Matrix Invariants of Topological Spaces
by
Stephen Watson
York University

Both the Sorgenfrey line and the Alexandroff duplicate are assymetric because a point x can be close to y while y is far from x. But we can investigate this assymetry more finely by considering three points x, y, z and studying which (ordered) pairs of points can be close while the other pairs are farther apart. To do this rigorously, we introduce the notion of a topological space avoiding a 3x3 matrix of real numbers. This concept then leads to a classification of the relevant 3x3 matrices which turns out to be somewhat finer than Jordan canonical form. Eduardo Santillan has recently showed that there are exactly 116 extremal types of these 3x3 matrices and calculated many numerical invariants such as eigenvalues for them. We can then prove theorems that show that, for each two spaces X and Y among spaces like beta N - N, omega_1, the Sorgenfrey line, the double arrow space, and the real line, there is a type (and often a Jordan canonical form) which X avoids while Y does not. At a coarser level, one can even prove that the Sorgenfrey line (unlike most spaces) must a non-real eigenvalue. This is a preliminary report and the nature of this unexpected interaction between topology and linear algebra is still a mystery. This work has been done jointly with Hans-Peter Kunzi and Eduardo Santillan.

Date received: February 13, 1998