TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS BJ"ORN SCHUSTER AND NOBUAKI YAGITA Abstract.The BP*-module structures of BP*(BG) for extraspecial 2-groups a* *re stud- ied using transfer and Chern classes. These give rise to p-torsion elemen* *ts in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained* * by Totaro. 1.Introduction Let G be a compact Lie group, e.g. a finite group, and h*(BG) a good generali* *zed co- homology theory on the classifying space BG of G. Here "good" shall mean that h* **(BU(m)) is isomorphic to h* H*(BU(m)) for the unitary groups U(m). Then we can define C* *hern classes in h*(BG) for complex representations of G, and also transfer maps. We * *are inter- ___ ested in the Mackey closure Chh(G) of the ring of Chern classes in h*(BG), name* *ly the subring of h*(BG) generated by transfers of Chern classes. For ordinary mod p cohomology, Green-Leary [G-L] showed that the inclusion map ___ i : ChHZ=p ,! H*(BG; Z=p) is an F-isomorphism, i.e., the induced map of varieti* *es is a p _ homeomorphism. Green-Minh [G-M ] however noticed that i= 0need not be an isomo* *rph- ism in general. Nect consider h = BP or h = K(n), the n-the Morava K-theory, at* * a fixed prime p. Following Hopkins-Kuhn-Ravenel [H-K-R], we shall call a group G "good* *" for h-theory if h*(BG) is generated (as an h*-module) by transferred Euler classes * *of represent- ations of subgroups of G. It is clear that if the Sylow p-subgroup of G is good* *, then so is G ___ and one has an isomorphism h*(BG) ~=Chh(G). Furthermore, it follows from [R-W-Y* * ] that G is good for BP if it is good for K(n) for all n. Examples for groups that are* * K(n)-good for all n are the finite symmetric groups. Another typical case are p-groups of* * p-rank at most 2 and p 5: in [Y ] it is shown that the Thom map ae : BP*(-) ! H*(-)(p)in* *duces an isomorphism BP*(BG) BP* Z(p)~=Heven(BG). Note however that I. Kriz claimed that K(n)odd(BG) 6= 0 for some p-groups G. ___________ 1991 Mathematics Subject Classification. Primary 55P35, 57T25; Secondary 55R3* *5, 57T05. Key words and phrases. BP-theory, transfer, Chern classes, Chow ring. 1 2 BJ"ORN SCHUSTER AND NOBUAKI YAGITA On the other hand, B. Totaro [T1] found a way to compare BP-theory to the Cho* *w ring. For a complex algebraic variety X, the groups CHi(X) of codimension i algebraic* * cycles P modulo rational equivalence assemble to the Chow ring CH*(X) = iCHi(X). Totaro constructed a map "ae: CH*(X) ! BP*(X) BP* Z(p)such that the composition ae: CHi(X)(p)-"ae!BP*(X) BP* Z(p)-ae!H*(X)(p) coincides with the cycle map. One of the main results of [T1] is that there exi* *sts a group G for which the kernel of aecontains p-torsion elements. To prove this, Totaro * *defined the Chow ring of a classifying space BG as Limm!1 CH*((Cm -S)=G) where G acts on Cm* * -S freely and codim(S) > 0. He then constructed a non-zero element x in Ker(ae) su* *ch that ___ * (1.1) x 2 ChBP(BG) \ (BP (BG) BP* Z(p)) : Since transfers and Chern classes also exist in the Chow ring CH*(BG), there is* * an element ___ 1+4 x 2 ChCH (G) that also lies in Ker(ae). The group Totaro uses is G = Z=2 x D+ * *, where D1+4+= D(2) is the extraspecial 2-group of order 32, which is isomorphic to the* * central product of two copies of the dihedral group D8 of order 8. He first proves that* * there exists an element x 2 BP*(BD(2)) satisfying (1.1)but which restricts to zero under the* * map aeZ=2: BP*(-) ! H*(-; Z=2), where he uses the computation