We exhibit an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes of associated complex line bundles. We use the existence of such a splitting to assist our computation of real and complex K-groups. Moreover, we compute the Sq2-homology of bounded flag manifolds to make use of relevant Atiyah-Hirzebruch spectral sequence of KO-theory. | Turk J Math 26 (2002) , 447 – 463. ¨ ITAK ˙ c TUB KO-groups of Bounded Flag Manifolds Yusuf Civan Abstract We exhibit an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes of associated complex line bundles. We use the existence of such a splitting to assist our computation of real and complex K-groups. Moreover, we compute the Sq2 -homology of bounded flag manifolds to make use of relevant Atiyah-Hirzebruch spectral sequence of KO-theory. Key Words: Bounded flag manifolds, Sq2 -homology, KO-theory, toric variety, stably complex structure 1. Introduction As explained by Buchstaber and Ray [2], the geometry of bounded flag manifolds plays an important role in complex cobordism, namely that they generate the double cobordism ring ΩDU ∗ . These objects were originally constructed by Bott and Samelson, and were introduced into complex cobordism by Ray [7]. Bounded flag manifolds also fit into the settings of toric geometry. We showed in [3] that they are smooth projective toric varieties associated to fans arising from crosspolytopes. By analogy with many stable splitting phenomena discovered in the 80s, we will carry out a programme of exhibiting an appropriate suspension of bounded flag manifolds as a wedge sum of Thom complexes of associated complex line bundles. We then use the existence of such a splitting to assist our computation of real and complex K-groups. More generally, Bahri and Bendersky[1] have announced a method for computing KOgroups of any toric manifold via the relevant Adam spectral sequence. Our first step overlaps with theirs in that we compute the Sq 2 -homology of bounded flag manifolds. We begin with introducing some notations. We follow combinatorial convention by writing [n] for the set of natural numbers {1, 2, . . . , n}, and an interval in the poset [n] 447 ˙ CIVAN has the form [a, b] for some 1 ≤ a ≤ b ≤ n which consists of all k satisfying a ≤ k ≤ b. Throughout, ω1 , . . . , ωn+1 will denote the .
