Dotted links, Heegaard diagrams, and colored graphs for PL 4-manifolds
A traditional way to study an oriented PL or smooth \(4\)-manifold is via a handle decomposition. These decompositions have a useful description as framed links drawn in the \(m\)-fold connected sum \(\#_{m}(S^1\times S^2)\); this connected sum is the boundary of the result of attaching \(m\) \(1\)-handles to a \(4\)-ball, and the link corresponds to the attaching instructions for additional \(2\)-handles. In order to produce a closed \(4\)-manifold, it is necessary that the boundary of the \(4\)-manifold represented by the framed link be a connected sum \(\#_{n}(S^1\times S^2)\), in which case we can additionally attach \(n\) \(3\)-handles and a \(4\)-handle. Since the attachment of \(3\)- and \(4\)-handles can be done in a unique way, up to homeomorphism, these handles do not need to be explicitly drawn in the framed link. A less well-known way of representing PL manifolds is through \textit{crystallization theory}, which depicts PL manifolds using edge-colored graphs. In general, an \(n\)-manifold uses \(n+1\) colors. This paper discusses how one may obtain crystallization graph descriptions of \(4\)-manifolds presented as framed links. Given a framed link representing a bounded \(4\)-manifold with only \(0\)-, \(1\)-, and \(2\)-handles, the author gives an algorithm to construct a \(5\)-colored graph representing the same manifold. In the case where one begins with a framed link describing a closed \(4\)-manifold, one may apply this algorithm to the union of the \(0\)-, \(1\)-, and \(2\)-handles, and she proves that one may modify the resulting graph to obtain a graph of the original closed \(4\)-manifold. This second result is not purely algorithmic, since the process requires one to recognize \(\#_{n}(S^1\times S^2)\) in a \(4\)-colored crystallization graph, and it is shown in the paper by counterexample that the graph-theoretic criterion used in the proof to recognize \(\#_{n}(S^1\times S^2)\) is not a necessary one.
scientific article; zbMATH DE number 2109945