of BP*(BSO(4)) from [* *K-Y ]. Let D(n) = 21+2n+denote the extraspecial 2-group of order 22n+1; it is isomor* *phic to the central product of n copies of D8. In this paper, we construct non-zero ele* *ments x 2 BP*(BD(n)) satisfying (1.1)but with aeZ=2(x) = 0 directly for each n. Let "W be a maximal elementary abelian 2-subgroup and N the center of D(n). F* *or a one-dimensional real representation e of "Wrestricting non-trivially to the cen* *ter, set = IndD(n)"W(e). This is the unique irreducible representation which acts non-triv* *ially on N. Then the i-th Stiefel-Whitney class wi() for i < 2n can be written as a polynomial i* *n variables w1(ej); 1 j 2n, for 1-dimensional representations ej of D(n)=N ([Q ], Remark * *5.13), i.e. wi() = wi(w1(e1); :::; w1(e2n)). Let e0Cdenote the complex representation induc* *ed from the real representation e0. Then we can prove that (1.2) x = c2n-1(C) - w2n-1(c1(e1C); : :;:c1(e2nC)) satisfies (1.1)together with aeZ=2(x) = 0, and furthermore conclude Ker(ae) 6= * *0 for G = Z=2 x D(n). TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 3 Secondly, we construct a non-nilpotent element (1.3) x 2 Ker(ae) \ (BP*(BG) BP* Z(p)) ___ which is not in ChBP (BG). However we do not know whether x comes from the Chow* * ring or not, and we only obtain the result for n = 3; 4. Set (1.4) x = [v1 w2n()] : to be the class represented by v1 w2n() in the E1 -page of the Atiyah-Hirzebru* *ch spectral sequence. We can prove that d2n+1(w2n()) = vn-1 Qn-1(w2n()) 6= 0 and v1 Qn-1(w2n()) 2 Im(d3). Furthermore, restricting to the center of D(n) we see* * that ___ x 62 ChBP (BG). However, it seems difficult to see that this cycle is permanent* *. For the case n = 3; 4, we use the BP-theory of BSpin(7) and BSpin(9) computed in [K-Y ]. These arguments do not seem to work for other extraspecial 2-groups nor 2-gro* *ups that have a cyclic maximal normal subgroup [S]. In Section 2, we recall the mod 2 cohomology of extraspecial 2-groups followi* *ng [Q ]. In particular, w2n-2i() is represented by the Dickson invariant Di, and we study t* *he action of the Milnor primitives Qj on Di. To see ae(x) 6= 0 in H*(BD(n); Z), we recall the integral cohomology in Secti* *on 3. In Section 4, we show that x satifies (1.1). In Section 5, we study how elements i* *n Ker(ae) are represented in the Atiyah-Hirzebruch spectral sequence, explaining the easiest * *case h* = ___ Z=2[vn-2; vn-1]. The element x in (1.4)is proved not to be in ChBP (BD(n)) in S* *ection 6. In the last section the element x in (1.4)is proved to be a permanent cycle* * in the Atiyah-Hirzebruch spectral sequence for n = 3; 4 by comparing the spectral sequ* *ence to the corresponding spectral sequence for H*(BSpin(2n + 1)). 2.Extraspecial 2-groups The extraspecial 2-group D(n) = 21+2n+is the central product of n copies of the* * dihedral group D8 of order 8. So there is a central extension (2.1) 0 ! N -! D(n) -ss!V ! 0 4 BJ"ORN SCHUSTER AND NOBUAKI YAGITA with N ~=Z=2 and V elementary abelian of rank 2n. Take a set of generators c; "* *a1; : :;:"a2n of D(n) such that c is a generator of N, the elements ai= ss("ai) form a Z=2-ba* *sis of V , and 8 < c ifj = 2i - 1 ["aj; "a2i] = : 0 else Using the Hochschild-Serre spectral sequence associated to extension (2.1), Qui* *llen [Q ] de- termined the mod 2 cohomology of D(n). Let ei denote the real 1-dimensional re* *pres- entation of D(n) given as the projection onto followed by the nontrivial c* *haracter ! {1} R, and e : "V odd! N ! {1} R where "V odd= is a maximal elementary abelian 2-subgroup of D(n). Define classes xi2 H1(D(n);* * Z=2), w2n 2 H2n(D(n); Z=2) as the Euler classes of the ei and of = IndD(n)"V(odde), * *respectively. The extension (2.1)is represented by the class f = x1x2+ . .