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Compact 4-manifolds admitting special handle decompositions

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of special handle decompositions for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connected PL 4-manifold with empty or connected boundary having colored triangulations that minimize the combinatorially defined PL invariants regular genus, gem-complexity or G-degree among all such manifolds with the same second Betti number.

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Notes

  1. Actually, the result holds in general dimension n, by simply substituting \(n+1\) to 5, and it is usually stated in terms of the associated graphs, by making use of the notion of crystallization, which gives the name to the whole combinatorial representation theory via edge-colored graphs: see [17], or Sect. 2 of the present paper.

  2. Note that, as their names suggest, the class of simple crystallization is a subclass of that of weak simple ones; actually, they both are particular cases of even larger classes of crystallizations, representing not simply-connected compact PL 4-manifolds, and still minimizing combinatorially defined PL invariants, i.e. weak semi-simple and semi-simple crystallizations. Precise definitions of all these classes may be found in [15], together with its references.

  3. Note that, as a consequence, the links of all h-simplices of the triangulation with \(h > 0\) are \((n-h-1)\)-spheres.

  4. The notion of pseudocomplex is a generalization of that of simplicial complex. Roughly speaking, in a pseudocomplex two n-simplexes may have more than one common \((n-1)\)-face; however identifications of faces of the same simplex are not allowed.

  5. Note that, in this case, each \(\{c_1,\dots ,c_h\}\)-residue (\(1\le h < n\)) of \(\varGamma \) represents an \((n-h-1)\)-sphere.

  6. Actually, weak semi-simple crystallizations turn out to admit also interesting properties with respect to the existence of trisections of the represented compact 4-manifolds, and to the computation of their trisection genus: see [11] and references therein.

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Acknowledgements

This work was supported by GNSAGA of INDAM and by the University of Modena and Reggio Emilia, project: “Discrete Methods in Combinatorial Geometry and Geometric Topology”

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  1. Department of Physics, Informatics and Mathematics, University of Modena and Reggio Emilia, Modena, Italy

    Maria Rita Casali & Paola Cristofori

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  1. Maria Rita Casali
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  2. Paola Cristofori
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Correspondence to Maria Rita Casali.

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Casali, M.R., Cristofori, P. Compact 4-manifolds admitting special handle decompositions. RACSAM 115, 118 (2021). https://doi.org/10.1007/s13398-021-01001-x

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