On homotopy elements represented by quotients of Lie groups
Haruo Minami
TL;DR
This work addresses representing new stable homotopy elements via quotients $G/B$ of a simply connected compact simple Lie group by carefully chosen subgroups, using extended framings induced from realified complex representations. The authors construct framed quotients $(G/B^r, (L^{k\lambda})_{B^r})$ and apply the Pontryagin–Thom framework, together with Euler-type invariants $e'_R$ and $e_C$, to produce explicit representatives for early stable homotopy classes such as $a[7]$, $a[9]$, $a[11]$, $a[15]$, and beyond, validating their nontriviality and determining their orders. By leveraging fibration structures, quaternionic line bundles, and LS/Ossa-type results, they extend the known table of elements in the stable homotopy groups $\pi^S_k$ with Lie-group–originating framings, including $b[19]$ and $a[20]$. The approach provides a constructive bridge between Lie group geometry and stable homotopy theory, enabling systematic generation of additional Lie-group–based representatives for stable homotopy elements in future work.
Abstract
Consider the quotient $G/B$ of a simple matrix Lie group $G$ by a subgroup $B$ isomorphic to a direct product of some of $S^1$s and $S^3$s such that its adjoint representation can be extended over $G$. Then it naturally inherits a stable framing from a twisted left invariant framing $\mathscr{L}^α$ of $G$ where $α$ is the realization of a complex representation of $G$. In this note we want to add some homotopy elements represented by such quotient framed manifolds to those presented in a table of [E. Ossa 1982].