+.x2n-1x2n, and one* * has (2.2) H*(BD(n); Z2) ~=Z=2[w2n] Z2[x1; : :;:x2n]=(f; Q0f; : :;:Qn-2f) where the Qiare Milnor's operations recursively defined by Q0 = Sq1 and Qi= [Sq* *2i; Qi-1]. The extension class f defines a quadratic form q : V ! Z=2 on V . A subspace W * * V is said to be q-isotropic if q(x) = 0 for all x 2 W. The maximal (elementary) abelian s* *ubgroups of D(n) are in one-to-one correspondence with the maximal isotropic subspaces of V* * . Indeed, if W is maximal isotropic, then "W:= ss-1(W) ~=N W is maximal (elementary) abe* *lian. Quillen also proved that the mod 2 cohomology of D(n) is detected on maximal el* *ementary abelian subgroups, i.e. the restrictions define an injective map Y (2.3) H*(BD(n); Z=2) ,! H*(W"; Z=2) where the product ranges over conjugacy classes of maximal elementary abelian s* *ubgroups. Since the restriction of to any such "Wis the real regular representation (see* * [Q ], Section 5), we have Y (2.4) ResW"(w2n) = (z + x) x2H1(W;Z=2) where z denotes the generator of H*(N; Z=2) dual to c. For simplicity, write w0* *= Res"Ww2n, and choose generators of H*(W; Z=2) ~=Z=2[x01; : :;:x0n]. It is well-known that* * the right hand side of (2.4)can be written in terms of Dickson invariants, n 2n-1 (2.5) w0= z2 + D1z + . .+.Dnz TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 5 where Dihas degree 2n- 2n-iand H*(W; Z=2)GL2(Z=2)~=Z=2[D1; : :;:Dn]. Using that* * the product of all the x0i's is clearly invariant and that the Milnor primitives ar* *e derivations, it is easy to see that the Dickson invariants may be written in terms of the Qias * *follows: Dn = Q0Q1: :Q:n-2(x01. .x.0n) (2.6) Di = (Q0: :^:Qn-i-1:Q:n:-1(x01. .x.0n))=Dn Lemma 2.1. The Milnor operations act by (1)Qn-1Di= DnDi; (2)Qn-j-1Dj= Dn; (3)QiDj= 0 for i < n - 1 and i 6= n - j - 1. Proof.First note that from (2.6)and Q2k= 0 we immediately get Qk(Dn) = 0 for k * *6= n - 1 and Qn-1Dn = Q0: :Q:n-1(x01. .x.0n) = D2n. Thus, for each 1 i n - 1, 0 = Qn-1(Q0: :^:Qn-i-1:Q:n:-1)(x01. .x.0n) = Qn-1(DiDn) = (Qn-1Di)Dn + DiQn-1Dn = (Qn-1Di)Dn + DiD2n whence (1). Similarly, (2) is implied by D2n = Qn-1: :Q:0(x01. .x.0n) = Qn-i-1(DiDn) = (Qn-i-1Di)Dn + DiQn-i-1Dn = (Qn-i-1Di)Dn : Finally, for k 6= n - i - 1 we get 0 = Qk(DiDn) = (QkDi)Dn + DiQkDn = (QkDi)Dn.* * |___| Corollary 2.2.Qn-1w0= Dnw0and Qkw0= 0 for k < n - 1. P n-1 n-i j+1 j+1 Proof.For j 6= n-1, we have Qjw0= i=1(QjDi)z2 +Qj(Dnz) = Dnz2 +Dnz2 = 0 : For j = n - 1, we get Qn-1w0= 0 + DnD1z2n-1+ . .+.DnDn-1z2 + Qn-1(Dnz) : The last term equals D2nz + Dnz2n; the claim follows. * * |___| Corollary 2.3.Qkw2n= 0 for 0 k n - 2, but Qn-1w2n6= 0. |___| 3.The integral cohomology The integral cohomology of D(n) is studied by Harada-Kono ([H-K ], also see [* *B-C]) by means of the Bockstein spectral sequence (3.1) E1 = H*(BG; Z=2) =) Z=2 H*(BG)=(2-torsion) : 6 BJ"ORN SCHUSTER AND NOBUAKI YAGITA The E2-page of this spectral sequence is the Q0-homology of H*(BG; Z=2), and E1* * ~=Z=2 for a finite group G. For 0 i n - 2, let R(i) = H*(BV ; Z=2)=(f; Qof; : :;:Qi* *f). Using the long exact sequence associated to the short exact sequence (3.2) 0 ! R(i - 1) Qif-!R(i - 1) -! R(i) ! 