On Dehn's Lemma and the Asphericity of Knots

The present paper contains a 'roof of Dehn's lemma and an analogous result that we call the sphere theorem, from which other theorems follow.' DEHN'S LEMMA. Let M be a 3-manifold, compact or not, with boundary which may be empty, and in M let D be a 2-cell with self-intersections (singularities), having as boundary the simple closed polygonal curve C and such that there exists a closed neighborhood of C in D which is an annulus (i.e. no point of C is singular). Then there exists a 2-cell Do with boundary C, semi-linearly imbedded in M. SPHERE THEOREM. Let M be an orientable 3-manifold, compact or not, with boundary which may be empty, such that 7r2(M) # 0, and which can be semi-linearly2 imbedded in a 3-manifold N, having the following property: the commutator quotient group of any non-trivial (but not necessarily proper) finitely generated subgroup of 7r,(N) has an element of infinite order (n.b. in particular this holds if 7r,(N) = 1). Then there exists a 2-sphere S semi-linearly imbedded in M, such that3 S X 0 in M. Dehn's lemma was included in a 1910 paper of M. Dehn 4 p. 147, but in 1928 H. Kneser 13 p. 260, observed that Dehn's proof contained a serious gap. In 1935 and 1938 appeared two papers by I. Johansson 11, 12, on Dehn's lemma. In the second one, p. 659, he proves that, if Dehn's lemma holds for all orientable 3-manifolds, it then holds for all non-orientable ones. We now prove in this paper that Dehn's lemma holds for all orientable 3-manifolds. Our proof makes use also of I. Johansson's first paper. As far as the sphere theorem is concerned we have to remark that, to the best knowledge of this author, the first one to attempt a theorem of this kind was H. Kneser in 1928, 13 p. 257; however his proof does not seem to be conclusive. In 1937 S. Eilenberg 5 p. 242, Remark 1, observed a relation between the nonvanishing of the second homotopy group and the existence of a non-contractible 2-sphere. Finally in 1939 J. H. C. Whitehead 25 p. 161, posed a problem which stimulated the author to prove the sphere theorem, stated above. We emphasize that, if 7r,(N) is a free group4 then the hypotheses of the sphere theorem are fulfilled, according to the following NIELSEN-SCHREIER THEOREM. Every subgroup of a free group is itself a free group.5