The $KO$-theory of toric manifolds

Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an $n$-dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they 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.