Cellular approximation theorem

Remarks

The proof proceeds cell by cell, by induction on the dimension. The key lemma concerns a map from a closed $k$-disk into a space obtained by attaching an $n$-cell with $n > k$. Such a map can be deformed so that it misses a point of the open cell, and then pushed off the cell by radial projection. Taking $A = \emptyset$ shows that every map is homotopic to a cellular map.

Example: give $S^n$ the CW structure with one $0$-cell and one $n$-cell, and let $k < n$. Every map $S^k \to S^n$ is homotopic to a cellular map, which sends $S^k = (S^k)^k$ into the $k$-skeleton of $S^n$, a single point. So $\pi_k(S^n) = 0$ for $k < n$.

The theorem also shows that the inclusion $X^n \to X$ induces an isomorphism on $\pi_k$ for $k < n$ and a surjection for $k = n$.

Going deeper

For a student

Cellular approximation reduces many homotopy questions to combinatorics of cells. It gives the vanishing of $\pi_k(S^n)$ for $k < n$. It shows that the fundamental group of a CW complex depends only on its $2$-skeleton. It also makes the maps induced on cellular chain complexes well defined. Its simplicial ancestor is the simplicial approximation theorem.