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Research on topology related to variational problems and application of Mathematica

变分问题相关拓扑研究及Mathematica应用

基本信息

批准号：
13640090
负责人：
KAMEKO Masaki
金额：
$ 2.05万
依托单位：
Toyama University of International Studies
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640090/
关键词：
classifying space exceptional Lie group loop group 複素多様体正則写像 Mathematica コンパクトリー群 Steenrod代数 holomorphic map complex manifold cohomology classifying space

项目摘要

Let G be a Lie group and let p be a prime. In the case the Lie group G is one of exceptional Lie groups F4, E6, E7, E8 and p=3 or in the case G=E8, p=5, the integral cohomology of G has p-torsion and the mod p cohomology of its classifying space BG is unknow or even if it has been already computed, the result is involved. There is a so-called Adams' conjecture on the mod p cohomology of the classifying spaces of compact Lie groups, which asserts that the Quillen homomorphism is a monomorphism for p an odd prime. If the the description of the mod p cohomology of classifying spaces in this form, it would be useful. In the study of the mod p cohomology of classifying spaces of exceptional Lie groups above, one of the most powerful tool is the Rothenberg-Steenrod spectral sequence whose E2 term was identified with the cotorsion product of the Lie group G.We computed certain rings of invariants, which could be done by computer calculation in some cases, and obtained the following results :(1) for (G,p>(F4,3), (E6,3), (E7,3), (E8,5), the Rothenberg-Steenrod spectral sequence converging to the mod p cohomology of BG collapses at the E2 term.(2) For (G,p>(E8,3), the Rothenberg-Steenrod spectral sequence does not collapse at the E2 level.Furthermore, the computation of certain cotorsion products is equivalent to the computation of the cyclic group C of order p with certain C-modules. We hope this computation would be applied to the computation of the mod p cohomology of classifying spaces of loop groups which are related to variational problems.

设G是李群，p是素数。当李群G是例外李群F_4、E_6、E_7、E_8中的一个且p=3或当G= E_8、p=5时，G的整上同调有p-挠，其分类空间BG的模p上同调未知或即使已计算过，也涉及到结果.关于紧李群的分类空间的模p上同调，有一个所谓的亚当斯猜想，它断言当p是奇素数时，Quillen同态是一个单同态。如果分类空间的模p上同调能用这种形式来描述，那将是很有用的。在研究上述例外李群分类空间的模p上同调时，最有力的工具之一是Rothenberg-Steenrod谱序列，它的E2项与李群G的余挠积一致。我们计算了某些不变量环，在某些情况下可以通过计算机计算完成，得到了以下结果：（1）对于（G，p>（F4，3），（E6，3），（E7，3），（E8，5），收敛于BG的模p上同调的Rothenberg-Steenrod谱序列在E2项处坍缩. (2)对于（G，p>（E8，3）），Rothenberg-Steenrod谱序列在E2水平上不坍缩.此外，某些余挠积的计算等价于p阶循环群C的某些C-模的计算.我们希望这种计算能应用于与变分问题有关的环群分类空间的模p上同调的计算。