Abstract <p>For a finite group <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation> and connected topological spaces <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource>  </InlineEquation> such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation> is endowed with a free left <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation>-action <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq8.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">\(\tau\)</EquationSource>  </InlineEquation>, we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the triple <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">\((X,Y;\tau)\)</EquationSource>  </InlineEquation>) to decide whether the Borsuk–Ulam property holds for based homotopy classes <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha\in[X,Y]_{0}\)</EquationSource>  </InlineEquation>, as well as for free homotopy classes <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha\in[X,Y]\)</EquationSource>  </InlineEquation>. Here a homotopy class <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq12.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(\alpha\)</EquationSource>  </InlineEquation> is said to satisfy the Borsuk–Ulam property if, for each of its representatives <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq13.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\in\alpha\)</EquationSource>  </InlineEquation>, there exists an <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation>-orbit, where <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq15.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(f\)</EquationSource>  </InlineEquation> fails to be injective. Our geometric characterization is attained by constructing an <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation>-equivariant map from <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation> to the classical configuration space <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq18.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(F_{|H|}(Y)\)</EquationSource>  </InlineEquation>. We derive an algebraic condition from the geometric characterisation, and show that former one is in fact equivalent to the latter one when <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation> and <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource>  </InlineEquation> are aspherical. We then specialize to the 1-dimensional case, i.e., when <InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(X\)</EquationSource>  </InlineEquation> is an arbitrary connected graph, <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(H\)</EquationSource>  </InlineEquation> is cyclic, and <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(Y\)</EquationSource>  </InlineEquation> is either a tree, a circle, or the connected graph <InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8220_Article_IEq24.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">\(S^{1}\vee I\)</EquationSource>  </InlineEquation> with two vertices and two edges one of which is a loop and the other is a closed interval. The graph-braid-group ingredient in our characterizations is then effectively controlled through the use of discrete Morse theory.</p>

Borsuk–Ulam Property for Graphs II: The \(\mathbb{Z}_{n}\)-Action

期刊论文
发表日期：

2025年3月1日

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D. L. Gonçalves,
J. González

摘要

Abstract

For a finite group \(H\) and connected topological spaces \(X\) and \(Y\) such that \(X\) is endowed with a free left \(H\) -action \(\tau\) , we provide a geometric condition in terms of the existence of a commutative diagram of spaces (arising from the triple \((X,Y;\tau)\) ) to decide whether the Borsuk–Ulam property holds for based homotopy classes \(\alpha\in[X,Y]_{0}\) , as well as for free homotopy classes \(\alpha\in[X,Y]\) . Here a homotopy class \(\alpha\) is said to satisfy the Borsuk–Ulam property if, for each of its representatives \(f\in\alpha\) , there exists an \(H\) -orbit, where \(f\) fails to be injective. Our geometric characterization is attained by constructing an \(H\) -equivariant map from \(X\) to the classical configuration space \(F_{|H|}(Y)\) . We derive an algebraic condition from the geometric characterisation, and show that former one is in fact equivalent to the latter one when \(X\) and \(Y\) are aspherical. We then specialize to the 1-dimensional case, i.e., when \(X\) is an arbitrary connected graph, \(H\) is cyclic, and \(Y\) is either a tree, a circle, or the connected graph \(S^{1}\vee I\) with two vertices and two edges one of which is a loop and the other is a closed interval. The graph-braid-group ingredient in our characterizations is then effectively controlled through the use of discrete Morse theory.