Abstract: Toric spaces have associated to them Borel constructions with respect to the actions of various tori. The cohomology (corresponding to complex-oriented theories) can be related to Stanley-Reisner rings, rings of piecewise polynomials and associated quotients. Examples exist for which the distinction between a ring of piecewise polynomials and the Stanley-Reisner ring mirrors that between the true orbit space and the Borel construction. The discussion will include also a short overview of work in progress on the KO-theory of these spaces.

The material is based on joint work with Matthias Franz and Nigel Ray and touches on additional joint work with Martin Bendersky, Fred Cohen and Sam Gitler.