Não disponível

The aim of this paper is to give conditions to reduces knot problems in S¹ x D² to knot problems in R³. For this we give the notions of small knot and trivial knot in a three manifold M. We say that K is a small knot in M if there existe a 3-ball in the interior of M, submanifold of M containing K. Let K be a knot in S¹ x D² and α ∈ π (S¹ x D² - T(K)) = π(K) the homotopy class, whose representative loop is the piecewise linear homeomorphism φ : Δ² → u₀ x C, where C is the boundary of the disc D², u₀ ∈ S¹ and T(K) is an open tubular neighbourhood of K. Then we have Main Theorem: α = 0 in π(K) iff K is a small knot To prove this theorem we use Dehn's lemma and the notion of polyhedral regular neighbourhood. In chapter I we, develop the theory of colapses and the theory of regular neighbourhoods from the polyhedral view point and we believe that we gave an original form of presentation of Polyhedral Manifolds Theory.