The KO-theory of toric manifolds

Toric manifolds, a topological generalization of smooth projective toric va rieties, are determined by an n-dimensional simple convex polytope and a fu nction from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiew icz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the t oric variety case. Recently it has been shown by Buchstaber and Ray that th ey generate the complex cobordism ring. We use the Adams spectral sequence to compute the KO-theory of all toric manifolds and certain singular toric varieties.