Three Kneser Lemmata

This note contains three versions of a result which is sometimes referred to as Kneser’s lemma and states that (with respect to a fixed triangulation) there is a bound on the number of disjointly embedded normal surfaces in a 3–manifold which satisfy an extra non–triviality condition. One often stipulates that no two of the surfaces be parallel in M . We relax this condition and merely assume that no two of the surfaces are normal isotopic, under which we understand that they do not have the same normal surface coordinates. The three versions address in turn compact, closed and cusped 3–manifolds (where we restrict ourselves to torus cusps). Techniques are borrowed from [4, 1] and lectures by Hyam Rubinstein. We also extend the main result of [1] to show that if T is the number of tetrahedra in a triangulation of a closed, irreducible 3–manifold M , then if a 2–sided, geometrically incompressible surface in a closed, irreducible 3–manifold M has more than or equal to 3 2 T components, then two of them must be parallel.