TITLE: (Path) connected set of matrices?
QUESTION [12 upvotes]: Let $N \in \mathfrak{M}_n(\mathbb{C})$ nilpotent, such that there exists $X \in \mathfrak M_n(\mathbb{C})$ with $X^2=N$ (take for instance $n>2$ and $N(1,n)=1$; $N(i,j)=0$ otherwise).
Denote by $\mathcal{S}_N$ the set of $X \in \mathfrak M_n(\mathbb{C})$ such that $X^2=N$.
Is $\mathcal{S}_N$ connected or path-connected ? What happens when we change $2$ by $3,4,\ldots $?

REPLY [2 votes]: This is a memorial to an incorrect solution that used to be here. Unfortunately, I can't delete it since it was accepted.