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

这三名调查人员威尔克森、史密斯和麦克卢尔是 参与了三个有关同伦的一般性研究 理论 威尔克森研究的主要目标是完成 有限圈空间作为紧空间的推广的分类 李群理论 这些工具是功能复合体， 计算与斯廷罗德代数回到工作的亚当斯- 威尔克森，米勒，卡尔森，和兰尼斯。 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 史密斯寻求一个明确的结构， 摩拉瓦K理论 这些同源性理论发挥了至关重要的作用， Devinatz-Hopkins-Smith零值定理 先前 建筑依赖于几何技术， 同伦解释 所得到的分类空间可以 真的只能构造"到同伦"，留下了许多问题 关于他们的本性的开放。 最后，在与S.雅科夫斯基 和r.奥利弗，麦克卢尔打算试图找到一个简化的证明， 沉浸猜想 他的其他三个问题都涉及到 拓扑Hochschild同调理论的某些方面。 这三个部分的细节各不相同，但都是有关的 或者通过减少对象的几何信息， 计算或完善一个主要的代数工具 用于此目的。 几何信息的本质 问题的症结在于。 虽然关于 长度、面积、角度、体积等等，几乎都在呼喊着， 如果我们把它简化为计算，它与我们所知道的大不相同。 几何物体的拓扑性质。 这些是 诸如连通性（所有的都是一体的）之类的属性， 结，没有洞，等等。 所有系统研究 例如，如何判断两个几何 对象在这些属性之一方面确实不同，或者 只是表面上的不同，或者如何分类的品种， 可能发生的差异，所有这些都只是真正的 当它们被简化为 计算.

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.