Homotopy Groups of the Space of Curves on a Surface

Abstract

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>1 for $F\neq S^2, RP^2$. It is also proved that $\pi_2(\mathcal F)=\Z$, and $\pi_n(\mathcal F)=\pi_n(S^2)\oplus\pi_{n+1}(S^2)$, n>2, for $F$ equal to $S^2$ or $RP^2$.