On homotopy elements represented by quotients of Lie groups

TL;DR

This paper investigates homotopy elements represented by quotients of simple matrix Lie groups with specific subgroup structures, extending known classifications by adding new quotient framed manifolds.

Contribution

It introduces a method to realize homotopy elements via quotient manifolds with stable framings derived from twisted invariant framings, expanding previous classifications.

Findings

Identifies conditions for extending the adjoint representation over G.

Constructs quotient manifolds with natural stable framings.

Adds new homotopy elements to existing classification tables.

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 $\mathscr{L}^\alpha$ of $G$ where $\alpha$ 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].