The surjection property and computable type (Q6592027)

The authors provide a detailed, further study of two properties of spaces and pairs of spaces, the surjection property and the \(\varepsilon\)-surjection property. These were introduced by them in [LIPIcs -- Leibniz Int. Proc. Inform. 229, Article 111, 16 p. (2022; Zbl 07870321)] to characterize the notion of computable type arising from computability theory. They state that, at least informally, a metrizable compactum has \textit{computable type} if for every embedded copy \(X\) of it in the Hilbert cube, if the set of metric balls that are disjoint from \(X\) can be enumerated by a Turing machine, then the set of metric balls intersecting \(X\) can be enumerated by a Turing machine. One would have to look further into the references, however, to find a complete definition of this notion. Let us review a definition from Section 2.\N\NDefinition 2.1. A pair \((X,A)\) has the \textit{surjection property} if every continuous function \(f:X\to X\) such that \(f|A\) is the identity is surjective. If \((X,d)\) is a metric space, \(A\subset X\) is a closed subset, and for all \(\varepsilon>0\), every continuous function \(f:X\to X\) with \(f|A\) equal the identity and \(d(f,\mathrm{id}_X)<\varepsilon\) is a surjection, then we say that \((X,A)\) has the \(\varepsilon\)-\textit{surjection property}. In the special case that \(A=\emptyset\), then one says that \(X\) has the \(\varepsilon\)-\textit{surjection property}.\N\NThe focus of this research is on studying the properties given in Definition 2.1. and in relating them to computability theory. Although this definition is given in a general form, its application will be to compact metric spaces \(X\) and closed, hence compact, subspaces \(A\subset X\). The organization of the paper goes as follows. Section 2 contains an investigation of general aspects of the \(\varepsilon\)-surjection property, such as its preservation under countable unions. The next section includes a result showing that for finite unions of cones, the \(\varepsilon\)-surjection property is equivalent to the surjection property on each cone. In Section 4, the authors consider the surjection property of cones and obtain precise relationships with homotopy and homology theories, and in Section 5, they apply these results to a family of spaces to obtain a source of counterexamples. It is in Section 6 that applications of the previously developed theory are given to the computable type property.\N\NSection 7 contains a summary of what has been accomplished and it states some applications that have resulted:\N\N1. The cone on a compact manifold has computable type.\N\N2. Each finite simplicial complex together with its odd boundary has computable type.\N\N3. The computable type property is not (in general) preserved under products.\N\NSection 7 is followed by four appendixes which are designed to accumulate certain theory concerning homotopies, cone and join, Hopf's extension and classification theorem, and coefficient groups.