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<p>Koenig introduced the notion of a trisection of a closed orientable 3-manifold. It is easy to see that given an open book of a closed orientable 3-manifold <i>M</i>, it induces a trisection of <i>M</i>. We give a characterization of a trisection of a closed orientable 3-manifold that is induced by an open book of the 3-manifold. Then, we show that every closed orientable 3-manifold with a given trisection admits a trisection embedding into a trivial trisection of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12044_2025_820_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(S^6\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>S</mi> <mn>6</mn> </msup> </math></EquationSource> </InlineEquation>. Finally, we discuss when two sectors in a trisection of a 3-manifold <i>M</i> can be combined to get a Heegaard splitting of <i>M</i>.</p>

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A note on trisections of \(\textbf{3}\)-manifolds

  • 期刊论文

  • 发表日期：

    2025年4月18日

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  • Abhijeet Ghanwat,
  • Suhas Pandit,
  • A Selvakumar

摘要

Koenig introduced the notion of a trisection of a closed orientable 3-manifold. It is easy to see that given an open book of a closed orientable 3-manifold M, it induces a trisection of M. We give a characterization of a trisection of a closed orientable 3-manifold that is induced by an open book of the 3-manifold. Then, we show that every closed orientable 3-manifold with a given trisection admits a trisection embedding into a trivial trisection of \(S^6\) S 6 . Finally, we discuss when two sectors in a trisection of a 3-manifold M can be combined to get a Heegaard splitting of M.