Some compact contractible manifolds containing disjoint spines
People have long wondered whether a Mazur-type contractible 4-manifold can possess a pair of disjoint spines. The question was extended to all compact contractible PL manifolds. In the present paper the author establishes the existence of this phenomenon in a class of compact contractible manifolds. Let \(L\) be a compact, acyclic, \(k\)-dimensional polyhedron in \(S^n\) and let \(M\) be the closure of the complement of a regular neighborhood of \(L\). If \(n \geq k + 3\), then \(M\) is contractible and \(\pi_1 (\partial M) = \pi_1 (L)\); if the latter group is nontrivial, then \(M\) is said to be a Newman contractible manifold. The main theorem of the present paper states that every such \(M\) with \(n > 4k\) contains a pair of disjoint spines. The author then applies his theory to settle a conjecture in the topological embedding theory. A subset \(K\) of a metric space \(X\) is said to be slippery if it can be pushed off itself by homeomorphisms of \(X\) arbitrarily close to the identity, and \(K\) is said to be sticky otherwise. It has been believed that if \(J\) is a circle and \(H\) is any non-simply-connected homology \((n - 2) \)-sphere, then \(J\) (and any subarc of \(J)\) is sticky in \(J*H = \Sigma^2 H \approx S^n\). However, here the author proves that \(J\) is slippery in \(S^n\) if \(H\) is the boundary of a compact contractible \((n - 1)\)-manifold containing a pair of disjoint spines, and thus his main theorem provides lots of counterexamples.
scientific article; zbMATH DE number 722020