Topology and its Applications
In this paper we investigate conditions under which a compact set in a piecewise linear 4-manifold has close neighborhoods with 1-dimensional spines. We prove that such neighborhoods exist in case the compactum satisfies the inessential loops condition and either has fundamental dimension 0 or has the shape of S1. Corollaries include two complement theorems and a weak flatness theorem for compacta in S4.
Venema, Gerard A., "Neighborhoods of compacta in 4-manifolds" (1989). University Faculty Publications. 554.