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<p>The purpose of the present paper is twofold: first to extend to non-orientable compact 4-manifolds the notion of <i>gem-induced trisection</i>, directly obtained from colored triangulations (or, equivalently, from colored graphs encoding them, called <i>gems</i>); second to prove that, both in the orientable and non-orientable case, if the boundary is homeomorphic to a connected sum of sphere bundles over <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2024_2790_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {S}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">S</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, gem-induced trisections naturally give rise to trisections of the corresponding closed 4-manifold. As a consequence, an estimation of the trisection genus of any closed orientable 4-manifold is obtained via colored triangulations, in terms of the combinatorial properties of a Kirby diagram representing it.</p>

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Trisections of PL 4-Manifolds Arising from Colored Triangulations

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  • 发表日期：

    2025年1月16日

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  • Maria Rita Casali,
  • Paola Cristofori

摘要

The purpose of the present paper is twofold: first to extend to non-orientable compact 4-manifolds the notion of gem-induced trisection, directly obtained from colored triangulations (or, equivalently, from colored graphs encoding them, called gems); second to prove that, both in the orientable and non-orientable case, if the boundary is homeomorphic to a connected sum of sphere bundles over \(\mathbb {S}^1\) S 1 , gem-induced trisections naturally give rise to trisections of the corresponding closed 4-manifold. As a consequence, an estimation of the trisection genus of any closed orientable 4-manifold is obtained via colored triangulations, in terms of the combinatorial properties of a Kirby diagram representing it.