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<p>We prove that if an <i>n</i>-dimensional space <i>X</i> satisfies certain topological conditions then any triangulation of <i>X</i> as well as any its representation as a simplicial set with contractible faces has at least <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> faces of dimension <i>n</i>. One example of such <i>X</i> is the <i>n</i>-dimensional torus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((S^1)^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mn>1</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>.</p>

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Topological Lower Bounds on the Sizes of Simplicial Complexes and Simplicial Sets

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月10日

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  • Sergey Avvakumov,
  • Roman Karasev

摘要

We prove that if an n-dimensional space X satisfies certain topological conditions then any triangulation of X as well as any its representation as a simplicial set with contractible faces has at least \(2^n\) 2 n faces of dimension n. One example of such X is the n-dimensional torus \((S^1)^n\) ( S 1 ) n .