Research on higher dimensional polyhedra and operads

高维多面体与运算体的研究

• 批准号: 13640080
• 负责人: HEMMI Yutaka
• 金额: $ 2.18万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640080/
• 关键词: Hopf space; H-map; A_n-space; A_n-map; projective space; associahedron; homotopy group; Whitehead product; ポップ空間; quasi p-regular; unstable higher order operation; 高位ヒモトピー可換性; 準Cp-空間; mod p torus theorem; 球面のホモトピー群; タイリング空間

The summary of reserch results is as follows.1. We study conditions for mod p finite H-spaces to be quasi p-regular, and we give a generalization to the theorem of Kumpel on p-regulraity. Our result includes results by Harper, McClearly and Wilkerson.2. We define unstable p-th order mod p cohomology operations for odd primes p. Then, by using the operations we show that there are no H-spaces with certain cohomology.3. Williams gave a definition of higher order homotopy commutativity of the multiplication on associative H-spaces. We generalize his definition to higher homotopy assocative H-spaces. Then, by using this definition we give the mod p torus theorem for H-spaces with finitely generated cohomology.4. We show that there is no even degree generators except for degree eight and twenty in the mod 3 cohomology of finite H-spaces. We also determine the cohomology strucre of them as algebras.5. We consider the inbedding of an An-space X to its loop space of the projective n-1-space ΩP_<n-1>X. Then we study relations between an A_<n-1>-cohomology class x∈H^*(X;F) and its lift x^^^∈H^*(ΩP_<n-1>X;F), where F is a field.6. It is well known that if a map f: X → Y between H-spaces is an H-map then the map induces a map between their projective planes. Moreover, it is also known that if Y is homotopy associative then the converse holds. We show that the assumption of homotopy associativity is necessary in showing the converse. We also generalize the result to the case of A_n-spaces.7. It is proved that the moduli space of a punctured Rieman sphere has cellular decomposition. We consider a decomposition of the associahedron by using a homotopy theoretical property, and then by using the decomposition we construct a handle decomposition of the moduli space which is a dual decomposition of the original decomposition.

研究结果总结如下：1。研究了模p有限h空间是拟p正则的条件，并推广了Kumpel关于p正则的定理。我们的结果包括Harper， mcclear和wilkerson的结果。定义了奇数素数p的不稳定p阶模p上同调运算，并利用这些运算证明了不存在具有一定上同调的h空间。Williams给出了结合h空间上乘法的高阶同伦交换性的定义。我们将其定义推广到高同伦关联h空间。然后，利用这个定义，给出了有限上同调h空间的模p环面定理。证明了在有限h空间的模3上同调中，除了8次和20次外没有偶数次产生子。并确定了它们作为代数的上同调结构。我们考虑一个an空间X嵌入到它的n-1射影空间ΩP_<n-1>X的循环空间。然后研究了A_<n-1>-上同类x∈H^*（x；F）与它的升力x^^∈H^*（ΩP_<n-1> x；F）之间的关系，其中F是一个域。众所周知，如果在h空间之间的映射f: X→Y是一个h映射，那么该映射在它们的投影平面之间会产生一个映射。此外，我们还知道，如果Y是同伦结合的，则反之成立。我们证明了同伦结合性的假设是证明逆性的必要条件。我们也将结果推广到a_n空间的情况。证明了穿孔Rieman球的模空间具有细胞分解性。利用同伦的理论性质考虑了共轭面体的分解，然后利用该分解构造了模空间的柄分解，该柄分解是原分解的对偶分解。

项目成果

Y.Kamiya, K.Shimomura: "The Picard group on E(2)-localized category"Contemporary Math.. (印刷中).

Y.Kamiya、K.Shimomura：“E(2) 本地化类别的皮卡德小组”当代数学..（正在出版）。

Y.Hemmi: "H-spaces as direct product factors of loop spaces"Topology and its Applications. (印刷中).

Y. Hemmi：“H 空间作为循环空间的直接乘积因子”拓扑及其应用（正在出版）。

Y.Kawamoto: "Higher homotopy commutativity of H-spaces with finitely generated co-homology"Pacific J. Math.. 204. 145-161 (2002)

Y.Kawamoto：“具有有限生成同调性的 H 空间的更高同伦交换性”Pacific J. Math.. 204. 145-161 (2002)

K. Morisugi and J. Mukai: "The Whitehead square of a lift of the Hopf map to a mod 2 Moore space"J. Math. Kyoto Univ.. Vol. 42. 331-336 (2002)

K. Morisugi 和 J. Mukai：“Hopf 升力的 Whitehead 平方映射到 mod 2 摩尔空间”J.

K.Shimomura, X.Wang: "The homotopy groups π_*(L_2S^0) at the prime 3"Topology. (印刷中).

K.Shimomura、X.Wang：“素数 3 处的同伦群 π_*(L_2S^0)”拓扑（正在出版）。