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<p>We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {C}P^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <msup> <mi>P</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, and the rational homology sphere <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(QS^4(2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <msup> <mi>S</mi> <mn>4</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations—which we deem interesting in their own rights.</p>

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Small Triangulations of 4-Manifolds and the 4-Manifold Census

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月25日

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  • Rhuaidi Burke,
  • Benjamin Burton,
  • Jonathan Spreer

摘要

We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, \(\mathbb {C}P^2\) C P 2 , and the rational homology sphere \(QS^4(2)\) Q S 4 ( 2 ) , where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations—which we deem interesting in their own rights.