CW complexes and cellular approximation

Start with a set of zero-cells and attach n-disks along continuous maps from their boundary spheres into the previous skeleton. Each disk interior becomes an open n-cell, and its characteristic map is a homeomorphism on that interior but can identify boundary points. A CW complex satisfies the closure-finiteness and weak-topology conditions, which ensure the cellwise construction has the intended global topology. A cell's closure need not be embedded as a closed ball: identifying its boundary is exactly how circles, projective spaces, and many other examples are built.

The n-skeleton $X^{(n)}$ contains all cells of dimension at most n. A subcomplex is a union of cells closed under taking the attached boundaries. Attaching maps are part of the data; recording only how many cells occur in each dimension loses essential information. Euler characteristic, when defined by a finite cell count, and cellular homology capture some consequences but not all homotopy information. The one-skeleton supplies loop generators, the two-cells supply relations, and higher cells affect higher-dimensional deformation questions.

A map between CW complexes is cellular when it takes $X^{(n)}$ into $Y^{(n)}$ for every n. The cellular approximation theorem says every continuous map is homotopic to a cellular map. If the original map is already cellular on a subcomplex A, the homotopy can keep A fixed. In particular, when chosen basepoints are zero-cells, based maps can be approximated through based homotopies. Preserving vertices alone is not enough for cellularity: each edge must land in the target one-skeleton, and the condition continues through all dimensions.

Give a sphere its standard structure with one zero-cell and one top-dimensional cell. A based map $S^k\to S^n$ with $k<n$ can be homotoped to a cellular map, whose entire image lies in the k-skeleton of the target, just the basepoint. Thus the homotopy group below the sphere dimension vanishes. The same theorem says a space's two-skeleton determines π₁, because loops and their disk homotopies can be brought into the needed low skeleta. It does not imply that every map into a lower-dimensional target is nullhomotopic; the Hopf map is a decisive warning.

A weak homotopy equivalence induces a bijection on path components and isomorphisms on all based homotopy groups in every component. Whitehead's theorem states that such a map between CW complexes is a homotopy equivalence. Both the map and the CW hypotheses are important. Merely knowing that two spaces have isomorphic abstract groups in each degree does not supply one compatible map inducing those isomorphisms. Nor does equality of homology groups substitute for the required homotopy-group information.

Outside the CW setting, weak equivalence need not be a homotopy equivalence, so one cannot erase these hypotheses after a successful calculation. CW approximation is a related theorem: a space admits a CW model mapping to it by a weak homotopy equivalence, but the map need not already be a homotopy equivalence for an arbitrary target. The cell viewpoint is therefore a controlled framework for inference, not a guarantee that every geometric space has been completely classified. Keep the attaching maps and the maps between models visible throughout the argument.