nt subset V ⊆ C n , whe r e H acts linearly on C n , with associated moment map Φ C n : C n → h ∗ . (2) Under t he map ψ, the s et C is identified with a level set of the induced H -moment map on C n . In other words, ψ ( C ) = Φ − 1 C n ( η ′ ) ⊆ C n for some regular val ue η ′ ∈ h ∗ . (3) Ther e exists ξ ∈ h such that Φ ξ C n | ψ ( C ) is pr op e r and bounded below . W e take a moment to discuss situ ations in equivariant symplectic geomet ry in which we may expect the above d efinition to be app licab le. Re call that the equivariant Darboux t h e orem s tates that, n e ar an isolated H -fixed point p ∈ M H , the re e xists an open ne ighborhood U p of p which is H -equivariantly symplectomorphic to an af fine s pace C n equipped with a linear H -action (her e p is identified with the origin 0 of C n ). Un d er s ome t echnical ass u mptions (cf. [18]) which a re not very restrictive in p r actice, it is also pos sible to arrange the s y mplectomorphism such that the H -isotypic decomposition C n ∼ = M α C α , where the sum is over we ights α ∈ h ∗ Z and C α denote s the subspace o f C n of weight α , has the property that the moment map Φ C n associated to this H -action has a component which is p r op er and bounded below . I t is then evident that a closed subset C of M which lies entirely inside such an equivariant neighborhood U p ∼ = C n near p ∈ M T , and which may be identified w ith a level set of Φ C n via the e quivariant Darboux the orem, is se milocally Delzant. Mor eo ver , simil ar statements could be made of s ubsets C ′ of M which lie entir ely in p roper coordina te subspaces of C n under the same equivaria nt identification with U p ⊆ C n . Info r mally , we may s ay that H -invariant closed 14 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM subsets which are “near eno ugh to an iso lated fixed point” can be semilocally Delzant as d escribed above. In particular , this p oint of view leads to concrete examples of symplectic quot ient construc- tions (e.g. of Hamiltonian H -spaces with isolated fixed points, such as those where the H -action is GKM) with critical sets C satisfying Definition 6.1 . Indeed , we note that a concr ete famil y of examples of Hamiltonian T -spaces with well-known such e quivariant neighborhoods are the flag varieties (coadjoint or bits ) G/B and G/P of compact connected Lie groups. The maximal t o rus T of t he compact connected Lie group G acts naturally on s uch homog eneous spaces, with fixed points corr espo nding to cosets W /W P . Moreover , G/B (similarl y G/P ) has a convenient ope n cover obtained by W e y l translates of t h e open Bruhat cell B w 0 B /B , whe re w 0 is the long est word in the W e yl group. By using [30, Proposition 2.8] and some knowledge of the T -orbit stratification of G/B , it is possible to identify a “la rge” open subset U of a Bruhat cell which provides such an equivaria nt Darboux neigh bo rhood. Moreover , the subset U can be concretely described in te rms of moment map data. If a closed s ubset C of G/B (similarly G/P ) may be s een t o lie entirely within such a s ubset U , the n the T -action ne ar C may be modelled by a linear T -action on C ℓ ( w 0 ) (here ℓ ( w 0 ) denote s the B ruhat lengt h o f w o ). W e illustrate a concrete example of s uch a situation, using a non-generic coadjoint orbit of Lie type B 3 in Example 6.4 below . Returning t o the discussion of orbifold K -the ory , we first note that it is immediate fr om Theo- rem 4.1 th at if C is semilocally Delzant, then K 0 H ( C ) has no additive t orsion and t hat K 1 H ( C ) = 0 . This leads t o t h e following. Theorem 6.2. Let M be a Hamiltonian T space, and let H ⊂ T be a conn ected subtorus. Let Z = Φ − 1 H ( η ) ⊂ M be a level set of the H -mo ment map Φ H : M → h ∗ and X = [ Z/H ] be the orbifold obtained as a symplectic quotient M / /H . Let ξ ∈ t be such that T ξ = T , and f = Φ ξ | Z the corres ponding moment m ap restr icted to Z . S uppose that (1) f is pr oper and bounded below , (2) for all t ∈ H , the set of connected components π 0 (Crit( f | Z t )) is finite, (3) for all t ∈ H , each connected component C of Crit( f | Z t ) is semi-locall y Delzant with respect to H . Then K orb ( X ) has n o add itive torsion. Furthermor e, K 1 H ( Z t ) = 0 for all t ∈ H . Pro of. By T h e orem 4.1, K orb ([ C /H ]) ha s no additive tors ion for each conne cted component C of C r it ( f | Z t ) for all t ∈ H . In particular , K 0 H ( C ) has no tors ion. Since we also have K 1 H ( C ) = 0 for all critical sets, we have satisfied the hypothe ses of Theorem 1.1 . He nce K orb ( X ) has no additive torsion.  Remark 6.3. W e note that if the level set Z itself is s emilocall y Delza nt, then by transfer r ing all analysis to the appropriate e quivaria nt Darboux neighborhood U ⊆ C n and using the same ar- gument as in S ection 4 , we i mmediately see t hat f or all t ∈ H, all conne cted compone nts C of Crit( f | Z t ) are semilocally Delzant with respect to H . Hence, in this case the hypo t hesis (3) above is automatical ly s atisfied. Example 6.4. W e close with an example of a symplectic quotient o f a type B 3 coadjoint orbit by a 2 -dimensional to rus. Sinc e the subtorus is d imension 2 , T heorem 5.1 does not apply , but we may use Theorem 6.2. Recall that the complex form of the compact Lie group of ty pe B 3 is P S O (7 , C ) . The maximal torus T is 3 -dimensional, and the root syste m is depicted in Figure 6.1. W e deno te the associated momen t map by Φ . TORSION IN THE FU LL ORBIFOLD K -T HEOR Y OF ABELIAN SYMPLECTIC QUOTIENTS 15 P S f r a g r e p l a c e m e n t s α 1 = L 1 − L 2 α 2 = L 3 α 3 = L 2 − L 3 L 1 = α 1 + α 2 + α 3 L 2 = α 2 + α 3 L 1 − L 3 = α 1 + α 3 L 1 + L 2 = α 1 + 2 α 2 + 2 α 3 L 1 + L 3 = α 1 + 2 α 2 + α 3 L 2 + L 3 = 2 α 2 + α 3 F I G U R E 6 . 1 . The r oot diagram for type B 3 with pos itive simple roots α 1 , α 2 , α 3 (for de- tails, s ee [14, § 19.3]). T h e e leme nt λ ∈ t ∗ lies on the pos itive span o f the po sitive root L 1 = α 1 + α 2 + α 3 . P S f r a g r e p l a c e m e n t s p 1 p 2 p 3 π T ′ Lie ( T ′ ) ∗ F I G U R E 6 . 2 . The GKM graph for M = O λ ∼ = P S O (7 , C ) /P α 2 ,α 3 . The t hick line and thick black dots schematically illustrate the (image u nder Φ of the) inverse images Z := (Φ T ′ ) − 1 ( η ) and the critical compo nents of Crit(Φ ξ | Z ) , r esp ectively . W e choose to work with a no n-generic coadjoint orbit O λ which may be identified with the com- plex homog eneous space P S O (7 , C ) /P α 2 ,α 3 where P α 2 ,α 3 is t he parabolic subgroup corresponding to the subset of the positive simple roots { α 2 , α 3 } . W e choose λ lying o n the po sitive span o f the positive r oot L 1 = α 1 + α 2 + α 3 as i n Figure 6.1 . The GKM graph of O λ is also schematica lly shown. The image o f the equivariant 1 -ske leton o f M = O λ includes the three 2 -dimensional in- terior quadrilaterals giv en b y the convex hull o f the r oots {± L 1 , ± L 2 } , {± L 2 , ± L 3 } , {± L 1 , ± L 3 } respectively . Let T ′ ⊂ T be the 2 -dimensional connecte d subtorus o f T corresponding to the 2 -plane spanned by the roots {± L 1 , ± L 2 } , with correspond ing projection π T ′ : t ∗ → Lie( T ′ ) ∗ . W e wish to compu t e K orb of the symplectic quotient O λ / / T ′ . The pr eimage π − 1 T ′ ( η ) ∩ ∆ in ∆ = Φ( O λ ) of a generic regular value η ∈ Lie( T ′ ) ∗ is depicted as the thick interval in Figure 6.2. W e wish now to sho w th at the full orbifold K -theory of t he qu otient O λ / / T ′ is fr ee of additive torsion by us ing Theo r em 6.2. There are sever al ways to proceed. T