ON KO THEORY OF GRASSMANNIANS
ON KO THEORY OF GRASSMANNIANS
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格拉斯曼主义者的ON KO理论
DOI：
10.1093/qmath/20.1.447
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发表时间：
1969
影响因子：
0.7
通讯作者：
S. G. Hoggar
中科院分区：
文献类型：
--
作者：
S. G. Hoggar
导言设KO(X)是空间X上实向量丛的Grothendieck环，KO*是相应的上同调理论。当F=R、C或四元数H时，保护空间FP*的大部分锦鲤环已被求出[例如见(1)、(8)、(12)、(13)]。本文从Fn中FC-平面的Grassmann流形OK(Fn)的KO~*问题入手，把注意力限制在F=C和H上，证明了商空间Gk{Fn)IGk(Fn-1)是OK_1(Fn-1)上某一丛的Thorn空间，但我们不能使用KO-理论Thorn同构，因为它的第二个Stiefel-Whitney类w2是非零的。在计算过程中，我们注意到了有限(JW-复形)范畴中一些有用的引理，这些引理的单元格仅为偶数维(记为@)。常见的自然变换有络合化c：KO*-*·K*，解复分解r：K*->；KO*，共轭·：K*-*·K*，关系式Re=2，cr=1+*。C和·是环同态，r是交换群的同态。我们的结果的一个结果是，对于所考虑的流形，复化得到一个单态c：KO(M)-+K(M)。这意味着由Atiyah(2)的Ito方法得到的这些流形的非嵌入和非浸入结果可以在(一般)更容易的复理论中同样好地计算。我们利用(9)中的这一事实证明了d=4N-8维的流形OT{Cn)既不嵌入Ri+id，也不嵌入Hd+id~1(然而，这一结果可以通过非常类似的上同调计算得到，如(2)所暗示的那样)。
Introduction LET KO (X) be the Grothendieck ring of real vector bundles on a space X, and KO* the associated cohomology theory. Most of the KOi rings of protective spaces FP* for F= R, C, or the quaternions H, have been worked out [see for example (1),(8),(12),(13)]. In this paper we make a start on the more general problem of KO* for Grassmann manifolds Ok (Fn) of fc-planes in Fn, restricting attention to F= C and H. It turns out that the quotient space Gk {Fn) IGk (Fn-1) is the Thorn space of a certain bundle over Ok_1 (Fn-1) but that we cannot use the KO-theory Thorn isomorphism because its second Stiefel-Whitney class w2 is non-zero. During the computation we note a number of useful lemmas for spaces in the category of finite (JW-complexes with cells only in even dimensions (denoted by@). There are the well-known natural transformations complexification c: KO*-*• K*, deoomplexifioation r: K*-> KO*, conjugation•: K*-*• K*, with relations re= 2, cr= 1+*. c and• are ring homomorphisms and r is a homomorphism of abelian groups. A consequence of our results is that, for the manifolds considered, complexification gives a monomorphism c: KO (M)-+ K (M). This means that non-embedding and non-immersion results for these manifolds obtained by the iTO-method of Atiyah (2) can equally well be computed in the (in general) easier complex theory. We make use of this fact in (9) to show that the manifold Ot {Cn) of dimension d= 4n—8 does not embed in Ri+ id, nor immerse in Hd+ id~ 1 (however, this result can be obtained by a very similar calculation with cohomology, as implied in (2)).