The KO*-rings of BT^m, the Davis-Januszkiewicz Spaces and certain toric manifolds

THE K O ∗ -RINGS OF B T m , THE D A VIS-JANUSZKIEWICZ S P A CES AND CER T AIN TORIC MANIF OLDS L. ASTEY, A. BAHRI, M. BENDERSKY, F. R. COHEN, D. M. DA VIS, S. GITLER, M. MAHOW ALD, N. RA Y, AND R. WOOD Abstract. This pap er contains a n explicit computation of the K O ∗ -ring structure of an m -fold pro duct o f C P ∞ , the Davis-Januszkiewicz spaces and of toric manifolds which hav e trivial Sq 2 -homology . A key ingredien t is the stable splitting of the Davis-Jan uszkiewicz spaces given in [6]. 1. Intr oduction The term “toric manifold” in this pap er refers to the top ological space ab out whic h de- tailed information may b e fo und in [14] and [10]; a brief description is giv en in Section 6. These spaces are called also “quasitoric manifolds” and includ e the class of all non-singular pro jectiv e toric v arieties. An n -torus T n acts o n a toric manifold M 2 n with quotien t space a simple p olytop e P n ha ving m co dimension-one faces (facets). Asso ciated to P n is a simplicial complex K P on v ertices { v 1 , v 2 , . . . , v m } with eac h v i correspo ndin g to a single facet F i of P n . The set { v i 1 , v i 2 , . . . , v i k } is a simplex in K P if and only if F i 1 ∩ F i 2 ∩ . . . ∩ F i k 6 = ∅ . The classifying space of the real n -torus T n is denoted b y B T n . Asso ciated to the torus action is a Borel-space fibration (1.1) M 2 n − → E T n × T n M 2 n p − → B T n . Of course here, B T n = C P ∞ × C P ∞ × . . . × C P ∞ , ( n factors). The homotop y type of the Borel sp ace E T n × T n M 2 n dep end s on K P only . It is referred to as the Da vis-Jan uszkiew icz space of K P and is denoted b y the sym b ol DJ ( K P ). More 1991 Mathematics Subje ct Classific ation. Primary: 5E45, 14M25 , 55N1 5, 55 T15 Secondary: 5 5P42, 55R20, 57N65. K ey wor ds and phr ases. Qua sitoric manifolds, Da vis-Ja n us z k iewicz Space, K O -theory , K -theor y , pro duct of pro jectiv e spac e s , Stanley- Reis ner ring, toric v arieties, toric ma nifolds, classifying space of a torus. 1 The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 2 generally , a Da vis-Jan uszkie wicz space exists for any simplicial complex K ; Section 5 con tains more details ab out this generalization. It is kno wn that for an y complex-orien ted cohomology theory E ∗ (1.2) E ∗ ( DJ ( K P )) = E ∗ ( B T m ) . I E S R where I E S R is a n ideal in E ∗ ( B T m ) described next. In this con text, the tw o-dimensional generators of the graded ring E ∗ ( B T m ) are denoted by { v 1 , v 2 , . . . , v m } . The ideal I E S R is generated b y all square-free monomials v i 1 v i 2 · · · v i s correspo ndin g t o { v i 1 , v i 2 , . . . , v i s } / ∈ K P . The ring (1.2) is called the E ∗ -Stanley-Reisner ring. F or a toric manifold M 2 n an argumen t, based on the collapse of the A tiy a h-Hirzeb ruc h- Serre sp ectral se quence for (1.1), yields an isomorphism of E ∗ -algebras (1.3) E ∗ ( M 2 n ) ∼ = E ∗ ( DJ ( K P )) . J E where the ideal J E is generated b y p ∗ ( E 2 ( B T n )) and therefore b y the E -theory Chern classes of certain associated line bundles, ([11], page 18 and [12], page 6). F or the case of non-singular compact pro jectiv e toric v arieties and E equal to ordinary singular cohomology with inte gral co efficien ts ( E = H Z ), this is the celebrated r esult of Danilo v and Jurkew icz, [13]. The E = H Z v ersion for the top ological generalization of compact smo oth toric v arieties, is due to Davis and Jan uszkiew icz [14]. F or certain singular toric v arieties, the results of [13] cov er also the case E = H Q . The question of an analogue of (1.3) for a non-complex-orien ted theory arises naturally . The o bvious first candidate is K O -theory . The ring structure of (1.4) K O ∗ ( B T m ) = K O ∗ ( m Y i =1 C P ∞ ) do es not seem to app ear in the literature for m > 2. The thesis of Dobson [16] deals with the case m = 2. The K O ∗ -algebra structure of K O ∗ ( C P n ) and K O ∗ ( C P ∞ ) app ears explicitly for the first time in [12] where it is discusse d in the con text of a theorem of W o o d whic h is men tioned in Section 2 b elo w. T w o differen t presen tations for the ring K O ∗ ( B T m ) are g iven in Sections 3 and 4 . F ollow - ing Atiy ah and Segal [3], these provi de a description of the completion of the represen tation ring R O ( T m ) a t the augmen tation ideal. The calculation herein may b e in terpreted in that The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 3 con t ext, along the lines of Anderson [2]. In particular, the fact that the complexification map and the realification map c : K O ∗ ( B T m ) − → K U ∗ ( B T m ) (1.5) r : K U ∗ ( B T m ) − → K O ∗ ( B T m ) (1.6) are injectiv e and surjectiv e resp ectiv ely , is used throughout. This follo ws f ro m the Bott exact sequenc e (1.7) . . . − → K O ∗ +1 ( X ) · e − → K O ∗ ( X ) χ − → K ∗ +2 ( X ) r − → K O ∗ +2 ( X ) − → . . . where χ is complexification (1.5) follow ed by m ultiplication b y v − 1 the Bott elemen t. Since K O ∗ ( B T m ) is concen trated in ev en degree, the Bott sequence implies that the realification map r is surjectiv e and compl exification c is injectiv e. They are related b y (1.8) ( r ◦ c )( x ) = 2 x and ( c ◦ r )( x ) = x + x. The complexit y of the calculation is a result of the fact that the realification map r is not a ring ho momorphism. The first presen tation generalizes the metho ds of [16 ]. A companion result is giv en for K O ∗ ( V m i =1 C P ∞ ). The second approac h pro duces generators b etter suited to the task of givin g a descrip tion of K O ∗ ( DJ ( K P )) in terms of K O ∗ ( B T m ). The results o f [4] are used then to giv e a description of K O ∗ ( M 2 n ) analogo us to (1.3) for any toric manifold whic h has no Sq 2 -homology . The group structure of K O ∗ ( V m i =1 C P ∞ ) is m uc h more access ible than the ring struc- ture. The A dams sp ectral sequence yields a concise description in terms of the groups K U ∗ ( V k i =1 C P ∞ ) with few er smash pro duct fa ctors. This is discus sed in the next section. 2. The gr oup structure of K O ∗ ( V m i =1 C P ∞ ) 2.1. The k o -homology. The calculation b egins with the determination of k o ∗ ( B T m ), the connectiv e k o -homology corresp onding to the sp ectrum bo . The main to ol is the Adams sp ectral sequence . It is used in conjunction with the fo llo wing equiv alence, whic h is w ell kno wn among homotop y theorists and extends a result of W o o d. Let bu denote the sp ectrum correspo ndin g to connectiv e complex k -theory . The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 4 Theorem 2.1. The r e is an e quivalenc e of sp e ctr a ∞ _ k =0 Σ 4 k + 2 bu − → bo ∧ C P ∞ . Pr o of. Bac kground information ab out the Adams sp ectral sequenc e in connection with k o - homology calculations may b e f o und in [4] or [8]. A c hange of rings theorem implies that the E 2 -term of the A dams sp ectral sequence f or k o ∗ ( C P ∞ ) = π ∗ ( bo ∧ C P ∞ ) dep ends on the A 1 -mo dule structure of H ∗ ( C P ∞ ; Z 2 ) where A 1 denotes the sub-algebra of t he Steenro d algebra A generated b y Sq 1 and Sq 2 . As an A 1 -mo dule, H ∗ ( C P ∞ ; Z 2 ) is a sum of shifted copies of H ∗ ( C P 2 ; Z 2 ). Conseq uen tly , the E 2 term of the sp ectral sequence is a sum of shifted copies of Ext s,t A 1 ( H ∗ ( C P 2 ; Z 2 ) , Z 2 ) and so has classes in ev en degree only . Hence, the spectral sequence collapses. A non-trivial class in eac h π 4 k + 2 ( bo ∧ C P ∞ ) is represe n ted in the E 2 -term by a generator of dimension 4 k + 2 in filtration zero. The η -exten sion on this class is trivial as the E 2 -term is zero in o dd degree. So the map (2.1) S 4 k + 2 − → bo ∧ C P ∞ extends ov er a (4 k + 4)- cell e 4 k + 4 attac hed by the Hopf map η . This giv es a map (2.2) Σ 4 k C P 2 = S 4 k + 2 ∪ η e 4 k + 4 − → bo ∧ C P ∞ whic h extends to (2.3) s : ∞ _ k =0 Σ 4 k C P 2 − → bo ∧ C P ∞ . Smashing with bo and comp osing with the pro duct map bo ∧ bo µ − → bo gives (2.4) bo ∧ ( ∞ _ k =0 Σ 4 k C P 2 ) 1 ∧ s − − → bo ∧ bo ∧ C P ∞ µ ∧ 1 − − → bo ∧ C P ∞ . The equiv alence of sp ectra Σ 2 bu → bo ∧ C P 2 , due to W oo d a nd cited in [1], is used next to giv e a map The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 5 (2.5) g : ∞ _ k =0 Σ 4 k + 2 bu − → bo ∧ C P ∞ . This map induce s an isomorphis m of stable homotopy groups and hence giv es the required equiv alence of sp ectra.  