Abstract
Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold [Formula: see text] admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, [Formula: see text] may be represented by a framed link yielding [Formula: see text], with exactly [Formula: see text] components ([Formula: see text] being the second Betti number of [Formula: see text]). As a consequence, the regular genus of [Formula: see text] is proved to be the double of [Formula: see text]. Moreover, the characterization of any such PL 4-manifold by [Formula: see text], where [Formula: see text] is the gem-complexity of [Formula: see text] (i.e. the non-negative number [Formula: see text], [Formula: see text] being the minimum order of a crystallization of [Formula: see text]), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).
Highlights
Introduction and main resultsFor any piecewise linear (PL) n-manifold M n, it is known the existence of a contracted triangulation, i.e. a pseudocomplex1 triangulating M n, whose 0-skeleton consists of exactly n + 1 vertices.Note that contracted triangulations of a PL n-manifold M may be seen as an intermediate notion, between simplicial complexes and singular triangulations.What makes contracted triangulations user-friendly is the possibility of representing them by means of their dual graphs, which turn out to be a special kind of edge-coloured graphs, called crystallizations
In this paper we show that simple crystallizations, recently introduced in [6], meet both requirements, and we study the properties of the PL 4-manifolds admitting such crystallizations
In virtue of Theorem 2, both the invariants gem-complexity and regular genus turn out to be additive with respect to connected sum within the class of all PL 4-manifolds admitting simple crystallizations: see Proposition 9
Summary
For any PL n-manifold M n, it is known the existence of a contracted triangulation, i.e. a pseudocomplex triangulating M n, whose 0-skeleton consists of exactly n + 1 vertices. The representation by crystallizations has allowed the definition of graph-defined PL-invariants: one of them, the regular genus ([24]), has yielded classification theorems of particular significance in dimension 4 and 5 (see, for example, [19], [9] and [20]). In virtue of Theorem 2, both the invariants gem-complexity and regular genus turn out to be additive with respect to connected sum within the class of all PL 4-manifolds admitting simple crystallizations (in particular: within the class of all “standard” simplyconnected PL 4-manifolds): see Proposition 9. From this viewpoint, our result about additivity for 4-manifolds admitting simple crystallizations appears to be significant, in connection with the problem of the existence of simple crystallizations for a given -connected PL 4-manifold (see Proposition 11 for some particular families yielding a negative answer and for relationships with 4-dimensional crystallization catalogues). We point out that simple crystallizations may be useful in order to prove algorithmically the PL-equivalence of different triangulations of the same (-connected) topological 4-manifold: for example, in [6, Section 1] and [18, Section 1], ongoing attempts to prove via simple crystallizations the conjecture of [30] concerning the K3-surface are described
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