Mathematical Sciences: Homotopy Theory of Spectra

数学科学：谱的同伦论

• 批准号: 9505130
• 负责人: Jeffrey Smith
• 金额: $ 5万
• 依托单位: Purdue Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9505130&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Homotopy; Theory; Spectra

9505130 Smith Spectra have been important in homotopy theory for many years. Recently the investigator has constructed a category of spectra with a symmetric monoidal smash product. The construction is elementary and will provide an introduction to spectra which will not scare away the novice. The strict associativity of the smash product means that many other definitions regarding spectra can be greatly simplified. For example, one can easily discuss ring spectra, commutative ring spectra, module spectra. The investigator is using this new definition of ring spectra in the study of topological Hochschild homology and topological cyclic homology. Spectra have been developed over the last thirty years as a bridge between the geometric problems that the algebraic topologist studies and the algebraic techniques which he uses. The earliest examples were given by Rene Thom in his work on the cobordism classification of manifolds, where he showed that manifolds, the geometric objects common to mathematics and physics, are classified up to cobordism by certain spectra. Over the years, as one needed spectra having more and more properties, many different constructions have been given. These constructions have became increasingly complicated, until only a few experts could understand them. However, recently the investigator has found a much easier way to construct spectra that nonetheless have all the properties one needs for the applications. This makes the field far more accessible, especially to graduate students, and should lead to much fuller appreciation of this powerful tool. ***

多年来，9505130史密斯光谱在同型理论中一直很重要。 最近，研究者构建了具有对称单型粉碎产品的光谱类别。 该构建是基本的，将为光谱提供介绍，不会吓到新手。 Smash产品的严格关联意味着可以大大简化有关光谱的许多其他定义。 例如，人们可以轻松地讨论环光谱，交换环光谱，模块光谱。 研究者在拓扑Hochschild同源性和拓扑循环同源性研究中使用了环光谱的新定义。 在过去的三十年中，光谱是作为代数拓扑学家研究的几何问题与他使用的代数技术之间的桥梁开发的。 最早的例子是Rene Thom在其关于歧管的恢复性分类的工作中给出的，在那里他表明，歧管（数学和物理学共同的几何对象）被某些光谱分类为COBORDISM。 多年来，由于需要越来越多的属性，因此提供了许多不同的结构。 这些结构变得越来越复杂，直到只有少数专家才能理解它们。 但是，最近，研究人员发现了一种更容易的方法来构建光谱，尽管如此，该光谱仍然具有某种应用所需的所有属性。 这使得该领域更容易获得，尤其是对于研究生而言，应该对此强大的工具产生更加的认识。 ***