number of roots on SO(3)

Question

Suppose we have a smooth map$ f:SO(3) → SO(3)$ of manifolds s.t.$ f(X)=X^2$. $I$ though since I is a regular value of this map and f is orientation preserving, to calculate degree of it, it is enough to check the number of roots of $X^2-I$ in SO(3). If X is a root of it, then i have that $X=VDV^T$ where $V$ is an orthogonal matrix and $D$ is a diagonal matrix. So far I found that $D$ have to be $diag(1,1,1), diag(-1,-1,1), diag(-1,1,-1), diag(1,-1,-1)$. Any suggestion would be thankful.

Answer 1

The equation $X^2=I$ simply means that $X$ is an involution, i.e. order 2 rotation about the origin. Such rotation is determined by its axis and, thus you get continuum of solutions. The point is that $I$ is not a regular value of $f$. Hint: Try to solve the equation $f(X)=R$, where $R$ is a fixed rotation by any angle different from $0, 2\pi/3$ and $\pi$. Every solution of this equation will commute with $R$ which will narrow down your options.

Answer 2

$I$ is not a regular value since $f^{-1}(I)$ is infinite. To construct an element of $f^{-1}(I)$, take a line $l$ and the plan $P$ or thogonal to the line. Consider the linear tranformation $r_l$ whose restriction to $l$ is the identity and whose restriction to $P$ is the rotation of angle $\pi$, $r_l$ is in $SO(3)$ and $r_l^2=I$.