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Intersections of curves on surfaces. (English) Zbl 0576.57009

In the terminology of the authors, a loop is a 1-gon on a surface F if it is homotopic to its vertex in its interior on F. Two arcs form a 2-gon if one is homotopic to the other through the interior on F with fixed endpoints. All curves used in the homotopy are assumed to be general position curves. The two main theorems are: 1. If f is an arc or loop on an orientable surface F which is homotopic to an embedding but is not embedded, then there is an embedded 1- or 2-gon on F bounded by arcs of f. A counterexample is given for the annulus for a similar statement involving two curves. 2. If f is an arc or loop on an orientable surface F with more than the minimum number of self-intersections in its homotopy class, then there is a singular 1- or 2-gon on F bounded by arcs of f. If the hypothesis of orientability is dropped, a similar theorem can be proved only if the restriction on the admissable homotopies is dropped. The paper contains many more results and examples and counterexamples, in particular a construction for surfaces of genus \(>2\) of a curve that is not homotopic to a curve on an incompressible subsurface. The authors also mention a number of open questions.
Reviewer: H.Guggenheimer

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
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References:

[1] Freedman, M. H.; Hass, J.; Scott, P., Closed geodesics on surfaces, Bull. London Math. Soc., 14, 385-391 (1982) · Zbl 0476.53026 · doi:10.1112/blms/14.5.385
[2] J. Hass and J. H. Rubinstein,One-sided closed geodesic on surfaces, University of Melbourne, preprint. · Zbl 0614.53035
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