Abstract: The M by N chessboard complex is the simplicial complex of non-threatening configurations of rooks on an M by N chessboard. Homotopy properties of this complex have been studied in topological combinatorics. In particular, it is known that these complexes are highly connected. If the chessboard is far enough from a square shape, the complex is even Cohen-Macauley (a strengthening of connectivity, which I will explain). The symmetric groups on M and N letters both act on this complex by permuting rows and columns of the chessboard. I will describe a series of complexes that relate to the chessboard complexes as the braid groups relate to the symmetric groups. These complexes are huge: every non-maximal simplex has an infinite link. I shall discuss connectivity and the Cohen-Macauley property for the new family of complexes.