R emark. An equiv a lence of the fo rm 2.4 f o llo ws also fro m t he metho ds of [17] and the fact that tw ice the Hopf bundle o v er C P ∞ is a Spin bundle and therefore bo -orien table. The next corollary follo ws imme diately . Corollary 2.2. Ther e is an isomorphism of gr ade d ab e lian g r oups ∞ M k =0 Σ 4 k + 2 ( f k u ∗ ( m ^ i =2 C P ∞ )) − → f k o ∗ ( m ^ i =1 C P ∞ ) . Notice here that the summands on the left hand side a re the underlying groups of a tensor pro duct of divided p o we r algebras eac h of whic h is dual to a p olynomial algebra. Recall next that there a r e class es e ∈ k o 1 , α ∈ k o 4 , β ∈ k o 8 so that (2.6) k o ∗ = Z [ e, α , β ] . h 2 e, e 3 , eα, α 2 − 4 β i and a class v ∈ k u 2 so that (2.7) k u ∗ = Z [ v ] Remark 2.3. An examination of the E 2 -term of the A dams sp ectral sequenc e for k o ∗ ( C P 2 ) rev eals that the action of k o ∗ on k o ∗ ( C P 2 ) ∼ = k u ∗ is give n b y (2.8) e · 1 = e · v = 0 , α · 1 = 2 v 2 , β · 1 = v 4 This coincides with the mo dule action of k o ∗ on k u ∗ giv en by the “complexification” map, ([16], page 16). The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 6 2.2. F rom k o -homology to K O -cohomology . One cons equence of the calculation ab o v e is that the Bott elemen t β acts as a monomorphism on k o ∗ ( V s i =1 C P ∞ ) and so can b e inv erted to get the p erio dic K O -homology of V s i =1 C P ∞ . Prop osition 2.4. Ther e is an isomorphism of ab elian gr oups ∞ M k =0 g K U ∗ + 4 k +2 ( m ^ i =2 C P ∞ ) − → g K O ∗ ( m ^ i =1 C P ∞ ) . Pr o of. The result follo ws from Corollary 2.2 and Remark 2.3  The pro of of Theorem 2.1 w orks equally w ell in the dual situation. Let D ( C P 2 n ) denote the S -dual of C P 2 n . Aside from a dimensional shift, H ∗ ( D ( C P 2 n ); Z 2 ), as an A 1 -mo dule, is isomorphic to a sum of suspended copies of H ∗ ( C P 2 ; Z 2 ). So, the A dams spectral sequence for π ∗ ( bo ∧ D ( C P 2 n )) collapses for dimensional reasons. The argumen t of Theorem 2.1 go es through essen tially unc hanged to give an equiv alence of sp ectra (2.9) g : n _ k =1 Σ − 4 k bu − → bo ∧ D ( C P 2 n ) . The next lem ma, which follows directly fr o m the discuss ion in Section 2.1, records the fact that (2.9) is natural with respect to the inclusion C P 2 n ⊂ − → C P 2( n +1) . Lemma 2.5. The fol lowing diagr a m c ommutes (2.10) n W k =1 Σ − 4 k bu g − − − → bo ∧ D ( C P 2 n ) x    φ ψ x    n +1 W k =1 Σ − 4 k bu g − − − → bo ∧ D ( C P 2( n +1) ) wher e the map φ c ol lap s es Σ − 4( n +1) bu to a p oint and ψ i s induc e d by the inclusion C P 2 n ⊂ − → C P 2( n +1) .  The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 7 The dualit y result from [1 , Proposition 5.6], implies (2.11) D ( m ^ i =1 C P 2 n ) ≃ m ^ i =1 D ( C P 2 n ) . F rom t his follows an isomorphism of abelian groups, analogous to Prop osition 2.4 fo r finite pro jectiv e spaces, (2.12) n M k =1 g K U ∗− 4 k  D ( m ^ i =2 C P 2 n )  − → g K O ∗  D ( m ^ i =1 C P 2 n )  and so an isomorphism (2.13) n M k =1 g K U ∗ + 4 k ( m ^ i =2 C P 2 n ) − → g K O ∗ ( m ^ i =1 C P 2 n ) . The next result extends (2 .13) t o V m i =1 C P ∞ . Prop osition 2.6. Ther e ar e isomorphisms g K O ∗ ( m ^ i =1 C P ∞ ) ∼ = lim ← − n g K O ∗ ( m ^ i =1 C P 2 n ) and ∞ M k =1 g K U ∗ +4 k ( m ^ i =2 C P ∞ ) ∼ = lim ← − n n M k =1 g K U ∗ +4 k ( m ^ i =2 C P 2 n ) . Pr o of. It follow s from the calculations ab o v e that the maps in the in v erse limit arising fr o m g K O ∗  m ^ i =1 C P 2( n +1)  − → g K O ∗  m ^ i =1 C P 2 n  and n +1 M k =1 g K U ∗ +4 k ( m ^ i =2 C P 2( n +1) ) − → n M k =1 g K U ∗ +4 k ( m ^ i =2 C P 2 n ) (induced from the maps ψ and φ of diagram (2.10)) are all surjectiv e. Th us the Mittag-Leffler condition is satisfie d and the lim ← − n 1 terms are zero.  The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 8 Finally , Lemma 2.5 implies that the isomorphisms (2.13) are compatible with the maps in the inv erse limits and so yield the main result of this section. Theorem 2.7. The r e is an isomorphis m of gr ade d ab e l i an gr oups ∞ M k =1 g K U ∗ + 4 k ( m ^ i =2 C P ∞ ) − → g K O ∗ ( m ^ i =1 C P ∞ ) . 3. The algebra K O ∗ ( B T m ) This section con tains the first of t wo descriptions of the the algebra K O ∗ ( B T m ). It extends the calculation done in [1 6 ] for the case m = 2. 3.1. Notation and statemen t of r esults. Here, as in Sec tion 1, B T m ∼ = m Y i =1 C P ∞ The tw o sets of generators presen ted for K O ∗ ( B T m ) hav e con trasting adv an tages. The first description yields generators whic h are sligh tly complicated but the relations among them are fairly straigh tforward. This situation is rev ersed in the sec ond description. The complexification and realification maps, (1.5) and (1.6), are denoted again by c and r respectiv ely . Let α ∈ K O − 4 and β ∈ K O − 8 b e the elemen ts arising from (2.6), for which α 2 = 4 β . Let v ∈ K U − 2 b e the Bott elemen t, whic h satisfies r ( v 2 ) = α and c ( α ) = 2 v 2 . The generators of K U 0 ( B T m ) are denoted b y x i for i = 1 , . . . , m s o that K U 0 ( B T m ) ∼ = Z [ [ x 1 , . . . , x m ] ]. More notation is established nex t. Definition 3.1. Consider the set N = { 1 , . . . , m } and let S ⊆ N . Then (1) set min( S ) = min { i : i ∈ S } , (2) let | S | denote the cardinalit y of S , (3) for s ∈ { 0 , 1 , 2 } , let X ( s ) S = r ( v s Y i ∈ S x i ) ∈ K O − 2 s ( B T m ), (4) let X S = X (0) S and X i = X { i } = r ( x i ), (5) for s ∈ { 0 , 1 } , let X ( s ) φ = 1 + ( − 1) s , The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 9 (6) for s ∈ { 0 , 1 } , let M ( s ) S = X ( s ) S · Y i ∈ N \ S X i and M S = M (0) S . (7) d B T m = V m i =1 C P ∞ and (8) Z [ [ − ] ] denotes the augmen tation ideal of a pow er series ring. Theorem 3.2. The r e is an isomorphis m of gr ade d rings K O ∗ ( B T m ) ∼ = Z [ γ ± 1 ] ⊗ Z [ [ X S , X (1) S : φ 6 = S ⊆ N ] ] . ∼ wher e γ is an eleme n t with | γ | = − 4 sa tisfying 2 γ = α and γ 2 = β . Her e ∼ r efers to the two families of r elations (I) and (I I) b elow . (I) If A , B and C ar e disjoint subsets of N and 0 ≤ s, t ≤ 1 , then X ( s ) A ∪ B X ( t ) A ∪ C = Y i ∈ A X i ·   X T ⊆ A X ( s + t ) T ∪ B ∪ C + ( − 1) s + | A ∪ B | X S ⊆ B ( − 1) | S | Y i ∈ S X i ! X ( s + t ) C ∪ B \ S   . Her e B , C , S , T may b e emp ty, X (2) S = γ X S and pr o ducts o ver empty sets ar e c onsider e d e qual to 1 . (I I) F or i < min( S ) , | S | > 1 and s ∈ { 0 , 1 } , X i X ( s ) S = ( − 1) s X T ⊆ S   ( − 1) | T | Y j ∈ S \ T X j ! · X ( s ) { i }∪ T   + X ( s ) { i }∪ S . A gain, T may b e empty. Remark 3.3. The elemen t γ is in tro duced here for notational con v enience. It arises nat- urally in the A dams sp ectral sequence and has the prop ert y that γ r ( x ) = r ( v 2 x ) for all x ∈ K U 0 ( B T m ). The use of γ ma y b e remov ed in Theorem 3.2 and in Corollary 3.4 (b elo w), b y allowi ng the choic e of the exponen t s in D efinition 3.1 to be unrestricte d. Relations (I) allow the elimination of all pro ducts X ( u ) U X ( v ) V with | U | and | V | b oth greater than 1, reducing ev erything to pro ducts of X i ’s times at most one X ( w ) W with | W | > 1 . Relations (I I) allo w the elimination of X i X ( w ) W with | W | > 1 and i < min( W ). Notice that for a pro duct X i 1 X i 2 · . . . · X i k X ( w ) W , (I I) need b e performed once only for just one X i j with minimal i j . The next corollary is now immediate. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 10 Corollary 3.4. Every ele ment of K O ∗ ( B T m ) c an b e expr esse d as a formal sum of terms fr om G 1 = ( γ j  m Y i = mi n ( S ) X e i i  X ( s ) S : S ⊆ N , S 6 = φ, e i ≥ 0 , j ∈ Z and s ∈ { 0 , 1 } ) . The example below follow s easily from Theorem 3.2. Example 3.5. F or s ∈ { 0 , 1 } , K O − (4 j +2 s ) ( B T 2 ) has a basis G 1 = n γ j X e 2 2 X ( s ) 2 , γ j X e 1 1 X e 2 2 X ( s ) 1 , γ j X e 1 1 X e 2 2 X ( s ) { 1 , 2 } : e 1 , e 2 ≥ 0 o . The fo llo wing relations dete rmine all pro ducts among these basis elemen ts. Here s ∈ { 0 , 1 } and i ∈ { 1 , 2 } . Recall X (2) S = γ X S . X { 1 , 2 } X { 1 , 2 } = X 1 X 2 ( X { 1 , 2 } + X 1 + X 2 + 4) X ( s ) { 1 , 2 } X (1) { 1 , 2 } = X 1 X 2 ( X ( s +1) { 1 , 2 } + X ( s +1) 1 + X ( s +1) 2 ) X (1) i X (1) i = γ ( X 2 i + 4 X i ) X ( s ) 1 X (1) 2 = 2 X ( s +1) { 1 , 2 } − X 2 X ( s +1) 1 X (1) 1 X (1) { 1 , 2 } = γ X 1 (2 X 2 + X { 1 , 2 } ) . The first t w o relations are of ty p e (I); the last three are of t yp e (I I). R emark. The case m = 2 is done in [16]. The example ab o ve agrees with Prop osition 8.2.20 in [16] aft er certain ty p ographical errors are corrected. (Thes e include replacing all the equal signs with min us signs and correctin g the formula for “ w 2 i w 2 j ” so that it is consisten t with Lemma 8.2.8 in the same do cume n t.) A closely-related result giv es K O ∗ ( d B T m ). Theorem 3.6. g K O − (4 j +2 s ) ( d B T m ) is a fr e e mo dule over Z [ [ X 1 , . . . , X m ] ] on n γ j M ( s ) S : 1 ∈ S ⊆ N o The pr o d uct M ( s ) S 1 M ( t ) S 2 c an b e c om pute d in terms of this b asis fr o m the r elations (I) and (I I) of The or em 3 .2. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 11 Relations (I) and (I I) o f Theorem 3.2 are prov ed next. This is f ollo w ed by an iden tification of the terms g iv en b y Theorem 3.6 with those app earing in Theore m 2.7. Finally , Theorem 3.2 is deriv ed from Theorem 3.6. 3.2. The pro of of relations (I) and (I I). The complexification map c is injectiv e and so it suffices to prov e relations (I) and (I I) after c is applied. F or con veni ence, the relations will b e v erified in the ring K U ∗ ( B T m ) with the classes { z i = √ 1 + x i : i = 1 , . . . , m } adjoined. c ( X ( s ) S ) = v s Y i ∈ S x i + v s Y i ∈ S x i = v s ( Y i ∈ S x i )  1 + ( − 1) s + | S | Y i ∈ S 1 1 + x i  = v s Y i ∈ S  x i √ 1 + x i  ·   Y i ∈ S √ 1 + x i + ( − 1) s + | S | Y i ∈ S 1 √ 1 + x i   = v s Y i ∈ S  z 2 i − 1 z i  ·   Y i ∈ S z i + ( − 1) s + | S | Y i ∈ S 1 z i   = v s Y i ∈ S  z i − 1 z i  ·   Y i ∈ S z i + ( − 1) s + | S | Y i ∈ S 1 z i   . More notation is in tro duced next. Definition 3.7. Let A , S , and T be disjoin t subsets of N . Then (1) Let w ( s ) A,S,T := ( Q j ∈ A z 2 j )( Q j ∈ S z j ) Q j ∈ T z j + ( − 1) s + | S ∪ T | Q j ∈ T z j ( Q j ∈ A z j 2 )( Q j ∈ S z j ) The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 12 (2) for A = φ , set w ( s ) S,T = w ( s ) φ,S,T and fo r T = φ , w ( s ) S = w ( s ) S,φ (3) set w i = w (0) { i } = z i − 1 z i . Notice that in this new notation, the calcul ation ab o v e is c ( X ( s ) S ) = v s  Y i ∈ S w i  w ( s ) S . Recall that If A , B and C are disjoin t subsets of N and 0 ≤ s, t ≤ 1, relation (I) is (3.1) X ( s ) A ∪ B X ( t ) A ∪ C = Y i ∈ A X i ·   X T ⊆ A X ( s + t ) T ∪ B ∪ C + ( − 1) s + | A ∪ B | X S ⊆ B ( − 1) | S | Y i ∈ S X i ! X ( s + t ) C ∪ B \ S   . Applying c and dividing both sides b y v s + t  Q i ∈ A w 2 i  Q i ∈ B ∪ C w i  mak es (3.1) equiv alen t to (3.2) w ( s ) A ∪ B w ( t ) A ∪ C = X T ⊆ A   Y i ∈ T w i  w ( s + t ) T ∪ B ∪ C  + ( − 1) s + | A ∪ B | X S ⊆ B ( − 1) | S |  Y i ∈ S w i  w ( s + t ) C ∪ B \ S . The left hand side of (3.2) is c hec k ed easily to satisfy (3.3) w ( s ) A ∪ B w ( t ) A ∪ C = w ( s + t ) A,B ∪ C,φ + ( − 1) s + | A ∪ B | w ( s + t ) C,B . The first term on the righ t hand side of (3.2) satisfie s (3.4) X T ⊆ A   Y i ∈ T w i  w ( s + t ) T ∪ B ∪ C  = X T ⊆ A  X R ⊆ T ( − 1) | T \ R | w ( s + t ) R,B ∪ C,φ  , where R ⊆ T is defined b y the fact that T \ R is the set of i ’s in T fo r whic h, in Q i ∈ T w i , is chose n the second term of w i = z i − 1 z i . With T satisfying R ⊆ T ⊆ A , the order of summation on the righ t hand side of (3.4) is rearranged to giv e The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 13 (3.5) X T ⊆ A  X R ⊆ T ( − 1) | T \ R | w ( s + t ) R,B ∪ C,φ  = X R ⊆ A w ( s + t ) R,B ∪ C,φ  X T ⊆ A ( − 1) | T \ R |  . No w (3.6) X T ⊆ A ( − 1) | T \ R | = | A \ R | X j =0 ( − 1) j | A \ R | j ! . Here, the binomial co efficien t on the righ t hand side coun ts the num b er of sets T , R ⊆ T ⊆ A satisfying | T \ R | = j . Notice that the right-hand side is zero unless A = R in whic h case it equals 1. So no w ( 3 .4) implies that the first term on the righ t of (3.2) is w ( s + t ) A,B ∪ C,φ whic h is the first term o n the right-hand side of (3.3). The second term on the righ t hand side of (3.2) is analyzed similarly . With S satisfying U ⊆ S ⊆ B , (3.7) ( − 1) s + | A ∪ B | X S ⊆ B ( − 1) | S |  Y i ∈ S w i  w ( s + t ) C ∪ B \ S = ( − 1) s + | A ∪ B | X S ⊆ B ( − 1) | S | X U ⊆ S ( − 1) | U | w ( s + t ) C ∪ B \ U,U ! where here S \ U is the set of i ’s in S for whic h is c hosen in Q i ∈ S w i the sec ond term of w i = z i − 1 z i . Contin uing as ab o v e, (3.8) ( − 1) s + | A ∪ B | X S ⊆ B ( − 1) | S | X U ⊆ S ( − 1) | U | w ( s + t ) C ∪ B \ U,U ! = ( − 1) s + | A ∪ B | X U ⊆ B " w ( s + t ) C ∪ B \ U,U  X S ( − 1) | S \ U |  # = ( − 1) s + | A ∪ B | X U ⊆ B   w ( s + t ) C ∪ B \ U,U | B \ U | X j =0 ( − 1) j | B \ U | j !   = ( − 1) s + | A ∪ B | w ( s + t ) C,B The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 14 whic h is the second term on the righ t hand side of (3.3). This completes the pro of of the relations (I). The v erification of relations (I I) is next. F or i < min( S ), | S | > 1 and s ∈ { 0 , 1 } , the second set of relations is X i X ( s ) S = ( − 1) s X T ⊆ S  ( − 1) | T |  Y j ∈ S \ T X j  · X ( s ) { i }∪ T  + X ( s ) { i }∪ S . Applying c to b oth sides and dividing by Q j ∈{ i }∪ S w j mak es relations (II) equi v alen t to (3.9) w i w ( s ) S = ( − 1) s X T ⊆ S  ( − 1) | T | w ( s ) { i }∪ T Y j ∈ S \ T w j  + w ( s ) { i }∪ S . Definition 3.7 implies immediately that (3.10) w i w ( s ) S = − w ( s ) S, { i } + w ( s ) { i }∪ S . and so it remains to sho w that (3.11) ( − 1) s X T ⊆ S  ( − 1) | T | w ( s ) { i }∪ T Y j ∈ S \ T w j  = − w ( s ) S, { i } . T o this end and using the fact that i < min( S ) implie s i / ∈ S , The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 15 ( − 1) s X T ⊆ S  ( − 1) | T | w ( s ) { i }∪ T Y j ∈ S \ T w j  = ( − 1) s X T ⊆ S  ( − 1) | T | X B ⊆ S \ T ( − 1) | B | w ( s ) { i }∪ S \ B ,B  = ( − 1) s X B ⊆ S  ( − 1) | B | w ( s ) { i }∪ S \ B ,B X T ⊆ S \ B ( − 1) | T |  = ( − 1) s X B ⊆ S   ( − 1) | B | w ( s ) { i }∪ S \ B ,B S \ B X j =0 ( − 1) j | S \ B | j !   = ( − 1) s + | S | w ( s ) { i } ,S = − w ( s ) S, { i } where, as in (3.6), S \ B X j =0 ( − 1) j | S \ B | j ! = 0 unles s S = B in whi c h case it equ als 1. 3.3. The pro of of Theorems 3.6 and 3.2. First, the additiv e generators app earing in Theorem 3.6 are iden tified with the K U generators giv en b y Theorem 2.7. Cho ose generators y i ∈ g K U 0 ( V m i =2 C P ∞ ) so that (3.12) g K U ∗ ( m ^ i =2 C P ∞ ) ∼ = Z [ v ± 1 ][ [ y 2 , . . . , y m ] ] · ( y 2 · · · y m ) . Theorem 2.7 can b e written as (3.13) g K O ∗ ( m ^ i =1 C P ∞ ) ∼ = ∞ M k =1 g K U ∗ + 4 k ( m ^ i =2 C P ∞ ) ∼ = Z [ v ± 1 ][ [ z , y 2 , . . . , y m ] ] · ( z y 2 · · · y m ) where z is giv en grading equal to 4. The fa ct that the realification map r increases filtra- tion in the Adams sp ectral sequence b y 1, v has filtration 1 and γ filtration 2, allo ws the determination of the Adams filtration of the generators desc rib ed in Theorem 3.6. This then mak es p ossi ble a comparison of generators via the A dams spectral sequenc e, mo dulo terms of higher filtration. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 16 Lemma 3.8. L et S b e a set satisfying 1 ∈ S ⊆ N . In the description (3.13), the element v 2 j + s z e 1 y 2 e 2 2 · . . . · y 2 e m m  Y i ∈ N \ S y i  z y 2 · . . . · y m ∈ g K U − (4 j +2 s )+ 4( e 1 +1) ( m ^ i =2 C P ∞ ) c orr esp on ds to the eleme n t X e 1 1 · . . . · X e m m γ j M ( s ) S ∈ g K O − (4 j +2 s ) ( d B T m ) = g K O − (4 j +2 s ) ( m ^ i =1 C P ∞ ) , mo dulo terms of highe r filtr ation in the A da ms sp e ctr al se quenc e.  Lemma 3.8 allow s no w the use of Theorem 2.7 to conclude that the generators giv en by Theorem 3.6 m ust span g K O − (4 j +2 s ) ( d B T m ) a nd be linearly indep enden t. The pro of of Theorem 3.2 from Theorem 3.6 use s the homotopy equiv alence (3.14) Σ( Y 1 × Y 2 × . . . × Y m ) − → Σ  _ S ⊆ N  ^ i ∈ S Y i   where Y i = C P ∞ for all i ∈ N . Theorem 3.6 is a pplied to each wed ge summand V i ∈ S Y i to con- clude that for S = { i 1 , i 2 , . . . , i | S | } , g K O − (4 j +2 s )  ^ i ∈ S Y i  is a free mo dule ov er Z [ [ X i 1 , . . . , X i | S | ] ] on n γ j M ( s ) T : 1 ∈ T ⊆ S o . Assem bling these generators o v er all S ⊆ N , S 6 = φ pro duces the generators in Theorem 3.2. The m ultiplicativ e relations (I) and (I I) ha v e been c hec k ed. 4. A second set of genera tors for the algebra K O ∗ ( B T m ) 4.1. Notation and statemen t of r esults. As usual, K U ∗ ∼ = Z [ v , v − 1 ] with v ∈ K U − 2 K O ∗ ∼ = Z [ e, α, β , β − 1 ] / (2 e, e 3 , eα, α 2 − 4 β ) where e ∈ K O − 1 , α ∈ K O − 4 and β ∈ K O − 8 . As in Section 3, denote b y x i , i = 1 , . . . , n , the generators of K U 0 ( B T n ). It is con v enien t to write The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 17 (4.1) K U 0 ( B T m ) ∼ = Z [ [ x 1 , . . . , x m , x 1 , . . . , x m ] ] . ( x i x i + x i + x i ) . Let I = ( i 1 , i 2 , . . . , i m ) and J = ( j 1 , j 2 , . . . , j m ) with i k ≥ 0 , j k ≥ 0 for k = 1 , . . . , m . F or s ∈ Z and r the realification map (1.6) , set [ I , J ] ( s ) := r  v s x i 1 1 x i 2 2 . . . x i m m ( x 1 ) j 1 ( x 2 ) j 2 . . . ( x m ) j m  in K O − 2 s ( B T m ). If s = 0, the notation [ I , J ] is used instead of [ I , J ] (0) . Theorem 4.1. The classes [ I , J ] ( s ) satisfy the r elations: (A) [ I , J ] ( s ) = ( − 1) s [ J, I ] ( s ) (B) [ I , J ] ( s ) = − [ I ′ , J ] ( s ) − [ I , J ′ ] ( s ) wher e, for I = ( i 1 , . . . , i k , . . . , i m ) , J = ( j 1 , . . . , j k , . . . , j m ) with i k · j k 6 = 0 , I ′ = ( i 1 , . . . , i k − 1 , . . . , i m ) and J ′ = ( j 1 , . . . , j k − 1 , . . . , j m ) . (C) [ I , J ] ( s ) · [ H, K ] ( t ) = [ I + H , J + K ] ( s + t ) + ( − 1) s [ J + H , I + K ] ( s + t ) wher e the pr o duct her e is in K O ∗ ( B T m ) . R emark. F orm ula (C) is symmetric b ecause relation (A) implies ( − 1) s [ J + H , I + K ] ( s + t ) = ( − 1) t [ I + K , J + H ] ( s + t ) . Pr o of of The or em 4.1. Relations (A) fo llo w immediately f ro m (1 .8) b y applying complexifi ca- tion then realification. Relations (B) follow b y recalling that x = − x (1 + x ) and decompo sin g x i x j − 1 as x i x j − 1 = x i − 1 xx j − 1 = x i − 1 x j − 1 ( − x (1 + x )) = − x i − 1 x j − x i x j The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 18 whic h giv es x i x j = − x i − 1 x j − x i x j − 1 . T o see relations (C), the complexifi cation monomor- phism c is applied to b oth sides. c  [ I , J ] ( s ) · [ H, K ] ( t )  = c  [ I , J ] ( s )  · c  [ H , K ] ( t )  = [ v s x i 1 1 x i 2 2 . . . x i m m ( x 1 ) j 1 ( x 2 ) j 2 . . . ( x m ) j m + ( − 1) s v s ( x 1 ) i 1 ( x 2 ) i 2 . . . ( x m ) i m x j 1 1 x j 2 2 . . . x j m m ] · [ v t x h 1 1 x h 2 2 . . . x h m m ( x 1 ) k 1 ( x 2 ) k 2 . . . ( x m ) k m + ( − 1) t v t ( x 1 ) h 1 ( x 2 ) h 2 . . . ( x m ) h m x k 1 1 x k 2 2 . . . x k m m ] = v s + t x i 1 + h 1 1 x i 2 + h 2 2 . . . x i m + h m m ( x 1 ) j 1 + k 1 ( x 2 ) j 2 + k 2 . . . ( x m ) j m + k m + ( − 1) s + t v s + t x j 1 + k 1 1 x j 2 + k 2 2 . . . x j m + k m m ( x 1 ) i 1 + h 1 ( x 2 ) i 2 + h 2 . . . ( x m ) i m + h m + ( − 1) s v s + t x j 1 + h 1 1 x j 2 + h 2 2 . . . x j m + h m m ( x 1 ) i 1 + k 1 ( x 2 ) i 2 + k 2 . . . ( x m ) i m + k m + ( − 1) t v s + t x i 1 + k 1 1 x i 2 + k 2 2 . . . x i m + k m m ( x 1 ) j 1 + h 1 ( x 2 ) j 2 + h 2 . . . ( x m ) j m + h m = c  [ I + H , J + K ] ( s + t )  + c  ( − 1) s [ J + H , I + K ] ( s + t )  = c  [ I + H , J + K ] ( s + t )  + c  ( − 1) t [ I + K , J + H ] ( s + t )  .  Remark 4.2. The elemen ts X ( s ) S of Definition 3.1 are related to the class es [ I , J ] ( s ) b y X ( s ) S = [( ǫ (1) , ǫ (2) , . . . , ǫ ( m )) , (0 , 0 , . . . , 0)] ( s ) where ǫ is the c haracteristic function of S . Next, a distinguished class of elemen ts [ I , J ] ( s ) ∈ K O − 2 s ( B T m ) is se lected. F or I = ( i 1 , i 2 , . . . , i m ) a nd J = ( j 1 , j 2 , . . . , j m ), with all i k ≥ 0, j k ≥ 0, set (4.2) G 2 := n [ I , J ] ( s ) : I · J = 0 and i l ≥ j l if i k + j k = 0 for k < l o where here, I · J denotes the dot pro duct of v ectors. The K O ∗ -mo dule structure is described easily . Recall that K O ∗ ∼ = Z [ e, α, β , β − 1 ] / (2 e, e 3 , eα, α 2 − 4 β ) . The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 19 Lemma 4.3. The K O ∗ -mo dule action on K O ∗ ( B T m ) is given by e · ([ I ] , [ J ]) ( s ) = 0 α · ([ I ] , [ J ]) ( s ) = 2([ I ] , [ J ]) ( s +2) β · ([ I ] , [ J ]) ( s ) = ([ I ] , [ J ]) ( s +4) . Pr o of. The complexific ation monomorphism c is a pplied to b oth sides of these relations. The result follo ws then from the identitie s c ( e ) = 0 , c ( β ) = v 4 and c ( α ) = 2 v 2 from [16], Lemma 2.0.3.  