0 ; Harada-Kono computed the E2-page for D(n) as follows: (3.3) H(H*(BD(n); Z=2); Q0) ~=(a; b1; : :;:bn-1) Z=2[w2n] where |a| = 3 and |bi| = 2i. Since E1 ~=Z=2, the first non-trivial differential* * must be da = b1, and there have to be subsequent differentials d(abi) = bi+1. Thus there app* *ear exactly n non-zero differentials in this spectral sequence. On the other hand, using co* *restriction arguments it is easy to see that the exponent of H*(BD(n)) is at most n + 1. Ba* *sed on these facts, Harada-Kono proved the following. Theorem 3.1. [H-K ] Let C(n)* = H*(BD(n))=H*(BV ). Then C(n)* H*(BD(n)), and there is an additive isomorphism 8 < Z=22(k) if2(k) n - 1 ; C(n)k = : Z=2n+1 if2(k) = n where 2(k) denotes the 2-adic valuation of k. * *|___| Let ck(n) denote a Z(2)-module generator of C(n)2k. Then cn(n) reduces to w2n* *modulo H*(BV ; Z=2). Consider the restriction map i : C(n)* ! C(n-1)*. Now cn-1(n-1) =* * w2n-1 mod H*(BV ; Z=2) implies i*cn(n) = cn-1(n - 1)2. Since the order of cn-1(n) is * *2n-1 and the order of cn-1(n - 1) is 2n, we know that i*cn-1(n) = 2scn-1(n - 1) for some* * s > 0. A corestriction argument now implies s = 1, since the index of D(n - 1) in D(n) i* *s 2. The elements a and bj are natural in the sense that i*(a) = a and i*(bj) = bj* * for 1 j n - 2, abusing notation. Thus i*cj(n) = cj(n - 1) for j < n - 1, and we o* *btain Corollary 3.2.If n 2, there is an additive isomorphism C(n)* ~=Z{1; 2wffl222.w.f.fln-12n-1| ffli= 0 or1}[w2n]=(2i+1w2i= 0 | 2 * *i n) where the w2iare the reductions of the elements w2iin H2i(BD(i)). * * |___| Remark.When n = 1, the element w2 2 H*(BD8; Z=2) does not lift to the integral * *cohomo- logy and C(1)* ~=Z[w22]=(4w22). TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 7 4.Brown-Peterson Cohomology of BD(n) Let BP*(-; Z=2) denote BP-theory mod 2 with coefficients BP*=(2) = Z=2[v1; v2* *; : :]:. We consider the Atiyah-Hirzebruch spectral sequence (4.1) E*;*2= H*(BD(n); Z=2) BP* =) BP*(BD(n); Z=2) : Lemma 4.1. The elements x2iand w22nare permanent cycles in the spectral sequenc* *e (4.1). Proof.These elements are the top Chern classes of the representations eiCand C,* * respect- ively. |___| It is well-known that some of the differentials of (4.1)are given by (4.2) d2i+1-1(x) = vi Qix mod (v1; : :;:vi-1) : Since Qn-1w2n6= 0 by Corollary 2.3, we know that w2ncannot be a permanent cycle* *, which implies w2n 62 Im(aeZ=2: BP*(BD(n)) ! H*(BD(n); Z=2)). Thus the integral lift w* *2n of w2ndoes not lie in the image of ae : BP*(BD(n)) ! H*(BD(n)), either. Let as above "Wdenote a maximal elementary abelian subgroup of D(n), and w() * *the total Stiefel-Whitney class of . Then Y n n-1 ResD(n)"W(w())= (1 + x + z) = (1 + z)2 + D1(1 + z)2 + . .+.Dn(1 + z) = 1 + D1+ . .+.Dn + ResD(n)"W(w2n) ; in particular, (4.3) ResD(n)"W(w2n-i()) = Di: Hence, by (2.2), we can choose polynomials "Di2 Z=2[x1; : :;:x2n] ~=H*(BV ; Z=2* *) with w2n-i= "Di. Theorem 4.2. There is a BP*-module generator x = c2n-1(C) - "D1(c1(e1C); : :;:c1(e2nC)) 2 BP*(BD(n)) such that (1)ae(x) = 2w2n mod H*(BV ), (2)aeZ=2(x) = 0 in H*(BD(n); Z=2). 