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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713313
• 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.

这一提议的主要主题是同伦理论的研究。我的程序将探索两个主要方向：(I)拓扑复杂性和L.-s-范畴；(Ii)非单连通空间。 (I)拓扑复杂性人们应该记住的图像是固定在装配厂X中的某个位置的机械臂的图像。问题是：是否有可能在手臂可以到达的任何两个给定点之间找到一条手臂将沿着的连续路径，即，对角线地图d：X-X是否包含适当的连续截面？不幸的是，地图d承认的唯一一家装配厂X是一个没有障碍物的装配厂(没有支撑屋顶的支柱，甚至连机器人本身都没有！)因此，我们不得不满足于次好的情况：空间X的连续局部截面，即拓扑复杂性TC(X)是最小自然数k，使得存在由k+1个开集U1，UK+1形成的X×X的覆盖，允许经典赋值纤维XI-&gt；X的连续局部截面si：ui-&gt；X x X与对角线映射d相关联。事实上，TC(X)是Lusternick-Schnirelmann范畴理论框架内的纤维的分段范畴的特例。在过去的40年里，对这些L.S.型变星的研究吸引了大量的资源，并一直是一个非常动态的学科，应用于同伦理论、微分几何、动力系统和工程。它是纯数学和/或应用数学中HQP的理想选择。我未来五年的目标是继续研究这些不变量，并将至少一名博士生和两名硕士生纳入我的研究。 (Ii)非单连通空间-Quillen和Sullivan构造了代数模型的有理同伦型单连通空间的理论。不幸的是，两者都不能处理像RP2这样的空间。为了纠正这种情况，Bousfield和Kan提出了如下建议：设X是容纳泛覆盖UX的空间，考虑伴随的纤维UX-&gt；X-&gt；B(pi1(X))，并应用纤维有理化法得到一个新的纤维UxO-&gt；X‘-&gt；B(pi1(X))，其中UxO是单连通空间UX的通常有理形式。该纤维的全空间X‘是原始空间X的有理化模型。下一步是寻找一个代数范畴来刻画X’，以体现Quillen和Sullivan的精神。Gomez-Tato，Halperin和Tanré构造了一个局部系统的代数范畴，它包含非常像Sullivan的极小模型和实现函子。他们理论的主要缺点是它只适用于具有有限类型万能覆盖的空间。我的目标是通过使用Quillen愿景来模拟UXO并建立那些本地系统，将这些想法扩展到所有通用覆盖范围。这个方向最适合包括一名博士后研究员和至少一名博士生。