2 ANDREW BAKER b) As an E algebra, E (MSp) = E [Q E k : k > ]; and moreover the natural morphism of ring spectra j : MSp?! MU induces an embedding of

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1 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp Andrew Baker Introduction. In this paper, we derive formul in Brown-Peterson homology at the prime 2 related to the family of elements ' n 2 MSp 8n?3 of N. Ray, whose central r^ole in the structure of MSp has been highlighted by recent work of V. Vershinin and other Russian topologists. In eect, we give explicit \chromatic" representatives for these elements, which were known to be detected in KO and mod 2 KU-homology, and are thus \v -periodic" in the parlance of [4] and [5]. In future work we will investigate further the v periodic part of MSp and discuss the relationship of our work with that of B. Botvinnik. I would like to thank Nigel Ray for many helpful discussions and large amounts of advice on MSp (including severe warnings!) over many years; in particular, x5 in this paper was prompted by his suggestions about the detection of ' n in the classical Adams spectral sequence. I would also like to thank Boris Botvinnik, Vassily Gorbunov and Vladimir Vershinin for discussions on the material of earlier versions of this paper both during and after the J. F. Adams Memorial Symposium and in particular for bringing to my attention Buhstaber's article [2] which contains related results. x Some algebraic results on E (MSp). Let E be a commutative ring spectrum, and let x E 2 E 2 (C P ) be a complex orientation in the sense of []. Then the results of the following Theorem are well known. Theorem (.). a) The natural map j : C P?! H P induces a split monomorphism and a split epimorphism E (H P ) j??! E (C P ); of modules over E. E (C P ) j??! E (H P ); Key words and phrases. Symplectic bordism, Brown-Peterson theory. The author would like to acknowledge the support of the Science and Engineering Research Council whilst this research was conducted. Published in `Adams Memorial Symposium on Algebraic Topology, Vol. 2' editors N. Ray & G. Walker, CUP (992), 263{80. [Version 5: /09/999] Typeset by AMS-TEX

2 2 ANDREW BAKER b) As an E algebra, E (MSp) = E [Q E k : k > ]; and moreover the natural morphism of ring spectra j : MSp?! MU induces an embedding of E algebras j : E (MSp) j?! E (MU): We will need explicit sets of generators for the E homology and cohomology of H P and MSp. Recall the canonical complex line bundle?! C P and its E-theory st Chern class c E () = x E ; we have E (C P ) = E [[x E ]]. Then the map j : C P?! H P classies the quaternionic line bundle C H?! C P, which has as its underlying complex bundle +, where ( ) denotes complex conjugation. We dene the E-theory st symplectic Pontrjagin class of a symplectic bundle?! X to be the negative of the E-theory 2nd Chern class of the complex bundle 0 underlying, } E () =?c 2 ( 0 ) 2 E 4 (X): In particular we set } E = } E () 2 E 4 (H P ), where?! H P is the canonical symplectic line bundle; this gives j i } E = } E ( + ) =?c E ()c E ( ) 2 E 4 (C P ): We also have (as graded algebras over E ) (.2) E (H P ) = E [[} E ]]: Now let the elements n E 2 E 2n (C P ), n > 0, form the standard E basis for E (C P ) as detailed in []; this basis is dual to that of the monomials (x E ) n in E (C P ) under the Kronecker pairing h ; i: (x E ) r ; s E = r;s ; where is the Kronecker delta function. Also for any T, let T = [?] E (T ) be the? series for the formal group law F E (X; Y ) 2 E [[X; Y ]] associated to the orientation x E as described in []; notice that x E = c E ( ). Now in E (H P ) we can dene a sequence of elements n 2 E 4n (H P ), n > 0, by requiring that these are dual to the (} E ) n : (.3) (} E ) r ; s = r;s : It is easily veried that a generating function for these elements is the series (.4) X n>0 n (?T T ) n = X n>0 j E n T n : Let i : H P ' MSp()?! 4 MSp be the standard map, then we have Theorem (.5). a) The elements n for n > 0 form an E basis for E (H P ). b) The elements i n+ for n > form a set of polynomial generators for the E algebra E (MSp).