8 BJ"ORN SCHUSTER AND NOBUAKI YAGITA Proof.Since x is defined via Chern classes, it is an element of BP*(BD(n)). Ass* *ertion (2) is immediate from (4.3). Since w2n 62 Im(ae), it suffices to prove (1) to show* * that x is a BP*-module generator. Let F = D(n); this is cyclic of order 4. By the d* *ouble coset formula, M g-1"V oddg ResD(n)FIndD(n)"V(oddeC)=IndFF\g-1"VRoddgesF\g-1"V(oddgg*eC) FgV"odd 2n-1M = IndFN(eC) since the elements g = affl22.a.f.fl2n2n, ffli= 0 or 1, form a complete set of * *double coset represent- atives. Notice that IndFN(eC) decomposes as eF -eF where eF is a faithful 1-di* *mensional complex representation of Z=4. Thus the total Chern class of C restricts to F as n-1 2 2n-1 * ResF(c(C)) = ((1 + u)(1 - u))2 = (1 - u ) withH (BF) ~=Z[u]=(4u) : Consequently, we have ResF(c2n-1(C)) = 2u2n-1in H*(F). Since ResF(c1(eiC)) = 2iu for some i2 Z=4, we immediately obtain ResF(D"i) = 0 and therefore (1). * * |___| Now recall the following lemma of Totaro. Lemma 4.3. ([T1]) Let p be a prime and X any space. If aeZ=p: BP*(X) BP* Z(p)! H*(X; Z=p) is not injective, then ae : BP*+2(X x BZ=p) BP* Z(p)! H*+2(X x BZ=p)* * is also not injective. |___| Let ae0: CH*(-) ! H*(-) denote the cycle map respectively ae0Z=2the cycle map* * followed by reduction modulo 2. Since Chow rings have Chern classes, we easily deduce Corollary 4.4.There is a non-zero element x0in CH2n(BD(n)) satisfying (1)ae0(x0) = 2w2n mod H*(BV ); (2)ae0Z=2(x0) = 0. Hence ae0: CH2n+2(B(D(n) x Z=2)) ! H2n+2(B(D(n) x Z=2)) is not injective. * * |___| Remark.First note that the above argument does not hold for n = 1. Indeed, in t* *hat case H*(BD8) Im(ae) moduloH*(BV ). Similar facts hold for 2-groups G which have a c* *yclic maximal normal subgroup [S], i.e. dihedral, semidihedral, quasidihedral, and g* *eneralized quaternion groups of order a power of 2. Moreover BP*(BG) is generated by Chern* * classes for these groups. The extraspecial 2-groups of order 22n+1are of two types. Qui* *llen calls them the real and the quaternionic type, where the real type corresonds to the * *groups D(n) TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 9 considered above, and the quaternionic group of order 2n+1is the central produc* *t of D(n-1) with the quaternion group Q8 of order 8. Consider now this second case, and den* *ote this group by D0(n); it also has center Z=2 with quotient V ~=(Z=2)2n. In Quillen's * *notation [Q ], this corresponds to h = n + 1 and r = 2. The quadratic form (extension c* *lass) is P n f = x21+ x1x2+ x22+ i=2x2i-1x2i, and the cohomology is given by H*(BD0(n); Z=2) ~=Z=2[w2n+1] Z=2[x1: :;:x2n]=(f; Q0f; : :;:Qn-1f) : Here the xiare as before the generators of H*(BV ; Z=2) inflated to D0(n), and * *w2n+1is the Euler class of the 2n+1-dimensional irreducible representation . The cohomology* * of D0(n) is also detected on subgroups W" ~=Q8 x W in one-to-one correspondence with max* *imal isotropic subspaces, i.e. there is an injection Y H*(BD0(n); Z=2) ,! H*(B(Q8x W); Z=2) W where W ranges over the maximal isotropic subspaces of V (which have dimension * *n - 1). The Stiefel-Whitney classes wj() are zero except for the following values of j * *([Q ], (5.6)): 8 < (D0)4 forj = 2h - 2h-i; 1 i n - 1 ResQ8xW(wj()) = : P in-22i 0 4 n+1 i=0e (Dn-i-1) forj = 2 where e 2 H4(Q8; Z=2) is the Euler class of the obvious 4-dimensional irreducib* *le represent- ation of Q8 and D0iis the degree (2n-1- 2n-1-i) Dickson invariant for rank n - * *1. Thus almost all arguments for D(n) work in this case, too, except for Qm wj() = 0. F* *or ex- ample, we can define x = c2n(eC)-(D"01)4 in BP*(BD0(n)); this class satisfies a* *e(x) = 2w2n+1 and aeZ=2(x) = 0. However, it seems that we can not prove that x is a BP*-modul* *e gen- 0(n) n erator of BP*(BD0(n)) because ResN(c2n(IndDZ=4W(eF))) = u2 and w2n+1() 2 Im(ae) mod (H*(BV )). 