\(\nu_1\)-periodic homotopy groups of the Dwyer-Wilkerson space (Q942245)

In [Homology Homotopy Appl. 5, No.~1, 297--324 (2003; Zbl 1031.55008)], the second author completed the determination of the \(p\)-primary \(v_1\)-periodic homotopy groups \(v_1^{-1}\pi_*(G)\) for all compact simple Lie groups \(G\) (for all \(p\)). Now, in the last 15 years, new spaces \((X,BX, X\, {\simeq}\,\Omega BX)\), called \textit{\(p\)-compact groups} have been defined. These are analogues of compact Lie groups and their classifying spaces, but also possess qualities such as \(BX\) being \(p\)-complete and \(H^*(X;\mathbb F_p)\) being finite. \textit{W. Dwyer} and \textit{C. Wilkerson} [J. Am. Math. Soc. 6, 37--64 (1993; Zbl 0769.55007)] defined an interesting \(2\)-compact group \(DI(4)\) with \(H^*(BDI(4);\mathbb F_2)\) isomorphic to the rank 4 mod 2 Dickson invariants (hence the notation \(DI(4)\)). Recently, K. Andersen and J. Grodal have shown that \((DI(4),BDI(4))\) is the only simple \(2\)-compact group \textit{not} arising as the \(2\)-completion of a compact connected Lie group. Therefore, \((DI(4),BDI(4))\) presents the next interesting case for the computation of \(v_1\)-periodic homotopy. In this paper, the \(2\)-primary groups \(v_1^{-1}\pi_*(DI(4))\) are computed. The proof proceeds by showing a non-obvious equivalence between two spectra, and this is accomplished by showing the corresponding Adams (operations) modules are isomorphic and then quoting the result that \(K/2\)-local spectra \(X\) having some \(K^i(X)=0\) are determined by the Adams module \(K^*_{CR}(X)\). (An interesting part of the paper is the determination of solutions to certain necessary number theory congruences found with the software Maple.)