On the triangulation of manifolds and the Hauptvermutung 论文

摘要

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 3 —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; Z2)=0; a homeomorphism h: MI—ÏMÎ of PL manifolds of dimension ^6 is isotopic to a PL homeomorphism provided H 3 (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 [ll] 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. (I) CLASSIFICATION THEOREM. Let M m be any topological manifold of dimension m}^6 (or ^5 if the boundary dM is empty). There is a natural one-to-one correspondence between isotopy classes of PL structures on M and equivalence classes of stable reductions of the tangent microbundle r(M) of M to PL microbundle. (There are good relative versions of this classification. See [7] and proofs in §2.) Explanations. Two PL structures 2 and 2 ' on M, each defined by a PL compatible atlas of charts, are said to be isotopic if there exists a