5.Permanent cycles This section is concerned with the following statement: Assumption 5.1.In the Atiyah-Hirzebruch spectral sequence converging to BP*(BD(* *n)), every element in (2; v1; : :;:vn-1) w2nis a permanent cycle. 10 BJ"ORN SCHUSTER AND NOBUAKI YAGITA Unfortunately, we can not prove this in full generality, only a rather weak v* *ersion (covering the cases n = 3; 4), which nevertheless seems to justify why we expect such cla* *sses to be permanent. Let BP*(-) denote the cohomology theory with coefficients BP* = Z=p[vn-2; vn-1]. Then there are natural transformations BP*(-) -! BP*(-) -! H*(-; Z=2) : Proposition 5.2.In the Atiyah-Hirzebruch spectral sequence converging to BP*(BD(n)), every element in (vn-2) w2n is a permanent cycle. Proof.First note that w2n is not in the image of Qn-2, which is easily seen by * *restricting to N: ResN(w2n) = z2n= Qn-2z2n-1+1, whereas the image of ResD(n)Nis generated b* *y z2n. This means that no element in (vn-2) w2n lies in the image of the first potent* *ially non- zero differential d2n-1+1. The next non-zero differential is d2n-3= v2n-2 -. Di* *mensional reasons easily imply that (vn-2) w2n62 Im(dr) for any r. Suppose first that drw2n= 0 for all r < 2n - 1. Then d2n-1w2n= vn-1 Qn-1w2n6=* * 0. Thus by naturality of the spectral sequence and Corollary 2.2, we see that Res"W(d2n-1w2n) = vn-1 Qn-1w0= vn-1 Dnw0: Since there are classes "Di2 H*(BD(n); Z=2) which restrict to Di on each W", Qu* *illen's detection result (2.3)shows (5.1) d2n-1w2n= vn-1 "Dnw2n: Furthermore, "Dnw2nis vn-2-torsion, since d2n-1"D1= vn-2D"n, whence vn-2w2npers* *ists to E*;*2n+1. So assume now that drw2n 6= 0 for some r < 2n - 1. For dimensional* * reasons again, the only possibility for such a differential is (5.2) d2n-3w2n= v2n-2 (aw2n+ b) witha; b 2 H*(BV ; Z=2) : Note that the subgroup of the automorphisms of G stabilizing the center N is th* *e orthogonal group O(V ) of V associated to the quadratic form q ([B-C], p. 216). Since is * *the unique irreducible representation which acts non-trivially on N, the element w2nis inv* *ariant under the orthogonal group ([Q ], Remark 4.7). Let as before V odd= Z=2{a2i-1| 1 i * *n}, and V ev= Z=2{a2i| 1 i n}. Both are maximal isotropic subspaces, and V = V odd V * *ev. TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 11 Then (5.3) O(V ) = {g tg-1| g 2 GLn(Z=2)} (where t is transposition). Any maximal q-isotropic subspace W has dimension n. Interchanging a2i-1with a* *2iif necessary, we can turn the projection pr : W V = V odd V ev-! V odd into an isomorphism (see [Q ], p. 201). Consider the commutative diagram V odd---- W ----! D(n) ? ? ? (5.4) g?y g?y g?y: V odd---- gW ----! D(n) Let a be as in (5.2)and suppose ResW(a) 6= 0. Since pr* : H*(BV odd; Z=2) ~=H*(* *BW; Z=2), we have (pr*)-1(a) 6= 0 in H*(BV odd; Z=2). The image of a in H*(BD(n); Z=2) is* * invariant under the action of O(V ). Hence (pr*)-1(a) is invariant under the GLn-action, * *by (5.3). But |a| = 2n - 3, and there is no invariant of that degree, whence a = 0. Using* * similar reasoning, we can prove (pr*)-1(b) = Dn-1Dn. But in the spectral sequence conve* *rging to BP*(BD(n)), there is the differential d2n-1-1(D"n-1"D1) = vn-2 "D* *n-1"Dn. Hence we can remove vn-2 b from (5.2). Finally, we shall prove dr(vn-1 w2n) = 0 for all r 2n. Suppose X dr(vn-1 w2n) = vkn-2vln-1akl for someakl2 H*(BD(n); Z=2) : Then each aklis invariant under the action of O(V ), and moreover the degrees o* *f the akl are odd. Arguing as above, we see that X (5.5) Res"V odd(akl) = DI withDI 2 (Dn) where DI = Di1. .D.im, 1 ij n + 1, identifying