Problems in higher dimensional topology

其中几个主题，这一项目将集中在一个（可能）新类型的不变量有限群行动的拓扑空间。 设G表示有限p-群，对某个素数p，F表示特征为p的域，G作用于有限CW复形K上的任何胞腔群h：GxK-K都在群环F（G）上产生链复形C（*）. C（*）的一个"元素链子复形"是任意链子复形E（*），使得对于某个整数i，边界映射E（i）-E（i-1）是主F（G）-模之间的同构，并且如果j不等于i，i-1，则E（j）= 0。 C（*）的一个链子复形D（*）称为C（*）的"极小核"，如果C（*）是D（*）与C（*）的一些元素链子复形的直和，并且如果D（*）没有任何元素链复形的直和项. 在最近的工作琼斯表明，一个最小的核心总是存在的，它的同构类型只取决于等变同伦类型的群作用h。 在未来的工作中，琼斯计划把重点放在分类（直到同构）的所有最小的核心在组环F（G）。在拓扑学（与本课题联系最紧密的数学领域）中，人们以一种非常松散的方式研究空间的结构。 拓扑学家研究的空间的例子无处不在-从理论物理到日常生活中发生的物体，如球或甜甜圈。 从拓扑学的角度来看，所有的球都是相同的（当被认为是抽象的拓扑空间时，它们具有相同的形状）;同样，任何两个甜甜圈都具有相同的“形状”;然而，球的“形状”不同于甜甜圈（因为甜甜圈有一个洞，但没有球有一个洞）。 拓扑学的主要目的是确定两个拓扑网空间何时具有相同的"形状"（这样的空间被称为"拓扑等价"）。一百多年来，解决这个问题的一个重要方法是将代数对象（如数、群、环等）联系起来。以这样的方式，每个空间，如果两个不同的空间是拓扑等价的，那么所有已知的相关代数对象必须是平等的。 这种方法非常成功，因为相关联的代数对象通常比空间本身更容易理解。 琼斯最近发现了一种与空间有关的新型代数对象。 目前，他正试图计算这个新的代数对象，并了解它是如何与许多旧的众所周知的代数对象相关联的空间。

One of the several subjects which this project will focus on is a(possibly) new type of invariant for finite group actions on toplogical spaces. Let G denote a finite p-group, for some prime integer p, and letF denote a field of characteristic p. Any cellular group actionh:GxK --- K by G on a finite CW complex K gives rise to a chain complexC(*) over the group ring F(G). An "elemental chain subcomplex" of C(*)is any chain subcomplex E(*) such that for some integer i the boundarymap E(i) --- E(i-1) is an isomorphism between principle F(G)-modules, and E(j)=0 if j is not equal to i,i-1. A chain subcomplex D(*) of C(*) is called a "minimal core" for C(*) if C(*) is the direct sumof D(*) and some elemental chain subcomplexes of C(*), and if D(*) does not have any elemental chain complex direct summands. In recent work Jones has shown that a minimal core always exists and that its isomorphism type depends only on the equivariant homotopy type of the group action h. In future work Jones plans to focus on the classification (up to isomorhism) of all minimal cores over the group ringF(G). In toplogy (that field of mathematics to which this project is most closely associated) one studies the structure of spaces in a veryloose manner. Examples of the spaces which topologists study occur everywhere ---- from theortical physics to objects occuring in everydaylife such as a ball or donut. From the point of view of toplologyall balls are the same (they have the same shape when considered asabstract toplological spaces); likewise any two donuts have the same"shape"; however a ball has a different "shape" than a donut (sincea donut has a hole but no ball has a hole in it). The main object of topology is the determination of when two differenet spaces have the same "shape" (such spaces are said to be "topologically equivalent").For well over one hundered years an important approach to this problemhas been to associate algebraic objects (such as numbers, groups, rings,etc.) to each space in such a way that if two different spaces are topologically equivalent then all their known associated algebraic objects must be equal. This approach has been very successful because the associatedalgebraic objects are generally much easier to understand then are the spaces themselves. Jones has recently discovered what seems to be a newtype of algebraic object associated to spaces. Currently he is trying calculate this new algebraic object and to understand how it is related to the many older well known algebraic objects associated to spaces.