Homotopy type of gauge groups of quaternionic line bundles over spheres
The authors study the homotopy types of groups of self-equivalences (gauge groups) \(\mathcal{G}_{f}\) of principal \(S^3\)-bundles \(P_{f}\) over \(S^{n}\) by methods introduced in [\textit{A. Kono}, Proc. R. Soc. Edinb., Sect. A 117, No. 3--4, 295--297 (1991; Zbl 0722.55008)]. Those bundles are classified by elements \([f]\in\pi_{n-1}(S^{3})\) and the authors use detailed knowledge on the low-dimensional homotopy groups of \(S^{3}\), mostly covered by \textit{H. Toda} [Composition methods in homotopy groups of spheres. Annals of Mathematics Studies. 49. Princeton, N. J.: Princeton University Press (1962; Zbl 0101.40703)]. In particular, they derive a formula for the connecting map in the homotopy sequence of the evaluation fibration \(\mathcal{G}_f\to S^{3}\). From this formula they draw their descriptions of the homotopy types of \(\mathcal{G}_{f}\) in the range \(5 \leq n \leq 25\) (the case \(n = 21\) the authors are not able to solve, they rather formulate a conjecture). The typical statement they give is like ``On \(S^n\) there are \dots principal \(S^{3}\)-bundles. The gauge groups of \(P_f\) and \(P_{f'}\) have the same homotopy type if and only if \([f ]\) and \([f']\) satisfy the relation\dots'' (where the dots have to filled out according to \(n\)).
scientific article; zbMATH DE number 5522082