Homotopy Groups of the Space of Curves on a Surface

Vladimir Tchernov

Homotopy Groups of the Space of Curves on a Surface

Geometric Topology 2007-05-23 v1 Differential Geometry

Authors: Vladimir Tchernov

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 $RP^2$ .

Keywords

homotopy groups space of curves immersed closed curves fundamental group surface mapping class group

Cite

@article{arxiv.math/9906123,
  title  = {Homotopy Groups of the Space of Curves on a Surface},
  author = {Vladimir Tchernov},
  journal= {arXiv preprint arXiv:math/9906123},
  year   = {2007}
}

Comments

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

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