The special unitary groups $SU(2n)$ as framed manifolds

Abstract

Let $[SU(2n), \mathscr{L}]$ denote the bordism class of $SU(2n)$ $(n\ge 2)$ equipped with its left invariant framing $\mathscr{L}$. Then it is well known that $e_\mathbb{C}([SU(2n), \mathscr{L}])=0$ where $e_\mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show that replacing $\mathscr{L}$ by the framing obtained by twisting it by a specific map the zero value of $e_\mathbb{C}([SU(2n), \mathscr{L}])$ can be transformed into a generator of $\mathrm{Im} \, e_\mathbb{C}$ which is isomorphic to a cyclic group. In addition we show that the same procedure affords an analogous result for a quotient of $SU(2n+1)$ by a circle subgroup which inherits a canonical framing from $SU(2n+1)$ in the usual way. .