Extension Types of Infinite Symmetric Products

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

Proposal: DMS-0072356 PI: Jerzy Dydak The principal investigator plans to work on a general theory of dimension called the 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 K means that K is an absolute extensor of X. In particular, dim(X) being at most the n-sphere is equivalent to the covering dimension of X being at most n. dim(X)=K means that K is minimal with respect to all L such that dim(X) is at most L. It turns out that infinite symmetric products play a crucial role in the whole theory. Their properties lead to a natural algebra with self-duality. That self-duality is an algebraic manifestation of the geometric duality between compact spaces and CW complexes. All infinite symmetric products equivalent to compact or finite-dimensional CW complexes are classified. In classical dimension theory one tries to attach a natural number n (or infinity) to every space. It turns out that the natural number n is simply a substitute for the n-sphere and saying that dim(X) is at most n reflects a certain relationship between the space X and the n-sphere. One can generalize the notion of dimension by considering the same relationship between the space X and a polyhedron K and that is stated as "dimension of X is at most K". For example, one can investigate if the dimension of X is at most the projective plane. It turns out that the dimension theory constructed that way is much closely connected to the mainstream of topology. In particular, one gets links to homological algebra and algebraic topology. One of the most interesting aspects of that theory is duality, a fundamental idea in the whole of mathematics. At the simplest level it means that not only are we trying to attach dimension (a polyhedron) to a space, but also we attach a space to a dimension.