Homotopy Groups of the Space of Curves on a Surface

We explicitly calculate the fundamental group of the space $\mathcal{F}$ of all immersed closed curves on a surface $F$. It is shown that $\pi_n(\mathcal F)=0$,n>1for$F\neq S^2, RP^2$\mathrm{.}Itisalsoprovedthat$\pi_2(\mathcal F)=\Z$,and$\pi_n(\mathcal F)=\pi_n(S^2)\oplus\pi_{n+1}(S^2)$,n>2,for$F$equalto$S^2$or$RP^2$.