TITLE: Smallest $\mathbb R^n$ into which $SO(3)$ can be topologically embedded?
QUESTION [7 upvotes]: I think there are theorems in differential geometry guaranteeing that $SO(3)$ can be embedded into $\mathbb R^n$ for some $n$. But do we know what particular $n$ this is for $SO(3)$, and do we know of a particular embedding?
This would be nice to know if $n$ happens to be small enough for us to "think about" the global topology of $SO(3)$ in a somewhat geometrical way.

REPLY [4 votes]: The standard reference I know is
Massey, W. S., On the imbeddability of the real projective spaces in Euclidean space, Pac. J. Math. 9, 783-789 (1959). ZBL0094.36003.
Namely, $RP^n$ can never embed topologically in $R^{n+1}$ for any $n\ge 2$. Massey credits Rene Thom with this result.
Alternatively (in the smooth setting), one can quote Theorem A in
Levine, J., Imbedding and immersion of real projective spaces, Proc. Am. Math. Soc. 14, 801-803 (1963). ZBL0117.17007.
The group $SO(3)$ is homeomorphic (actually, diffeomorphic) to the projective space $RP^3$, hence, it cannot embed topologically in $R^4$. At the same time, every 3-dimensional manifold embeds smoothly in $R^5$. This theorem is due to Hirsch in the orientable case (and $RP^3$ is orientable):
Hirsch, Morris W., On imbedding differentiable manifolds in Euclidean space, Ann. Math. (2) 73, 566-571 (1961). ZBL0123.16701.
and, independently, Rokhlin and Wall for non-orientable manifolds.
An explicit embedding $RP^3\to R^5$ (and much more) was constructed in
James, I. M., Some embeddings of projective spaces, Proc. Camb. Philos. Soc. 55, 294-298 (1959). ZBL0090.13201.
However, I find it extremely unlikely that you would gain better understanding of the topology of $RP^3$ using such an embedding. For instance, I am unaware of anybody who understands better the real projective plane using its embedding in $R^4$ (or immersion in $R^3$)  comparing to the standard descriptions of real-projective spaces.