Cohomogeneity one action on $S^7$-bundles over $S^8$

Question

Is it known if the total space of an $S^7$-bundles over $S^8$ with structure group $SO(8)$ admits a cohomogeneity one action?

Answer 1

I don't think the argument is written down anywhere, but not all $S^7$-bundles over $S^8$ admit cohomogeneity one actions.

Focusing on those with Euler class $\pm 1$, the resulting total space $E$ is a homotopy sphere. Homotopy spheres admitting cohomogneity one actions have been classified by Straume in

Compact connected Lie transformation groups on spheres with low cohomogeneity I. Mem. Amer. Math. Soc., 119(569), 1996.

In particular, Straume shows that in dimensions not of the form $4k+1\geq 5$, that a homotopy sphere admitting a cohomogeneity one action is diffeomorphic to the standard sphere. (In dimension $4k+1$, one can also get Brieskorn varieties.)

So, in dimension $15$, the only homotopy sphere which admits a cohomogeneity one action is the standard sphere. Since there are exotic spheres which are the total space of an $S^7$-bundle over $S^8$ (as shown by Shimada in

Differentiable structures on the 15-sphere and Pontrjagin classes of certain manifolds, Nagoya Math. J. 12 (1957), 59–69. MR0096223 (20 #2715) Zbl 0145.20303,

the result follows.