On the triangulation of manifolds and the hauptvermutung (1968).

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Abstract	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
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Publisher Date	1968-01-01
Access Restriction	Open
Subject Keyword	Topological Manifold Triangulation Manifold Hauptvermutung Classification Theorem Pl Structure Pl Compatible Atlas New Method Stable Homeomorphism Conjecture Pl Manifold Main Result Homotopy Group First Author Pl Microbundle Stable Reduction Good Relative Version Isotopy Class Homotopy Type Immersion Theorem Compact Topological Manifold Boundary Dm Top Pl Piecewise Linear Manifold Structure Simple Homotopy Equivalence Pl Homeomorphism Structure Theorem Natural One-to-one Correspondence Triangulation Problem Positive Result Tangent Microbundle Piecewise Linear Finite Complex Compact Pl Manifold Single Obstacle Equivalence Class Lashofs Paper High Dimension
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