CW Complex - Examples

Examples

• The space has the homotopy-type of a CW-complex (it is contractible) but it does not admit a CW-decomposition, since it is not locally contractible.
• The Hawaiian earring is an example of a topological space that does not have the homotopy-type of a CW-complex.
• The standard CW-structure on the real numbers has 0-skeleton the integers and as 1-cells the intervals . Similarly, the standard CW-structure on has cubical cells that are products of the 0 and 1-cells from . This is the standard cubical lattice cell structure on .
• A polyhedron is naturally a CW-complex.
• A graph is a 1-dimensional CW-complex. Trivalent graphs can be considered as generic 1-dimensional CW-complexes. Specifically, if X is a 1-dimensional CW-complex, the attaching map for a 1-cell is a map from a two-point space to X, . This map can be perturbed to be disjoint from the 0-skeleton of X if and only if and are not 0-valence vertices of X.
• The terminology for a generic 2-dimensional CW-complex is a shadow.
• The n-dimensional sphere admits a CW-structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from to 0-cell. There is a popular alternative cell decomposition, since the equatorial inclusion has complement two balls: the upper and lower hemi-spheres. Inductively, this gives a CW-decomposition with two cells in every dimension k such that .
• The n-dimensional real projective space admits a CW-structure with one cell in each dimension.
• Grassmannian manifolds admit a CW-structure called Schubert cells.
• Differentiable manifolds, algebraic and projective varieties have the homotopy-type of CW-complexes.
• The one-point compactification of a cusped hyperbolic manifold has a canonical CW-decomposition with only one 0-cell (the compactification point) called the Epstein-Penner Decomposition. Such cell decompositions are frequently called ideal polyhedral decompositions and are used in popular computer software, such as SnapPea.