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 the left invariant framing $\mathscr{L}$. Then it is well known that $e_\mathbb{C}([SU(2n), \mathscr{L}])=0$ in $\mathbb{O}/\mathbb{Z}$ where $e_\mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show that replacing $\mathscr{L}$ by the twisted framing by a specific map it can be transformed into a generator of $\mathrm{Im} \, e_\mathbb{C}$. In addition to that we also 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.