On homotopy elements represented by quotients of Lie groups

Haruo Minami

On homotopy elements represented by quotients of Lie groups

Algebraic Topology 2025-11-21 v1

Authors: Haruo Minami

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].

Keywords

homotopy elements Lie groups quotients framed manifolds stable framing complex representation

Cite

@article{arxiv.2511.16121,
  title  = {On homotopy elements represented by quotients of Lie groups},
  author = {Haruo Minami},
  journal= {arXiv preprint arXiv:2511.16121},
  year   = {2025}
}

Comments

7 pages; introduces a framed quotient of Lie groups by a torus subgroup with some circle components replaced by quaternion circles,thereby determining the homotopy elements in $\pi^S_n$ $(\le n\le 20)$

On homotopy elements represented by quotients of Lie groups

Abstract

Keywords

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