Recall there are multiple ways to define the unit sphere bundle of a vector bundle. One is by constructing a fiberwise vector space metric and declaring the sphere bundle to have fibers the unit spheres in each of the vector space fibers. The other way is to use the equivalence of vector bundles and principal $O(n)$ bundles and then since $O(n)$ acts faithfully on $S^{n-1}$ we may replace the fiber to obtain a sphere bundle.


There is, however, a distinction between vector bundles and $\mathbb{R}^n$ bundles, that is fiber bundles with fiber $\mathbb{R}^n$ and structure group $Homeo(\mathbb{R}^n)$. It is not always possible to assign a coherent vector space structure to the fibers to make it a vector bundle. Is there a notion of an associated sphere bundle in this context?


Now I tend to just believe fiberwise constructions are always possible, so I would believe someone if they said we could pick a metric on the fibers of any $\mathbb{R}^n$ bundle and define its unit sphere bundle in the same way as for vector bundles. But on the principal bundle side of things, this seems to me to be asserting that $Homeo(\mathbb{R}^n)$ has a subgroup $H$ so that the inclusion $H \rightarrow Homeo(\mathbb{R}^n)$ is a weak equivalence, and $H$ preserves the unit sphere $S^{n-1}$ while acting faithfully on it. This seems very difficult to believe.


I will add that this is an unstable question. We can always form the fiberwise one point compactification of a $\mathbb{R}^n$ bundle since homeomorphisms extend to the one point compactification. If the unit sphere bundle exists, it's fiberwise suspension should coincide with this sphere bundle.



Do $\mathbb{R}^n$ bundles have unit sphere bundles?
It is not true in general that there is a subgroup $H$ of $Homeo(\mathbb{R}^n)$ such that the inclusion is a homotopy equivalence and $H$ preserves the unit sphere. If there was, then $H$ would also preserve the unit disk and thus every topological $\mathbb{R}^n$-bundle would contain a $D^n$-bundle. But it is not the case by a result of Browder, Theorem 1 of [Open and Closed Disk Bundles](https://www.jstor.org/stable/1970428).


