PL 4-manifolds admitting simple crystallizations: framed links and regular genus
| Autor: | Casali, M. R., Cristofori, P., Gagliardi, C. |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | J. Knot Theory Ramifications 25, 1650005 (2016) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1142/S021821651650005X |
| Popis: | Simple crystallizations are edge-coloured graphs representing PL 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In the present 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 $\mathbb S^3$, with exactly $\beta_2(M)$ components ($\beta_2(M)$ being the second Betti number of $M$). As a consequence, the regular genus of $M$ is proved to be the double of $\beta_2(M)$. Moreover, the characterization of any such PL $4$-manifold by $k(M)= 3 \beta_2(M)$, 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). Comment: 14 pages, no figures; this is a new version of the former paper "A characterization of PL 4-manifolds admitting simple crystallizations" |
| Databáze: | arXiv |
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