3 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp 3 From now on we set Q E n = i n+ 2 E 4n (MSp). We also need some information on the image of the ring homomorphism j : E (MSp)?! E (MU). We have the two generating functions Q E (T ) = X n>0 Q E n T n+ 2 E (MSp)[[T ]]; j Q E (T ) = X n>0 j Q E n T n+ 2 E (MU)[[T ]]: Recall from [] the standard generators B E n 2 E 2n (MU), (for n > 0 and B 0 = ) for which E (MU) = E [B E n : n > ]; and let B E (T ) = P n>0 BE n T n+. The proof of the next result is left to the reader. Proposition (.6). The series j Q E (?T T ) 2 E (MU) satises the equation Q E (?T T ) =?B E (T )B E (T ): x2 Symplectic Pontrjagin classes in E ^ MSp theory. Recall that the ring spectrum MSp is universal for orientations for quaternionic bundles. The universal orientation is induced from the class We also have } MSp : H P ' MSp()?! 4 MSp 2 MSp 4 (H P ): (2.) MSp (H P ) = MSp [[} MSp ]]: Now let E be a complex oriented ring spectrum as in x. Then the class } E will serve as a universal orientation for quaternionic line bundles in E theory. We can consider the representable cohomology theory (E^MSp) ( ) on either of the categories of CW spectra or spaces. As E (MSp) is free over E, we have a Boardman isomorphism (2.2) (E ^ MSp) ( ) = E (MSp)b E E ( ) where b denotes the completed tensor product with respect to the skeletal topology for innite complexes. From x and standard arguments about the map i : H P?! 4 MSp, we have Proposition (2.3). In (E ^ MSp) (H P ) = E (MSp)b E E (H P ) we have the identity } MSp = Q E (} E ): For later convenience we also introduce the series N E (T ) = X n>0 N E n T n+ 2 E (MSp)[[T ]] determined by the equation (2.4) N E (Q E (T )) = T: Notice that N E n?q E n modulo decomposables, and hence we can take the N E n to be polynomial generators for E (MSp).

4 4 ANDREW BAKER x3 Detecting Ray's element ' n with a complex oriented cohomology theory. In this section we again let E be a complex oriented commutative ring spectrum. We will also require the following assumption to hold: TF: E is torsion free. This condition is satised for the following spectra: E = MU, BP, KU, H Z, which include all those which we will be explicitly considering in this paper. Now consider the space RP. Let?! RP be the canonical real line bundle and = R C?! RP be its complexication. Let w E = c E () 2 E 2 (RP ). Then we have (3.) E (RP ) = E [[w E ]]=([2] E (w E )); where [2] E T denotes the 2 series of the formal group law associated to x E. Now we will consider the space RP H P and apply the cohomology theory (E ^ MSp) ( ) to it. Since both E (MSp) and E (H P ) are free modules over E, we have the following isomorphisms (E ^ MSp) (RP H P ) =E (MSp)b E E (RP )b E E (H P ); (E ^ MSp) (RP C P ) =E (MSp)b E E (RP )b E E (C P ): Let?! H P be the canonical quaternionic line bundle. Then the quaternionic bundle R = C?! RP H P is dened and so has a st symplectic Pontrjagin class in each of the theories represented by the spectra MSp, E and E ^ MSp. In the ring (E ^ MSp) (RP H P ), we have expressions of the form (3.2) } E ( R ) = } E + w 2 + X n> b E n (} E ) n+ ; and (3.3) } MSp ( R ) = } MSp + w 2 + X n> b MSp n (} MSp ) n+ ; where be n 2 E 4n (RP ) and bmsp n 2 MSp 4n (RP ). Upon applying the split monomorphism j these yield the following equations in (E ^ MSp) (RP C P ): (3:2 0 ) (3:3 0 ) } E (( C ) C H ) = j } MSp (( C ) C H ) = j 2 4 X 3 } E + w 2 + b n E (} E ) n+ 5 ; n> 2 4 } MSp + w 2 + X n> b MSp n (} MSp ) n+ 3 5 ;

