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 MR, 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 A.1/–modules.
Acknowledgements The authors would like to thank Tony Bahri for the introduction and discussions on the KO \u2013theory of toric manifolds, and thank Anton Ayzenberg for the discussions on torsion-free small covers. Cai was supported by National Natural Science Foundation of China (grant no. 11801457). Choi was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1A2C2010989). Park was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2019R1G1A1007862).
