  Let $M$ be a connected open Riemann surface. We prove that the space
$\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of
$M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic
contact structure, is weakly homotopy equivalent to the space $\mathscr
C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb
S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy
equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb
C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows
that $\mathscr L(M,\mathbb C^{2n+1})$ is $(4n-3)$-connected.
