Hauptvermutung

About: Hauptvermutung is a research topic. Over the lifetime, 47 publications have been published within this topic receiving 1942 citations.

Foundational Essays on Topological Manifolds, Smoothings, and Triangulations.

Abstract: Since Poincare's time, topologists have been most concerned with three species of manifold. The most primitive of these--the TOP manifolds--remained rather mysterious until 1968, when Kirby discovered his now famous torus unfurling device. A period of rapid progress with TOP manifolds ensued, including, in 1969, Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between 1968 and 1970. Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.

On the triangulation of manifolds and the Hauptvermutung

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—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.

On the Hauptvermutung for manifolds

Abstract: The "Hauptvermutung" is the conjecture that homeomorphic (finite) simplicial complexes have isomorphic subdivisions, i.e. homeomorphic implies piecewise linearly homeomorphic. I t was formulated in the first decade of this century and seems to have been inspired by the question of the topological invariance of the Betti and torsion numbers of a finite simplicial complex. The Hauptvermutung is known to be true for simplicial complexes of dimension 4 (Milnor, 1961). The Milnor examples, K and L, have two notable properties: (i) K and L are not manifolds, (ii) K and L are not locally isomorphic. Thus it is natural to restrict the Hauptvermutung to the class of piecewise linear w-manifolds, simplicial complexes where each point has a neighborhood which is piecewise linearly homeomorphic to Euclidean space R or Euclidean half space R\. We assume that Hz(M, Z) has no 2-torsion.