A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )
A Direct Summand in H ∗ (MO 〈8 〉, Z 2 )
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H ∗ (MO 〈8 〉, Z 2 ) 中的直接被加数
DOI：
10.2307/2042276
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发表时间：
1980
影响因子：
0.9
通讯作者：
M. Mahowald
中科院分区：
文献类型：
--
作者：
A. Bahri;M. Mahowald
H*(MO, Z2)作为Steenrod代数上的模，被证明有一个直接和a //A2。在这篇笔记中，我们证明，作为Steenrod代数a上的一个模，H*(MO, Z2)有一个从维数0开始的直接和。证明很简单，但与Giambalvo[3]定理相矛盾。回想一下，MO是由7连通覆盖的投影p: BO -* BO在BO上由正则束导出的束的Thom空间。用通常的方法把MO看作是一个谱，就得到了共协理论。对于部分计算和进一步的细节，读者可以参考[2]。设A2是由{Sq？， Sq', Sq2, Sq4}。用A//A2表示商代数A/AA2。设U为H*(MO)中的Thom类。所有同调群的系数都是Z2。定理。A//A2* U是H*MO的直接求和。证明。这个论证与Priddy证明K（Z2）是Thom谱[5]的论证类似。设X表示BO的15-骨架，i: X -* BO表示包涵体。由于BO是一个双环空间，因此存在一个诱导双环映射X: - 222x U222BO -* BO，其中第一个映射是Q22:2i，第二个映射是单位双环的伴随。设a: a //A2 -* H*MO表示在Thom类上的求值，P*: H*BO -> H*MO，即a的对偶——Thom同构a*是代数在a*上的态射，即Steenrod代数的对偶。现在，*W*H* 22:2X是H* MO的a*上的子代数，因为它等于r* H* M(pw)，其中M（pw）是与pw: 2222XBO相关的Thom谱，r: M(pw) -* MO是W诱导的映射。为了证明这个定理，足以证明a*: P*W*H* U22:2X _(a IA2)*是一个代数同构，其中(a //A2)*是a //A2的对偶。要做到这一点，我们需要知道w*的像。编辑于1979年1月15日收到。AMS （MOS）学科分类（1970年）。主要57 d90;二次55 g10。第二作者得到了NSF Grant MCS 76-07051的部分支持。吗?1980 American Mathematical Society 0002-9939/80/0000-0083/$02.00 295此内容于2016年10月11日星期二04:19:18 UTC从157.55.39.58下载。P. bahri和m. e. mahowald引理。C* H g212X Z2[P8, P12, P14, Qg(p15), n >]，其中pi是HiBO中的非零元，Qn是定义在双环空间上的Dyer-Lashof操作的第n次迭代。问吗?=恒等式，dim Qn(p15) = 2n-4 _ 1。证明。H*BO的结构由strong[6，定理A]计算得到，并由H*BO H*K(Z, 8)/ASq2 0& Z2[，i]给出，其中9i是H*BO中具有模可分解的类。第一个是16维的016。由此可见，X是一个四单元格复体，其维度为8、12、14和15的单元格分别对应于X8 = E8、X12 = SqVE8、X14 = Sq E8 = Sq2Sqt E8、X15= Sq'E8 = Sq'Sq2Sq'E8，其中E8是H* K（Z, 8）中的第一类。我们用表示与的同调对偶的类。我们现在可以利用Browder[1]的定理3得出H* 2222X是由四种元素生成的Z2上的多项式环。分别是i = 8 12 14 15；Q1 (pi), n >0 0；我(π,y);Q（4,1(p,,j)）在这里我们已经确定了pi和它在包含H*X c H* 2222X下的像。4J是在[1]中的双循环空间上定义的浏览器操作，y是H* 2222X中包含4'的迭代产品。它们是通过给出H*2X在其张量代数中生成的渐变李代数的基来确定的。详情请参阅？IV的[1]。现在我们将依次分析在W*下上述每个元素会发生什么。映射x sa2z x BO，其中y是Y2X上恒等式的伴随，只是包含了15-骨架，我们可以安全地将pi与它在w*下的像等同起来。这些给出了H*BO中的基本元素。由于(是一个双环映射，根据4的自然性，我们有，在双环空间的范畴中，W*4I(pi,yj) = Apl(pi, W*(yj)) = 0。因为右边的操作是在BO中，BO是一个三循环空间，必须有4'相同的零。Browder操作的这种“不稳定性”直接来自[1]中的定义。w* Qln'（, yj）也都是零，因为Qn相对于w是自然的。斯强定理A的一个结果是映射p*: H*BO H*BO是映上的，因此p*是单态的。我们现在用它来确定*Q n（）由于BO -> BO是一个双环映射，我们有P*(*Qn (Pi)) = Qn(P*(Pi)) （n >0）。所有的内容都使用http://about.jstor.org/terms A DIRECT SUMMAND IN H*(MO, Z2) 297。Kochman[4，推论35]在i = 8,12,14时RHS为零，在i = 15时RHS为非零，所以w*Q （pi）为i = 8,12,14时为0，在i = 15时为0。因此，我们已经证明，在w*条件下，只有P8P12、P14、P5和Q （p15）产生H* 2:2X。为了完成引理，我们需要证明这些元素生成一个多项式环。我们会这样做，同时完成定理的证明。现在（A //A2）_ Z2[8', 24, (2, f4, 45....I where dim (I = 2' 1 and Z2[t,, 42'(3,44，…]是Steenrod代数的对偶。我们知道，在每个维度中，w *H*j22:2X的秩不大于(A//A2)*的秩，因此定理由下一个引理推出。引理。H*MO -*(A/A2)* i A
H*(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