5 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp 5 Now recall from (2.3) that } MSp = Q E (} E ). Thus we obtain } MSp (( C ) C H ) = Q E (j } E + w 2 + X n> b E n (j } E ) n+ ): Recall from [6] the following construction. Consider the inclusion of the bottom cell S = RP,?! RP. Then we can restrict ;?! RP to ;?! RP and obtain the classes and its image under j, } MSp ( R ) 2 MSp 4 (RP H P ); } MSp ( C ( + )) 2 MSp 4 (RP C P ): By denition, Ray's elements n 2 MSp 4n?3 are given by } MSp ( R )? } MSp = X n> n (} MSp ) n in MSp (RP H P ), and we also have } MSp (( C ) C H )? } MSp = X n> n (} MSp ) n in (E ^ MSp) (RP C P ). Stably, the smash product RP ^ H P is a retract of RP H P, and this allows us to identify the above expressions with elements of MSp (RP ^ H P ) and (E ^ MSp) (RP ^ C P ). We will also use the notation ' n = 2n of [6]. We can factor the inclusion RP,?! RP as RP,?! RP 2,?! RP, and since RP 2 is a Z=2 Moore space, we see that 2 n = 0. Under our assumption TF together with (3.), we thus have that n 7?! 0 under the E theory Hurewicz homomorphism MSp?! E (MSp). Equivalently, the class } MSp (( C ) C H )?} MSp maps to 0 under the natural homomorphism MSp (RP ^ H P )?! (E ^ MSp) (RP ^ H P ). Let } MSp ( 2 C ) = } MSp + X n> e n (} MSp ) n 2 (E ^ MSp) (RP 2 H P ) where e n 2 MSp (RP 2 ^ H P ) (E ^ MSp) (RP 2 ^ H P ) is the image of b MSp under the map induced by the inclusion RP 2,?! RP. Now for any spectrum F with F torsion free there is an isomorphism F k (RP 2 ) = F 2?k =2F 2?k induced from Spanier-Whitehead duality. In particular, this applies to the cases F = E ^ MSp; MSp that we are dealing with, and thus we can interpret n e as an element of MSp 4n?2 =(2) E 4n?2 (MSp)=(2). Let w 2 2 E 2 (RP 2 ) be the restriction of the generator w E 2 E 2 (RP ); note that w 2 = c E ( 2 ). We can now give our main calculational result.

6 6 ANDREW BAKER Theorem (3.4). Let u = x E and u = x E, satisfying and?uu = N E (j } MSp )?uu ; u + u 2 im [E (H P )?! (E ^ MSp) (H P )] : Then under assumption TF together with (3.), we have the following identity in (E ^ MSp) (RP 2 ^ C P ): } MSp ((( 2 C ) C H )) + uu = w 2 N E 0 (j }MSp ) u log E 0 (u ) + u log E 0 (u) and moreover this element is equal to j } MSp ( 2 R )? j } MSp and lies in the image of the natural map MSp (RP 2 ^ H P )?! (E ^ MSp) (RP 2 ^ H P ). Proof. We need to recall some relevant facts. Firstly, we have w 2 2 = 0 = 2w 2 : Hence only the rst order terms in w 2 are required. Secondly, we have the following formula for formal derivatives: Q E 0 (N E (T )) = N E 0 (T ) : Finally, by denition, } E (( 2 C ) C H ) =?c E ( 2 C + 2 C ): But expanding this using the Cartan formula, we obtain } E (( 2 C ) C H ) =?c E ( 2 C )c E ( 2 C ) =?F E (w 2 ; u)f E (w 2 ; u ) =?F E (w 2 ; u)f E (w 2 ; u ); using the fact that 2 2 is a trivial line bundle. In this, we are setting u = c E () and u = c E ( ). We also require the following well known Lemma. Let log E (X) 2 E Q[[X]] denote the logarithm series of the formal group law F E. Lemma (3.5). The formula holds in E [[X]], where F2 E (X; Y ) = coecients in E. log E 0 (X) = F E 2 (X; 0) Proposition (3.4) now follows from F E (X; Y ). Hence the logarithm of F E, log E (X), has ;

