Compact contractible \(n\)-manifolds have arc spines \((n\geq 5)\)
The following two theorems were motivated by questions about the existence of disjoint spines in compact contractible manifolds. Theorem 1. Every compact contractible \(n\)-manifold \((n\geq 5)\) is the union of two \(n\)-balls along a contractible \((n-1)\)-dimensional submanifold of their boundaries. A compactum \(X\) is a spine of a compact manifold \(M\) if \(M\) is homeomorphic to the mapping cylinder of a map from \(\partial M\) to \(X\). Theorem 2. Every compact contractible \(n\)-manifold \((n\geq 5)\) has a wild arc spine. Also a new proof is given that for \(n\geq 6\), every homology \((n-1)\)- sphere bounds a compact contractible \(n\)-manifold. The implications of arc spines for compact contractible manifolds of dimensions 3 and 4 are discussed in section 5. The questions about the existence of disjoint spines in compact contractible manifolds which motivated the preceding theorems are stated in section 6.
scientific article; zbMATH DE number 779060