We have proposed a complete set of basis Euler operators for updating cell complexes in arbitrary dimensions, which can be classified as homology-preserving and homology-modifying. Here, we define the effect of homology-preserving operators on the incidence graph representation of cell complexes. Based on these operators, we build a multi- resolution model for cell complexes represented in the form of the incidence graph, and we compare its 2D instance with the pyramids of 2-maps, designed for images.