h e first metho d, which depends on Remark 6.3, is to observe that the full level set Z is semilocall y Delzant. In this case, we may apply [3 0, Proposition 2.8] to see t hat the thick vertical line in F igure 6.2 lies i n an equivari ant Darboux neighborhood of the T -fixed point p corr esp o nding to t he root L 1 = α 1 + α 2 + α 3 . The T - action and corresponding moment map Φ restricted to this ne igh bo rhood may be identified with that o f a linear T -action on C 5 with we ights {− L 1 , − L 1 ± L 2 , − L 1 ± L 3 } o n the coordinates. The T ′ -action is the restriction of this linear T -action, hence Z is se milocal ly Delzant with r es pect t o T ′ . By Remark 6.3 we may immediately apply Theorem 6.2, as desired. In o r de r to illustrate t he concrete, straightforward nature of our met h o d of computation, for this example we also briefly sketch the explicit analysis of each of the components of Crit( f ) for appropriate f = Φ ξ | Z . Analysis of Crit( f | Z t ) , for t 6 = 1 , would of course be similar . W e begin by choosing ξ g eneric such that Crit( f ) consists of the three components schematically indicated in Figure 6.2. 16 REBECCA GOLDIN, MEGU MI HARADA, AND T ARA S. HOLM Observe that the situations o f the t wo exterior points p 1 , p 3 in π − 1 T ′ ( η ) ∩ ∆ lying o n t he boundary ∂ ∆ are evidently symmetric, so it suffices t o do the computations for only one of t hem. W e begin with the t op e xterior point p 1 . A straightforward a nalysis of the li near T -action on t he B ruhat cell de scribed above shows that Φ − 1 ( p 1 ) ⊆ O λ consists o f a single T -orbit dif feomorphic t o T ′ . Moreover , the inter s ection of the stabiliz er of the Bruhat cell with T ′ is trivial, so p 1 corresponds to a free T ′ -orbit. Hence the contr ibution to the full orbifold K -theo ry coming from p 1 is the (ordina ry) K -theory of a po int, and is hence t orsion-free. W e now proceed with the interior point p 2 . (One way to view this computation is to recall that t he h o rizontal quadrilateral obtained as the convex hull of the roots {± L 1 , ± L 2 } corr espo nds to a subvariety of P S O (7 , C ) /P α 2 ,α 3 which may be ident ified with the homog eneous space o f P S O (5 , C ) of t y pe B 2 studied in a p revi ous example, although this is not necessary for the com- putation.) Another straightforward analysis of linear T -actions, using the explicit list of T -weights given above, yields that the corresponding symplectic quotient is the “teardr op” orbifold, i.e. the weighted projective space P (1 , 2) (following notation of [15]). He nce the contr ibution to the full orbifold K -theory of O λ / / µ T ′ coming from the inte rior p oint p 2 is that associated to P (1 , 2) , w hich is explicitly comput ed in [15], and has no additive torsion. R E F E R E N C E S [1] A. Adem and Y . Ruan. T wisted or bifold K -theory Comm. Math. Phys. , 237(3): 533–556 , 200 3. [2] A. Al Amr ani. 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M A T H E M AT I C A L S C I E N C E S M S 3 F 2 , G E O R G E M A S O N U N I V E R S I T Y , 4 4 0 0 U N I V E R S I T Y D R I V E , F A I R FA X , V I R G I N I A 2 2 0 3 0 , U S A E-mail address : rgold in@math.gmu.edu URL : ht tp://math.gmu. edu/ ˜ rgoldin/ D E PA R T M E N T O F M AT H E M AT I C S A N D S TAT I S T I C S , M C M A S T E R U N I V E R S I T Y , 1 2 8 0 M A I N S T R E E T W E S T , H A M I L - T O N , O N TA R I O L 8 S 4 K 1 , C A N A D A E-mail address : Megum i.Harada@math.m cmaster.ca URL : ht tp://www.math. mcmaster.ca/Meg umi.Harada/ D E PA R T M E N T O F M A T H E M AT I C S , M A L O T T H A L L , C O R N E L L U N I V E R S I T Y , I T H A C A , N E W Y O R K 1 4 8 5 3 - 4 2 0 1 , U S A E-mail address : tsh@m ath.cornell.edu URL : ht tp://www.math. cornell.edu/ ˜ tsh/