Research on the higher homotopy commutativity and higher dimensional polytopes

研究结果总结如下：1。将Williams给出的关于高同伦组合h空间的乘法的高同伦交换性的定义推广到高同伦组合h空间。然后给出了有限生成上同环的h空间的模p环面定理。结果显示在T.ran.Amer.Math.Soc.356.2中。研究了Williams引入的c_n结构的高同伦交换性的变化。然后考虑了具有高同伦交换性的多面体。特别地，我们考虑了由固定shuffle的逆组成的第n个对称群的子集，并证明了由这个子集给出的表示高同伦交换性的多映体称为shuffle多映体。此外，我们还证明了用于证明Williams的c_n空间是一个拟c_n空间的分解与Kapranov和Voevodsky3给出的分解是相同的。我们得到了一个与Devados的镶嵌相对应的穿孔黎曼球的柄分解。通过扩展Boardman Vogt.4的关联面分解，给出了句柄分解。如果h空间之间的映射是一个h映射，那么它就会导出h空间的投影空间之间的映射。在映射的目标空间是同伦结合的假设下，Converse也成立。证明了在证明其逆性时，需要假设目标空间的同伦并置性。我们也将这个结果推广到a_n空间之间的映射。结果出现在广岛数学35.5。研究了对奇素数p具有高同伦交换性的有限a - p空间的模p上同调的化简幂运算的作用。研究结果将发表在《几何与拓扑》专著上。

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.