Homotopy Simplicial Faces and the Homology of Realizations of Simplicial Topological Spaces

The notion of a differential module with homotopy simplicial faces is introduced, which is a homotopy analog of the notion of a differential module with simplicial faces. The homotopy invariance of the structure of a differential module with homotopy simplicial faces is proved. Relationships between the construction of a differential module with homotopy simplicial faces and the theories of $A_\infty$-algebras and $D_\infty$-differential modules are found. Applications of the method of homotopy simplicial faces to describing the homology of realizations of simplicial topological spaces are presented.