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

TL;DR

This paper investigates how specific framings of the special unitary groups $SU(2n)$ and certain quotients can produce nontrivial elements in the image of the complex Adams $e$-invariant, revealing new generator constructions.

Contribution

It demonstrates that twisting the framing of $SU(2n)$ can transform the zero $e$-invariant into a generator of its cyclic image, extending the result to certain quotients.

Findings

Twisted framings produce nontrivial $e$-invariant elements.

The zero $e$-invariant can be turned into a generator.

Results apply to quotients of $SU(2n+1)$ by circle subgroups.

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. .