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<k, 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<k, 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.

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