7 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp 7 As a sample application of this, we consider the case of E = KU, i.e., complex KU-theory. This allows us to determine the KO-theory Hurewicz images of Ray's elements n 2 MSp 8n?3. Recall that we have KU = Z[t; t? ], where t 2 KU 2 is the Bott generator; this theory clearly satises the condition TF. Corresponding to the standard complex orientation in KU ( ), the formal group law F KU (X; Y ) = X + Y + txy ; this has as its logarithm the series. X log KU (X) = t? (?t) k? X k ln( + tx) = : k k> Thus we have log KU 0 (X) = ( + tx) : Now the formula of (3.4) becomes } MSp ((( 2 C ) C H )) = } MSp + w 2 u N KU 0 (} MSp ) log KU 0 (u ) + u log KU 0 (X) : We also have [?] KU (X) =?X ( + tx) : Thus we obtain } MSp ((( 2 C ) C H )) = } MSp + = } MSp + w 2 N KU (} MSp [u( + tu ) + u ( + tu)] ) w 2 N KU 0 (} MSp )?tu 2 ( + tu) = } MSp + tw 2} KU N KU 0 (} MSp ) = } MSp + tw 2N KU (} MSp ) N KU 0 (} MSp ) : The coecient of w 2 (} MSp ) n in this series is e n. Notice that we can write (3.6) N(} MSp ) Pr> N KU 0 (} MSp ) }MSp + N 2r?(} MSp ) 2r Ps>0 N 2s(} MSp ) 2s (mod 2): From this result, we can immediately deduce that e 2r? = 0 if r >. This suces to prove the following result which was conjectured in [6] and also follows from an unpublished result of F. Roush which actually shows that 2r? = 0 when r >. Let ko: MSp?! KO (MSp) be the KO-theory Hurewicz homomorphism.

8 8 ANDREW BAKER Proposition (3.7). For r >, we have in KO 8r?7 (MSp), ko( 2r? ) = 0: Proof. Let M(2) = S 0 [ 2 e denote the mod 2 Moore spectrum and let : S?! S 0 denote the non-trivial element of S together with the map S ^ KO?! KO it induces; let M() = S 0 [ e 2 be the mapping cone. Recall from [] \the Theorem of Reg Wood": there is a cobre sequence S ^ KO?! KO?! KO ^ M() ' KU: Now consider the exact diagram of abelian groups in which the rows are induced from the cobre sequence for and the columns from the cobre sequence for multiplication by 2: KO 8r?7 (MSp)????! KO 8r?7 (MSp ^ M(2))????! KO 8r?8 (MSp)? yepi? y? yepi KO 8r?6 (MSp)????! KO 8r?6 (MSp ^ M(2))????! KO 8r?7 (MSp)? y KU 8r?6 (MSp ^ M(2)) Now 2r? e = 0 in the group KU 8r?6 (MSp^M(2)) appearing in this diagram, and therefore we can lift ko( 2r? ) in the group KO 8r?7 (MSp) to an element of the torsion group KO 8r?7 (MSp^M(2)). But this means that ko( 2r? ) is the image of an element of KO 8r?8 (MSp), a torsion free group. The only way that both of these can be true is if ko( 2r? ) = 0. We can also see that for each n >, e' n = e 2n tn 2n? (mod decomposables): This of course shows that in the ring KU (MSp)=(2), the elements e' n are algebraically independent over KU =(2). x4 Some calculations in BP (MSp). We now consider the case of E = BP, the Brown-Peterson spectrum at the prime 2; we will assume the reader to be familiar with [] and [5] which contains detailed information on BP. We begin with some algebraic Lemmas on the formal group law for BP. Recall that the logarithm of F BP is the series (4.) log BP (X) = X n>0 `nx 2n 2 (BP Q)[[X]]; where `n 2 BP 2(2n?)Q. The Hazewinkel X generators are then dened recursively by the formula (4.2) v n = 2`n? `kvn?k 2k for n >. 6k6n? We also have the following relation for the 2-series [2] BP (X): (4.3) [2] BP (X) XBP n> v n X 2n (mod 2); where as usual the symbol P BP indicates formal group summation. We will need the following well known facts about the formal derivative of the logarithm. The next result may be known to others but we know of no reference.

