Computability of glued manifolds

Abstract

Abstract We examine conditions under which a semicomputable set in a computable topological space is computable. In particular, we examine topological spaces $\varDelta $ that have computable type, which means that any semicomputable set homeomorphic to $\varDelta $ is computable. It is known that each compact manifold has computable type. In this paper, we examine compact manifolds $M$ and $N$ and a space $M\cup _{\gamma }N$ obtained by gluing $M$ and $N$ together by way of a homeomorphism $\gamma :A\rightarrow B$, where $A$ and $B$ are closed subspaces of $M$ and $N$, respectively. We show that $M\cup _{\gamma }N$ in general need not have computable type. We prove that $M\cup _{\gamma }N$ has computable type under the additional assumption that $A$ and $B$ are contained in regular submanifolds of $M$ and $N$. We also show that the same holds for a space obtained by gluing finitely many manifolds, but not for infinitely many.