Functorial Calculus and Manifolds

这个项目主要关注的是研究代数k理论所无法触及的流形拓扑的那些方面的长期目标。该方法是几种“函子演算”的结合。一项任务是研究这些结石之间的一些相互关系，以加强一般方法。这将涉及到庞加莱对偶复合体的一些基础工作。另一个长期的任务是研究代数k理论的二阶模拟；它是由流形现象的一种二次镇定性引起的。就短期而言，有一项工作是构造一种循环同调的二阶类似物，它应该与上述内容相关，因为循环同调与代数k理论相关。至少有一点希望，这将产生类似于众所周知的解析定义的Seiberg-Witten不变量的拓扑定义的四流形不变量。之所以使用“微积分”一词，是因为它与普通的函数微积分有一个吸引人的类比。当然，有时候解决数值问题的方法是把答案看作一个函数的值；借助强大的微积分工具，可以根据函数的性质计算出这个数。同样，关于某些几何上定义的物体的事实，有时最好的证明方法是将该物体看作函子的一个特定值，并使用某种函子演算的工具。这个比喻或许能显示出这部作品的某种味道；内容很难向非专业人士描述，因为大多数“几何”对象都是通过相当长的抽象概念链与日常现实联系在一起的。*＊*

9806981 Goodwillie This project is mainly concerned with the long-range goal of investigating those aspects of the topology of manifolds that lie beyond the reach of algebraic K-theory. The method is a combination of several kinds of "functor calculus." One task is to investigate some interrelationships between these calculi with an eye to strengthening the general method. This is expected to involve some foundational work on Poincare duality complexes. Another task for the long haul is to look at what may be called a second-order analogue of algebraic K-theory; it arises by a kind of quadratic stabilization of manifold phenomena. For the short haul there is the job of constructing a kind of second-order analogue of cyclic homology that should be related to the above as cyclic homology is related to algebraic K-theory. There is at least a slight hope that this would yield topologically defined invariants for four-manifolds that are akin to the well-known analytically defined Seiberg-Witten invariants. The word "calculus" is used because of an attractive analogy with the ordinary calculus of functions. Sometimes, of course, the way to settle a numerical question is to see the answer as a value of a function; by means of the powerful apparatus of calculus, properties of the function can lead to a computation of the number. Likewise, a fact about some geometrically defined object is sometimes best proved by viewing the object as a particular value of a functor and using the apparatus of some kind of functor calculus. This analogy may show something of the flavor of the work; the content is harder to describe to the non-expert, because most of the "geometric" objects in question are connected to everyday reality by rather long chains of abstract ideas. ***