A big chunk of Algebra including Algebraic Geometry and Galois theory is related to factorizations of polynomials over commutative rings. We know a lot about such factorizations. Much less is known about factorizations of polynomials over noncommutative rings, e.g. polynomials with matrix coefficients. Unlike their commutative counterparts, noncommutative polynomials admit many different factorizations which makes their theory much harder and more interesting. In 1995 I. Gelfand and the speaker constructed $n!$ different factorizations of a "generic" noncommutative polynomial in one variable with $n$ distinct roots. Later with R. Wilson we studied "algebras of pseudo-roots" or "noncommutative splitting algebras" associated with such factorizations. Such algebras can be described in terms of special directed graphs called layered graphs. To any cell complex one can also associate a layered graphs and a "splitting algebra" defined by this graph. There are surprising connections between properties of cell complexes and related splitting algebras. In my talk I will construct a bridge between noncommutative algebra related to factorizations of polynomials and combinatorial topology.