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 $?



(Path) connected set of matrices?
This is a memorial to an incorrect solution that used to be here. Unfortunately, I can't delete it since it was accepted.


