Extension Types of Infinite Symmetric Products

提案：DMS-0072356 PI：Jerzy Dydak首席研究员计划研究一种称为扩展维的维的一般理论。在这个理论中，dim（X）不是自然数，而是CW复形，dim（X）至多为K意味着K是X的绝对扩张。特别地，dim（X）至多为n-球面等价于X的覆盖维数至多为n。dim（X）=K表示K关于所有L最小使得dim（X）至多为L.结果表明无穷对称积在整个理论中起着至关重要的作用.它们的性质导致了一个具有自对偶的自然代数。这种自对偶性是紧空间与CW复形之间的几何对偶性的代数表示.所有等价于紧有限维CW复形的无穷对称积都被分类.在经典维数论中，人们试图把自然数n附加到一个自对偶性中.（或无穷大）到每一个空间。事实证明，自然数n只是一个替代品的n-球，并说，dim（X）至多n反映了空间X与n-球面之间的某种关系，我们可以把空间X与多面体K之间的这种关系推广为“X的维数至多为K”。例如，我们可以研究X的维数是否至多是射影平面，结果证明，这样构造的维数理论与拓扑学的主流有着密切的联系，特别是与同调代数和代数拓扑有着密切的联系。这个理论最有趣的方面之一就是对偶性，这是整个数学的一个基本思想。在最简单的层面上，它意味着我们不仅试图将维度（多面体）附加到空间上，而且还将空间附加到维度上。

Proposal: DMS-0072356PI: Jerzy DydakThe principal investigator plans to work on a general theory of dimension calledthe extension dimension. In this theory, dim(X) is not a natural number.Instead, dim(X) is a CW complex and dim(X) being at most Kmeans that K is an absolute extensor of X. In particular,dim(X) being at most the n-sphere is equivalent to the covering dimensionof X being at most n. dim(X)=K means that Kis minimal with respect to all L such that dim(X) is at most L.It turns out that infinite symmetric products play a crucial rolein the whole theory. Their properties lead to a natural algebrawith self-duality. That self-duality is an algebraic manifestationof the geometric duality between compact spaces and CW complexes.All infinite symmetric products equivalent to compactor finite-dimensional CW complexes are classified.In classical dimension theory one tries to attacha natural number n (or infinity) to every space.It turns out that the natural number n is simply a substitutefor the n-sphere and saying that dim(X) is at most n reflectsa certain relationship between the space X and the n-sphere.One can generalize the notion of dimension by consideringthe same relationship between the space X and a polyhedron Kand that is stated as "dimension of X is at most K". For example, one caninvestigate if the dimension of X is at most the projective plane.It turns out that the dimension theory constructed that wayis much closely connected to the mainstream of topology.In particular, one gets links to homological algebraand algebraic topology. One of the most interesting aspects ofthat theory is duality, a fundamental idea in the whole ofmathematics. At the simplest level it means that not onlyare we trying to attach dimension (a polyhedron) to a space,but also we attach a space to a dimension.