2005 Fiscal Year Final Research Report Summary
Study on automorphic functors on the stable homotopy category through homotopy groups of finite complexes
| Project/Area Number |
16540073
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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 | Kochi University |
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Principal Investigator |
SHIMOMURA Katsumi Kochi University, Department of Mathematics, Professor, 理学部, 教授 (30206247)
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| Co-Investigator(Kenkyū-buntansha) |
HEMMI Yutaka Kochi University, Department of Mathematics, Professor, 理学部, 教授 (70181477)
KOMATSU Kazushi Kochi University, Department of Mathematics, Associate Professor, 理学部, 助教授 (00253336)
OHKAWA Tetsusuke Hiroshima Institute of Technology, Faculty of Engineering, Associate Professor, 工学部, 助教授 (60116548)
HIKIDA Mizuho Prefectural University of Hiroshima, Faculty of Life and Environmental Science, Professor, 生命環境学部, 教授 (80156570)
NAKAI Hirofumi Oshima National College of Maritime Technology, Lecturer, 講師 (80343739)
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| Project Period (FY) |
2004 – 2005
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| Keywords | stable homotopy category / finite spectrum / Adams-Novikov spectral sequence / Johnson-Wilson spectrum / Bousfield localization / homotopy groups of spheres / invertible spectrum / Picard group |
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| Research Abstract |
In this research, we study the automorphic functor on the Bousfield localized stable homotopy category L_n with respect to the Johnson-Wilson spectrum E(n) through homotopy groups of finite complexes. We show that such an automorphic functor is represented by a spectrum, which has the same E(n)_*-homology as the one of the sphere spectrum, and that the set of such spectra is contained in the direct sum of the E_r-terms E_r^<r,r-1> of the E(n)-based Adams-Novikov spectral sequence converging to the homotopy groups of spheres. Besides, we show that in many cases, the set consists of spheres. In the case where n=1 at the prime two, it contains a spectrum other than the spheres, and it is well known. We show that the next non-trivial case is n=2 at the prime three by constructing a spectrum other than spheres. We also show that there are at most two such spectra. If we consider the Toda-Smith spectrum V(1), which is a finite complex, then the automorphic functor X:L_2→L_2 given by X induces an isomorphism X∧V(1)=Σ^<48>:L_2∧V(1)→L_2∧V(1). Here, Σ denotes the functor represented by the sphere. A similar argument with the Moore spectrum V(0) shows that X∧V(0) is not Σ^<48>. Furthermore, we show the existence of spectra other than X∧V(0) with E(2)_*-homology same as the one of spheres. The difference appears only in the dimensions of the homotopy groups of them. In order to study the other cases, we determine homotopy groups of the smash product of the Ravenel spectrum T(m) and finite spectra by use of the E(n)-based Adams-Novikov spectral sequence. Moreover, by investigating Picard groups and invertible spectra, without using Adams-Novikov spectral sequence, we show that there is no spectrum other than spheres that induces the automophic functors on the Bousfield localized category with respect to a connected spectrum E.
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