What are sufficient conditions for finitely many equivalence classes of slice contours of surfaces?

Question

Apologies in advance for imprecision of the question. Thanks for improving it. Let M be a compact, connected, orientable surface in three dimensional Euclidean space without boundary and without self-intersections, such as a sphere, or a torus. Define a slice contour C of M to be the non-empty intersection of M with a plane. (For example, the equator of a sphere is a slice contour; there are different ways that a torus admits slice contours consisting of two disjoint circles.) Define two slice contours C and D to be equivalent if there exists a continuous homotopy H from the inclusion map of C into M to the inclusion map of D into M, such that the inclusion map at every stage of H is an injective homeomorphism. This is an equivalence relation on the set of all slice contours. What are sufficient conditions for the following statements to be jointly true: (1) a slice contour is a finite 1-dimensional CW complex; (2) there is a finite number of equivalence classes; (3) there exists at least two equivalence classes with exactly one member; (4) there exists at least one equivalence class with a continuum of members.

Answer 1

A theorem and proof without additional assumptions except smoothness might proceed roughly along the following lines. First, a finite 2-dimensional simplicial complex in space consisting of points, closed line segments for 1-simplices, and closed triangles for 2-simplicies could satisfy the conditions that two 2-simplices meet at most along a 1-simplex, every 1-simplex is the intersection of exactly two 2-simplices, and every 2-simplex may be oriented so that any two 2-simplices meeting at a 1-simplex induce opposite orientations on it. I am guessing that the union of the simplices in such a complex satisfy a three-dimensional version of the Jordan Curve Theorem. Second, I am guessing that such a union satisfies the slice contour conditions (1)-(4) by induction on the number of 2-simplices. Second, if the surface M is sufficiently smooth, with an upper bound on its curvature, then it may be approximated by such a union with its vertices in M, and its 1-simplices very close to geodesics in M connecting the vertices, in the sense that the distance between a 1-simplex and its corresponding geodesic path in M is the supremum of the Euclidean distances between appropriately parametrized moving points. If epsilon is the upper bound on those suprema over all 1-simplices, then the finite number N of equivalence classes depends on epsilon, and third, N(epsilon) is bounded as epsilon goes to zero by some argument about curvature of M.