Invariant theory and the K-theory of the Dwyer-Wilkerson space (Q2707513)

Let \(G\) denote the exotic 2-compact group \(BDI(4)\) constructed by \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} [J. Am. Math. Soc. 6, No. 1, 37-64 (1993; Zbl 0769.55007)]. The authors determine the 2-adic \(K\)-theory of its classifying space \(BG\). The result is \(K^*(BG; \mathbb{Z}_{\widehat 2})\cong \mathbb{Z}_{\widehat 2}[[\xi, \lambda^2(\xi), \lambda^2 (\lambda^2(\xi))]]\) where \(\xi\) is formally the adjoint representation of \(G\). By Dwyer and Wilkerson's construction of \(G\) it follows that there is a monomorphism of \(\text{Spin} (7)_{\widehat 2}\) into \(G\) of 2-compact groups such that the composition of this with the maximal torus inclusion gives an explicit maximal torus \(T\) of \(G\). Previously the first author and Jeanneret have shown that \(K^*(BG; \mathbb{Z}_{\widehat 2})\) is isomorphic to the ring of invariants \(K^*(BT; \mathbb{Z}_{\widehat 2})^W\) of the Weyl group action. This fact and all the information about \(B\text{Spin} (7)\) play, of course, a crucial role in the determination of \(K^*(BG; \mathbb{Z}_{\widehat 2})\). However additional extra information about \(BG\) is needed to lead to the full computation. The authors succeed in obtaining the result by analyzing in detail two kinds of spectral sequences of Bockstein and Atiyah-Hirzebruch associated to \(BG\) with the help of Mathematica.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].