 TOPICS  # Cellular Approximation Theorem

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

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

Cellular Map, CW-Complex

This entry contributed by Rasmus Hedegaard

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Hedegaard, Rasmus. "Cellular Approximation Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CellularApproximationTheorem.html