In algebraic topology, the cellular approximation theorem states that 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 : X → Y 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 for all n. The cellular approximation theorem states that any continuous map f : X → Y 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.

Idea of proof

The proof can be given by induction after n, with the statement that f is cellular on the skeleton Xn. For the base case n=0, notice that every path-component of Y must contain a 0-cell. The image under f of a 0-cell of X can thus be connected to a 0-cell of Y by a path, but this gives a homotopy from f to a map which is cellular on the 0-skeleton of X.

Assume inductively that f is cellular on the (n1)-skeleton of X, and let en be an n-cell of X. The closure of en is compact in X, being the image of the characteristic map of the cell, and hence the image of the closure of en under f is also compact in Y. Then it is a general result of CW-complexes that any compact subspace of a CW-complex meets (that is, intersects non-trivially) only finitely many cells of the complex. Thus f(en) meets at most finitely many cells of Y to be a cell of highest dimension meeting f(en, the map f is already cellular on en, since in this case only cells of the n-skeleton of Y meets f(en), so we may assume that k>n. It is then a technical, non-trivial result (see Hatcher) that the restriction of f can be homotoped relative to Xn-1 to a map missing a point p∈ek. Since Yk{p} deformation retracts onto the subspace Yk-ek, we can further homotope the restriction of f to a map, say, g, with the property that g(en) misses the cell ek of Y, still relative to Xn-1. Since f(en) met only finitely many cells of Y miss all cells of Y of dimension larger than n.

We repeat this process for every n-cell of X, fixing cells of the subcomplex A on which f is already cellular, and we thus obtain a homotopy (relative to the (n1)-skeleton of X and the n-cells of A) of the restriction of f to Xn to a map cellular on all cells of X of dimension at most n. Using then the homotopy extension property to extend this to a homotopy on all of X, and patching these homotopies together, will finish the proof. For details, consult Hatcher.

Applications

Some homotopy groups

The cellular approximation theorem can be used to immediately calculate some homotopy groups their canonical CW-structure, with one 0-cell each, and with one and one Any base-point preserving is then homotopic to a map whose image lies in the which consists of the base point only. That is, any such map is nullhomotopic.

Cellular approximation for pairs

Let f:(X,A)→(Y,B) be a map of CW-pairs, that is, f is a map from X to Y under f sits inside B. 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. Use the homotopy extension property to 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.

As a consequence, we have that a CW-pair (X,A) is n-connected have dimension strictly greater than n→(X,A) is homotopic to a cellular map of pairs, and since the n-skeleton of X sits inside A, any such map is homotopic to a map whose image is in A. is n-connected, so it follows from the long exact sequence.

CW approximation

For every space X one can construct a CW complex Z and a weak homotopy equivalence that is called a CW approximation to X. CW approximation, being a weak homotopy equivalence, induces isomorphisms on homology and cohomology groups of X. Thus one often can use CW approximation to reduce a general statement to a simpler version that only concerns CW complexes.

CW approximation is constructed inducting on skeleta by attaching (i+1)-cells that (for all basepoints)

• (and are mapped to X by the contraction of the corresponding spheroids)

• are attached by constant mappings and are mapped to X ).

is evident from the second step of the construction.

References