Piecewise Linear Topology

2nd derived neighborhood A₁ ambient isotopy annulus theorem B₁ barycentric boundary collar C₁ chain complex choose codimension compact P. L. contained Corollary Definition deformation retract derived subdivisions dual cells exists a PL exists an ambient f and g F₁ face fixed h-cobordism h₁ Hence homeo homotopy equivalence identity induction isotopy H joinable K₁ L₁ Lemma Let F Let h level preserving linearly link(A locally finite locally unknotted m-ball m-manifold M₁ map f meets morphism N₁ N₂ non-degenerate P. L. embedding P. L. homeomorphism P. L. manifold P. L. map P. L. spaces P. L. subspace P₁ PL homeomorphism PL manifold PL subspace polyhedra polyhedron Proof proper ball pair r-handles regular neighborhood sequence simplex simplicial complex star star(v sunny collapses Suppose Theorem topological triangulation uniqueness of regular vertex vertices W₁