A General Theory of Relative Regular Neighborhoods
Marshall M. Cohen · Transactions of the American Mathematical Society · 1969
Regular neighborhoods of polyhedra in manifolds have been discussed in many papers, most notably, in the present context, in [W], [H-Z], [H] and [S].We study regular neighborhoods of X mod Y in Z where (Z, X, Y) is an arbitrary triad of finite dimensional, locally compact polyhedra.When applied to manifolds our theory generalizes previous results in that we make no compactness assumptions and place no restrictions on the polyhedra X and Y to be considered.The paper is organized as follows:1. Definitions and notation 2. Simplicial preliminaries 3. The uniqueness theorem 4. The characterization of collared subpolyhedra 5. Cone-retracting and collaring regular neighborhoods 6.The stellar neighborhood theorem 7. Relationships between regular neighborhoods 8. Collapsibility and collapsible retractions 9. Regular neighborhoods via collapsibility.V is defined to be a regular neighborhood of X mod Y in Z if there exists a full triangulation (/, K, L; h) of (Z, X, Y) in which V underlies a relative first derived neighborhood, V=hN(K-L, /').Thus, on the one hand, a given regular neighborhood can be presented in an explicit manner with a fixed triangulation.On the other hand, regular neighborhoods are piecewise linear rather than simplicial objects ("there exists a triangulation") and there are uncountably many regular neighborhoods of X mod Y in Z implicit in the choice of triangulating complex and triangulating homeomorphism.We develop the theory by first exploiting the seeming concreteness ( § §2-5) and then developing the implicit generality ( § §6-9).After presenting the basic simplicial data we turn immediately to the uniqueness theorem (3.1).(Existence is automatic.)This asserts that, given two regular neighborhoods of X mod Y in Z, there is an ambient piecewise linear isotopy taking one onto the other and keeping X u Y fixed.The proof is quite direct and carries a great deal of extra information.For example (3.4) if Z is an ^-manifold the isotopy can be realized by a sequence of 2n "moves relative to X u F."