THE COMPLEX COBORDISM OF BSOn KOUICHI INOUE AND NOBUAKI YAGITA 1.Introduction The complex cobordism of the classifying space of the n-th orthogonal group was computed by W.S.Wilson [Wi], which is the simplest possible result that we can expect ; MU*(BOn) ~=MU*[[c1, ..., cn]]=(c1 - c*1, ..., cn - c*n) where ck is the Conner-Floyd Chern class of complexification map O(n) ! U(n) and c*kis its complex conjugate. The next problem is the case BSOn. When n is odd, there is the isomorphism On ~=SOn x Z=2 and we get MU*(BSOodd) directly from the Wilson's result, MU*(BSO2m+1) ~=MU*(BO2m+1)=(F1) where F1 is the image of c1 under Bdet* : MU*(Z=2) ! MU*(BSO2m+1). Kono,Yagita and Inoue [K-Y], [In] computed MU*(BSO2n) for n 3 by us- ing the Atiyah-Hirzebruch spectral sequence. The results are also simple but the Atiyah-Hirzebruch spectral sequence is very complicated even n = 3 (see [In]). On the other hand, Molina and Vistoli [Mo-Vi] recomputed Chow rings CH*(BG) for classical groups G (e.g., GLn, On, SOn, ...) by using the stratification me* *thod, introduced by Vezzosi [Ve]. Applying this method to MU*(-) theory, we get the following theorems Theorem 1.1. There is an element ym 2 MU2m (BSO2m) with y2m= (-1)m 22m-2c2m mod(v1, ...) such that there is the MU* algebra isomorphism MU*(BSO2m) ~=MU*[[c2, c4, ..., c2m]]{ym } MU*(BO2m)=(F1) with c2i-1ym = 0 mod(v1, ...) for 1 i m. Theorem 1.2. The following Kuneth formula holds for all ni 1 and 1 i s MU*(BSOn1x ... x BSOns) ~=MU*(BSOn1) MU* ... MU* MU*(BSOns). Let *(X) be the algebraic cobordism defined by Levine and Morel [M-L1,2] and MGL2*,*(X) the (2*, *)-dimensional parts of MGL*,*(X) ([Mo-Vo],[Vo]) the motivic cobordism defined by Voevodsky. Theorem 1.3. For all n 1, there are isomorphism *(BSOn) ~=MGL2*,*(BSOn) ~=MU2*(BSOn). In this paper we use BP -theory assuming p = 2 instead of MU-theory because there is the isomorphism MU*(X)(p)~=MU*(p) BP* BP *(X). ____________ 2000 Mathematics Subject Classification. Primary 55P35, 57T25; Secondary 55R* *35, 57T05. Key words and phrases. MU-theory, BSOn, stratification methods. 1 2 K.INOUE AND N.YAGITA 2. stratification method We recall in this section the arguments by Molina and Vistoli [Mo-Vi]. (See also [Gu],[Vi].) For a smooth algebraic set X over a field k of ch(k) = 0, let A*(X) = CH*(X) be the Chow ring generated by algebraic cocycles modulo ra- tional equivalence. Let G be an algebraic group over k. Suppose G acts on X. Let A*G(X) be the equivariant Chow ring ( the Borel cohomology) defined by Edidin and Graham [E-G]( and by Totaro [To]) as follows. For each i 0, choose a representation V of G with an open algebraic set U on which G acts freely, and codimV (V - U) > i. Then quotient (X x U)=G exists as a smooth algebraic space, and we can define AiG(X) = Ai((U x X)=G) and this definition is independent on the choice of such V and U. Of course we identify A*G= A*G(pt.) = A*(BG). For a subgroup H of G, by the definition, we see A*G((X x G)=H) ~=A*H(X). One of the most important properties for A*G(-)-theory is the localization ex* *act sequence ; if Y is closed G-equivariant algebraic subset of X of codimension s,* * and denote by i : Y X and j : X -Y X the inclusions, then the following sequence is exact * A*-sG(Y ) i*!A*G(X) j!A*G(X - Y ) ! 