Higher homotopy algebras in transformation groups

该计划的长期目标是研究同伦格斯滕哈伯代数和其他结构直到更高的同伦，以及它们在拓扑和特别是变换群中出现的问题中的应用。本计划的起点是我最近在预印本“齐次空间的上同环”（arxiv:1907.04777）和“光滑环变异体的上同环和矩角复合体的商”（arxiv:1907.04791）中取得的突破。通过确定这些上同环，我回答了一个已经开放了50年的问题（齐次空间），并纠正了一个已经在文献中存在了20年的错误（矩角复数的部分商）。证明中的一个关键要素是同伦格斯滕哈伯代数（hga）的概念。迄今为止，齐次空间上同调中的杯积仅在给定系数环上2可逆的假设下成立。我的第一个目标是将它扩展到任意系数。另一个难题是如何将群的分类空间的证明推广到多项式上同调的任意空间。这两个目标将成为该方案的指导性问题。迄今为止所取得的进展是基于对同伦格斯滕哈伯代数和所谓的强同伦交换代数的改进研究。我有一些“普通”的证明，需要非常长的时间，但不是很有启发性的计算。然而，这些解的形式表明，这些公式背后可能潜藏着更深层次的结构。在这个方向上获得更好的理解是另一个目标。到目前为止，另一个关键因素是环面空间分类的hga形式结果。它已经扩展到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别。在群体行为的研究中，还会出现或预期出现同伦结构的其他实例，解决它们是进一步的目标。让我们提一下实环空间的情况，总秩猜想及其在自由环空间和弦拓扑中的应用。与J. Minác教授一起，我想把我最近的成果应用到伽罗瓦理论的问题上。对同伦结构进行人工计算很快就会变得单调乏味，因为公式的大小增加得非常快。该计划的第二个目标是开发软件工具来促进这些计算（并确定合适的计算机代数系统来实现这些工具）。

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).