Cellular Approximation Theorem

Let X and Y be CW-complexes, and let f:X->Y be a continuous map. Then the cellular approximation theorem states that any such f is homotopic to a cellular map. In fact, if the map f is already cellular on a CW-subcomplex A of X, then the homotopy can be taken to be stationary on A.

A famous application of the theorem is the calculation of some homotopy groups of k-spheres S^k. Indeed, let n<k and bestow on both S^n and S^k their usual CW-structure, with one 0-cell, and one n-cell, respectively one k-cell. If f:S^n->S^k is a continuous, base-point preserving map, then by cellular approximation, it is homotopic to a cellular map g. This map g must map the n-skeleton of S^n into the n-skeleton of S^k, but the n-skeleton of S^n is S^n itself, while the n-skeleton of S^k is the zero-cell, i.e., a point. This is because of the condition n<k. Thus g is a constant map, whence pi_n(S^k)=0.

See also

Cellular Map, CW-Complex

This entry contributed by Rasmus Hedegaard

Explore with Wolfram|Alpha

Cite this as:

Hedegaard, Rasmus. "Cellular Approximation Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications