Contractible n-Manifolds and the Double n-Space Property

Abstract

We are interested in contractible n-manifolds M which decompose or split as M = A union B where A,B, and A intersect B are all homeomorphic to Euclidean n-space or A,B, and A intersect B are all homeomorphic to the n-dimensional unit ball. We introduce a 4-manifold M containing a spine which splits as A union B with A,B, and A intersect B all collapsible which in turn implies M splits as the union of two 4-balls whose intersection is also a 4-ball. From M we obtain a countably infinite collection of distinct 4-manifolds all of which split this way. Connected sums at infinity of interiors of manifolds from sequences contained in this collection constitute an uncountable set of open 4-manifolds each of which splits as the union of two 4-spaces with intersection also a 4-space.