The existence of certain types of manifolds

Abstract

Introduction. The first part of the paper gives a class of polyhedral 3manifolds in the 4-sphere S4 and with homology groups H(M) —H2(M) =0, but with 7ri(M) 5^0. Thus some Poincare spaces are imbeddable in Si. Also, necessary and sufficient conditions are given for a group to be Hi(M) for some 3-manifold M in S4. In Part Two a negative answer is found for the following question: Since the metric cases of 2-gcms reduce to the classical types [12, IX], and the 3-gcms generally do not, can one state that a spherelike 3-gcm in S4 must be a classical manifold? We construct 3-gcms in Si which have the same homology and character as the 3-sphere (globally and locally), but which are not locally euclidean. In part three of the paper a negative answer is found for a question proposed by Griffiths [6] by showing the existence of a 3-dimensional homotopy manifold which is not locally euclidean. This is a space previously obtained by R. H. Bing by identifying the points on a certain set of tame arcs in S3. Part I. Let P be a connected 2-polyhedron which is contained in S4 as a subcomplex of some simplicial decomposition 2 of Si. Before anything else is done we make a barycentric subdivision of 2. This is to make sure that the resulting decomposition of P will be complete in the sense that if all of the vertices of a simplex of .S4 lie in P, then the simplex itself is in Now every simplex of St(P) — P is the join of a simplex of P with a simplex in Si — St(P). Hence each xE(St(P)—P) may be assigned a number ^(x) between 0 and 1, its distance from P. Newman [8] has shown that the set M= [x|/(x) = l/2] is a manifold. The set M is clearly 3-dimensional, poly