Recherche en théorie de l'homotopie

同伦理论研究

• 批准号: RGPIN-2018-06133
• 负责人: Parent, PaulEugène
• 金额: $ 1.17万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675768
• 关键词: Recherche; en; th; orie; de

The major theme of this proposal is research in homotopy theory. My program will explore two main directions: (I) topological complexity and L.-S.-category; and (II) non-simply connected spaces. ***(I) Topological complexity The image that one should keep in mind is the one of a robotic arm positioned somewhere fixed in an assembly plant X. The problem is: is it possible to find a continuous path that the arm will follow between any two given points that the arm can reach, i.e., does the diagonal map d:X -> X x X admit an appropriate continuous section? Unfortunately the only assembly plant X for which the map d admits a global section is one with no obstruction (no pillard to hold up the roof, not even the robot itself!). Hence we have to settle for the next best thing: continuous local sections, i.e., the topological complexity, TC(X), of a space X is the least natural number k such that there exists a covering of X x X formed by k+1 open sets U1,,Uk+1, admitting continuous local sections si:Ui->XI of the classical evaluation fibration XI->X x X associated to the diagonal map d. In fact TC(X) is a special case of the sectional category of a fibration within the Lusternick-Schnirelmann category theory framework. The study of these L.S.-type inavariants have attracted massive amount of resources over the last 40 years and continues to this date to be a very dynamic subject with applications to homotopy theory, differential geometry, dynamical systems and engineering. It is ideal for HQP's in both pure and/or applied mathematics. My goal for the next five years is to pursue the study of these invariants and to incorporate to my research at least one Ph.D. student and two Master's students. ***(II) Non-simply connected spaces - Quillen and Sullivan constructed theories to algebraically model the rational homotopy type of simply-connected spaces. Unfortunately, both can't handle spaces like RP2. To remedy this situation, Bousfield and Kan proposed the following: Let X be a space admitting a universal cover UX and consider the associated fibration UX->X->B(pi1(X)) and apply fibrewise rationalization to obtain a new fibration UXo->X'->B(pi1(X)), where UXo is the usual rationalization of the simply-connected space UX. The total space X' of that fibration is the model for the rationalization of the original space X. The next step is to find an algebraic category to characterize X' in the spirit of Quillen and Sullivan. Gomez-Tato, Halperin and Tanré constructed an algebraic category of local systems that admitted minimal models and realization functors very much like Sullivan. The main drawback of their theory is that it only applies to spaces having a finite type universal cover. My goal is to expand these ideas to all universal covers by using the Quillen vision to model UXo and build those local systems. This direction is ideal to incorporate a post-doctoral fellow and at least one Ph.D. student.

该提案的主题是同伦理论的研究。我的计划将探索两个主要方向：（一）拓扑复杂性和L。S.-类;（II）非单连通空间。* （I）拓扑复杂度我们应该记住的是一个机器人手臂的形象，它位于装配工厂X的某个固定位置。问题是：是否有可能找到一个连续的路径，手臂将遵循之间的任何两个给定的点，手臂可以达到，即，对角映射d：X -> X x X是否允许一个适当的连续截面？不幸的是，地图d中唯一允许全局截面的装配厂X是一个没有障碍物的装配厂（没有支柱支撑屋顶，甚至没有机器人本身！）。因此，我们不得不退而求其次：连续的局部部分，即，空间X的拓扑复杂度TC（X）是最小的自然数k，使得存在由k+1个开集U1，Uk+1形成的X × X的覆盖，允许与对角映射d相关联的经典评估纤维化XI->X × X的连续局部部分si：U1->XI。事实上TC（X）是Lusternick-Schnirelmann范畴理论框架中纤维化的截面范畴的一个特例。对这些LS的研究-型不变在过去的40年里吸引了大量的资源，并继续到今天是一个非常动态的主题，应用同伦理论，微分几何，动力系统和工程。它是理想的HQP的在纯数学和/或应用数学。我在未来五年的目标是继续研究这些不变量，并将至少一个博士学位纳入我的研究。学生和两个硕士生。* （II）非单连通空间-奎伦和沙利文构建了理论来代数地模拟单连通空间的有理同伦类型。不幸的是，两者都不能处理像RP 2这样的空间。为了解决这个问题，Bousfield和Kan提出了如下的方法：设X是一个允许泛覆盖UX的空间，考虑相关的纤维化UX->X->B（pi 1（X）），并应用纤维化方法得到一个新的纤维化UXo-> X '->B（pi 1（X）），其中UXo是单连通空间UX的通常的有理化。该纤维化的总空间X'是原空间X的合理化的模型。下一步是找到一个代数范畴来刻画X'在奎伦和沙利文的精神。Gomez-Tato、Halperin和Tanré构造了一个局部系统的代数范畴，它承认极小模型和实现函子，非常像Sullivan。他们理论的主要缺点是它仅适用于具有有限型泛覆盖的空间。我的目标是通过使用Quillen的愿景来建模UXo并构建这些本地系统，将这些想法扩展到所有通用覆盖。这个方向是理想的纳入博士后研究员和至少一个博士学位。学生.