9 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp 9 Lemma (4.4). We have the following congruence in BP [[X]]: log BP 0 (X) X n>0 v 2n? X 2n? (mod 2): Proof. Let log BP 0 (X) = X n>0 L n X 2n? where L n = 2 n`n is an element of BP 2 n+?2. We will show by induction upon n that L n v 2n? (mod 2): For n = 0 and, the result is clear. Suppose X that it holds for some n. Then we have 2 n v n+ = L n+? 2 n?k L k vn+?k 2k 6k6n v 2n+? (mod 2) by the inductive hypothesis. Hence, the desired result holds for n + and the Lemma follows. Our next result allows us to estimate the series [?] BP (X) modulo 2. Again it is quite possibly known to others. Lemma (4.5). The following congruences hold in BP [[X]]: [?] BP (X) XBP n>0 X + BP [2 n ] BP (X) (mod 2) XBP n> r ;::: ;r n > v r v 2r r 2 v 2r ++r n? r 2 X 2r ++r n where in each case the right hand side is an X-adically convergent series. (mod 2); Proof. The sequence [2 n?] BP (X) is Cauchy with respect to the X-adic topology on BP =(2)[[X]] since the formal group dierences of successive terms have the form [2 n ] BP (X) and by (4.3) the leading term of this is that in X 2n. Hence, we see that the limit of this sequence is giving the desired result. XBP n>0 [2 n ] BP (X); We are now in a position to calculate } MSp ((( 2 C ) C H )) as an element of (BP ^ MSp) (RP 2 H P ). Recall that we have from x3, } MSp ((( 2 C ) C H )) = } MSp w 2 u + N BP 0 (} MSp ) log BP 0 (u ) + u log BP 0 : (X) The ring of power series in u and u which is symmetric with respect to the automorphism interchanging these two elements is easily seen to be the power series ring on?uu, hence we can express this last quantity as an element of (BP ^ MSp) (RP 2 )[[} BP ]] by making use of the fact that } BP =?uu. Of course, this will be hard to do explicitly in general, but we can extract sucient information from our formula to enable us to make some interesting deductions. Amongst these we have the following, which we leave the reader to verify. Recall that in BP, the ideal I r = (v 0 ; v ; : : : ; v r ) is invariant and prime (here as usual we set v 0 = 2).

10 0 ANDREW BAKER Theorem (4.6). For r > we have in (BP ^ MSp) (RP 2 H P ) the congruence: } MSp ((( 2 C ) C H )) } MSp + This allows us to deduce that w 2 hv N BP 0 (} MSp r (} MSp ) 2r? + v r+ (} MSp ) 2ri ) (mod I r + (} MSp ) 2r + ): e' = v 2 + v N ; e' 2 = v 3 + v 2 N 2 mod I 2 + decomposables; and in general e' 2 r+ = v r+ + v r N 2 r? mod I r? + decomposables; where the term \decomposables" refers to decomposables in the BP algebra BP [N k : k > ]. x5 Detecting Ray's elements ' n in mod 2 homology. In this section we will give a formula for the element in the group Ext 4m?2 A (F 2 ; H F 2 (MSp)) (a part of the E 2 term of the classical Adams Spectral Sequence for (MSp)) which detects N. Ray's element m 2 MSp 4m?3 ; here, A denotes the dual Steenrod algebra at the prime 2, and H F 2 is mod 2 ordinary homology. For the case of m = 2n, we have ' n = 2n. Our approach is to use the formul we obtained in x4 to determine the image of e m under the composition Ext 0 BP (BP) (BP ; BP (MSp)=(2))?! Ext BP (BP) (BP ; BP (MSp))?! Ext A (F 2 ; H F 2 (MSp)) ; in which denotes the coboundary (of bidegree (; 0)) induced from the exact sequence of comodules 0?! BP (MSp)??! 2 BP (MSp)?! BP (MSp)=(2)?! 0; and denotes the reduction map which has bidgree (0; 0) and is induced by the morphism of comodules (BP (BP); BP (MSp))?! (H F 2 (H F 2 ) = A ; H F 2 (MSp)): From x4, we have as the generating function of the e m the series e(} MSp ) = X m> e m (} MSp ) m = u N BP 0 (} MSp ) log BP 0 (u ) + u log BP 0 (u)

