Mathematical Sciences: Function Complexes And The Steenrod Algebra In Homotopy Theory

威尔克森、史密斯和麦克卢尔这三位研究者参与了同伦理论中三个相互关联的一般调查。Wilkerson研究的一个主要目标是完成有限循环空间的分类，作为紧李群理论的推广。工具是函数复数和计算，Steenrod代数可以追溯到亚当斯-威尔克森、米勒、卡尔松和兰尼斯的工作。部分工作将关于群作用和不动点的已知事实推广到同伦不动点。史密斯寻求对莫拉瓦K理论进行明确的构建。这些同调理论在Devinatz-Hopkins-Smith幂零定理中起着至关重要的作用。以前的构造依赖于几何技术，很难用同伦来解释。由此产生的分类空间实际上只能“达到同伦”，留下了许多关于它们的性质的问题。最后，在与S·杰克夫斯基和R·奥利弗的合作中，麦克卢尔试图找到浸入猜想的简化证明。他的另外三个问题都涉及到拓扑Hochschild同调理论的某些方面。这三个部分的细节各不相同，但都涉及将几何信息归结为用于计算的主题，或完善用于此目的的主要代数工具之一。涉及的几何信息的性质是困难的症结所在。虽然关于长度、面积、角度、体积等的问题实际上需要归结为计算，但它与几何对象的拓扑属性有很大的不同。这些属性包括连通性(完好无损)、多节、无洞等。所有对这些性质的系统研究，例如，如何区分两个几何物体在这些性质中的一个是否真的不同，或者仅仅是表面上的不同，或者如何对可能发生的各种不同进行分类，所有这些只有在归结为计算的问题时才被真正理解和掌握。

The three investigators, Wilkerson, Smith, and McClure, are involved in three connected general investigations in homotopy theory. A major goal of Wilkerson's research is to complete the classification of finite loop spaces as a generalization of compact Lie group theory. The tools are function complexes and calculations with the Steenrod algebra going back to work of Adams- Wilkerson, Miller, Carlsson, and Lannes. Part of the work generalizes known facts about group actions and fixed points to homotopy fixed points. Smith seeks an explicit construction of the Morava K-theories. These homology theories play a vital role in the Devinatz-Hopkins-Smith nilpotence theorem. Previous constructions rely on geometric techniques which are hard to interpret homotopically. The resulting classifying spaces can really be constructed only "up to homotopy," leaving many questions about their nature open. Finally, in joint work with S. Jackowski and R. Oliver, McClure intends to try to find a simplified proof of the immersion conjecture. His other three problems all involve some aspect of the theory of topological Hochschild homology. The details of these three parts vary, but all are concerned either with reducing geometric information to a subject for calculation or to perfecting one of the principal algebraic tools used for this purpose. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation.