Divisible groups in the \(K\)-theory completion of \(SU(n)\)
The article under review studies the unstable homotopy groups of the special unitary group SU\((n)\). The main tool is the unstable Adams spectral sequence based on the generalized cohomology theory \(E(1)\), see [\textit{M. Bendersky} and \textit{R. D. Thompson}, Am. J. Math. 122, No. 3, 599--635 (2000; Zbl 0963.55010)]. Here \(E(1)\) is the usual summand of complex \(K\)-theory \(KU\). This unstable Adams spectral sequence converges to the homotopy groups of the \(K\)-theory completion of SU\((n)\). Another key tool is the fibration \[ \mathrm{SU}(n-1) \rightarrow \mathrm{SU}(n) \rightarrow S^{2n-1}, \] which allows for arguments by induction on \(n\). The article states the results of explicit calculations in an easy-to-use format. These results rely on previous related calculations, including results from [\textit{M. Bendersky}, Topology 31, No. 1, 47--64 (1992; Zbl 0751.55011)] and [\textit{D. M. Davis}, J. Lond. Math. Soc., II. Ser. 43, No. 3, 529--544 (1991; Zbl 0760.55013)].
scientific article; zbMATH DE number 6509434