On the $KO$–groups of toric manifolds

Abstract

We consider the real topological $K$–groups of a toric manifold $M$, which turns out to be closely related to the topology of the small cover $M_{ℝ}$, the fixed points under the canonical conjugation on $M$. Following the work of Bahri and Bendersky (2000), we give an explicit formula for the $KO$–groups of toric manifolds, and then we characterize the two extreme classes of toric manifolds according to their mod $2$ cohomology groups as $\mathcal{𝒜}\left(1\right)$–modules.