0. R.Field [Fi] computed the Chow ring of BSO2m Theorem 2.1. (Field) The Chow ring A*SO2m= CH*(BSO2m) is isomorphic to Z[c2, ..., c2m, ym ]=(y2m- (-1)m 22m-2cn, 2codd, ym codd). By using the Vezzosi's stratification method [Ve], Molina and Vistoli [Mo-Vi] give very clear explanation of A*Gfor classical groups G. In particular, outlin* *e of their arguments for G = SO2m is following. Let G = SOn, n = 2m. Recall that SOn is defined as the subgroup of SLn generated by elements which invariant the quadratic form q(x1e1 + ... + xnen) = x21+ ... + x2m- x2m+1- ... - x2n for the basis e1, ..., en of V = An. Hence the sets B = {x 2 An|q(x) 6= 0}, C = {x 2 An - {0}|q(x) = 0} and An - {0} are all SOn-invariant sets. Thus we have the localization exact sequence (1) A*-nG({0}) i1*-!A*G(An) -! A*G(An - {0}) ! 0 (2) A*-1G(C) i2*-!A*G(An - {0}) -! A*G(B) ! 0. Then the stabilizer of e1 2 B for the SOn-action is isomorphic to SOn-1. Its or* *bit is SOn . e1 = {x 2 An|q(x) = 1} B. It is proven (the detailed proof is given * *for On in [Mo-Vi]) that B ~=(Gm xZ=2On)=On-1 ~=(Gm x SO2m)=(Z=2 x SO2m-1) MU*(BSOn) 3 where Gm = A* = A - {0} is the multiplicative group. Hence we have the isomor- phism A*SOn(B) ~=A*SOn((Gm x SO2m)=(Z=2 x SO2m-1)) ~=A*Z=2xSO2m-1(Gm ). By using facts that Gm ~=A - {0} and A*Z=2~=Z[y]=(2y), and the localization sequence again, we see A*SOn(B) ~=A*Z=2xSOn-1(Gm ) ~=A*SOn-1. Next consider A*SOn(C). The stabilizer of the pair (e1, em+1 ) is isomorphic * *to SOn-2 and the action is transitive. The stabilizer of the one point e1 contains elements in SOn which are represented by transformations em+1 7! a2e2 + ... + anen. ai2 A. Thus it is proven that (see x4 in [Mo-Vi]) C ~=SOn=(An-1 o SOn-2) where o means the semidirect product. Since A*An-1oG~=A*G, we have the isomor- phisms A*SOn(C) ~=A*An-1oSOn-2~=A*SOn-2. Moreover we know ym = -i2*(ym-1 ) by Lemma 5.5 in [Mo-Vi] and i1*(1) = cn. By induction, we see that A*Gis multiplicatively generated by c2, ..., cn, ym .* * Then the Field's theorem is proved by considering restriction to A*TGfor the maximal torus TG of G. These arguments work for A*(X) = *(X) the algebraic cobordism defined by Levine and Morel ([M-L1,2]) or A*(X) = MGL2*,*(X) the (2*, *)-dimensional parts of MGL*,*(X) ([Mo-Vo],[Vo]) the motivic cobordism defined by Voevodsky. It is still known [Mo-Le 2] that *(X) * Z ~=CH*(X) and we may not have new information directly from the above arguments. However if we can show the main theorem Theorem 1.1 , then Theorem 1.3 is immediate. Next consider the case A*(-) = BP *(-) the Brown-Peterson cohomology. In general BP odd(X) 6= 0 and there does not exist the local exact sequence (, in general, j* is not epic). Moreover BPZodd=2xSOn-1(Gm ) 6= 0. However we will pr* *ove the main theorem in the introduction by using assumption BPSoddOn0= 0 for n0< n in the next sections. 3. BP -theories of BOn and BSOodd In this section, we apply the stratification methods to BP *-theory for G = On and G = SOodd by using the result of Wilson [Wi]. Of course we consider the case k = A = C the complex number field for BP *(BG). Note that there is the Totaro's cycle map "cl[To] such that the composition "cl 2* 2* A*(X)(p)! BP (X) BP* Z(p)! H (X)(p) is the usual cycle map. We will see "clare isomorphic for cases X = BOn, BSOn. At first we consider the case G = On. For a compact Lie group G, we mainly consider its