Cellular Approximation

Cellular Approximation

In algebraic topology, in the cellular approximation theorem, a map between CW-complexes can always be taken to be of a specific type. Concretely, if X and Y are CW-complexes, and f : XY is a continuous map, then f is said to be cellular, if f takes the n-skeleton of X to the n-skeleton of Y for all n, i.e. if for all n. The content of the cellular approximation theorem is then that any continuous map f : XY between CW-complexes X and Y is homotopic to a cellular map, and if f is already cellular on a subcomplex A of X, then we can furthermore choose the homotopy to be stationary on A. From an algebraic topological viewpoint, any map between CW-complexes can thus be taken to be cellular.

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Applications - Cellular Approximation For Pairs
... Then f is homotopic to a cellular map (X,A)→(Y,B) ... To see this, restrict f to A and use cellular approximation to obtain a homotopy of f to a cellular map on A ... extend this homotopy to all of X, and apply cellular approximation again to obtain a map cellular on X, but without violating the cellular property on A ...