Higher homotopy algebras in transformation groups

变换群中的高等同伦代数

• 批准号: RGPIN-2020-06458
• 负责人: Franz, Matthias
• 金额: $ 1.31万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741225
• 关键词: Higher; homotopy; algebras; transformation; groups

The long-term goal of the programme is to study homotopy Gerstenhaber algebras and other structures up to higher homotopies as well as their applications to problems arising in topology and in transformation groups in particular. Starting point of the programme is the recent breakthrough achieved in my preprints "The cohomology rings of homogeneous spaces" (arxiv:1907.04777) and "The cohomology rings of smooth toric varieties and quotients of moment-angle complexes" (arxiv:1907.04791). By determining these cohomology rings, I answer a question that has been open for 50 years (homogeneous spaces) and correct a mistake that has been in the literature for 20 years (partial quotients of moment-angle complexes). A key ingredient in the proofs is the notion of a homotopy Gerstenhaber algebra (hga). So far, the cup product in the cohomology of a homogeneous space is only established under the assumption that 2 is invertible in the given coefficient ring. My first objective is to extend this, ideally to arbitrary coefficients. Another hard problem is to generalize the proof obtained so far from classifying spaces of groups to arbitrary spaces with polynomial cohomology. These two objectives will serve as a guiding problems for the programme. The progress obtained so far rests on a refined study of homotopy Gerstenhaber algebras and also of so-called strongly homotopy commutative (shc) algebras. I have "pedestrian" proofs that require extremely long, but not very illuminating calculations. However, the form of the solutions suggest that there might be a deeper structure lurking behind the formulas. Obtaining a better understanding in this direction is another objective. Another crucial ingredient so far is an hga formality result for classifying spaces of tori. It has already been extended to Davis-Januszkiewicz spaces. I want to study further extensions of hga formality or possibly weaker notions to classifying spaces of other groups or more general classes of spaces. There are also other instances where up-to-homotopy structures appear or are expected to appear in the study of group actions, and addressing them are further objectives. Let us mention the case of real toric spaces, the toral rank conjecture and applications to free loop spaces and string topology. With Prof. J. Minác I want to apply my recent results to problems in Galois theory. Manual calculations with up-to-homotopy structures soon become tedious because the size of the formulas increases very quickly. A secondary objective of the programme is to develop software tools to facilitate these computations (and to identify suitable computer algebra systems to implement the tools).

该计划的长期目标是研究同伦Gerstenhaber代数和其他结构，以更高的同伦以及它们的应用所产生的问题，在拓扑结构和特别是在变换群。该计划的起点是最近取得的突破，我的预印本“上同调环的齐次空间”（arxiv：1907.04777）和“上同调环的光滑环面品种和concurents的时刻角复合”（arxiv：1907.04791）。通过确定这些上同调环，我回答了一个问题，已经开放了50年（齐次空间），并纠正了一个错误，已在文献中20年（部分矩角复形）。证明中的一个关键要素是同伦Gerstenhaber代数（hga）的概念。到目前为止，齐型空间的上同调中的杯积仅在给定系数环中2可逆的假设下成立。我的第一个目标是将其推广到任意系数。另一个困难的问题是推广到目前为止得到的证明从分类空间的群体，以任意空间的多项式上同调。这两个目标将作为方案的指导问题。到目前为止所取得的进展依赖于同伦Gerstenhaber代数和所谓的强同伦交换（shc）代数的精细研究。我有“行人”证明，需要非常长的时间，但不是很有启发性的计算。然而，解的形式表明，公式背后可能隐藏着更深层次的结构。在这方面取得更好的理解是另一个目标。到目前为止，另一个关键因素是hga对环面空间进行分类的形式结果。它已经被推广到Davis-Januszkiewicz空间。我想研究hga形式的进一步扩展，或者可能更弱的概念来分类其他群的空间或更一般的空间类。在群作用的研究中，也有上同伦结构出现或预期会出现的其他例子，解决这些问题是进一步的目标。让我们提到真实的复曲面空间的情况、toral秩猜想以及对自由循环空间和弦拓扑的应用。我想和米纳克教授一起把我最近的成果应用到伽罗瓦理论中的问题上。由于公式的大小增加得非常快，因此使用上至同伦结构的手动计算很快变得乏味。该方案的第二个目标是开发软件工具，以促进这些计算（并确定适当的计算机代数系统来实施这些工具）。