Study on unstable homotopy groups of finite complexes

有限复形不稳定同伦群的研究

基本信息

批准号：
15540067
负责人：
MUKAI Juno
金额：
$ 0.9万
依托单位：
Shinshu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540067/
关键词：
projective space suspension order homotopy type Hopf construction Hopf homomorphism Toda bracket Whitehead product m-twisted Adams写像ホモトピー群 Hopf準同形写像実射影空間 Moore空間戸田の積

项目摘要

Summary of research results is the following.1.Concerning the formula about the Hopf invariant of some Toda bracket, there exist some restrictions. Inoue and I generalize the formula applicable for the case of finite complexes.2.Let denote by P^n the real n dimensional projective space and by γ_{n-1}:S^{n-1}→P^{n-1} the attaching map of the top n cell. In 2000, some homotopy groups of a suspension EP^3 were determined and a suspension map Eγ_4 was represented as the composite of three maps. This time, the fourth multiple element of the identity class of EP^6 is takes as a representative of some Toda bracket, and this relates with the composite of Eγ_5 and the Hopf map. And these procedure gave an affirmative solution to the suspension order conjecture of P^{2n}.3.We Set M^n=E^{n-2}P^2, which is called a mod 2 Moore space. Let i_n : S^{n-1}→M^n be the inclusion. The problem whether the Whitehead product of i_{n+1} and itself generates the direct summand Z/2Z in the homotopy group π_{2n-1}(M^{n+1}) was proposed by Spokenkov at Suzue University. We gave a partial answer.4.By use of the method solving the suspension order conjecture of P^{2n}, Miyauchi and I determined the group of self-homotopies of a double suspension of P^6.5.By the jointed work with K. Yamaguchi, the homotopy type of the complex 4 dimension m twisted projective space was determined according as the case m=0, m : odd and m is a multiple of 8. Furthermore, it was proven that there exists no homotopy type in the case m is even and m is not divisible by 8.6.Inoue and I proved that ε_3 and μ_3 are lifted to the Moore Spaces M^{11} and M^{12}, respectively.7.To develop the joint work with Golasinski, I tried determining the orders of the Whitehead products of iota_n with alphainpi^n_{n+k} for n>k+1, k<25.Investigators, except for Abe and Golasinski, could obtain no result.

研究结果总结如下：1.关于某些Toda括号的Hopf不变量的公式，存在一些限制，Inoue和我推广了适用于有限复数情况的公式。2.用P^n表示。实数 n 维射影空间和由 γ_{n-1}:S^{n-1}→P^{n-1} 顶部 n 单元的附着映射 2000 年，一些同伦群。确定了悬浮图 EP^3 的值，并将悬浮映射 Eγ_4 表示为三个映射的复合。这次，EP^6 的恒等类的第四个多重元素被视为某个 Toda 括号的代表，这涉及到。结合Eγ_5和Hopf图，这些过程给出了P^{2n}.3的悬浮阶猜想的肯定解。我们设M^n=E^{n-2}P^2，即称为 mod 2摩尔空间。设 i_n : S^{n-1}→M^n 为包含问题。 i_{n+1} 与其本身的怀特海积是否生成同伦群 π_{2n 中的直被加数 Z/2Z 。 -1}(M^{n+1})是由铃江大学的Spokenkov提出的，我们给出了部分答案。4.利用求解P^{2n}的悬浮阶猜想的方法， Miyauchi 和我确定了 P^6.5 双悬置的自同伦群。通过与 K. Yamaguchi 的共同工作，根据 m=0 的情况确定了复 4 维 m 扭曲射影空间的同伦类型， m ：奇数且 m 是 8 的倍数。此外，证明了在 m 为偶数且 m 不能被 8.6 整除的情况下不存在同伦型。Inoue 和我证明了ε_3 和 μ_3 分别提升到摩尔空间 M^{11} 和 M^{12}。7.为了与 Golasinski 开展联合工作，我尝试用 alphainpi^n_{n 确定 iota_n 的 Whitehead 乘积的阶数+k} 表示 n>k+1，k<25。除了 Abe 和 Golasinski 之外，其他研究人员无法获得任何结果。