Homotopical Methods in Manifold Theory

PI将研究流形代数拓扑中的各种问题。研究的领域是等变同局部交理论、高Reidemeister扭转、庞加莱对偶空间周期族的构造以及光滑嵌入空间的多重分离问题。在这些项目中所采用的方法都有一个共同的线索，包括等变和纤维同伦理论。拟议中的活动将加强5所大学（布朗大学、圣母大学、布法罗大学、阿尔图纳大学和韦恩州立大学）个人之间的数学合作。它还将促进韦恩州立大学代数拓扑学研究生的培训。“流形”是满足齐次性的拓扑空间。局部来说，所有流形都是相似的，因为在任何一点上都可以看到欧几里得空间的副本。流形的整体结构使它们成为有趣的研究对象。通常，代数拓扑学家通过给流形分配某些称为“不变量”的代数量来研究流形，这些代数量测量流形的全局拓扑结构。具有不同不变量的流形可以彼此区分。流形在物理、化学和生物学中自然出现，作为一组适当的“漂亮”代数方程的解的空间，模拟了研究的科学对象（时空、原子、动力系统等）。流形在数学中起着核心作用。通常情况下，流形的数学问题可以用流形相关空间间的参数化函数族来表述。同伦理论是一门旨在解决这类函数族问题的学科。PI提出研究一类可以从同伦理论工具箱中分析的流形问题。

The PI will work on diverse problems in the algebraic topology of manifolds. The area of investigations are equivariant homotopical intersection theory, higher Reidemeister torsion, the construction of periodic families of Poincare duality spaces, and multiple disjunction problems for spaces of smooth embeddings. The methods to be employed on each of these projects have a common thread, involving equivariant and fiberwise homotopy theory. The proposed activity will strengthen collaborative mathematics between individuals at 5 universities (Brown, Notre Dame, Buffalo, Altoona and Wayne State). It will also foster the training of graduate students in algebraic topology at Wayne State.A "manifold" is a topological space that satisfies a homogeneity property. Locally speaking, all manifolds are alike in that at any point one sees a copy of Euclidean space. It is the global structure of manifolds that makes them interesting objects of study. Typically, algebraic topologists study manifolds by assigning certain algebraic quantities, called "invariants," to them, which measure their global topological structure. Manifolds having different invariants can then be distinguished from one another. Manifolds arise naturally in physics, chemistry and biology as spaces of solutions of a suitably "nice" set of algebraic equations modeling the scientific object of study (space-time, atoms, dynamical systems, etc.) . Manifolds play a central role in mathematics. It is often the case that mathematical questions about manifolds can be formulated in terms of parametrized families of functions between spaces associated with manifolds. Homotopy theory is a subject designed to tackle questions about such families of functions. The PI proposes to study certain kinds of manifold questions which can be analyzed from the homotopy theoretic toolbox.