complexification GC but not G itself, since BG and BGC is homotopi* *c. 4 K.INOUE AND N.YAGITA Hence hereafter the group G means its complexification GC but not the original (real) Lie group. For example, On is identified as the subgroup of GLn(C) generated matrixes A with AtA = In here At is the transposed matrix, namely A is matrixes which preserve the quadratic form q(x1e1 + ... + xnen) = x21+ ... + x2n for the basis e1, ..., en of Cn as described in x2. The topological counter parts of the local exact sequence given in x2 is the following long exact sequence. Let Y be a closed G-equivariant complex manifold of G-complex manifold X of codimension s. It is well known that each complex bundle is MU*(-) orientable ( page 400 in [Sw]) and so it is BP *(-) orientable. Hence we have the Thom isomorphism BP *-2s(Y ) ~=BP *(T hY (X)) ~=BP *(X=(X - Y )) where T hY (X) is the Thom space for the normal bundle induced from Y X. By the definition BPG*(X) = BP *((U x X)=G)), its G-equivariant version follows from non-equivariant version. Thus we have the long exact sequence * ! BPG*-2s(Y ) i*!BPG*(X) j!BPG*(X - Y ) ! BPG*-2s+1(Y ) ! . By Wilson's result, we know BPOoddn= 0. The BP -version of the exact sequence (1) in the proceeding section is given by (1)0 0 ! BPO2*-1n(Cn - {0}) ! BPO2*-2nn({0}) cn!BP 2* n 2* n On(C ) ! BPOn(C - {0}) ! 0. Next we will study the BP -version of the exact sequence (2) in the preceding section. As C = {x 2 Cn - {0}|q(x) = 0}, we have the similar results (*) BPO*n(C) ~=BPC*n-1oOn-2~=BPO*n-2. As for B = {x 2 Cn - {0}|q(x) 6= 0}, we have the isomorphism BPO*n(B) ~=BPZ*=2xOn-1(C*) similarly from the the isomorphism for A*SOn(B). This isomorphism induces the long exact sequence ! BPO*-1n(B) ! BPZ*-2=2xOn-1({0}) i*!BPZ*=2xOn-1(C) j!BPO*n(B) ! . Here we recall that BP *(BZ=2) ~=BP *[[y]]=([2](y)) with |y| = 2 where y = c1 ; the first Chern class of the induced bundle from the natural inc* *lusion Z=2 C* = GL1(C), and [2](y) = 2y +v1y2+... is the sum of the formal group law for BP *-theory. Since this BP *-module satisfies the condition of the Landweber exact functor theorem [Ko-Ya], we know that BPZ*=2xOn-1~=BPO*n-1[[y]]=([2](y)). We also see that i*(x) = y . x in the above exact sequence. Hence we have the isomorphisms ( BPO* [[y]]=([2]y, y) ~=BPO* , for * = even (**) BPO*n(B) ~= n-1*-1 *-1 n-1 BPOn-1{[2](y)=y} ~=BPOn-1 for * = odd. MU*(BSOn) 5 The BP -version of the exact sequence (2) is written as ! BPO2*-1n(Cn - {0}) ! BPO2*-1n(B) * ! BPO2*-2n(C) i2!BPO2*n(Cn - {0}) ! BPO2*n(B) ! . From the isomorphisms (*), (**), we have (2)0 0 ! BPO2*-1n(Cn - {0}) ! BPO2*-2n-1 * ! BPO2*-2n-2i2!BPO2*n(Cn - {0}) ! BPO2*n-1! 0. Lemma 3.1. BPO2*n(Cn - {0}) ~=BPO2*n-1, and Ker(cn)|BPO2*n~=BPO2*+2n-1n(Cn - {0}) ~=Ker(BPO2*+2n-2n-1! BPO2*+2n-2n-2). Proof.From (1)0and (2)0we see the existence of epimorphisms BPO*n=(cn) ! BPO*n(Cn - {0}) ! BPO*n-1. By Wilson we still know that BPO*n=(cn) ~= BPO*n-1. Hence we have the first isomorphism. From the first isomorphism, the map i*2= 0 in (2)0. This implies the last_ isomorphism. The second isomorphism follows from (1)0. |__| Since BPO*n-1is generated by BPO*n-2and cn-1, it is immediate Ker(BPO*n-1! BPO*n-2) ~=Ideal(cn-1) BPO*n-1. Hence we have the following corollary. Corollary 3.2.The kernel Ker(cn)|BPO*nis isomorphic to BP *[[c1, ..., cn-1]]{cn-1}=((c1 - c*1, ..., cn-1 - c*n-1) \ Ideal(cn-1)). Next consider the BP -theory for BSO2m+1. Recall that B ~=(C* xZ=2O2m+1)=O2m ~=(C* x SO2m+1)=O2m. Hence we see BPS*O2m+1(B) ~=BPO*2m(C*). We consider the exact sequence ! BPO*2m({0}) xF1!BPO*2m(C1) ! BPO*2m(C*) ! . Here we note that F1 = Bdet*(y) under Bdet* : BPZ*=2! BPO*2mbecause O2m acts on C* by det : Om ! Z=2 in the above isomorphism. So we have the isomorphism ( * BPSO2m+1(B) ~= BPO2m=(F1)*+for1* = even BPO2m=(F1){[2](F1)=F1} for * = odd. Thus we get the exact sequence for n = 2m + 1 0 ! BPS2*-1On(Cn - {0}) ! BPO2*-2n-1=(F1) * ! BPO2*-2n-2=(F1) i2!BPS2*On(Cn - {0}) ! BPO2*n-1=(F1) ! 0. 6 K.INOUE AND N.YAGITA Lemma 3.3. BPS2*On(Cn - {0}) ~=BPO2*n-1=(F1), and Ker(cn)|BPS2*On~=Ker(BPO2*+2n-2n-1! BPO2*+2n-2n-2)=(F1). Here we notice that there exists the quotient map q : B ~=C* xZ=2O2m+1=O2m ~=(C* x SO2m+1)=O2m ! SO2m+1=SO2m by definding q(t, s) = s for t 2 C*, s 2 SO2m+1 because SO2m+1 \ O2m = SO2m. This quotient map induces maps of BPS*On(-) theories BPS*On(pt) ! BPS*On(SOn=SOn-1) = BPS*On-1 q*!BP * * SOn(B) = BPOn-1=(F1). So we get ; Lemma 3.4. The map q* : BPS*O2m! BPO*2m=(F1) is a split epimorphism. Proof.Of course each ci is in BPS*O2m, and from the above map q*, we have the composition map BPO*2m=(F1) ! BPS*O2m! BPO*2m=(F1) which is the identity. |___| 4. BP -theories of BSO2m Now we study BPS*Onfor n = 2m. By induction on m, we assume BPS*On-2~=BPO*n-2=(F1) BP *[[c2, ..., c2m-2]{ym-2 }. For ease of notations, let us write BP *[[ceven]]{yk} = BP *[[c2, c4, ..., c2k]* *]{yk}. By this assumption BPSoddOn-2= 0 and the arguments similar to the case (2)0, and we have the BPSOn-version of the exact sequence (2)00 0 ! BPS2*-1On(Cn - {0}) ! BPS2*-2On-1 * ! BPS2*-2On-2i2!BPS2*On(Cn - {0}) ! BPS2*On-1! 0. We also write the long exact sequence (1)00 ! BPS*-1On(Cn - {0}) ! BPS*-2nOn({0}) !cnBP * n * n SOn(C ) ! BPSOn(C - {0}) ! . Here we note Lemma 4.1. BP *[[ceven]]{ym } BPS*On. Proof.From (1)00, we know BPS*On=(cn) BPS*On(Cn - {0}). Let us define i2*(ym-1 ) = ym 2 BPS*On(Cn - {0}). We still know that ym 2 CH*(BSOn) from the argument in x2 (Field's theorem). By the Totaro's cycle map, we can take ym 2 BPS*Onbut only decided with mod(cn, v1, ...). Moreover considering the restriction to the BP *-free algebra BP *(BTSOn) ~=BP * H*(BTSOn), for the maximal torus TSOn, we see BP *[[ceven]]{ym } BPS*On. |_* *__| MU*(BSOn) 7 Lemma 4.2. ( Ker(BPO*-1 ! BPO*-1)=(F1) if * = odd BPS*On(Cn - {0}) ~= n-1 n-2 BPO*n-1=(F1) BP *[[ceven]]{ym }=(cn) otherwise. Proof.Consider the exact sequence (2)00. For the element 1 2 BPS*On-2, the image i2*(1) = 0 since so in BPO*n-2. Recall that Ker(BPS*On-1! BPS*On-2) ~=Ideal(cn-1) BPS*On-1. From (2)00and BPS*On-1~=BPO*n-1=(F1), we have the isomorphism for * = odd. When * = even, the right hand formula in this lemma is contained in the lefth* *and side formula by Lemma 4.1 and (1)00. Since i2*(ym-1 ) = ym and i2*(1) = 0 in_(2* *)00, we see the isomorphism for * = even. |__| From the above lemma, we show that the map BPS2*On(Cn) ! BPS2*On(Cn - {0}) in (1)00is an epimorphism. Lemma 4.3. In (1)00, the map BPS2*-1On(Cn - {0}) ! BPS2*-2nOn({0}) is injectiv* *e. Proof.By the naturality for SOn On, of course, there is the map r : Ker(cn|BPO*n)=(F1) ! Ker(cn|BPS*On). From Lemma 3.4, its composition map Ker(cn|BPO*n)=(F1) ! Ker(cn|BPS*On) ! BPO*n=F1 is injective. So the map r itself is injective. On the other hand, from Lemma 3.1 and 4.2, we see the isomorphisms Ker(cn|BPO*n)=(F1) ~=Ker(BPO*n-1! BPO*n-2)=(F1) ~=BPSoddOn(Cn - {0}). Hence BPS*-1On(Cn - {0}) ! Ker(cn|BPS*On) is injective. |___| From the above lemma and (1)00, we have 0 ! BPSoddOncn!BPSoddOn! 