Problems in higher dimensional topology

本课题将重点研究拓扑空间上有限群作用的一种（可能的）新型不变量。对于素数p，设G表示有限p群，设F表示特征为p的域。在有限CW复形K上的任何胞群作用：GxK—K by G，在群环F(G)上产生一个链复形c（*）。C（*）的“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）是原理F(G)-模之间的同构，如果j不等于i，i-1，则E(j)=0。C（*）的链子配合物D（*）称为C（*）的“最小核心”，如果C（*）是D（*）与C（*）的一些元素链子配合物的直接和，并且D（*）没有任何元素链复数直接和。在最近的工作中，Jones已经证明了最小核总是存在的，并且它的同构类型仅取决于群作用h的等变同伦类型。在未来的工作中，Jones计划关注群环f (G)上所有最小核的分类（直到同构）。在拓扑学（与本课题联系最为密切的数学领域）中，人们以一种非常松散的方式研究空间结构。拓扑学家研究的空间例子无处不在----从理论物理到日常生活中发生的物体，如球或甜甜圈。从拓扑的角度来看，所有球都是相同的（当它们被视为抽象拓扑空间时，它们具有相同的形状）；同样，任何两个甜甜圈都有相同的“形状”；然而，球的“形状”与甜甜圈不同（因为甜甜圈有一个洞，而球却没有洞）。拓扑学的主要目标是确定两个不同的空间何时具有相同的“形状”（这样的空间被称为“拓扑等效”）。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）与每个空间联系起来，如果两个不同的空间拓扑相等，那么它们所有已知的相关代数对象必须相等。这种方法非常成功，因为相关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相关的新型代数对象。目前，他正在尝试计算这个新的代数对象，并了解它与许多与空间相关的旧的众所周知的代数对象之间的关系。

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.