The groups $Sp(4n+1)$ and $Spin(8n-2)$ as framed manifolds
Haruo Minami
TL;DR
The paper extends the construction of left-invariant framings twisted by a representation to realize generators of the image of the complex Adams e-invariant for two families of simply-connected compact Lie groups: $Sp(4n+1)$ and $Spin(8n-2)$ across $n\ge 1$. By decomposing the associated complex line bundle $E$ over a circle bundle $S\to G\to G/S$ into tensor products of standard line bundles, the authors reduce the problem to computing Chern classes via Prop. 2.1 of LS and obtain explicit Bernoulli-number expressions for $e_\mathbb{C}$ under carefully chosen framings: $\mathscr{L}^{2n\rho}$ for $Sp(4n+1)$ and $\mathscr{L}^{(2n-1)\Delta}$ for $Spin(8n-2)$. The key technique involves constructing embeddings through Hopf-like bundles from products of spheres (symplectic case) or Clifford-algebra-based decompositions (spin case), enabling precise evaluation of $e_\mathbb{C}$ and demonstrating untwisted framings yield zero while the twisted framings yield the stated generators. These results parallel the known $SU(2n)$ case and illustrate a unified approach to generating the image of $e_\mathbb{C}$ via framing twists for broader Lie groups.
Abstract
We consider a compact Lie group as a framed manifold equipped with the left invarianat framing $\mathscr{L}$. In a previous paper we have proved that the Adams $e_\mathbb{C}$-invariant value of $SU(2n)$ $(n\ge 2)$ gives a generator of the image of $e_\mathbb{C}$ by twisting $\mathscr{L}$ by a certain map. In this note we show that in a similar way we can obtain analogous results for $Sp(4n+1)$ and $Spin(8n-2)$ $(n\ge 1)$.