Study on automorphic functors on the stable homotopy category through homotopy groups of finite complexes

通过有限复形同伦群研究稳定同伦范畴的自守函子

基本信息

批准号：
16540073
负责人：
SHIMOMURA Katsumi
金额：
$ 2.24万
依托单位：
Kochi University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540073/
关键词：
stable homotopy category finite spectrum Adams-Novikov spectral sequence Johnson-Wilson spectrum Bousfield localization homotopy groups of spheres invertible spectrum Picard group

项目摘要

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.

在这项研究中，我们通过有限复合物的同型组群研究了Bousfield局部稳定同型类别L_N的自动型函数L_N。 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 spectra, 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 spectrum sequence converging to the homotopy groups of spheres.此外，我们表明，在许多情况下，该集合由球体组成。如果在素数为prime的情况下，它包含除球体以外的其他频谱，并且众所周知。我们表明，下一个非平凡情况是在三分之二的n = 2，除了球以外的其他频谱。我们还表明，最多有两个这样的光谱。如果我们考虑是有限复合物的Toda-Smith Spectrum V（1），则X给出的自动形函数X：L_2→L_2诱导同构X∧V（1）=σ^<48>：L_2∧V（1）→L_2∧V（1）。在这里，σ表示由球表示的函子。摩尔频谱V（0）的类似参数表明X∧V（0）不是σ^<48>。此外，我们显示了与e（2）_* - 同源性相同的光谱（0）以外的光谱的存在。差异仅出现在它们的同型组的尺寸中。为了研究其他情况，我们通过使用基于E（N）基于E的Adams-Novikov光谱序列来确定Ravenel Spectrum t（M）和有限光谱的粉碎产物的同型组。此外，通过研究PICARD组和可逆光谱，在不使用Adams-Novikov光谱序列的情况下，我们表明，除了球形以外，没有其他光谱可以诱导Bousfield局部类别的自动型函数相对于连接光谱E。