New Approaches to Global Homotopy Theory

全局同伦理论的新方法

• 批准号: 9704761
• 负责人: Gregory Arone
• 金额: $ 3.52万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 1999-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704761&HistoricalAwards=false
• 关键词: New; Approaches; Global; Homotopy; Theory

9704761 Arone The investigator's general area of specialization is algebraic topology. His recent work has been concerned with achieving a better understanding of the global structure of homotopy theory. This work combines two hitherto disjoint approaches, the "chromatic approach," and the "calculus of functors" approach. In joint work with M. Mahowald, the investigator examined the Goodwillie derivatives of unstable homotopy theory and used them to prove a new result about unstable homotopy of spheres. In other work, the investigator studied the Taylor polynomials (as opposed to derivatives) of unstable homotopy theory and provided an explicit space-level description of these. One of the key ingredients of the investigator's contribution to his joint work with Mahowald was a combinatorial analysis of the poset of partitions of a finite set, which turned out to have consequences for the chromatic filtration of spheres. Further investigation of the combinatorics involved has led the investigator to discover seemingly new and mysterious recurrence patterns. He believes that some of the deep periodic phenomena in homotopy theory can be studied via this combinatorial periodicity. In very recent work, the investigator utilized this combinatorial periodicity to construct certain "self maps" on the derivatives of unstable homotopy theory, which imply, among other things, the existence of finite complexes whose cohomology realizes certain finite subalgebras of the Steenrod algebra. The investigator believes that these finite complexes will be useful for localization and periodicity. In the investigator's area of Algebraic Topology, he is interested in how to analyze the space of continuous functions between two topological spaces. It turns out that an analogue of the theory of Taylor series can be developed, and one can study the space of continuous maps via its "polynomial approximations" (an idea due to T. Goodwillie). The investigato r combined this theory with more classical methods in topology, and he used it (in joint work with M. Mahowald) to prove a new result on the global structure of an important algebraic tool of the trade, the so-called unstable homotopy of spheres. He now intends to use the aformentioned polynomial approximations to construct certain spaces with interesting (cohomological) properties, spaces whose mere existence should lead to further insights in topology. His work has a strong combinatorial flavor (specifically, one of the main gadgets in his work is the lattice of partitions of a finite set), and he hopes that some of his results will be of interest to combinatorialists. ***

9704761 Arone 调查员的一般专业领域是代数拓扑。 他最近的工作致力于更好地理解同伦理论的整体结构。 这项工作结合了两种迄今为止不相交的方法，即“色法”和“函子微积分”方法。 在与 M. Mahowald 的合作中，研究人员检查了不稳定同伦理论的 Goodwillie 导数，并用它们证明了关于球体不稳定同伦的新结果。 在其他工作中，研究人员研究了不稳定同伦理论的泰勒多项式（而不是导数），并提供了它们的明确的空间级描述。 研究人员与马霍瓦尔德共同工作的贡献之一是对有限集划分偏序集的组合分析，结果证明这对球体的色过滤产生了影响。 对所涉及的组合学的进一步研究使研究人员发现了看似新的和神秘的复发模式。 他认为同伦论中的一些深层周期现象可以通过这种组合周期性来研究。 在最近的工作中，研究者利用这种组合周期性在不稳定同伦理论的导数上构建了某些“自映射”，这意味着有限复形的存在，其上同调实现了 Steenrod 代数的某些有限子代数。 研究人员认为，这些有限复合体对于定位和周期性很有用。 在代数拓扑研究领域，他对如何分析两个拓扑空间之间的连续函数空间感兴趣。 事实证明，可以开发泰勒级数理论的类似物，并且可以通过其“多项式近似”（T. Goodwillie 提出的想法）来研究连续映射的空间。 研究人员将该理论与更经典的拓扑方法相结合，并用它（与 M. Mahowald 合作）证明了该行业重要代数工具的全局结构的新结果，即所谓的球体不稳定同伦。 他现在打算使用上述多项式近似来构造某些具有有趣（上同调）属性的空间，这些空间的存在本身就应该导致对拓扑学的进一步见解。 他的工作具有强烈的组合味道（具体来说，他工作中的主要小玩意之一是有限集的划分格子），他希望他的一些结果能够引起组合主义者的兴趣。 ***