A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )
1980; American Mathematical Society; Volume: 78; Issue: 2 Linguagem: Inglês
10.2307/2042276
ISSN1088-6826
Autores Tópico(s)Advanced Topics in Algebra
ResumoH*(MO , Z2) as a module over the Steenrod algebra is shown to have a direct summand A//A2. U. In this note we show that, as a module over the Steenrod algebra A, H*(MO , Z2) has a direct summand beginning in dimension 0. The proof is easy but contradicts the theorem of Giambalvo [3]. Recall that MO is the Thom space of the bundle induced from the canonical bundle over BO by p: BO -* BO the projection of the 7-connected covering. A cobordism theory a results from considering MO as a spectrum in the usual way. For some partial computations and further details the reader is referred to [2]. Let A2 be the augmentation ideal of the Hopf subalgebra of A generated by {Sq?, Sq', Sq2, Sq4}. Denote by A//A2 the quotient coalgebra A/AA2. Let U be the Thom class in H*(MO ). All homology groups are to have Z2 coefficients. THEOREM. A//A2* U is a direct summand in H*MO . PROOF. The argument follows similar lines to Priddy's proof that K(Z2) is a Thom spectrum [5]. Let X denote the 15-skeleton of BO and i: X -* BO the inclusion. Since BO is a double loop space there is an induced double loop map X: -222X U222BO -* BO where the first map is Q22:2i and the second is the adjoint of the identity double looped. Let a: A//A2 -* H*MO denote evaluation on the Thom class and P*: H*BO -> H*MO the Thom isomorphism a*, the dual of a, is a morphism of algebras over A*, the dual of the Steenrod algebra. Now *W*H * 22:2X is a subalgebra over A* of H* MO since it is equal to r* H* M(pw) where M(pw) is the Thom spectrum associated with pw: 2222XBO and r: M(pw) -* MO is the map induced by w. To prove the theorem it will be enough to show that a*: P*W*H* U22:2X _(A IA2)* is an algebra isomorphism where (A//A2)* is the dual of A//A2. To do this we need to know about the image of w*. Received by the editors January 15, 1979. AMS (MOS) subject classifications (1970). Primary 57D90; Secondary 55G10. The second author was supported in part by NSF Grant MCS 76-07051. ? 1980 American Mathematical Society 0002-9939/80/0000-0083/$02.00 295 This content downloaded from 157.55.39.58 on Tue, 11 Oct 2016 04:19:18 UTC All use subject to http://about.jstor.org/terms 296 A. P. BAHRI AND M. E. MAHOWALD LEMMA. C* H g212X Z2[P8, P12, P14, Qg(p15), n > 0] where pi is the nonzero primitive element in HiBO and Qn is the nth iterate of the Dyer-Lashof operation defined on a double loop space. Q? = the identity and dim Qn(p15) = 2n-4 _ 1. PROOF. The structure of H*BO has been computed by Stong [6, Theorem A] and is given by H*BO H*K(Z, 8)/ASq2 0& Z2[,i] where the 9i are classes in H*BO wi mod decomposables. The first is 016 in dimension 16. It follows then that X is a four-cell complex with cells in dimensions 8, 12, 14 and 15 corresponding to the classes X8 = E8, X12 = SqVE8, X14 = Sq E8 = Sq2Sqt E8, X15= Sq'E8 = Sq'Sq2Sq'E8, where E8 is the first class in H* K(Z, 8). We will denote by pi the class in homology dual to xi. We may now use Theorem 3 of Browder [1] to conclude that H* 2222X is a polynomial ring over Z2 generated by four types of elements. These are pi, i = 8, 12, 14, 15; Q1 (pi), n > 0; I (pi,,y); Q( 4,1(p,,j)) where here we have identified pi with its image under the inclusion H*X c H* 2222X. 4J is the Browder operation defined on a double loop space in [1] and the y1's are iterated products involving 4' in H* 2222X. They are determined by giving a basis for the graded Lie algebra generated by H*2X in its tensor algebra. Full details may be found in ?IV of [1]. We will now analyse in turn what happens to each of the above elements under W*. The map x sa2z x BO , where y is the adjoint of the identity on Y2X, is just the inclusion of the 15-skeleton and we may safely identify pi with its image under w*. These give primitive elements in H*BO . Since ( is a double loop map, we have, by the naturality of 4, in the category of double loop spaces, that W*4I(pi,yj) = Apl(Pi, W*(Yj)) = 0. since the operation on the right is in BO which is a triple loop space and must have 4' identically zero in it. This 'instability' of the Browder operations follows immediately from their definition in [1]. The w* Qln'(pi, yj) are also all zero since the Qn are natural with respect to w. One of the consequences of Stong's Theorem A is that the map p*: H*BO H*BO is onto and hence thatp* is a monomorphism. We will now use this to determine the *Q n(pi). Sincep: BO -> BO is a double loop map we have P*(*Qn (Pi) = Qn(p*(Pi)) (n > 0). This content downloaded from 157.55.39.58 on Tue, 11 Oct 2016 04:19:18 UTC All use subject to http://about.jstor.org/terms A DIRECT SUMMAND IN H*(MO , Z2) 297 The RHS is zero for i = 8, 12, 14 and nonzero for i = 15 by Kochman [4, Corollary 35], so w*Q (pi) = 0 for i = 8, 12, 14 and # 0 for i = 15. We have shown therefore that the only generators of H* 2:2X which survive under w* are P8P12, P14, P5 and Q (p15) for n > 0. To finish the lemma we need to show that these elements generate a polynomial ring. We will do this and complete the proof of the theorem at the same time. Now (A //A2)_ Z2[8', 24, (2, f4, 45.... I where dim (i = 2' 1 and Z2[t,, 42' (3, 44, . . . ] is the dual of the Steenrod algebra. We know that, in each dimension, the rank of w *H*j22:2X is not greater than that of (A//A2)* and so the theorem follows from the next lemma. LEMMA. H 0222X H*MO -*(A/A2)* i a
Referência(s)