Cycle-free chessboard complexes and symmetric homology of algebras

Chessboard complexes and their relatives have been an important recurring theme of topological combinatorics. Closely related “cycle-free chessboard complexes” have been recently introduced by Ault and Fiedorowicz in [S. Ault, Z. Fiedorowicz, Symmetric homology of algebras. arXiv:0708.1575v54 [math.AT] 5 Nov 2007; Z. Fiedorowicz, Question about a simplicial complex, Algebraic Topology Discussion List (maintained by Don Davis) http://www.lehigh.edu/~dmd1/zf93] as a tool for computing symmetric analogues of the cyclic homology of algebras. We study connectivity properties of these complexes and prove a result that confirms a strengthened conjecture from [S. Ault, Z. Fiedorowicz, Symmetric homology of algebras. arXiv:0708.1575v54 [math.AT] 5 Nov 2007].