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Research on the higher homotopy commutativity and higher dimensional polytopes

高同伦交换性和高维多胞形的研究

基本信息

批准号：
15540083
负责人：
HAMMI Yutaka
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：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

The summary of research results is as follows.1.We extend the definition of the higher homotopy commutativity of the multiplications given for associative H-spaces by Williams to the one for higher homotopy associative H-spaces. Then we gave the mod p torus theorem the H-spaces with finitely generated cohomology ring. The result appears in T.ran.Amer.Math.Soc.356.2.We studied variety of higher homotopy commutativity related to the C_n-structure introduced by Williams. Then we considered polytopes representing the higher homotopy commutativity. In particular, we considered the subset of the n-th symmetric group consisting of the inverse of fixed shuffles, and showed that the polytope which represents the higher homotopy commutativity given by this subset is so-called shuffle polytope. Moreover, we showed that the decomposition used to show that a C_n-space of Williams is a quasi C_n-space by Hemmi is the same as the one by given by Kapranov and Voevodsky3.We obtained a handle decomposition of a punctured Rieman sphere corresponding to the tessellation by Devados. The handle decomposition is given by extending the decomposition of associahedra by Boardman Vogt.4.If a map between H-spaces is an H-map then it induces a map between projective spaces of the H-spaces. Converse also holds under the assumption that the target space of the map is homotopy associative. We showed that the assumption of the homotopy assocaativity of the target space is needed when we show the converse. We also extend this results for maps between A_n-spaces. The result appears in Hiroshima J Math 35.5.We studied the action of the reduced power operations on the mod p cohomology of finite A_p-spaces with higher homotopy commutativity for an odd prime p. The result is to appear in Geometry and Topology Monographs.

1.将Williams关于结合H-空间乘法的高同伦交换性的定义推广到关于高同伦结合H-空间的定义。然后给出了具有有限生成上同调环的H-空间的mod p环面定理。这一结果发表在T.ran.amer.Math.Soc.356.2上。我们研究了Williams引入的与C_n-结构有关的各种高同伦交换性。然后，我们考虑了代表较高同伦交换性的多面体。特别地，我们考虑了由固定改组的逆组成的第n个对称群的子集，并证明了由这个子集所给出的表示较高的同伦交换性的多面体称为混洗多面体。此外，我们还证明了由Hemmi证明Williams的C_n-空间是拟C_n-空间的分解与Kapranov和Voevodsky给出的相同。4.如果H-空间之间的映射是H-映射，则它诱导出H-空间的射影空间之间的映射。在映射的目标空间是同伦结合的假设下，逆也成立。反之，我们证明了目标空间的同伦关联性假设是必要的。我们还将这一结果推广到A_n-空间之间的映射。研究了对奇素数p具有较高同伦交换性的有限A_p-空间的模p上同调的作用，结果发表在《几何与拓扑专著》上。