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Contractibility and self-intersections of curves on surfaces

Laat, D. de (2009) Contractibility and self-intersections of curves on surfaces. Bachelor's Thesis, Mathematics.

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We discuss whether closed curves on closed orientable surfaces are contractible, and for noncontractible curves whether they are homotopic to a curve having no self-intersections. We prove that the minimal number of self-intersections of an (m,n)-torus curve is gcd(m,n)-1. We discuss Dehn's algorithm for solving the word problem in the fundamental group, which is the algebraic equivalent of the contractibility problem, and Poincaré's solution of the problem concerning intersection-free curves. Before doing this we develop the theory of curves on surfaces. By triangulating the surfaces we can associate them with normal form schemata and use this to realize the surfaces geometrically. We construct a locally isometric map from the spherical, Euclidean, or hyperbolic covering 2-spaces onto the surfaces and construct a group of isometries on this covering space. This map is used to give a characterization of homotopy and we prove that the fundamental group, consisting of the homotopy equivalence classes, is isomorphic to the covering isometry group. The Cayley graph explains the structure of the fundamental group and is important in the development of Dehn's algorithm. The geometric properties of the covering isometry group are important in Poincaré's approach.

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:28
Last Modified: 15 Feb 2018 07:28

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