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By the classification of Dold and Whitney, linear $S^2$-bundles over $S^4$ are classified by their first Pontryagin class $p_1$, which takes the value $4\lambda$ for the bundle corresponding to $\lambda\in\mathbb{Z}\cong\pi_3(\mathrm{SO}(3))$.

Is there an analogous classification for linear $S^{2k}$-bundles over $S^{4k}$ in terms of $p_k$?

As a first step one could try to determine the group $\pi_{4k-1}(\mathrm{SO}(2k+1))$. By using the long exact sequence of homotopy groups for the fibration $\mathrm{SO}(m)\hookrightarrow\mathrm{SO}(m+1)\to S^m$ and by using Serre's result that there are finite groups $F_k$, so that $$\pi_{m+n}(S^m)\cong\begin{cases} \mathbb{Z},\quad & n=0,\\\mathbb{Z}\oplus F_k,\quad &m=2k,n=2k-1,\\0,\quad &\text{else},\end{cases}$$ an inductive argument shows that $\pi_{4k-1}(\mathrm{SO}(2k+1))\cong \mathbb{Z}\oplus F_k^\prime$ for $F^\prime_k$ finite. Is it known what the value of $p_k$ is for a primitive element of infinite order in $\pi_{4k-1}(\mathrm{SO}(2k+1))$?

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  • $\begingroup$ Off the top of my head I know that $F'_k$ is 2-torsion (since $\pi_{4k-1}SO_{2k+1}\cong\pi_{4k-1}SO_{2k+1}\cong_{(\frac{1}{2})}\pi_{4k-1}Sp_k$, and this latter group is easily seen to be $\mathbb{Z}$). Inspection seems to indicate that $F'_k$ is generally nontrivial. $\endgroup$
    – Tyrone
    Commented Mar 11, 2022 at 15:54
  • $\begingroup$ It comes down to computing the Hurewicz map for $BSO(2k+1)$, right? $\endgroup$
    – user43326
    Commented Mar 13, 2022 at 22:40

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