On 3-dimensional manifolds
C. E. Clark · Bulletin of the American Mathematical Society · 1942
Let P be a 3-dimensional manifold.1 Let Q be a 2-dimensional manifold imbedded in P.Moreover, let P and Q admit of a permissible simplicial division K, that is, a simplicial division of P such that some subcomplex of K, say L, is a simplicial division of Q.Let Ki and L; denote the it\\ normal subdivisions of K and L, respectively.We define the neighborhood Ni of Li to be the simplicial complex consisting of the simplexes of Ki that have at least one vertex in Li together with the sides of all such simplexes.By the boundary Bi of Ni we mean the simplicial complex consisting of the simplexes of Ni that have no vertex in Li.Our purpose is to prove the following theorem.THEOREM.The boundary B 2 is a two-fold but not necessarily connected covering of Q, and change of permissible division K replaces B2 by a homeomorph of itself.PROOF.The neighborhood N x is the sum of a set of 3-dimensional simplexes.Some of these 3-simplexes, say ax, a 2 , • • • , have exactly one vertex in L x , others, say b x , b 2 , • • • , have exactly two vertices in L x , while the remaining, say c\, c 2 , • • • , have three vertices in L x .Since K x is a normal subdivision of K, the intersection of L\ and bi or Ci is a 1-simplex or 2-simplex, respectively.Let a», j3», and 7» be the intersections of B 2 and a», bi, and Ci, respectively.We shall regard on and 7» as triangles with vertices on the 1-simplexes of ai and Ci.Also we shall regard j3» as a square with vertices on the 1-simplexes of bi.Any 2-simplex of L x , say ABC, is incident to exactly two of the d.Let C\ = ABCM.There is a unique 3-simplex of N x , say a, that is incident to ABM and different from c x .This a is either a Ci, say c 2 , or a &;, say fr 2 .If