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 M admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, M may be represented by a framed link yielding S^3, with exactly β_2(M) components (β_2(M) being the second Betti number of M). As a consequence, the regular genus of M is proved to be the double of β_2(M). Moreover, the characterization of any such PL 4-manifold by k(M)=3β_2(M), where k(M) is the gem-complexity of M (i.e. the non-negative number p−1, 2p being the minimum order of a crystallization of M), 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).
PL 4-manifolds admitting simple crystallizations: framed links and regular genus / Casali, Maria Rita; Cristofori, Paola; Gagliardi, Carlo. - In: JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS. - ISSN 0218-2165. - STAMPA. - 25:1(2016), pp. 1-14. [10.1142/S021821651650005X]
PL 4-manifolds admitting simple crystallizations: framed links and regular genus
CASALI, Maria Rita;CRISTOFORI, Paola;GAGLIARDI, Carlo
2016
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 M admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, M may be represented by a framed link yielding S^3, with exactly β_2(M) components (β_2(M) being the second Betti number of M). As a consequence, the regular genus of M is proved to be the double of β_2(M). Moreover, the characterization of any such PL 4-manifold by k(M)=3β_2(M), where k(M) is the gem-complexity of M (i.e. the non-negative number p−1, 2p being the minimum order of a crystallization of M), 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).File | Dimensione | Formato | |
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