Spines of topological manifolds
Erik Kjær Pedersen · Commentarii Mathematici Helvetici · 1975
In this paper we prove that a closed 2-connected topological manifold has a PL-spine, i. e. there is a locally tamely embedded complex such that a regular neighborhood of this complex is the manifold with a disc deleted (dimension is assumed to be at least 6). This “spine method ” together with the relative edition of regular neighborhoods of complexes in topological manifolds [5] makes it easy to use general position arguments in topological manifolds. This will be used in a forthcoming paper to extend various embedding theorems to the topological category. The methods we use are PL-approximation theorems due to Cernavskii, Connally, Miller, Rushing... as quoted in [5, theorem 2] and blocktransersality for PL complexes and PL submanifolds as was first considered by C. Morlet [4] and later extended by D. Stone [6]. Definition 1. A spine of a topological manifold M with ∂M 6 = ∅ is a locally tamely embedded complex K ⊂ M so that K is a strong deformation retract of M and K ⊂ M is a simple homotopy equivalence. In case ∂M = ∅ by the spine of M we mean a spine of M with a disc deleted. Theorem 2. Let (M,∂−M,∂+M) be a triad of topological manifolds dim(M) = m, and assume m ≥ 6 and pij(M,∂+M) = 0 for j < m − r, r ≤ m − 3. Further assume there is a PL-complex P locally tamely embedded in the interior of ∂+M, dim(P) = p and m−p ≥ 4. Then there is a complex K of dimension max(p+1, r, 2), locally tamely embedded in M such that