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Abstract <p> The complement of an arrangement of diagonal subspaces <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x_{i_1}=\dots=x_{i_k}\)</EquationSource> </InlineEquation> in the real space is defined by a simplicial complex <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal K\)</EquationSource> </InlineEquation>. In this paper, we prove that the complement of a diagonal subspace arrangement is homotopy equivalent to a subcomplex <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathrm{Perm}(\mathcal K)\)</EquationSource> </InlineEquation> of faces of the permutohedron. The product in the cohomology ring of the complement of a diagonal arrangement is then described via Saneblidze and Umble’s cellular approximation of the diagonal map in the permutohedron. We consider the projection from the permutohedron to the cube and prove that the Saneblidze–Umble diagonal is mapped to the diagonal constructed by Li Cai for describing the product in the cohomology of a real moment–angle complex. </p>

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Permutohedral Complex and Complements of Diagonal Subspace Arrangements

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Taras Panov,
  • Vsevolod Tril

摘要

Abstract

The complement of an arrangement of diagonal subspaces \(x_{i_1}=\dots=x_{i_k}\) in the real space is defined by a simplicial complex \(\mathcal K\) . In this paper, we prove that the complement of a diagonal subspace arrangement is homotopy equivalent to a subcomplex \(\mathrm{Perm}(\mathcal K)\) of faces of the permutohedron. The product in the cohomology ring of the complement of a diagonal arrangement is then described via Saneblidze and Umble’s cellular approximation of the diagonal map in the permutohedron. We consider the projection from the permutohedron to the cube and prove that the Saneblidze–Umble diagonal is mapped to the diagonal constructed by Li Cai for describing the product in the cohomology of a real moment–angle complex.