<p>Let <i>X</i> and <i>Y</i> be two simply connected rational CW-complexes, and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1554_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(n \in \mathbb{N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. We study the homotopy set <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1554_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">\([P^nX, P^nY]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <msup> <mi>P</mi> <mi>n</mi> </msup> <mi>X</mi> <mo>,</mo> <msup> <mi>P</mi> <mi>n</mi> </msup> <mi>Y</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, where <i>P</i><sup><i>n</i></sup><i>X</i> and <i>P</i><sup><i>n</i></sup><i>Y</i> are the <InlineEquation ID="IEq178"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1554_Article_IEq178.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>n</mi> </math></EquationSource> </InlineEquation>-th Postnikov sections of <i>X</i> and <i>Y</i>, respectively. An equivalence relation is defined on this set, revealing connections with the cohomology groups. This approach uses rational homotopy theory to uncover new structural insights.</p>

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On the homotopy sets of simply connected rational CW-complexes

  • M. Benkhalifa

摘要

Let X and Y be two simply connected rational CW-complexes, and \(n \in \mathbb{N}\) n N . We study the homotopy set \([P^nX, P^nY]\) [ P n X , P n Y ] , where PnX and PnY are the \(n\) n -th Postnikov sections of X and Y, respectively. An equivalence relation is defined on this set, revealing connections with the cohomology groups. This approach uses rational homotopy theory to uncover new structural insights.