Research on stable homotopy groups of finite complexes

有限复形稳定同伦群的研究

基本信息

批准号：
10640082
负责人：
SHIMURA Katsumi
金额：
$ 2.11万
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：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640082/
关键词：
homotopy groups spheres finite complexes spectra Morava K-theories Bousfield localization Johnson-Wilson spectrum Adams-Novikov spectral sequence Adams-Novikov スペクトル系列 Bousfield 局所化 Jonson-Wilson スペクトラム Morava K 理論

项目摘要

In this research, we aimed two subjects. One is to make a more deep understanding of the Bousfield localization of finite complexes with respect to the Morava K-theories, and the other is to determine the homotopy groups πィイD2*ィエD2(LィイD22ィエD2M) of the Bousfield localized Moore spectrum LィイD22ィエD2M with respect to K(2).For the first one, we obtain some information on the localization with respect to K(1) from the view point of KOィイD2*ィエD2-quasi equivalence. We understand that the K(n)ィイD2*ィエD2 homologies reflect many properties of the space itself by observing Lie groups and H-spaces. Yagita and his coauthors show that some properties obtained from the structure of Morava K(n)ィイD2*ィエD2-homology imply the similar properties obtained from the one of BPィイD2*ィエD2.-homology, whose converse is the standard philosophy when we study this kind of homotopy theory.For the second, we determined the homotopy groups πィイD2*ィエD2(LィイD22ィエD2M) in the first year. In the second year we concentrate to determine the homotopy groups πィイD2*ィエD2(LィイD22ィエD2SィイD10ィエD1) at the prime 3 and succeeded. Note that πィイD2*ィエD2( LィイD22ィエD2SィイD10ィエD1) has already determined at a prime > 3. The results made Hopkins set up the chromatic splitting conjecture, and the result at the prime 3 gives a counter example of it. This shows that the structure of the homotopy groups πィイD2*ィエD2(LィイD22ィエD2SィイD10ィエD1) of the localized sphere spectrum at the prime 3 is different from the one at a prime > 3. At the prime 2, the structure seems very complex and we determine only the EィイD22ィエD2-term of the Adams-Novikov spectral sequence converging to πィイD2*ィエD2(LィイD22ィエD2M).

在这项研究中，我们的目标有两个：一是更深入地理解有限复形相对于 Morava K 理论的 Bousfield 局域化，二是确定 Bousfield 局域化 Moore 的同伦群 (LiD22D2M)。相对于 K(2) 的频谱 LiD22D2M。对于第一个，我们获得了一些有关定位的信息从 KOiiD2*ieD2*ieD2 准等价的角度来看 K(1)，通过观察 Lie 群和 H 空间，我们知道 K(n)iiD2*ieD2 同调反映了空间本身的许多性质。他的合著者表明，从 Morava K(n)iiD2*ieD2 同源性结构中获得的一些性质意味着从以下之一获得的相似性质BPIID2*IED2.-同伦，其逆是我们研究这种同伦理论时的标准哲学。第二年，我们在第一年确定了同伦群ED2M)，第二年我们集中精力确定同伦群。 π-D2*D2(L-D22-D2S-D10-D1) 在素数 3 处并且成功请注意 π-D2*D2(L-D22-D2S-D1) 在素数 3 处并且成功。在素数 > 3 处已经确定。这个结果使霍普金斯成立了色分裂猜想，并且素数 3 处的结果给出了它的反例。表明素数 3 处定域球面的同伦群 πIID2*D2(LIID22D2SIID10D1) 的结构与素数 > 3 处的同伦群不同。在素数 2 处，结构看起来非常复杂，我们仅确定 E-D22 - Adams-Novikov 谱序列的项收敛为πIID2*IED2(LIID22IED2M)。