### A proof and extension of Dehn's lemma

Arnold Shapiro, J. H. C. Whitehead
1958 Bulletin of the American Mathematical Society
1. Introduction. C. D. Papakyriakopoulos [2] has recently proved Dehn's lemma [l]. His proof has the merit that the basic construction (the tower) and the crucial lemmas apply to the sphere theorem as well as Dehn's lemma. However, if one is content with Dehn's lemma the proof can be simplified. In this note we first give a simplified proof of Dehn's lemma and then prove an analogous theorem for surfaces with more than one boundary curve. By a surface of type (p } r) we mean a connected,
more » ... , orientable surface of genus p with a boundary consisting of r 1-spheres. Thus the Euler characteristic of such a surface is 2(1 -p)-r. A surface of type (0, 1) is called a disc. A Dehn surface of type (p, r) is a polyhedral singular surface of type (p y r) with no singularities on the boundary. Our extension of Dehn's lemma refers to surfaces of type (0, r). In order to state it we need two more definitions. Let M be a connected 3-manifold and let M be a universal cover of M". The manifold M is orientable and the set of elements of Ti(M) = 7Ti(Af, x 0 ) which correspond, in the usual way, to orientationpreserving covering transformations of M is a sub-group of index 2. We denote it by o>(ikf). Thus M is orientable if, and only if, oo (M)