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 $F\ne S^2,R{P}^2$. It is also proved that ${\pi }_2(\mathcal{F})=\mathbb{Z}$, and ${\pi }_n(\mathcal{F})={\pi }_n(S^2)\oplus {\pi }_{n+1}(S^2)$, n>2, for $F$ equal to $S^2$ or $R{P}^2$.

8 pages, 1 figure This paper will appear in Math. Scand. probably in Vol. 86, no. 1, 2000