On the K-Ring of the Classifying Space of the Dihedral Group

Abstract

We describe K(BS 4 ) and make a connection of the order of the bundle induced from the standart representation over the four dimensional skeleton of BS 4 with the stable homotopy group π s 3 = Z 24 explaining the reasons of this connection by pulling this bundle over lens spaces.

On the $K$-ring of the classifying space of the generalized quaternion group

“…In the ring KO(BZ 2k ), we know that the main relation is ψ k+1 (w) − ψ k−1 (w) = 0, [6]. Furthermore, we deduce that ψ k+1 (ϕ) − ψ k−1 (ϕ) = 0 in the ring K(BQ 4k ) too.…”

“…Let us recall from [6] the effect of (the real) Adams operation of degree i, on the main generator w of KO(BZ 2k ) :…”

“…We sum up everything in: Theorem 1 K(BQ 2 n ) is generated by v 1 , v 2 , and ϕ with the minimal set of relations (1), (2), (4), (5), and (6) above.…”

“…A quick survey for the K -rings of the classifying spaces of cyclic and dihedral groups can be found in [6]. The complete result for the dihedral groups is also published before this paper, in [5], which surprisingly uses the results of this paper for the complicated even case of its problem.…”