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Classifying Pl 5-Manifolds by Regular Genus: The Boundary Case

Published online by Cambridge University Press:  20 November 2018

Maria Rita Casali
Maria Rita Casali*
Affiliation:
Dipartimento di Matematica, Via Campi 213 B, I-41100 MODENA, Italy e-mail: [email protected]
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Abstract

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In the present paper, we face the problem of classifying classes of orientable PL 5-manifolds M5 with h ≥ 1 boundary components, by making use of a combinatorial invariant called regular genusG(M5). In particular, a complete classification up to regular genus five is obtained: where denotes the regular genus of the boundary ∂M5 and denotes the connected sumof h ≥ 1 orientable 5-dimensional handlebodies 𝕐αi of genus αi ≥ 0 (i = 1, . . . ,h), so that .

Moreover, we give the following characterizations of orientable PL 5-manifolds M5 with boundary satisfying particular conditions related to the “gap” between G(M5) and either G(∂M5) or the rank of their fundamental group rk(π1(M5)): Further, the paper explains how the above results (together with other known properties of regular genus of PL manifolds) may lead to a combinatorial approach to 3-dimensional Poincaré Conjecture.

Keywords

57N1557Q1505C10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 2 , 01 April 1997 , pp. 193 - 211
Copyright
Copyright © Canadian Mathematical Society 1997

References

[B] Barden, D., Simply connected five-manifolds, Ann. of Math. 82(1965), 365–385.Google Scholar
[C1] Casali, M.R., A combinatorial characterization of 4-dimensional handlebodies, Forum Math. 4(1992), 123–134.Google Scholar
[C2] Casali, M.R., A note on the characterization of handlebodies, Europ. J. Combinatorics 14(1993), 301–310.Google Scholar
[C3] Casali, M.R., On the regular genus of 5-manifolds with free fundamental group, to appear.Google Scholar
[CG] Casali, M.R. and Gagliardi, C., Classifying PL 5-manifolds up to regular genus seven, Proc.Amer.Math. Soc. (1) 120(1994), 275–283.Google Scholar
[CM] Casali, M.R. and Malagoli, L., Handle-decompositions of PL 4-manifolds, Cahiers de Topologie etGeom. Diff. Cat., to appear.Google Scholar
[Cav] Cavicchioli, A., On the genus of smooth 4-manifolds, Trans. Amer.Math. Soc. 31(1992), 203–214.Google Scholar
[CH] Cavicchioli, A. and Hegenbarth, F., On the determination of PL-manifolds by handles of lower dimension, Topology Appl. (2) 53(1993), 111–118.Google Scholar
[CP] Chiavacci, R. and Pareschi, G., Some bounds for the regular genus of PL-manifolds, Discrete Math. 82(1990), 165–180.Google Scholar
[Co] Costa, A., Coloured graphs representing manifolds and universal maps, Geometriae Dedicata 28(1988), 349–357.Google Scholar
[FG1] Ferri, M. and Gagliardi, C., The only genus zero n-manifold is 𝕊 n, Proc. Amer. Math. Soc. 85(1982), 638–642.Google Scholar
[FG2] Ferri, M., A characterization of punctured n-spheres, Yokohama Math. J. 33(1985), 29–38.Google Scholar
[FGG] Ferri, M., Gagliardi, C. and Grasselli, L., A graph-theoretical representation of PL-manifolds. A survey on crystallizations, Aequationes Math. 31(1986), 121–141.Google Scholar
[G1] Gagliardi, C., Extending the concept of genus to dimension n, Proc. Amer.Math. Soc. 81(1981), 473–481.Google Scholar
[G2] Gagliardi, C., Regular genus: the boundary case, Geometriae Dedicata 22(1987), 261–281.Google Scholar
[GG] Gagliardi, C. and Grasselli, L., Representing products of polyhedra by products of edge-colored graphs, Journal of Graph Theory (5) 15(1993), 549–579.Google Scholar
[HW] Hilton, P.J. and Wylie, S., An introduction to algebraic topology - Homology theory, Cambridge Univ. Press, Cambridge, 1960.Google Scholar
[L] Laudenbach, F., Topologie de la dimension trois: homotopie et isotopie, Soc. Math. de France, Asterisque 12, 1974.Google Scholar
[LP] Laudenbach, F. and Poenaru, V., A note on 4-dimensional handlebodies, Bull. Soc. Math. France 100(1972), 337–344.Google Scholar
[LM] Lins, S. and Mandel, A., Graph-encoded 3-manifolds, Discrete Math. 57(1985), 261–284.Google Scholar
[M] J.M.Montesinos, Heegaard diagrams for closed 4-manifolds, Geometric Topology (Cantrell, J. Ed.), Proc. 1977 Georgia Conference, Academic Press, 1979. 219–237.Google Scholar