11 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp with the notational conventions of our earlier sections. The coecients of this series in } MSp lie in Ext 0 BP (BP) (BP ; BP (MSp)=(2)), and our rst task is to calculate the series whose coecients are the elements (e m ) 2 BP (BP) BP BP (MSp)=(2) (mod I 2 ); where : BP (MSp)?! BP (BP) BP BP (MSp) is the (left) coaction. To do this, we view the above series as lying in the ring (BP ^ MSp) (H P )=(2) and the coaction as being a ring homomorphism (BP ^ MSp) (H P )?! (BP ^ BP ^ MSp) (H P ) = BP (BP) BP (BP ^ MSp) (H P ): We view the two factors of BP in the latter object as left (L) and right (R) indexed by these letters. Then we obtain e(} MSp ) = X m> e m (} MSp ) m = N BP L 0 (} MSp ) " u L log BP L 0 (u L ) + u L log BP L 0 (u L ) But now in (BP ^ BP) (C P ), the left x BP L and right orientations x BP R are related by the following formula to be found in [5]: # : (5.) x BP L = BP X L k>0 t k (x BP R ) 2k X k>0 t k (x BP R ) 2k (mod I): Hence we have u L = BP X L k>0 t k (u R ) 2k : Now we also need to estimate log BP 0 (X) modulo I 2, where I = (v k : k > 0) / BP is the ideal generated by all the v k. Proposition (5.2). In the ring BP [[X]] we have the congruence log BP 0 (X) + v X (mod I 2 ): Proof. We proceed as in the proof of Lemma (4.4) and use the same notation. We must show that for n > 2, we have L n 0 (mod I 2 ):

12 2 ANDREW BAKER For n = 2, this is an easy consequence of the formula 2v 2 = L 2? v 3 dening the Hazewinkel generator v 2. An induction on n now gives general result. At this stage we have established that (5.3) e (} MSp ) N BP L 0 (} MSp ) N BP L 0 (} MSp ) " " u L ( + v u L ) + u L ( + v u L ) u L P ( + v k>0 t k(u R ) 2k ) + ul ( + v Pk>0 t k(u R ) 2k ) # # (mod I 2 ); where the second congruence is a consequence of (5.). Notice that we have (5.4) e (} MSp ) N BP L 0 (} MSp ) " Pk>0 t k(u R ) 2k ( + v P P k>0 t k(u R ) 2k ) + k>0 t k(u R ) 2k ( + v Pk>0 t k(u R ) 2k ) # (mod I); since X X + X k>0 v k X 2k (mod I 2 ) by (4.5). We still need to deal with the term N BP L 0 (} MSp ), which has to be expressed in terms of N BP R 0 (} MSp ). By the formula of (.6), we have Q BP (T 2 ) Q BP (?T T ) B BP (T ) 2 (mod I) X k>0 Bk BP 2 T 2k+2 (mod I) in the ring BP (MU)[[T ]]. The coaction on this series is given by (Q BP (T 2 )) = X k>0 B BP;BP (T ) 2k+2 BP Bk BP 2 where the series B BP;BP (T ) 2 BP (BP)[[T ]] is composition inverse of XBP k>0 t k T 2k X k>0 t k T 2k (mod I):

13 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp 3 As N BP (Z) is the composition inverse of Q BP (Z), we have (N BP (Z)) = X k>0 t 2 k (N BP (Z)) 2k (mod I); and hence Using this result together with (5.4), we have (N BP 0 (Z)) = N BP 0 (Z) (mod I): e(} MSp ) N BP R 0 (} MSp ) " u L ( + v P k>0 t k(u R ) 2k ) + u L ( + v Pk>0 t k(u R ) 2k ) # (mod I 2 ): Expanding carefully the term inside the square brackets yields u L + u L X k>0 v k (u R ) 2k (mod I 2 ) and therefore e(} MSp ) Pk>0 v k(n BP (u R )) 2k N BP R 0 (} MSp ) (mod I 2 ) since N BP R (} MSp ) = } BP R (u R ) 2 mod I 2. Having calculated this, we are at last in a position to determine (e m ) = 2 (? Id)( e m ) (mod I): We need one further fact, namely that the right unit on the v n has the form R (v n ) X 06k6n v k t pk n?k (mod I 2 ): We therefore obtain the following generating function: X (e m )(} MSp ) m = m> Pk>0 t k(u R ) 2k N BP 0 (} MSp ) (mod I) which when pushed into A H F 2 (MSp) yields Pk>0 2 k N H F 2 (} MSp ) 2k N H F 2 0 (} MSp ; ) where A = F 2 [ k : k > ] with k being the conjugate of Milnor's generator k 2 A 2k?.

