×

Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary. (English) Zbl 1487.57034

A crystallization \((\Gamma, \gamma)\) of a connected compact PL \(d\)-manifold is a certain type of edge-colored graph (with \((d + 1)\) colors) which represents the manifold. Minimising the number of vertices over all crystallisations of a fixed PL manifold \(M\) yields an invariant called the gem-complexity \(k(M)\) of \(M\). The regular genus of a crystallisation \((\Gamma,\gamma)\) is defined to be the smallest genus of a surface into which the graph \((\Gamma,\gamma)\) embeds regularly. Minimising the regular genus over all crystallisations of a fixed PL manifold \(M\) yields another invariant, denoted by \(\mathcal{G}(M)\), called the regular genus of \(M\). The regular genus of a PL manifold is a generalisation of the concept of genus into higher dimensions: for orientable surfaces it coincides with the ordinary genus; for orientable 3-manifolds it coincides with the Heegaard genus of the manifold.
B. Basak and M. R. Casali [Forum Math. 29, No. 4, 761–773 (2017; Zbl 1371.57020)] have established the following lower bounds in the case of closed PL 4-manifolds: If \(M\) is a closed connected PL 4-manifold then \(k(M)\geq 3 \chi(M)+10m-6\) and \(\mathcal{G}(M)\geq 2\chi(M)+5m-4\), where \(m\) is the rank of \(\pi_1(M)\). In the article under review, the authors consider PL 4-manifolds with boundaries. They show that if \(M\) is a connected compact PL 4-manifold with \(h\) boundary components, its gem-complexity \(k(M)\) and regular genus \(\mathcal{G}(M)\) satisfy the following: \begin{align*} &k(M)\geq 3\chi(M)+7m+7h-10, \\ &k(M)\geq k(\partial M)+3\chi(M)+4m+6h-9, \\ &\mathcal{G}(M) \geq 2\chi(M) + 3m + 2h - 4, \\ & \mathcal{G}(M) \geq \mathcal{G}(\partial M) + 2\chi(M) + 2m + 2h - 4, \end{align*} where \(m\) is the rank of the fundamental group of the manifold \(M\). They also show that these bounds are sharp for a large class of PL 4-manifolds with boundary.

MSC:

57Q15 Triangulating manifolds
57Q05 General topology of complexes
57K41 Invariants of 4-manifolds (including Donaldson and Seiberg-Witten invariants)
05C15 Coloring of graphs and hypergraphs

Citations:

Zbl 1371.57020
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] B. Basak, Genus-minimal crystallizations of PL 4-manifolds, Beitr. Algebra Geom. 59 (2018), no. 1, 101-111. · Zbl 1396.57036
[2] B. Basak, Regular genus and gem-complexity of some mapping tori, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019), no. 3, 2479-2493. · Zbl 1430.57022
[3] B. Basak and M. Binjola, Minimal crystallizations of 3-manifolds with boundary, preprint (2020), https://arxiv.org/abs/2001.10214v2.
[4] B. Basak and M. R. Casali, Lower bounds for regular genus and gem-complexity of PL 4-manifolds, Forum Math. 29 (2017), no. 4, 761-773. · Zbl 1371.57020
[5] B. Basak and B. Datta, Minimal crystallizations of 3-manifolds, Electron. J. Combin. 21 (2014), no. 1, Paper No. 1.61. · Zbl 1297.57055
[6] B. Basak and J. Spreer, Simple crystallizations of 4-manifolds, Adv. Geom. 16 (2016), no. 1, 111-130. · Zbl 1346.57022
[7] A. Björner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984), no. 1, 7-16. · Zbl 0538.06001
[8] J. A. Bondy and U. S. R. Murty, Graph Theory, Grad. Texts in Math. 244, Springer, New York, 2008. · Zbl 1134.05001
[9] M. R. Casali, A combinatorial characterization of 4-dimensional handlebodies, Forum Math. 4 (1992), no. 2, 123-134. · Zbl 0743.57013
[10] M. R. Casali, An infinite class of bounded 4-manifolds having regular genus three, Boll. Unione Mat. Ital. A (7) 10 (1996), no. 2, 279-303. · Zbl 0867.57014
[11] M. R. Casali, Classifying PL 5-manifolds by regular genus: The boundary case, Canad. J. Math. 49 (1997), no. 2, 193-211. · Zbl 0880.57008
[12] M. R. Casali and P. Cristofori, A catalogue of orientable 3-manifolds triangulated by 30 colored tetrahedra, J. Knot Theory Ramifications 17 (2008), no. 5, 579-599. · Zbl 1163.57017
[13] M. R. Casali and P. Cristofori, Cataloguing PL 4-manifolds by gem-complexity, Electron. J. Combin. 22 (2015), no. 4, Paper No. 4.25. · Zbl 1332.57020
[14] M. R. Casali and P. Cristofori, Gem-induced trisections of compact PL 4-manifolds, preprint (2019), https://arxiv.org/abs/1910.08777v1.
[15] M. R. Casali and C. Gagliardi, Classifying PL 5-manifolds up to regular genus seven, Proc. Amer. Math. Soc. 120 (1994), no. 1, 275-283. · Zbl 0801.57016
[16] M. R. Casali and L. Malagoli, Handle-decompositions of PL 4-manifolds, Cah. Topol. Géom. Différ. Catég. 38 (1997), no. 2, 141-160. · Zbl 0880.57007
[17] A. Cavicchioli, On the genus of smooth 4-manifolds, Trans. Amer. Math. Soc. 331 (1992), no. 1, 203-214. · Zbl 0762.57014
[18] A. Cavicchioli and C. Gagliardi, Crystallizations of \rm PL-manifolds with connected boundary, Boll. Unione Mat. Ital. B (5) 17 (1980), no. 3, 902-917. · Zbl 0453.57006
[19] R. Chiavacci, Colored pseudocomplexes and their fundamental groups, Ricerche Mat. 35 (1986), no. 2, 247-268.
[20] R. Chiavacci and G. Pareschi, Some bounds for the regular genus of PL-manifolds, Discrete Math. 82 (1990), no. 2, 165-180. · Zbl 0701.57004
[21] M. Ferri and C. Gagliardi, The only genus zero 𝑛-manifold is S^n, Proc. Amer. Math. Soc. 85 (1982), no. 4, 638-642. · Zbl 0522.57021
[22] M. Ferri, C. Gagliardi and L. Grasselli, A graph-theoretical representation of PL-manifolds—a survey on crystallizations, Aequationes Math. 31 (1986), no. 2-3, 121-141. · Zbl 0623.57012
[23] C. Gagliardi, A combinatorial characterization of 3-manifold crystallizations, Boll. Unione Mat. Ital. A (5) 16 (1979), no. 3, 441-449. · Zbl 0414.57004
[24] C. Gagliardi, Extending the concept of genus to dimension 𝑛, Proc. Amer. Math. Soc. 81 (1981), no. 3, 473-481. · Zbl 0467.57004
[25] C. Gagliardi, Cobordant crystallizations, Discrete Math. 45 (1983), no. 1, 61-73. · Zbl 0513.05060
[26] C. Gagliardi, Regular genus: The boundary case, Geom. Dedicata 22 (1987), no. 3, 261-281. · Zbl 0618.57009
[27] C. Gagliardi and L. Grasselli, Representing products of polyhedra by products of edge-colored graphs, J. Graph Theory 17 (1993), no. 5, 549-579. · Zbl 0781.05020
[28] J. L. Gross, Voltage graphs, Discrete Math. 9 (1974), 239-246. · Zbl 0286.05106
[29] M. Pezzana, Sulla struttura topologica delle varietà compatte, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 23 (1974), no. 1, 269-277. · Zbl 0314.57005
[30] S. Stahl, Generalized embedding schemes, J. Graph Theory 2 (1978), no. 1, 41-52. · Zbl 0396.05013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.