0 and BPSoddOn= 0. The proof of Theorem 1.1.By (1)00and BPSoddOn= 0, we see that BPS*Onis multi- plicatively generated by c1, ..., cn and ym . Hence there is the epimorphism r : BP *[[ceven]]{ym } BPO*n=(F1) ! BPS*On. From Lemma 3.4 and Lemma 4.1, we see this map is also isomorphism. |___| The arguments work also P (n)* and K(n)* theories for P (k)* = Z=p[vk, ...] and K(k)* = Z=p[vk, v-1k]. In particular, for all k P (k)odd(BSOn) = 0 and K(k)odd(BSOn) = 0. Thus Theorem 1.2 is immediate from the main theorem [K-Y], [R-W-Y]. Theorem 1.3 follows from the fact that *(X) (and MGL2*,*(X)) is generated by elements in CH*(X) as a MU*-module [M-L2]. 8 K.INOUE AND N.YAGITA 5. BP *-orientability Recall that an n-dimensional vector bundle p : E ! X is BP *-orientable if there is an element (Thom class) th 2 BP n(T hX (E)) such that for each inclusi* *on i : pt ! X the restriction image i*(th) 2 BP *(T hpt.(p-1(pt))) ~=BP *(Sn) is a BP *-module generator. If p : E ! X is a BP *-orientable, we have the Thom isomorphism BP *(X) ~=BP *+n(T hX (E)) by the standard argumets using Mayer-Vietoris sequence. It is well known that each complex bundle is BP *-orientable as stated in x3,* * of course there are SOn-bundles which are not BP *-orientable. Note that BOn ~=U xOn Dn, BOn-1 ~=U xOn On=On-1 ~=U xOn Sn-1 where U is a On-free contractible space, Dn is the n-dimensional disk. Hence we can identify T hBOn (U xOn Dn) ~=BOn=BOn-1. Similar fact also happens for SOn. Let us write by MOn = BOn=BOn-1 (resp. MSOn = BSOn=BSOn-1) the Thom space of BOn (resp. BSOn) for the universal bundle. The cofibering BOn-1 ! BOn ! MOn induces the exact sequence 0 BPO*n-1 BPO*n B"P*(MOn) 0. Hence we know ([Wi]) that B"P*(MOn) ~=Ker(BPO*n! BPO*n-1) ~=Ideal(cn) BPO*n. Theorem 5.1. There are isomorphisms 0 ! B"P*+2n-2(MOn-1) i!BPO*nTh!B"P*+2n(MOn) ! 0. 0 ! B"P*+2n-2(MOn-1)=(F1) i!BPS*OnTh!B"P*+2n(MSOn) ! 0. Proof.Consider the short exact sequence 0 ! Ker(cn) ! BPO*ncn!Ideal(cn) ! 0. By Lemma 3.1, we still know Ker(cn|B*On) ~=B"P*(MOn-1). Moreover we know Ideal(cn) ~=B"P*(MOn ). |___| The Ker(T h) is generated by only one element i(cn-1) 2 BPS*On, which is still computed in [Wi] also X i(cn-1) = vis2i-1= v2c3 + .... mod(2, v1, ...)2 P 2i-1 P where s2i-1 = xi identifying cj = xi1...xij i-th elementary symmetric polynomial. This element gives an obstruction for BP *-orientability. Proposition 5.2.Let p : E ! X be an SOn bundle and f ! 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On the cohomology and the chow ring of the classifying space * *of PGLp. Math.AG/0505052 (2005), May. [Vo] V. Voevodsky. The Milnor conjecture. Preprint (1996). [Wi] W. S. Wilson. The complex cobordism of BOn. J.London Math.Soc. 29 (1984)* *, 352-366. Department of Mathematics, Musashi Institute of Technology, Tamazutsumi, Seta- gaya, Tokyo, Japan, Department of Mathematics, Faculty of Education, Ibaraki Un* *i- versity, Mito, Ibaraki, Japan E-mail address: inoue@ma.ns.musashi-tech.ac.jp, yagita@mx.ibaraki.ac.jp