Study on unstable homotopy groups of finite complexes

研究结果总结如下：对于某些Toda括号的Hopf不变量公式，存在一些限制。我和Inoue推广了适用于有限复形情况的公式。用P^n表示实n维投影空间，用γ_{n-1}表示S^{n-1}→P^{n-1}表示最上面n个单元格的附加映射。在2000年，确定了悬架EP^3的几个同伦群，并将悬架映射Eγ_4表示为三个映射的复合。这一次，取EP^6的单位类的第四个倍数元素作为某个Toda括号的代表，这与e ^ _5与Hopf映射的复合有关。这些方法给出了P^{2n} 3的悬序猜想的一个肯定解。我们设M^n=E^{n-2}P^2，这被称为模2摩尔空间。设i_n: S^{n-1}→M^n为包含项。在同伦群π_{2n-1}（M^{n+1}）中，i_{n+1}与自身的Whitehead积是否产生直接和Z/2Z的问题是由苏苏大学的Spokenkov提出的。我们给出了不完整的答案。Miyauchi和我利用求解P^{2n}的悬架阶猜想的方法，确定了P^6.5的双悬架的自同伦群。通过与K. Yamaguchi的联合工作，在m=0， m为奇数，m为8的倍数的情况下，确定了复4维m扭曲射影空间的同伦类型。此外,它是证明不存在同伦类型在8.6 m和m不整除。Inoue和I证明了ε_3和μ_3分别被提升到摩尔空间M^{11}和M^{12}。为了开展与Golasinski的联合工作，我尝试用alphainpi^n_{n+k}确定iota_n的Whitehead积的阶数，当n>k+1, k<25时。除了安倍和戈拉辛斯基，调查人员没有得到任何结果。

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.