Theorem 4.4. Every element of K O ∗ ( B T m ) c an b e expr es se d as a formal sum of terms fr om G 2 . Pr o of. The class es v s x i 1 1 x i 2 2 . . . x i m m ( x 1 ) j 1 ( x 2 ) j 2 . . . ( x m ) j m generate K U ∗ ( B T m ) as a p o w er se- ries ring. The realification map r is on to by (1.7). So, the classe s [ I , J ] ( s ) ∈ K O − 2 s ( B T m ) generate K O ∗ ( B T m ) as a K O ∗ -mo dule. R elations (A) a nd (B) in Theorem 4.1 imply that ev ery elemen t [ I , J ] ( s ) ∈ K O − 2 s ( B T m ) can b e written as a linear combination of elemen ts in G 2 . A pro duct of t w o elemen ts in G 2 is not giv en explicitly in terms of elemen ts of G 2 b y relation (C) but rep eated a pplications of relation (A) and (B) reduc e the result of (C) to a linear comb ination of elemen t s of G 2 .  R emark. Lemma 4.3 and the pro of of Theorem 4.4 describ e the K O ∗ -algebra structure of K O ∗ ( B T m ). In particular, as noted in Section 1, this result and Theorem 3.2 b oth describe the completion of the repres en tation ring RO ( T m ) a t the augmen tation ideal. Prop osition 4.5. No finite r elations exist among the elements of G 2 . Pr o of. F or m = 1 , a finite relation among elemen ts of G 2 w o uld pro duces a relation of the form r ( p ( x 1 )) = 0 where p ( x 1 ) is a p olynomial. The Bott sequence implies then that a formal p o wer series θ ( x 1 ) exists in K U 0 ( C P ∞ ) satisfying θ ( x 1 ) − θ ( x 1 ) = p ( x 1 ) . It is straightforw ard to c heck that no suc h relation can o ccur in K U 0 ( C P ∞ ). The case m > 1 reduces easily to the case m = 1.  On the other hand, infinite relations do o ccur in K O − 2 s ( B T m ) a mong the generators G 2 . Consider a relation of the form (4.3) ∞ X k =0 a k [ I k , J k ] ( s ) = 0 , a k ∈ Z , [ I k , J k ] ( s ) ∈ G 2 The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 20 When applied to (4.4), the complex ification monomorphism pro duce s a relation among the generators x 1 , . . . , x m , x 1 , . . . , x m in K U − 2 s ( B T m ) whic h m ust then b e a consequen ce of the relations x i x i + x i + x i = 0 i = 1 , . . . , m. F or m = 1, t here is the ex ample (4.4) 2[1 , 0] + ∞ X n =2 ( − 1) n − 1 [ n, 0] = 0 , whic h is just the realification map r applied to the relation x 1 = − x 1 1 + x 1 = − x 1 + x 2 1 − x 3 1 + x 4 1 + . . . in K U 0 ( B T 1 ). Remark 4.6. The tw o generating sets G 1 and G 2 , describ ed in Corollary 3.4 and (4.2) respectiv ely , a re distinguished as follows. Although in b oth cases, an infinite sum of allow able generators will become finite under an y restriction (4.5) K O ∗ ( m Y i =1 C P ∞ ) − → K O ∗  C P k 1 × C P k 2 × · · · × C P k m  , relations of the t yp e (4.4) in the generators G 2 will pro duce finite linear relations. 5. The D a vis-Januszkiewicz s p aces 5.1. The Davis- Jan uszkiewicz space asso ciated to a simplicial complex. In Section 1, the Da vis-Jan uszkiewicz space D J ( K P ), associated to simple p olytop e P , w as defined in terms of a toric manifold M 2 n . More generally , a Davis-Jan uszkiewicz space DJ ( K ) can b e constructed for any simplicial complex K b y means of the generalized momen t-angle complex construction Z ( K ; ( X , A )) of [14 ], [10], [15] and [6]. A descrip tion of the space DJ ( K ) follows. Definition 5.1. Let K b e a simplicial complex with m v ertices. Iden tify simplices σ ∈ K as increas ing subseq uences of [ m ] = (1 , 2 , 3 , . . . , m ). The Dav is-Jan uszkiew icz space DJ ( K ) is defined b y DJ ( K ) = Z ( K ; ( C P ∞ , ∗ )) ⊆ B T m = m Y i =1 C P ∞ where ∗ repres en ts the basep oin t and The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 21 Z ( K ; ( C P ∞ , ∗ )) = [ σ ∈ K D ( σ ) with (5.1) D ( σ ) = m Y i =1 W i , where W i = ( C P ∞ if i ∈ σ ∗ if i ∈ [ m ] − σ. A toric manifold M 2 n is sp ecifie d by a simple n -dimensional p o lytope and a char acteristic function on its facets as des crib ed in [14]. Equiv alen tly , M 2 n can be realized as a quotient. The c haracteristic function corresp onds to a sp ecifi c c ho ice of sub-torus T m − n ⊆ T m whic h acts freely on the moment-angle complex Z ( K P ; ( D 2 , S 1 )) to giv e M 2 n ∼ = Z ( K P ; ( D 2 , S 1 )) . T m − n . This description of M 2 n yields an eq uiv alence of Borel construc tions (5.2) E T m × T m Z ( K P ; ( D 2 , S 1 )) ≃ E T n × T n  Z ( K P ; ( D 2 , S 1 )) /T m − n  ∼ = E T n × T n M 2 n = DJ ( K P ) . Moreo v er, for a ny simplicial complex K , there is an equiv alence ([14], [10] and [15]), (5.3) E T m × T m Z ( K ; ( D 2 , S 1 )) ∼ = Z ( K ; ( C P ∞ , ∗ )) . It follo ws that for K = K P , the three descriptions of DJ ( K P ) g iv en b y (5.2) and (5.3) agree up to homotop y equiv alence 5.2. The K O ∗ -rings of the Da vis-Jan uszkiewicz spaces. It is w ell kno wn that (1.2) extends to DJ ( K ) and so, for any complex-orien ted cohomology theory E ∗ (5.4) E ∗ ( DJ ( K )) ∼ = E ∗ ( B T m ) . I E S R where I E S R is the Stanley-Reisner ideal describ ed in Section 1. A related but more general result can b e found in [6], Theorem 2.35. Also f r o m [6], the follo wing geometric results The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 22 will prov e useful for the computation of K O ∗ ( DJ ( K )) in this section. Belo w, increasin g subseq uences of [ m ] = (1 , 2 , 3 , . . . , m ) are denoted b y σ = ( i 1 , i 2 , . . . , i k ) , τ = ( i 1 , i 2 , . . . , i t ) and ω = ( i 1 , i 2 , . . . , i s ) and X i j = C P ∞ for all i j . Theorem 5.2. The Davis-Januszkiewic z sp ac e DJ ( K ) de c om p ose s stably as fol low s . Σ  DJ ( K )  ≃ − → Σ  _ σ ∈ K X i 1 ∧ X i 2 ∧ . . . ∧ X i k  . Mor e over, ther e is a c ofibr ation se quenc e Σ  DJ ( K )  i − → Σ  _ τ ∈ [ m ] X i 1 ∧ X i 2 ∧ . . . ∧ X i t  q − → Σ  _ ω / ∈ K X i 1 ∧ X i 2 ∧ . . . ∧ X i s  , where the map i is split. A particular case of (5 .4 ) is giv en b y E ∗ = K U ∗ , so (5.5) K U ∗ ( DJ ( K )) ∼ = K U ∗ ( B T m ) . I K U S R Remark 5.3. No tice that in the represen tat io n of K U 0 ( B T m ) give n in (4.1), the monomials generating the ideal I K U S R could eq ually w ell con tain a generator x i or its conjugate x i . Theorem 5.2 and the results of section 2 imply that K O ∗ ( DJ ( K )) is concen trated in ev en degrees. The Bot t sequence (1.7) implies then that the realification map r : K U ∗ ( DJ ( K )) − → K O ∗ ( DJ ( K )) is onto and that the complexification map c : K O ∗ ( DJ ( K )) − → K U ∗ ( DJ ( K )) is a monomorphism. The goa l o f t he remainde r of this section is to use the generators G 2 of Section 4 to describ e the ring K O ∗ ( DJ ( K )). Let K b e a simplic ial complex on m v ertices. F or I = ( i 1 , i 2 , . . . , i m ) as in Section 4, set The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 23 ǫ ( I ) = { k : i k 6 = 0 } ⊆ [ m ] and let S R K O denote the ideal in K O ∗ ( B T m ) g enerated b y the set (5.6) { [ I , J ] ( s ) ∈ G 2 : ǫ ( I ) ∪ ǫ ( J ) / ∈ K } , where again, simplices of K are iden tified as increasing subsequ ences of [ m ] = (1 , 2 , 3 , . . . , m ). The notation S R K O for the K O Stanley-Reisner ideal is more appropriate than I K O S R as it is structurally differen t from that for a complex-orien ted theory . Next, the ideal S R K O is related to r ( I K U S R ). The non-multiplic ativit y of the map r mak es necessary a preliminary lemma. Lemma 5.4. The ab elian gr o up r ( I K U S R ) is an ide al in K O ∗ ( B T m ) . Pr o of. With reference to the notation of Theorem 5.2, set c X τ = X i 1 ∧ X i 2 ∧ . . . ∧ X i t and c X ω = X i 1 ∧ X i 2 ∧ . . . ∧ X i s . Recall here that each X i j = C P ∞ . The split cofibration of Theorem 5.2 giv es rise to the diagram followin g (5.7) g K U − 2 s  DJ ( K )  i ∗ ← − − − g K U − 2 s  W τ ∈ [ m ] c X τ  q ∗ ← − − − g K U − 2 s  W ω / ∈ K c X ω     y r    y r    y r g K O − 2 s  DJ ( K )  i ∗ ← − − − g K O − 2 s  W τ ∈ [ m ] c X τ  q ∗ ← − − − g K O − 2 s  W ω / ∈ K c X ω  . The maps i ∗ are o nto and so g K O ∗  DJ ( K )  is a quotien t of g K O ∗ ( B T m ). A diagram c hase is needed next. Let x ∈ g K O − 2 s  W τ ∈ [ m ] c X τ  b e suc h that i ∗ ( x ) = 0. Then, y ∈ g K O − 2 s  W ω / ∈ K c X ω  exists satisfying q ∗ ( y ) = x . Since r is on to, z ∈ g K U − 2 s  W ω / ∈ K c X ω  exists with r ( z ) = y . Then r ( q ∗ ( z )) = q ∗ ( r ( z )) = x. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 24 No w q ∗ ( z ) ∈ I K U S R whic h implies that x ∈ r ( I K U S R ). Conv ersely , the comm utativit y of the left-hand half of (5.7) implies that if x ∈ r ( I K U S R ) then i ∗ ( x ) = 0. Hence r ( I K U S R ) is the k ernel of the map (5.8) i ∗ : g K O ∗ ( B T m ) − → g K O ∗ ( DJ ( K )) . In particular, r ( I K U S R ) is an ideal in K O ∗ ( B T m ) as require d.  The next prop osition allows a c haracterization of this important ideal in terms of the condition (5.6). Prop osition 5.5. A s ide als in K O ∗ ( B T m ) r ( I K U S R ) = S R K O . Pr o of. Let [ I , J ] ( s ) ∈ S R K O . Since ǫ ( I ) ∪ ǫ ( J ) / ∈ K , (5.9) [ I , J ] ( s ) = r  v s y α 1 y α 2 . . . y α k m  where, in the ligh t of Remark 5.3, eac h y α j = x α j or x α j , { α 1 , α 2 , . . . , α k } / ∈ K and m is a monomial in the classes x 1 , . . . , x m , x 1 , . . . , x m . (Notice here that the choic e of α 1 , α 2 , . . . , α k in (5.9) ma y not b e unique.) No w v s y α 1 y α 2 . . . y α k m ∈ I K U S R and so [ I , J ] ( s ) ∈ r ( I K U S R ) . Con- v ersely , an elemen t in r ( I K U S R ) is a K O ∗ -sum of elemen ts each of the f o rm r  v s y α 1 y α 2 . . . y α k n  again with each y α j = x α j or x α j , { α 1 , α 2 , . . . , α k } / ∈ K and n is a monomial in the classes x 1 , . . . , x m , x 1 , . . . , x m . Now r  v s y α 1 y α 2 . . . y α k n  = [ I ′ , J ′ ] ( s ) for some I ′ and J ′ and moreo v er, ǫ ( I ′ ) ∪ ǫ ( J ′ ) / ∈ K b ecause { α 1 , α 2 , . . . , α k } / ∈ K . It follows that r ( I K U S R ) ⊂ S R K O , pro ving the conv erse.  The main theorem of this section follows. Theorem 5.6. The r e is an isomorphis m of gr ade d rings K O ∗ ( DJ ( K )) ∼ = K O ∗ ( B T m ) . S R K O Pr o of. Prop osition 5.5 iden tifies S R K O as r ( I K U S R ), whic h is the k ernel of the map i ∗ of (5.8) in the pro of of Lemma 5.4.  The examples followin g illustrate calculations in K O 0 ( DJ ( K )) based on Theorem 5.6. The relations of Theorem 4.1 are used with s = t = 0 and t he elemen ts [ I , J ] are to b e in terpreted mo dulo the ideal of relations S R K O . The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 25 Examples 5.7. (1) Let K = n { v 1 } , { v 2 } o b e the simplicial complex consis ting of t wo distinct v ertices. Classes of the form [( i, 0) , (0 , 0)] and [ (0 , h ) , (0 , 0 )] represen t G 2 generators of K O 0 ( C P ∞ × ∗ ) and K O 0 ( ∗ × C P ∞ ) resp ectiv ely in K O 0 ( B T 2 ) a s described b y (4.2). Now, for i and h not b oth zero, [( i, 0) , (0 , 0)] · [(0 , h ) , (0 , 0)] = [( i, h ) , (0 , 0)] + [ (0 , h ) , ( i, 0)] = 0 by (5.6) whic h is consisten t with the fact that DJ ( K ) = C P ∞ ∨ C P ∞ in this case. (2) Let L = n { v 1 } , { v 2 } , { v 3 } , { v 4 } , { v 1 , v 2 } , { v 2 , v 3 } , { v 3 , v 4 } , { v 2 , v 4 } , { v 2 , v 3 , v 4 } o b e the simplicial complex consisting of a 1-simplex w edged to a 2-simplex at the verte x v 2 . Here, classes of the form [( i 1 , i 2 , 0 , 0) , (0 , 0 , 0 , 0)] and [(0 , h 2 , h 3 , 0) , (0 , 0 , 0 , 0)] represen t G 2 genera- tors of K O 0 ( C P ∞ × C P ∞ × ∗ × ∗ ) and K O 0 ( ∗ × C P ∞ × C P ∞ × ∗ ) resp ectiv ely in K O 0 ( B T 4 ). No w [( i 1 , i 2 , 0 , 0) , (0 , 0 , 0 , 0)] · [(0 , h 2 , h, 3 , 0) , (0 , 0 , 0 , 0)] = [( i 1 , i 2 + h 2 , h 3 , 0) , (0 , 0 , 0 , 0)] + [(0 , h 2 , h 3 , 0) , ( i 1 , i 2 , 0 , 0)] = 0 by (5.6) reflecting the fact that { v 1 , v 2 , v 3 } / ∈ L . Moreo v er (5.10) [( i 1 , i 2 , 0 , 0) , (0 , 0 , 0 , 0)] · [ ( l 1 , l, 2 , 0 , 0) , (0 , 0 , 0 , 0)] = [( i 1 + l 1 , i 2 + l 2 , 0 , 0) , (0 , 0 , 0 , 0)] + [ ( l 1 , l 2 , 0 , 0) , ( i 1 , i 2 , 0 , 0)] . Rep eated a pplic ation of relations (A) a nd (B) in Theorem 4.1 reduce the right hand side of (5.10) to a sum of terms o f the form [( ∗ , ∗ , 0 , 0 ) , ( ∗ , ∗ , 0 , 0) ] eac h of whic h satisfies the G 2 condition for K O 0 ( C P ∞ × C P ∞ ). This is consisten t with the fact that K O ∗ ( C P ∞ × C P ∞ ) is a K O ∗ -subalgebra of K O ∗ ( DJ ( K )) corresp onding to the simplex { v 1 , v 2 } ∈ L . 5.3. The ca t ( K ) approac h. Definition 5 .1 expresses DJ ( K ) as the colimit of an exponen- tial diagram B T K ([20], where D ( σ ) is written B T σ ), o v er the category ca t ( K ) asso ciated to the p osets of faces of K . Since B T K is a cofibran t diagram, its homotop y colimit is homotop y equiv alen t to DJ ( K ) also. The K O ∗ v ersion of the Bousfield-Kan sp ectral sequenc e, studied in [1 9 , Section 3 ], applies in this case and give s an alternativ e calculation of K O ∗ ( DJ ( K )) in terms o f the ca t op ( K )-diagram K O ∗ ( B T K ) whose v a lue on each face σ ∈ K is K O ∗ ( D ( σ )). The argumen ts of [19] apply unchange d and are similar to those of Section 5.2. They imply The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 26 that the sp ectral seque nce collapses at the E 2 -term and is concen tr a ted en tirely along the v ertical axis. So the edge homomorphis m giv es an isomorphism (5.11) K O ∗ ( DJ ( K )) ∼ = − → lim K O ∗ ( B T K ) of K O ∗ -algebras, b y analogy with [19, Corollary 3.1 2]. Informally , the elemen t s of lim K O ∗ ( B T K ) are considered as finite sequences ( u σ ) whose terms u σ ∈ K O ∗ ( B T σ ) are compatible under the inclusions i : B T σ − → B T τ for ev ery τ ⊃ σ . More precisely , the isomorphism (5.1 1 ) leads to the conclus ion follo wing. Theorem 5.8. A s K O ∗ -algebr as, K O ∗ ( DJ ( K )) is isomorphic to n ( u σ ) ∈ Y σ ∈ K K O ∗ ( B T σ ) : i ∗ ( u τ ) = u σ for every τ ⊃ σ o wher e the multiplic ation and K O ∗ -mo dule structur e ar e define d termwise.  Theorem 5.8 extends to E ∗ ( DJ ( K )) for any arbitrary cohomology theory . The corollary follo wing is compleme n tary to Theorem 5.6. Corollary 5.9. The natur al h o momorphism ℓ : K O ∗ ( B T m ) − → lim K O ∗ ( B T K ) is onto with kernel e qual to the i d e al S R K O of The or em 5.6. Pr o of. The homomorphism ℓ is induce d b y the pro jections K O ∗ ( B T m ) → K O ∗ ( B T σ ) as σ ranges ov er the faces o f K , hence it is onto. Theorem 4.4 describ es eac h summand K O ∗ ( B T σ ) of K O ∗ ( B T m ) as g ene rated ov er K O ∗ b y those elemen ts [ I , J ] ( s ) of G 2 for whic h ǫ ( I ) ∪ ǫ ( J ) ⊆ σ . Moreo ver, Theorem 5.8 implies that ℓ ([ I , J ] ( s ) ) = 0 if and only if [ I , J ] ( s ) satisfies ǫ ( I ) ∪ ǫ ( J ) / ∈ K as in (5.6). So, ℓ maps non-trivial fo rmal sums of ele men ts in G 2 to zero if and only if they lie in S R K O .  Corollary 5.9 generalizes to an arbitrary cohomology theory and establishes an isomor- phism E ∗ ( B T m ) . k er ℓ − → E ∗ ( DJ ( K )) of E ∗ -algebras. It is instructiv e to revisit Example s 5.7 f ro m this complemen tary viewp oin t. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 27 Examples 5.10. (1) If K = {{ v 1 } , { v 2 }} , then ca t ( K ) con t a ins the ( − 1)-simplex ∅ a nd tw o 0-simplices. The- orem 5.8 give s K O ∗ ( DJ ( K )) as the K O ∗ -algebra K O ∗ ( B T { v 1 } ⊕ K O ∗ ( B T { v 2 } . The homo- morphism ℓ of Corollary 5.9 maps the elemen ts [( i, 0) , (0 , 0)] and [(0 , h ) , (0 , 0)] of K O 0 ( B T 2 ) to the elem en ts  [( i ) , (0)] , 0  and  0 , [( h ) , (0)]  ; in particular, their pro duct is zero. (2) If L = n { v 1 } , { v 2 } , { v 3 } , { v 4 } , { v 1 , v 2 } , { v 2 , v 3 } , { v 3 , v 4 } , { v 2 , v 4 } , { v 2 , v 3 , v 4 } o , then ca t ( L ) con t a ins the ( − 1)-simp lex ∅ , four 0-simplic es, four 1-simplic es and one 2-simplex . Theorem 5.8 expresses K O ∗ ( DJ ( K )) as a certain K O ∗ -subalgebra of K O ∗ ( B T { v 2 } ) × K O ∗ ( B T { v 1 ,v 2 } ) × K O ∗ ( B T { v 2 ,v 3 ,v 4 } ) , whic h ma y b e iden tified as the pullback (5.12) K O ∗ ( B T { v 1 ,v 2 } ) ⊕ K O ∗ ( B T { v 2 } ) K O ∗ ( B T { v 2 ,v 3 ,v 4 } ) . The elemen ts of (5.12) consist of ordered pairs ( u, w ), for whic h u ∈ K O ∗ ( B T { v 1 ,v 2 } ) and w ∈ K O ∗ ( B T { v 2 ,v 3 ,v 4 } ) share a common restriction to K O ∗ ( B T { v 2 } ). P airs are m ultiplied co ordinate-wise; pro ducts of the form ( u, 0) · (0 , w ) giv e (0 , 0 ) = 0. F or i 1 and h 3 nonzero, the homomorphism ℓ of Corollary 5.9 maps the elemen ts [( i 1 , i 2 , 0 , 0) , (0 , 0 , 0 , 0)] and [(0 , h 2 , h 3 , 0) , (0 , 0 , 0 , 0)] of K O 0 ( B T 4 ) to the pairs  [( i 1 , i 2 ) , (0 , 0)] , 0  and  0 , [( h 2 , h 3 , 0) , (0 , 0 , 0)]  respectiv ely . Their pro duct is zero as required. Similarly , ℓ maps [ ( i 1 , i 2 , 0 , 0) , (0 , 0 , 0 , 0)] and [( l 1 , l, 2 , 0 , 0) , (0 , 0 , 0 , 0)] to the pairs  [( i 1 , i 2 ) , (0 , 0)] , 0  and  [( l 1 , l 2 ) , (0 , 0)] , 0  respectiv ely and, their pro duct is  [( i 1 + l 1 , i 2 + l 2 ) , (0 , 0)] + [( l 1 , l 2 ) , ( i 1 , i 2 )] , 0  . 6. Toric manif olds 6.1. Bac kground. Briefly , a toric manifold M 2 n is a manifold cov ered by lo cal c harts C n , eac h with the standard T n action, compatible in such a w a y that the quotien t M 2 n . T n has the structure o f a simple p olytop e P n . A simpl e p olytop e P n has the prop ert y that at eac h v ertex, exactly n facets in t ersect. Unde r the T n action, eac h cop y of C n m ust pro ject to an R n + neigh b orho o d o f a v ertex of P n . The construction of Da vis and Janus zkiewicz ([14], section 1 . 5) realizes all suc h manifolds a s f ollo ws. Let The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 28 F = { F 1 , F 2 , . . . , F m } denote the set of facets of P n . The fact that P n is simple implies that ev ery co dimension- l face F can b e written uniquely as F = F i 1 ∩ F i 2 ∩ · · · ∩ F i l where the F i j are the facets con taining F . Let λ : F − → Z n b e a function in to an n -dimensional inte ger la t t ice satisfying the condition that whenev er F = F i 1 ∩ F i 2 ∩ · · · ∩ F i l then { λ ( F i 1 ) , λ ( F i 2 ) , . . . , λ ( F i l ) } span an l -dimensional submo dule of Z n whic h is a direct summand. Next, regarding R n as the Lie algebra of T n , λ asso ciates to eac h co dimension- l face F of P n a rank- l subgroup G F ⊂ T n . F inally , let p ∈ P n and F ( p ) be the unique face with p in its relativ e interior. Define an equiv a lence relation ∼ on T n × P n b y ( g , p ) ∼ ( h, q ) if and only if p = q and g − 1 h ∈ G F ( p ) ∼ = T l . Then M 2 n ∼ = M 2 n ( λ ) = T n × P n . ∼ and, M 2 n is a smo oth, closed, connected, 2 n -dimensional manifold with T n action induced b y left translation ([14], page 42 3). The pro jection π : M 2 n → P n is induced fr o m the pro jection T n × P n → P n . It is noted in [14] that ev ery smo oth pro jectiv e tor ic v ar iety has this description. The goal of this section is an analog ue of (1.3) f o r the K O ∗ -rings of certain toric manifolds M 2 n . F or toric manifolds determined b y a simple p olytop e and a c haracteristic map on its facets, a desc ription of the K O ∗ -mo dule structure of K O ∗ ( M 2 n ) was giv en in [4] in terms of H ∗ ( M 2 n ; Z 2 ) as a mo dule ov er Sq 1 and Sq 2 . A more refined computation of the K O ∗ -mo dule structure, f o r certain families of manifolds M 2 n , is presen ted in [18 ]. The K O ∗ -ring structure for families of toric manifolds kno wn as Bo tt tow ers ma y b e found in [12], without reference to K O ∗ ( DJ ( K )). The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 29 6.2. The Steenro d algebra structure of toric manifolds. As in Section 2, let A 1 denote the subalgebra of the Steenro d algebra generated by Sq 1 and Sq 2 . Let S 0 denote the A 1 - mo dule consisting o f a single class in dimension 0 and the trivial action of Sq 1 and Sq 2 . Denote by M the A 1 -mo dule with a class x in dimension 0, a class y in dimension 2 and the action given b y Sq 2 ( x ) = y . A ccording to (1.3), H ∗ ( M 2 n ; Z 2 ) is concen trated in ev en degree and so, as a n A 1 -mo dule, m ust b e isomorphic to a direct sum of suspended copies of the modules S 0 and M . That is, there is a decomp osition (6.1) H ∗ ( M 2 n ; Z 2 ) ∼ = n M i =0 s i Σ 2 i S 0 ⊕ n − 1 M j =0 m j Σ 2 j M , s i , m j ∈ Z . The n um b ers s i and m j w ere lab elled “BB-n umbers” in [12, Section 5]. The Sq 2 -homology of M 2 n , H ∗ ( M 2 n ; Sq 2 ), is zero precisely when s j = 0 for all j . Examples 6.1. The toric manifolds C P 2 k are Sq 2 -acyclic fo r an y p ositiv e in teger k . Examples 6.2. The toric manifolds C P 2 k + 1 ha v e s i = 0 for i ≤ k and s k +1 = 1, for any p ositiv e in teger k . The terminal ly o dd Bott tow ers of [12, Section 5] hav e s 1 = 1 and s i = 0 for i ≥ 2; the total ly even to w ers hav e m j = 0 for ev ery j . Examples 6.3. The non-singular toric v arieties X n ( r ; a r , . . . , a n ) constructed in [18] and satisfying 2 ≤ r ≤ n , a j ∈ Z and n − r o dd are all Sq 2 -acyclic These v ar ieties correspond to n -dimensional fans ha ving n + 2 ra ys . R emark. The preprin t [7] con tains a construction of families of toric manifolds deriv ed fro m a given one. W ork is in progress to confirm that this construction can b e done in suc h a w ay that the family of deriv ed toric manifolds will eac h b e Sq 2 -acyclic, though this prop ert y migh t not be satis fied b y the initial one. The next prop o sition is an immediate consequ ence of t he calculation in [4] . Prop osition 6.4. If M 2 n is Sq 2 -acyclic, then the gr ade d ring K O ∗ ( M 2 n ) is c onc entr ate d in even de gr e e and has no additive torsion.  6.3. The K O -rings of Sq 2 -acyclic toric manifolds. Recall from (1.1) the Borel fibration for toric manifolds, (6.2) M 2 n i − → E T n × T n M 2 n p − → B T n with total space DJ ( K ). The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 30 Theorem 6.5. F or any Sq 2 -acyclic toric manifold M 2 n , ther e is an isom orphism K O ∗ ( M 2 n ) ∼ = K O ∗  DJ ( K )   r ( J K U ) of K O ∗ -algebr as, wher e r is the r e alific ation map and J K U is the ide al define d in (1 . 