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Let $X$ be a topological space with basepoint $x_0$. Define a map $s\colon\pi_1(X,x_0)\to\mathbb{Z}_{\geq 0}$ by $$s([\gamma])=\min\{\text{number of self-intersections of }\gamma'\colon \gamma'\in [\gamma]\}.$$ Has the map $s$ been studied before? Is there some well-established theory about it?

To be specific, for example, how would one prove that if $A$ is annulus, then $$s(n)=\left\{ \begin{array}{ll} 0 & \mbox{if } n = 0 \\ |n|-1 & \mbox{if } n \neq 0 \end{array} \right.$$

Filanto Renika
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    When $X$ is a manifold, if its dimension is greater than 2, the $\gamma \mapsto 0$ for any $\gamma$. But if it's a 2-manifold, the keyword is the "intersection number"; more generally "oriented intersection theory". There's a nice discussion of all of this in Guillemin and Pollack's book on differential topology. I have no idea about the story for non-manifolds. –  Sep 03 '14 at 23:42
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    By the way, there's no guarantee (for $X$ arbitrary) that you should be able to homotopy $\gamma$ so it never intersects itself. Consider $z \mapsto z^2$ on $S^1$. So it should be a map to $\Bbb N \cup {\infty}$. –  Sep 03 '14 at 23:44
  • @MikeMiller: in addition to the oriented intersection number, there is also the "geometric intersection number", which is exactly what OP is asking about in the surface case. This number is much studied in the theory of surfaces, starting with Thurston's work. – Moishe Kohan Sep 05 '14 at 16:36
  • @studiosus You're quite right that the oriented intersection number is the incorrect concept here; sorry for bringing it up and thanks for the keyword. –  Sep 05 '14 at 17:38

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The invariant you are interested in, is called the geometric intersection number. (As opposed to the "oriented" or "algebraic intersection number" mentioned by Mike.) It is interesting when $X$ is a surface (although it has generalizations in other dimensions, but you use submanifolds of other dimensions, not loops). The geometric intersection number plays critical role in Thurston's theory of hyperbolic surfaces. See for instance "Thurston's work on surfaces" (section 3.3 and onward) or/and F.Luo and R.Stong "Measured lamination spaces on surfaces and geometric intersection numbers", Topology and its Applications, Volume 136, 2004, p. 205–217.

Moishe Kohan
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