Homotopy types and torsion in homology of generalized chessboard complexes

Abstract

Multiple chessboard complexes are deleted joins of skeleta of a simplex. We classify their homotopy types for at most three rows and extend this classification to arbitrary column capacities, with the $2$-torus as the only exception to a wedge of spheres. We determine explicit homotopy decompositions for two families with four and five rows and use a common capacity redistribution map to construct infinite families with $2$- and $3$-torsion in integral homology. Together, these results give a negative answer to the Segre part of the problem posed by Reiner and Roberts on characteristic-free resolutions of Segre and Veronese modules and identify three as the largest number of rows for which all generalized chessboard complexes have torsion-free integral homology. Our $3$-torsion construction applies to every square ordinary chessboard complex with at least five rows. Combined with earlier results, it completes the proof of the Shareshian-Wachs conjecture on torsion in chessboard complexes.

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BibTeX RIS

BibTeX

@misc{limonchenko2026homotopy,
  title = {{Homotopy types and torsion in homology of generalized chessboard complexes}},
  author = {Ivan Limonchenko and Seonghyeon Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37827},
  eprint = {2610.37827},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37827},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Homotopy types and torsion in homology of generalized chessboard complexes
AU  - Ivan Limonchenko
AU  - Seonghyeon Yu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37827
UR  - https://arxiv.org/abs/2610.37827
AB  - Multiple chessboard complexes are deleted joins of skeleta of a simplex. We classify their homotopy types for at most three rows and extend this classification to arbitrary column capacities, with the $2$-torus as the only exception to a wedge of spheres. We determine explicit homotopy decompositions for two families with four and five rows and use a common capacity redistribution map to construct infinite families with $2$- and $3$-torsion in integral homology. Together, these results give a negative answer to the Segre part of the problem posed by Reiner and Roberts on characteristic-free resolutions of Segre and Veronese modules and identify three as the largest number of rows for which all generalized chessboard complexes have torsion-free integral homology. Our $3$-torsion construction applies to every square ordinary chessboard complex with at least five rows. Combined with earlier results, it completes the proof of the Shareshian-Wachs conjecture on torsion in chessboard complexes.
ER  -

Abstract

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