Noncommutative Koszul algebras from combinatorial topology

Associated to any uniform finite layered graph G there is a noncommutative graded quadratic algebra A(Gamma) given by a construction due to Gelfand, Retakh, Serconek and Wilson. It is natural to ask when these algebras are Koszul. Unfortunately, a mistake in the literature states that all such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW-complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology groups H(X)(n,k), generalizing the usual cohomology groups H(n)(X). Along with several other results, our methods give a new and primarily topological proof of the main result of [12] and [7].