Cellular Approximation Theorem

Let  and  be CW-complexes, and let  be a continuous map. Then the cellular approximation theorem states that any such  is homotopic to a cellular map. In fact, if the map  is already cellular on a CW-subcomplex  of , then the homotopy can be taken to be stationary on .

A famous application of the theorem is the calculation of some homotopy groups of -spheres . Indeed, let  and bestow on both  and  their usual CW-structure, with one -cell, and one -cell, respectively one -cell. If  is a continuous, base-point preserving map, then by cellular approximation, it is homotopic to a cellular map . This map  must map the -skeleton of  into the -skeleton of , but the -skeleton of  is  itself, while the -skeleton of  is the zero-cell, i.e., a point. This is because of the condition . Thus  is a constant map, whence .