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Self maps of the suspension of H-spaces

H 空间悬浮的自映射

基本信息

批准号：
10640077
负责人：
MORISUGI Kaoru
金额：
$ 1.47万
依托单位：
Wakayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

项目摘要

The summary of research results is as follows.(a)Let X be an Hopf space. Oshima showed that the homotopy set of self maps of X [X, X] forms a group when X has at most 3-cells and he also determined the group structure of [X, X] for such spaces X.(b)We studied the group structure [ΣX, ΣX], where ΣX is the suspension space of X. We determined the group structure in case that X = SU(3) and Sp(2). They are typical example of above space X. In general, the suspension map [X, X] → [ΣX, ΣX] is not a homomorphism.(c)For general Hopf spaces, we can consider the Samelson products in [Y, X] when Y is a CW-complexes. The we found a formula which relates the Samelson product in [Y, X] and its suspension in [Y, ΩΣX] by using the generalized Hopf construction. The above item is an application of this formula.(d)It is important to know what nilpotency the group [X, X] for a Hopf space X. However another important is to study the composition structure of the group [X, X]. We determined this composition structure for X = SU(3) and Sp(2). This structure looks like "Square ring" which Baues in Germany studied.(e)Let MィイD1nィエD1 = SィイD1n-1ィエD1 UィイD22ィエD2eィイD1nィエD1 be the mod 2 Moore space. There is a canonical projection map P : MィイD1nィエD1 → SィイD1nィエD1. Given an element α∈ πィイD2κィエD2(SィイD1nィエD1), we studied when α has a lift to MィイD1nィエD1. One of our results is that the Whitehead square [ιィイD2nィエD2, ιィイD2nィエD2] does not have a lift for n ≠1, 3, or 7. This is the joint work with J.Mukai in Shinshu University.

研究结果总结如下：(A)设X为Hopf空间。Oshima证明了当X至多有3个胞格时，X[X，X]的自映射同伦集构成一个群，并确定了X的群结构。(B)我们研究了群结构[ΣX，ΣX]，其中ΣX是X的悬挂空间。我们确定了当X=SU(3)和Sp(2)时的群结构。一般地，悬挂映射[X，X]→[ΣX，ΣX]不是同态.(C)对于一般的Hopf空间，当Y是CW-复形时，我们可以考虑[Y，X]中的Samelson积.利用广义ΩΣ构造，得到了[Y，Hopf X]中Samelson积与[Y，Hopf X]中Samelson积之间的关系公式。上面的项是这个公式的一个应用。(D)知道群[X，X]对于Hopf空间X的幂零是很重要的。然而，另一个重要的是研究群[X，X]的合成结构。我们确定了X=SU(3)和Sp(2)的这种合成结构。(E)设MィイD_1nィエD_1=SィイD_1n-1ィエD_1UィイD_(22)ィエD_2eィイD_1nィエD_1为mod_2摩尔空间。有一个典型的投影图P：MィイD1nィエD1→SィイD1nィエD1.给定一个元素α∈πィイD2κィエD2(SィイD1nィエd1)，我们研究了α何时有提升到MィイD1nィエd1。我们的结果之一是怀特黑德正方形[ιィイD2nィエD2，ιィイD2nィエD2]对于n≠1，3或7没有升力。这是与新宿大学J.Mukai的合作成果。