Homotopy types and torsion in homology of generalized chessboard complexes

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.