First we will state another theorem, Whitehead’s Theorem: If a map between connected CW complexes induces isomorphisms for all , then is a homotopy equivalence. If is the inclusion of a subcomplex , we have an even stronger conclusion: is a deformation retract of .
The main theorem discussed in this post is the Cellular Approximation Theorem: Every map of CW complexes is homotopic to a cellular map. If is already cellular on a subcomplex , the homotopy may be taken to be stationary on . This theorem can be viewed as the CW complex analogue of the Simplicial Approximation Theorem.
Corollary: If , then .
Proof: Consider and with their canonical CW-structure, with one 0-cell each, and with one n-cell for and one k-cell for . Let , where is a base-point preserving map. By the Cellular Approximation Theorem, is homotopic to a cellular map , where cells map to cells of same or lower dimension.
Since , the n-cell can only map to the 0-cell in . The 0-cell in (the basepoint) is also mapped to the 0-cell in . Thus is the constant map, hence .