On the triangulation of manifolds and the Hauptvermutung

1. The first author's solution of the stable homeomorphism conjecture [5] leads naturally to a new method for deciding whether or not every topological manifold of high dimension supports a piecewise linear manifold structure (triangulation problem) that is essentially unique (Hauptvermutung) cf. Sullivan [14]. At this time a single obstacle remains—namely to decide whether the homotopy group 7T3(TOP/PL) is 0 or Z2. The positive results we obtain in spite of this obstacle are, in brief, these four: any (metrizable) topological manifold M of dimension ^ 6 is triangulable, i.e. homeomorphic to a piecewise linear ( = PL) manifold, provided H*(M; Z 2 ) = 0 ; a homeomorphism h: MI—IMI of PL manifolds of dimension ^ 6 is isotopic to a PL homeomorphism provided H(M; Z2) = 0 ; any compact topological manifold has the homotopy type of a finite complex (with no proviso) ; any (topological) homeomorphism of compact PL manifolds is a simple homotopy equivalence (again with no proviso). R. Lashof and M. Rothenberg have proved some of the results of this paper, [9] and [ l0] . Our work is independent of [ l0 ] ; on the other hand, Lashofs paper [9] was helpful to us in that it showed the relevance of Lees' immersion theorem [ l l ] to our work and reinforced our suspicions that the Classification theorem below was correct. We have divided our main result into a Classification theorem and a Structure theorem.