Dn+1 with w2n. From Qn-1Di= DiDn we have Qn-1DI = mDIDn. Hence if Qn-1DI 6= 0, the number of factors must be odd. Thus we assume the number of Di's in (5.5)is even. But for each such DI there * *is a corresponding D"I2 H*(BD(n); Z=2), and all those D"Iare (vn-2; vn-1)-torsion. * *Thus dr(vn-1 w2n) = 0 for dimensional reasons. |__* *_| 12 BJ"ORN SCHUSTER AND NOBUAKI YAGITA 6. Transfers of Chern classes To study Chern classes, we consider the restriction to the center N ~=Z=2 of * *D(n). Let I denote the ideal (2; v1; v2; : :):in BP*. Then aeZ=2: BP*BN =I ~=Z=2[z2] H*(BN; Z=2) : Since the image of the restriction H*(BD(n); Z=2) ! H*(BN; Z=2) is generated by* * w2n62 Im(aeZ=2), we see that n (6.1) Im[BP*(BD(n)) ! BP*(BN)=I] = Z=2[u2 ] ; where u denotes the obvious generator in degree 2. Let be a complex representa* *tion of D(n); it restricts to N as the sum of m copies (say) of the nontrivial characte* *r eC plus some trivial representations. Then there is an element u0 u mod I in BP*(BN) with (6.2) ResN(c()) = (1 + u0)m where c() denotes as usual the total Chern class of . Then um 2 Im[BP*(BD(n)) ! BP*(BN)=I], whence m has to be divisible by 2n. Proposition 6.1.Suppose Assumption 5.1 holds and n 3. Then the permanent cycles [v1w2n]; : :;:[vn-1w2n] are not represented by Chern classes. Proof.Suppose [viw2n] is the Chern class of some representation , which must sa* *tisfy (6.2) for some m = 2nm0. Then n-1 2 viu2 = ResN(c2n-1-2(2i-1)()) mod I : But the restriction of the total Chern class of is given by n-1 2 2n ResN(c()) = 1 + 2m0(u0)2 mod (I ; u ) n-1+1 2n-1+2i-1 2 2n = 1 + m0(v1u2 + . .+.viu + . .). mod (I ; u ) which does not contain the term viu2n-1, a contradiction. * * |___| Theorem 6.2. Suppose Assumption 5.1 holds and n 3. Then [v1w2n]; : :;:[vn-2w2n* *] are not represented by transfers of Chern classes. TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 13 Proof.Let H be a subgroup of D(n), and suppose [vjw2n] = TrD(n)H(x) for some x 2 BP*(BH). By the double coset formula, X g-1Hg (6.3) ResD(n)NTrD(n)H(x) = TrNg-1Hg\NResg-1Hg\N(g*x) HgN where the sum ranges over double coset representatives g of H\G=N. If H inters* *ects N trivially, then so does any conjugate of H. Hence we need only consider subgro* *ups H containing the center, and the double coset formula evaluates to |D(n)=H| . Res* *N(x). Since this element is represented by n-1 2 ResN [viw2n] = viu2 6 0 mod I ; we get |D(n)=H| = 2 and thus H ~=D(n - 1) x Z=2. The total Chern class c(i) of any representation i of D(n - 1) restricts as n-1m 2n-1 2n ResN(c(i)) = (1 + u0)2 = 1 + mu mod (I; u ) : Hence we have n-1 ResN(2c(i)) = 2 + 2mu2 i 2n-1+1 2n-1+2i-1 2 2n = (v1u2+ . .+.viu2 + . .).+ m(v1u + . .+.viu + . .). mod (I ; u )* * ; which does not contain viu2n-1. This is a contradiction. * * |___| 7.BP*(BSpin(7)) The mod 2 cohomology of BSpin(n) was computed by Quillen [Q ]: (7.1) H*(BSpin(n); Z=2) ~=Z=2[w2h()] Z=2[w2; : :;:wn]=(w2; Q0w2; : :;:Qh-1w2) where is a spin representation of Spin(n) and 2h the Radon-Hurwitz number (see* * [Q ] x6). This is proved by calculating the Serre spectral sequence of the fibration (7.2) BZ=2 -! BSpin(n) -! BSO(n) : We consider the case n = 7. Then h = 3 and the mod 2 cohomology of BSpin(n) is* * a polynomial algebra on the Stiefel-Whitney classes w4; w6; w7; w8 of a spin repr* *esentation, i.e. (7.3) H*(BSpin(7); Z=2) ~=Z=2[w4; w6; w7; w8] : 14 BJ"ORN SCHUSTER AND NOBUAKI YAGITA Recall that Spin(7) has the exceptional Lie group G2as a