The Hauptvermutung for 3-complexes

Abstract

1. Introduction. In two fundamental papers, [13] and [14], Poincare introduced the notions of a complex, and of a triangulation. The attempt to study certain topological spaces by means of a triangulation runs into an obvious problem, one must show that invariants defined in terms of the triangulation are independent of the choice of triangulation. For most invariants it is relatively easy to show that they remain unchanged when the triangulation is subdivided, but when an entirely new triangulation is substituted the situation is far from clear. Thus, for example, Tietze in [17] concerns himself with, among other things, showing that the Betti numbers, the torsion coefficients, the fundamental group, etc., are topological invariants. In [16] Steinitz takes, possibly for the first time, the point of view that complexes will be treated in their own right and that the notion of isomorphism will be combinatorial equivalence. This seems to be the reason that Steinitz gets the blame for the Hauptvermutung. Cases of the problem were treated by several people after that, an excellent account is given in [4]. Cairns was probably not aware when he wrote his article that Papakyriakopoulos in [12] had proved the Hauptvermutung for two dimensional complexes. In a series of papers E. Moise proved both the triangulation theorem and the Hauptvermutung for 3-manifolds without boundary in [9], and