From factorizations of noncommutative polynomials to combinatorial topology

In 1995 I. Gelfand and the speaker constructed n! different factorizations of a generic polynomial of degree n in one variable with noncommuting coefficients. Later we defined and studied "noncommutative splitting algebras" associated with such factorizations. These algebras can be described in terms of ranked quivers (layered graphs). Such quivers can be associated to any regular cell complex. I will describe some surprising connections between properties of cell complexes and the related splitting algebras.