Smallest $\mathbb R^n$ into which $SO(3)$ can be topologically embedded?

Question

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.

Answer 1

As stated in the comment of @MoisheKohan, the smallest such $n$ is $5$, but the way things work is not that one finds the global topology of $SO(3)$ via some "nice embedding" into $\mathbb R^5$. On the contrary, one first observes that, as a manifold, $SO(3)$ is diffeomorphic to the real projective space $\mathbb RP^3$. This projective space is obtained by identifying antipodal points in the three-sphere $S^3$ (similarly as the projective plane is obtained from the two-sphere). Alternatively, you can describe it as the space of lines through the origin in $\mathbb R^4$. For projective spaces it is known that they are particularly difficult to embed into $\mathbb R^n$, in particular, $\mathbb RP^3$ embeds into $\mathbb R^5$ but not into $\mathbb R^4$.

The identifiacation of $SO(3)$ winth $\mathbb RP^3$ can be obtained in a rather direct way by realizing $SO(3)$ as a quotient group of either the unit quaternions as mentioned in the comment of @Kajelad or of $SU(2)$, in both cases by identifying each element with its negative. The unit quaternions are the unit sphere in $\mathbb H\cong\mathbb R^4$ on the nose and hence an $S^3$. For $SU(2)$ one easily verifies that mapping $A\in SU(2)$ to its first column vector identifies $SU(2)$ with unit vectors in $\mathbb C^2\cong\mathbb R^4$, so again this is a $3$-sphere.

Answer 2

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.