2004 Fiscal Year Final Research Report Summary
Study on unstable homotopy groups of finite complexes
| Project/Area Number |
15540067
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Shinshu University |
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Principal Investigator |
MUKAI Juno Shinshu University, Faculty of Science, Professor, 理学部, 教授 (50029675)
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| Co-Investigator(Kenkyū-buntansha) |
KACHI Hideyuki Shinshu University, Faculty of Science, Professor, 理学部, 教授 (50020657)
ABE Kojun Shinshu University, Faculty of Science, Professor, 理学部, 教授 (30021231)
MATSUDA Toshimitsu Shinshu University, Faculty of Science, Associate Professor, 理学部, 助教授 (70020667)
KAMIYA Hisao Shinshu University, Faculty of Science, Lecturer, 理学部, 講師 (80020676)
GOLASINSKI Marek Copernicus University, Professor (80020676)
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| Project Period (FY) |
2003 – 2004
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| Keywords | projective space / suspension order / homotopy type / Hopf construction / Hopf homomorphism / Toda bracket / Whitehead product |
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| Research Abstract |
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.
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Research Products
(13 results)
