Problems in higher dimensional topology

Suny At Stony Brook, Stony Brook NY

Investigators

Abstract

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 let F denote a field of characteristic p. Any cellular group action h:GxK ---> K by G on a finite CW complex K gives rise to a chain complex C(*) over the group ring F(G). An "elemental chain subcomplex" of C(*) is any chain subcomplex E(*) such that for some integer i the boundary map 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 sum of 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 ring F(G). In toplogy (that field of mathematics to which this project is most closely associated) one studies the structure of spaces in a very loose manner. Examples of the spaces which topologists study occur everywhere ---- from theortical physics to objects occuring in everyday life such as a ball or donut. From the point of view of toplology all balls are the same (they have the same shape when considered as abstract toplological spaces); likewise any two donuts have the same "shape"; however a ball has a different "shape" than a donut (since a 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 problem has 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 associated algebraic objects are generally much easier to understand then are the spaces themselves. Jones has recently discovered what seems to be a new type 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.