Cellular Approximation Theorem and Homotopy Groups of Spheres

First we will state another theorem, Whitehead’s Theorem: If a map $f:X\to Y$ between connected CW complexes induces isomorphisms $f_*:\pi_n(X)\to\pi_n(Y)$ for all $n$, then $f$ is a homotopy equivalence. If $f$ is the inclusion of a subcomplex $X\to Y$, we have an even stronger conclusion: $X$ is a deformation retract of $Y$.

The main theorem discussed in this post is the Cellular Approximation Theorem: Every map $f:X\to Y$ of CW complexes is homotopic to a cellular map. If $f$ is already cellular on a subcomplex $A\subset X$, the homotopy may be taken to be stationary on $A$. This theorem can be viewed as the CW complex analogue of the Simplicial Approximation Theorem.

Corollary: If $n, then $\pi_n(S^k)=0$.

Proof: Consider $S^n$ and $S^k$ with their canonical CW-structure, with one 0-cell each, and with one n-cell for $S^n$ and one k-cell for $S^k$. Let $[f]\in\pi_n(S^k)$, where $f:S^n\to S^k$ is a base-point preserving map. By the Cellular Approximation Theorem, $f$ is homotopic to a cellular map $g$, where cells map to cells of same or lower dimension.

Since $n, the n-cell $S^n$ can only map to the 0-cell in $S^k$. The 0-cell in $S^n$ (the basepoint) is also mapped to the 0-cell in $S^k$. Thus $g$ is the constant map, hence $\pi_n(S^k)=0$.