Cellular Approximation For Pairs
Let f:(X,A)→(Y,B) be a map of CW-pairs, that is, f is a map from X to Y, and the image of under f sits inside B. Then f is homotopic to a cellular map (X,A)→(Y,B). To see this, restrict f to A and use cellular approximation to obtain a homotopy of f to a cellular map on A. Use the homotopy extension property to extend this homotopy to all of X, and apply cellular approximation again to obtain a map cellular on X, but without violating the cellular property on A.
As a consequence, we have that a CW-pair (X,A) is n-connected, if all cells of have dimension strictly greater than n: If, then any map →(X,A) is homotopic to a cellular map of pairs, and since the n-skeleton of X sits inside A, any such map is homotopic to a map whose image is in A, and hence it is 0 in the relative homotopy group .
We have in particular that is n-connected, so it follows from the long exact sequence of homotopy groups for the pair that we have isomorphisms → for all and a surjection →.
Read more about this topic: Cellular Approximation, Applications