<p>Let <i>X</i> be a finite, 2-dimensional cell complex. The curvature invariants <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3177_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho _\pm (X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mo>±</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3177_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _\pm (X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mo>±</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> were defined in [<CitationRef CitationID="CR13">13</CitationRef>], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3177_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\rho _\pm (X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ρ</mi> <mo>±</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3177_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$\sigma _\pm (X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mo>±</mo> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> are the extrema of explicit rational linear-programming problems. As a result they are rational, realised, and can be computed algorithmically.</p>

Rationality theorems for curvature invariants of 2-complexes

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2025年5月20日

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Henry Wilton

摘要

Let X be a finite, 2-dimensional cell complex. The curvature invariants $\rho _\pm (X)$ $ρ_{\pm} (X)$ and $\sigma _\pm (X)$ $σ_{\pm} (X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities $\rho _\pm (X)$ $ρ_{\pm} (X)$ and $\sigma _\pm (X)$ $σ_{\pm} (X)$ are the extrema of explicit rational linear-programming problems. As a result they are rational, realised, and can be computed algorithmically.