Lower bounds on the number of homotopy types of $\mathrm{SU}(n)$-gauge groups over spheres

Abstract

We explore the classical question of enumerating the homotopy types of gauge groups of principal $\mathrm{SU}(n)$-bundles over $S^{2m+1}$ in the unstable range $m\ge n$. Using computations of certain $p$-local Samelson products, we obtain lower bounds on the number of such homotopy types. We also present a discussion of the last stable case.