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UBC Theses and Dissertations

Homogeneity of combinatorial spheres Walker, Alexander Crawford

Abstract

The object of this thesis is to cover the results of [1] from a piecewise linear point of view. The principal result of [1] is the theorem on the homogeneity of spheres, i.e. the complement of a combinatorial n-cell in a combinatorial n-sphere is a combinatorial n-cell. A piecewise linear proof of this theorem by a "long induction" using regular neighbourhoods and collapsing was given in [4]. A direct piecewise linear proof appeared recently in [2]; it is based on the existence of a "collar" for the boundary of a combinatorial manifold with boundary. Our proof is similar to the proof in [2]. We proceed by induction on dimensions, proving simultaneously the existence of a collar for the boundary of a combinatorial manifold with boundary and the homogeneity theorem. From [2] we adopted an argument which eliminates a certain combinatorial technique applied in [1] and involving induction on the length of stellar subdivisions. The results of [1] were previously interpreted in piecewise linear topology by use of a theorem in [3] stating that piecewise linearly homeomorphic simplicial complexes have subdivisions which are combinatorially equivalent in the sense of [1]. The thesis is divided into three parts. The first gives definitions and basic properties relating to simplicial complexes. The second concerns combinatorial manifolds, and in the third we present our proof of the piecewise linear homogeneity of spheres.

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