The 23-rd and 24-th homotopy groups of the n-th rotation group

TL;DR

This paper determines the 23rd and 24th homotopy groups of the n-th rotation group, focusing on their 2-primary components, using fibration sequences and Toda's composition methods.

Contribution

It provides explicit group structures for the 23rd and 24th homotopy groups of rotation groups, a previously unresolved problem in algebraic topology.

Findings

Computed the 2-primary components of _{k}(R_n) for k=23,24.

Applied fibration sequences and Toda's methods to determine group structures.

Extended understanding of homotopy groups of rotation groups.

Abstract

We denote by $\pi_k(R_n)$ the $k$-th homotopy group of the $n$-th rotation group $R_n$ and $\pi_k(R_n:2)$ the 2-primary components of it. We determine the group structures of $\pi_k(R_n:2)$ for $k = 23$ and $24$ by use of the fibration $R_{n+1}\overset{R_n}{\longrightarrow}S^n$. The method is based on Toda's composition methods.