Immersions and embeddings of projective spaces
Immersions and embeddings of projective spaces
复制标题
投影空间的沉浸和嵌入
DOI：
10.1090/s0002-9939-1972-0321111-8
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发表时间：
1972
期刊：
影响因子：
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通讯作者：
D. Segal
中科院分区：
文献类型：
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作者：
S. Feder;D. Segal
主题是复杂和四元射影空间的嵌入和沉浸。结果是通过使用自旋表示并将其与它们出现的各种 X 理论联系起来获得的。获得了非嵌入和非沉浸的数值结果。本笔记的目的是修改和完善[1]和[2]的结果。读者可以参考上述文章来了解问题、符号和技术的说明。我们只记得 ^ 和 <£ 分别代表“不沉浸”和“不嵌入”； CPn 表示复数，HPn 表示四元数射影空间，KG 表示稳定 G 丛理论。完整的结果说明如下： 定理 1. 对于所有 n，CPn$ R*»-*«i » »-1。定理 2. CP„<£ R*n~!la<^ 对于所有 n。 定理 3. HPnï /p»-«««»)-*; HPnt R»"-2*«* //a(«)=0 (4)1; HPn$ R8n-Mn)-3; HPn£RSn-2aU)-2 ¡f «(„)=£ J (4) 和 ffPn$ /?8»-2«<»)-l ¡f x(n) = 0or3 (4). a(n) 是 n 的二进展开式中 l 的个数。所有结果都是使用 K 理论即自旋表示获得的。这些改进来自于知道在各个维度中复数自旋表示是定理 1 已经被证明为 n 奇数。为了消除这个限制，我们考虑 /u=H+v，其中 v 是 CPn^R2n+2k+l 的正规丛（结果比考虑 CP^^CP,,,）现在 ¡u 是一个自旋丛，x 是 4 月 22 日编辑收到的子代数的生成元。 1971. AMS 1970 科目分类。小学 57A35、中学 55B15。
The topic is embedding and immersions of complex and quaternionic projective spaces. The results are obtained using spin representations and relating these with the various X-theories in which they occur. A numerical result on nonembeddings and and nonimmersions is obtained. The purpose of this note is to revise and complete the results of [1] and [2]. The reader is referred to the aforementioned articles for a statement of the problem, notation and techniques. We recall only that ^ and <£ stand for "does not immerse" and "does not embed" respectively; CPn denotes the complex and HPn the quaternion projective space and KG denotes the theory of stable G-bundles. The complete statement of results would be as follows: Theorem 1. CPn$ R*»-*«i » »-1 for all n. Theorem 2. CP„<£ R*n~!la<^ for all n. Theorem 3. HPnï /p»-«««»)-*; HPnt R»"-2*«* //a(«)=0 (4)1; HPn$ R8n-Mn)-3; HPn£RSn-2aU)-2 ¡f «(„)=£ J (4) and ffPn$ /?8»-2«<»)-l ¡f x(n) = 0or3 (4). a(n) is the number of l's in the dyadic expansion of n. All the results are obtained using K-theory, namely the spin representation. The improvements come from the knowledge that in various dimensions the complex spin representations are restrictions of quaternionic ones or come from real representations. Theorem 1 has been proven for n odd. To remove this restriction we consider /u=H+v where v is the normal bundle of CPn^ R2n+2k+l (the result is better than that obtained considering CP^^CP,,). Now ¡u is a spin bundle. We let x=H+H—2, x is the generator of the subalgebra of Received by the editors April 22, 1971. AMS 1970 subject classifications. Primary 57A35, 57D40; Secondary 55B15.