3). Remark 6.6. Notice that r ( J K U ), whic h is the realification of the ideal generated b y the image of K U ∗ ( B T n ) p ∗ − → K U ∗  DJ ( K )  , is not the same as J K O whic h is the ideal generated b y p ∗  K O ∗ ( B T n )  ; this represen ts a significan t departure from the situation for complex- orien t ed E ∗ ( M 2 n ). As in Lemma 5.4, the non-m ultiplicativit y of the map r implies that r ( J K U ) is not in general an ideal but Theorem 6.5 confirms that K O ∗  DJ ( K )   r ( J K U ) is m ultiplicativ ely closed. Pr o of of The o r em 6.5. The Bott se quences (1.7) for M 2 n , D J ( K ) and B T n link together to giv e the comm utativ e diagram. (6.3) K O ∗− 2 ( M 2 n ) i ∗ ← − − − K O ∗− 2  DJ ( K )  p ∗ ← − − − K O ∗− 2 ( B T n )    y χ χ    y χ    y K U ∗ ( M 2 n ) i ∗ K U on to ← − − − − − K U ∗  DJ ( K )  p ∗ ← − − − K U ∗ ( B T n )    y r onto    y r ont o    y r ont o K O ∗ ( M 2 n ) i ∗ ← − − − K O ∗  DJ ( K )  p ∗ ← − − − K O ∗ ( B T n ) Recall no w that Proposition 6.4 implies that K O ∗ ( M 2 n ) is concen trated in ev en degrees and so a ll the Bott sequences are short exact. The low er left comm utativ e square in (6.3) implies that the maps i ∗ are on to. A diagram c hase is needed next to iden tify the k ernel o f i ∗ . Let z ∈ K O ∗  DJ ( K )  and supp ose i ∗ ( z ) = 0. Since r is onto, y ∈ K U ∗  DJ ( K )  exists with r ( y ) = z . Then r ( i ∗ K U ( y )) = i ∗ ( z ) = 0. The exactness of the leftmost Bott sequence implies now t hat x ∈ K O ∗− 2 ( M 2 n ) exists with χ ( x ) = i ∗ K U ( y ). The map i ∗ is onto so w ∈ K O ∗− 2  DJ ( K )  satisfying i ∗ ( w ) = x . Then The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 31 i ∗ K U ( y − χ ( w )) = i ∗ K U ( y ) − i ∗ ( χ ( w )) = i ∗ K U ( y ) − χ ( i ∗ ( w )) = i ∗ K U ( y ) − χ ( x ) = 0 . So y − χ ( w ) ∈ D p ∗  K U ∗ ( B T n ) E b y (1.3) for E = K U . Finally , r ( y − χ ( w )) = r ( y ) = z and so z ∈ r D p ∗ ( K U ∗ ( B T n )) E as required.  6.4. F urther examples. A few simple exampl es illustrate the fact that the situation is considerably more difficult when M 2 n is not Sq 2 -acyclic. In all whic h follows, the nu m b er s i and m j are those defined b y (6 .1). Manifolds M 2 n for whic h all m j = 0, as is the case for the totally ev en Bott to w ers of Examples 6.2, hav e K O ∗ ( M 2 n ) a free K O ∗ -mo dule. P articularly rev ealing is the most basic case M 2 n = Q n k =1 C P 1 with n = 1. Recall from Section 4 that K O ∗ ∼ = Z [ e, α, β , β − 1 ] / (2 e, e 3 , eα, α 2 − 4 β ) with e ∈ K O − 1 , α ∈ K O − 4 and β ∈ K O − 8 . Example 6.7. The classes X ( s ) 1 ∈ K O − 2 s ( C P ∞ ) and X (0) i = X i , sp ecified in Definition 3.1, restrict to classes in K O − 2 s ( C P 1 ) whic h a lso will b e denoted b y X ( s ) 1 and X 1 . The K O ∗ - algebra K O ∗ ( C P 1 ) is isomorphic to K O ∗ [ g ] . ( g 2 ) where g ∈ g K O 2 ( C P 1 ) is the generator arising from the unit o f the sp ectrum K O . In particular e 2 g = X 1 ∈ K O (0) ( C P 1 ) and 2 β g = X (3) 1 ∈ K O − 6 ( C P 1 ) . No w C P 1 is the smo oth toric v ariet y asso ciated to the simplicial complex K = n { v 1 } , { v 2 } o in the manner described in Sec tion 1. The fibration (6.2) sp ecializes to C P 1 i − → S ( η ⊕ R ) p − → B T 1 , the to ta l space o f whic h is the sphere bundle of η ⊕ R . So DJ ( K ) is homotop y equiv alen t to C P ∞ ∨ C P ∞ . (Of course, this agrees with the description give n b y (5.3).) The map i includes C P 1 in to each w edge summand b y pinc hing the equator. It follow s from [12, Section 4] that (1) i ∗ is an epim orphism on to K O d ( C P 1 ) for all d ≡ / 1 , 2 mo d 8. (2) If d = 1 − 8 t then eβ t g has o rder 2 but K O d ( C P ∞ ∨ C P ∞ ) = 0. (3) If d = 2 − 8 t then 2 β t g ∈ Im ( i ∗ ) but β t g / ∈ Im( i ∗ ). The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 32 These details com bined with diagram (6.3) confirm that Im( i ∗ ) ∼ = K O ∗  DJ ( K )   r ( J K U ) in dimensions ≡ / 3 , 4 mod 8. Example 6 .7 extends to an analysis of v a rious toric manifolds with a single non-zero s i but unrestricted m j . Example 6.8. The pro jectiv e space C P 4 k + 1 has s 2 k + 1 = 1 and all other s i = 0. It is the smo oth toric v ariet y asso ciated to the simplicial complex K whic h is the b oundary of t he simplex ∆ 4 k + 1 . The K O ∗ -algebra K O ∗ ( C P 4 k + 1 ) admits K O ∗ ( S 8 k + 2 ) as an additiv e summ and, generated b y h ∈ K O 8 k + 2 ( C P 4 k + 1 ) such that h 2 = 0. In particular, e 2 β k h = X 2 k + 1 1 ∈ K O 0 ( C P 4 k + 1 ) and 2 β k +1 h = X 2 k 1 X (3) 1 ∈ K O − 6 ( C P 4 k + 1 ) . It fo llows from Example 6.7 that i ∗ : K O d (  DJ ( K )  − → K O d ( C P 4 k + 1 ) is an epimorphism for d ≡ / 1 , 2 mo d 8. So the cok ernel of i ∗ is isomorphi c to the Z / 2 v ector space generated b y the elemen ts eβ t h and β t h , whereas Im( i ∗ ) ∼ = K O ∗  DJ ( K )   r ( J K U ) in dimensions ≡ / 3 , 4 mod 8. Example 6.9. A terminally o dd Bott t ow er M 2 n has s 1 = 1 and all other s i = 0 . In this case the simplicial complex K is the b oundary of an n -dimensional cross-p olytope. As in Example 6.7, it follow s that the cok ernel of i ∗ is isomorphic to the Z 2 -v ector space generated b y the elem en ts eβ t g and β t g whereas Im( i ∗ ) ∼ = K O ∗  DJ ( K )   r ( J K U ) in dimensions ≡ / 3 , 4 mod 8. A cknow le dg m ent. The authors are grateful to Matthias F ranz for stim ulating and helpful con vers ations ab out momen t-a ngle complexes , D a vis-Jan uszkiew icz spaces and toric mani- folds. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 33 References [1] J. F. Adams, Stable Homotopy and Gener alize d Homolo gy , Univ ersity of Chica go Press, 1 974. [2] D. W. Anderson, The Re al K -the ory of Classi fying Sp ac es , PNAS 1964 51 : 63 4 –636. [3] M. F. A tiyah and G. B . Sega l Equivariant K -the ory and Completion , J. Differential Geometry 3 (1969 ): 1–18. [4] A. Bahr i, M. Benders ky , The K O -the ory of t ori c manifolds , T rans . AMS, 200 0 352 , no. 3 , 1191– 1202 [5] A. Bahri, M. Bendersky , F. R. 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V ogt, Colimits, Stanley- R eisner Alge br as, and L o op Sp ac es , In Algebra ic T op ology: Categorical Deco mposition T echniques, P rogress in Mathematics 215 :261 –291. Birkhä user, Basel, 20 03. The K O ∗ -rings of B T m , the Da vis-Januszkiew icz Spaces and certain toric manifolds 34 Dep ar tment of Ma thema tics, Cinvest a v, San P edr o Zaca tenco, Mexico, D.F. CP 07 360 Ap ar t ado P ost al 14-740, Mexico E-mail addr ess : a stey@math .cinvestav.mx Dep ar tment of Ma thema tics, Rider U niversity, La wrenceville, NJ 08 648, U. S.A. E-mail addr ess : b ahri@ride r.edu Dep ar tment of Ma thema tics CUN Y, East 695 P ark A venue N ew York, NY 10 065, U. S.A. E-mail addr ess : m benders@x ena.hunter.cuny.edu Dep ar tment of Ma thema tics, University of Rochester, R ochester, NY 14625, U.S .A. E-mail addr ess : c ohf@math. rochester.edu Dep ar tment of Ma thema tics, Lehigh University, Bethlehem, P A 18015, U.S .A. E-mail addr ess : d md1@lehig h.edu Dep ar tment of Ma thema tics, Cinvest a v, San P edr o Zaca tenco, Mexico, D.F. CP 07 360 Ap ar t ado P ost al 14-740, Mexico E-mail addr ess : s gitler@ma th.cinvestav.mx Dep ar tment of Ma thema tics, Nor thwestern U niversity, No r thwestern University 2033 Sheridan Ro ad Ev anston, IL 6020 8-2730, USA E-mail addr ess : m ark@math. northwestern.edu School of Ma thema tics, University of Manchester, O xf ord R oad Manchester M13 9PL, United Kingdom E-mail addr ess : n ige@maths .manchester.ac.uk School of Ma thema tics, University of Manchester, O xf ord R oad Manchester M13 9PL, United Kingdom E-mail addr ess : R eg.Wood@m anchester.ac.uk