subgroup. G2contains a* * rank three elementary abelian 2-subgroup, and its mod 2 cohomology is isomorphic to the ra* *nk three Dickson invariants, i.e. H*(BG2; Z=2) ~=Z=2[D1; D2; D3]. Here we may identify t* *he Dickson invariants with the Stiefel-Whitney classes of the restriction of the spin repr* *esentation to G2, namely D1 = w4, D2 = w6, and D3 = w7. In particular, we have H*(BSpin(7); Z* *=2) ~= Z=2[D1; D2; D3] Z=2[w8]. The Brown-Peterson cohomology of BSpin(7) is given in [K-Y ]. Consider the A* *tiyah- Hirzebruch spectral sequence (7.4) E*;*2= H*(BSpin(7); Z) BP* =) BP*(BSpin(7)) : Since Q0w6 = w7 and since there is no higher 2-torsion, the E2-term is isomorph* *ic to BP*[w4; w26; w7; w8]=(2w7). It is shown in [K-Y ] that all non-zero differenti* *als are of the form d2m-1 = vm-1 Qm-1. Indeed, d3w4 = v1w7; d7w7 = v2w27; d7w8 = v2w7w8; d15(w7w8) = v3w27w28: Thus * * 2 2 E*;*16~=BP {1; 2w4; 2w8; 2w4w8; v1w8} BP =(2; v1; v2)[w7]{w7} 2 2 2 BP*=(2; v1; v2; v3)[w27]{w27w28} Z(2)[w4; w6; w8] : This page is generated by even degree elements, hence E16= E1 . Note that the r* *eason for the permanency of v1w8 is that d7w8 = v2w7w8 but d3(w4w8) = v1w7w8. Remark.The terms B=(2; v1){w8} in the formulas (6.10) and (6.11) of [K-Y ] are * *typing errors and should be replaced with B(2; v1){w8}. Lemma 7.1. The element [v1w8] is not represented by a Chern class, i.e. there * *is no representation ae with [v1w8] = c2(ae). Proof.This follows from Proposition 6.1 by looking at the commutative diagram 0 ----! Z=2 ----! Spin(7)----! SO(7) ----! 1 x x x =?? ?? ?? 0 ----! Z=2 ----! D(3) ----! (Z=2)6----! 0 whose rows are central extensions. |_* *__| The same diagram gives the following theorem as a consequence of Theorem 6.2. TRANSFERS OF CHERN CLASSES IN BP-COHOMOLOGY AND CHOW RINGS 15 Theorem 7.2. The element [v1 w8] is not represented by the transfer of a Chern * *class in BP*(BD(3)). |___| Similar arguments work for Spin(9) and D(4); in this case the Radon-Hurwitz n* *umber is 16. Theorem 7.3. In BP*(BD(4)), the elements [v1 w16] and [v2 w16] are not transfer* *s of Chern classes in BP*(BD(n)), n < 4. References [B-C] D. J. Benson and J. F. 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Lecture Notes in Math. 1509 (1992* *), Springer, 186-209. [K-Y] A. Kono and N. Yagita. Brown-Peterson and ordinary cohomology theories of* * classifying spaces for compact Lie groups. Trans. Amer. Math. Soc. 339 (1993), 781-798. [K] I. Kriz. Morava K-theory of classifying spaces: some calculations. Topolo* *gy 36 (1997), 1247-1273. [Q] D. G. Quillen. The mod 2 cohomology rings of extra-special 2-groups and t* *he spinor groups. Math. Ann. 194 (1971), 197-212. [R-W-Y]D. C. Ravenel, W. S. Wilson and N. Yagita. Brown-Peterson cohomology fro* *m Morava K-theory. To appear. [S] B. Schuster. On the Morava K-theory of some finite 2-groups. Math. Proc. * *Cambridge Philos. Soc. 121 (1997), 7-13. [T1] B. Totaro. Torsion algebraic cycles and complex cobordism. J. Amer. Math.* * Soc. 10 (1997), 467- 493. [T2] B. Totaro. The Chow ring of classifying spaces. To appear. [Y] N. Yagita. Cohomology for groups of rankp(G) = 2 and Brown-Peterson cohom* *ology. J. Math. Soc. Japan 45 (1993), 627-644. 16 BJ"ORN SCHUSTER AND NOBUAKI YAGITA FB 7 Mathematik, Bergische Universit"at-Gesamthochschule Wuppertal, Wuppertal* *, Germany E-mail address: schuster@math.uni-wuppertal.de Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ib* *araki, Japan E-mail address: yagita@mito.ipc.ibaraki.ac.jp