We describe the K-ring of the classifying space of the generalized quaternion group in terms of generators and the minimal set of relations. We also compute the order of the main generator in the truncated rings. | Turk J Math (2014) 38: 846 – 850 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article On the K -ring of the classifying space of the generalized quaternion group ¨ ˙ Mehmet KIRDAR∗, Sevilay OZDEM IR Department of Mathematics, Faculty of Arts and Sciences, Namık Kemal University, Tekirda˘ g, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: We describe the K -ring of the classifying space of the generalized quaternion group in terms of generators and the minimal set of relations. We also compute the order of the main generator in the truncated rings. Key words: Topological K -theory, representation theory, generalized quaternion group 1. Introduction The K -ring of the classifying space BQ2n of the generalized quaternion group Q2n , n ≥ 3, is described classically in [4] and [7]. In this note, we describe these rings in a simpler way, by a minimal set of relations on a minimal set of generators. We also make connections between these computations and those done for the lens spaces. e -order In particular, we compute the order of the main generator of that ring in its truncations, . the K of the main vector bundle over the corresponding spherical forms, in a much shorter way than is done in [7]. The reader may find more about the geometric meaning of these orders and quaternionic spherical forms in [4] and [7]. The description of the K -ring is done from the representation ring of the group Q2n via the Atiyah–Segal Completion Theorem (ASCT), which says that the K -ring of the classifying space of a group is the completion of the representation ring of this group at its augmentation ideal. Most importantly, we also check the minimality of the relations we found, through the Atiyah–Hirzebruch Spectral Sequence (AHSS), and this will also guarantee that the required completion of the representation ring mentioned in ASCT .
