Research on homotopy groups of localized finite complexes

定域有限复形的同伦群研究

基本信息

批准号：
12640077
负责人：
SHIMOMURA Katsumi
金额：
$ 2.3万
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：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

项目摘要

In this research, we aimed two subjects. One is to make a more deep understanding of finite complexes itself and the Bousfield localization of finite complexes with respect to the Morava K-theories, and the other is to determie the homotopy groups π_*(L_<K(2)>S^O) of the Bousfield localized sphere spectrum L_2S^O with respect to K(2).For the first one, Hemmi showed that even dimensional generator of the cohomology ring of a finite H-space appears only at dimension 8 and 20. This reflects an important feature of finite complexes. Komatsu studied finite real projective spaces through the bundle structure. Some information on the localization with respect to K(1) was obtained by Yosimura from the view point of KO_*.-quasi equivalence. Yagita obtained a result on the non-commutativity of the homotopy groups, and Ohkawa studied the Bousfield classes in stable homotopy categories.For the second, we determined the homotopy groups π_*(L_K (2)S^O) at the prime 3 in the first year. The groups for the prime p > 3 was determined before. We also determine the E_2-term of the Adams-Novikov spectral sequence converging to the homotopy groups π_*(L_2 S^O) at the prime 2. Since the computation of the differentials of the spectral sequence is too difficult to make, we studied, in the second year, the homotopy groups of the Ravenel spectra T(m), and considered the Picard groups consisting of the invertible spectra in a stable homotopy, which is closely related to the homotopy groups. We obtained the homotopy groups π_*(L_nT(m)ΛV(n-1)) for m > n^2-n, and showed that there is no invertible spectrum in the stable homotopy category of E-local spectra as long as E is connective.

本研究的目的有两个：一是更深入地理解有限复形本身以及有限复形相对于 Morava K 理论的 Bousfield 定域性，二是确定同伦群 π_*(L_)。布斯菲尔德定域球谱 L_2S^O 相对于 K(2) 的 <K(2)>S^O)。对于第一个，Hemmi 证明了上同调的偶维生成元有限 H 空间的环仅出现在 8 维和 20 维。这反映了 Komatsu 通过丛结构研究有限实射影空间的一些信息。 Yosimura从KO_*.-拟等价的角度得到了同伦群的非交换性结果，Ohkawa研究了稳定同伦范畴中的Bousfield类。其次，我们在第一年确定了素数 3 处的同伦群 π_*(L_K (2)S^O)。我们还确定了 Adams- 的 E_2 项。诺维科夫谱序列在素数 2 处收敛于同伦群 π_*(L_2 S^O)。由于谱序列的微分计算太困难，我们研究了第二年，拉弗内尔谱T(m)的同伦群，并考虑稳定同伦中由可逆谱组成的皮卡德群，它与同伦群密切相关，我们得到了同伦群π_*(L_nT(。 m)ΛV(n-1)) 对于 m > n^2-n，并表明只要 E 为 E 局域谱的稳定同伦范畴，就不存在可逆谱连接词。