14 4 ANDREW BAKER The element m is detected in the classical mod 2 Adams spectral sequence for (MSp) by an element which originates in the E 2 term in the group Ext 4m?2 A (F 2 ; H F 2 (MSp)) : For low even values of m, these detecting elements are given in the following Table, obtained using the symbolic algebra package MAPLE. Note that = [ 2 ] and 2n = ' n. Of course, for n >, 2n+ is zero and hence detected by 0! The notation is that implied by the cobar construction. ' : [ 2 N ] TABLE ' 2 : [ 2 N N N N N ] ' 3 : [ 2 N N N N N N 3 N N 2 N N N 2 ] ' 4 : [ 2 N N N N N 5 N N 3 N N 3 N N N N 2 N N 2 N N N N N ]

15 SOME CHROMATIC PHENOMENA IN THE HOMOTOPY OF MSp 5 TABLE (continued) ' 5 : [ 2 N 5 N N 5 N N N N N N N N2 2 N N N N 3 N N 3 N N 7 N N N 2 N N 2 N N3 2 N N N N 4 N N 2 ] ' 6 : [ 2 N 7 N N N2 2 N N N 2 N N 3 N2 2 N N N N 3 N N + 2 N 7 N N 9 N N 5 N N 3 N N 3 N N 5 N N N N N 2 N2 2 N N N 2 N N3 2 N N3 2 N N 2 N N 2 N N2 2 N N N 4 N N 4 N N N N 4 ] ' 7 : [ 2 N N N 9 N N 9 N N N N N2 4 N N N N 3 N N 3 N N 7 N N 5 N N N N N2 2 N N 3 N2 2 N N 3 N 2 N N 5 N2 2 N N 5 N N 5 N N N N N 7 N N5 2 N N 2 N 2 N N 2 N N 2 N N3 2 N N N3 2 N N 2 N2 2 N N 2 N N N2 2 N N 4 N N 4 N N N 6 ]

16 6 ANDREW BAKER TABLE (continued) ' 8 : [ 2 N 3 N N 3 N N 9 N N 9 N N 7 N2 2 N N 7 N N 5 N N N 7 N N 7 N N 5 N 2 N N 5 N N N N N N 3 N2 2 N N 5 N2 2 N N 3 N N N2 4 N N N4 2 N N N2 2 N N N 2 N N 3 N N 3 N N N N N N 3 N2 4 N N 2 N N3 2 N N3 2 N2 2 N N3 2 N N5 2 N N3 2 N N5 2 N N 2 N2 4 N N 2 N2 2 N N 2 N N 2 N N 2 N N N N N 4 N N 4 N N 4 N2 2 N N 4 N N N2 2 N N N N N N N2 2 N N ] References [] J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, 974. [2] V. M. Buhstaber, Topological applications of the theory of two-valued formal groups, Math. USSR Izvestija 2 (978), 77. [3] V. Gorbunov & N. Ray, Orientations of Spin bundles and symplectic cobordism, to appear in Publ. RIMS Kyoto University. [4] H. R. Miller, D. C. Ravenel & W. S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Annals of Math. 06 (977), 469{56. [5] D. C. Ravenel, Complex Cobordism and the Stable Homotopy Groups of Spheres, Academic Press, 986. [6] N. Ray, Indecomposables in Tors MSp, Topology 0 (97), 26{70. Manchester University, Manchester, M3 9PL, England. Current address: Glasgow University, Glasgow G2 8QW, Scotland. address: a.baker@maths.gla.ac.uk