CW-complex

cellular decomposition

A cell complex $ X $ satisfying the following conditions: (C) for any $ x \in X $ the complex $ X (x) $ is finite, that is, consists of a finite number of cells. (For any subset $ A $ of a cell complex $ X $, $ X (A) $ is the notation for the intersection of all subcomplexes of $ X $ containing $ A $.) (W) If $ F $ is some subset of $ X $ and if for any cell $ t $ in $ X $ the intersection $ F \cap \overline{t}\; $ is closed in $ \overline{t}\; $( and therefore in $ X $ as well), then $ F $ is a closed subset of $ X $. In this connection, each point $ x \in X $ belongs to a definite set $ t _ {x} $ of $ X $, and $ X (x) = X ( t _ {x} ) = X ( \overline{t}\; _ {x} ) $.

The notation CW comes from the initial letters of the (English) names for the above two conditions — (C) for closure finiteness and (W) for weak topology.

A finite cell complex $ X $ satisfies both conditions (C) and (W). More generally, a cell complex $ X $ each point $ x $ of which is contained in some finite subcomplex $ Y (x) $ is a CW-complex. Let $ F $ be a subset of $ X $ such that $ F \cap \overline{t}\; $ is closed in $ \overline{t}\; $ for each cell $ t $ in $ X $. Then for any $ x \in Y $ the intersection $ F \cap Y (x) $ is closed in $ X $. If the point $ x $ does not belong to $ F $, then the open set $ U _ {x} = X \setminus ( F \cap Y (x) ) $ contains $ x $ and does not intersect $ F $. The set $ ( X \setminus F ) = \cup _ {x \in X \setminus F } U _ {x} $ is open and $ F $ is closed.

The class of CW-complexes (or the class of spaces of the same homotopy type as a CW-complex) is the most suitable class of topological spaces in relation to homotopy theory. Thus, if a subset $ A $ of a CW-complex $ X $ is closed, then a mapping $ f $ from the topological space $ A $ into a topological space $ Y $ is continuous if and only if the restrictions of $ f $ to the closures of the cells of $ X $ are continuous. If $ C $ is a compact subset of a CW-complex $ X $, then the complex $ X (C) $ is finite. There exists for every cell $ t $ of a CW-complex $ X $ a set $ D $ that is open in $ \overline{t}\; $ and has $ \overline{t}\; \setminus t $ as a deformation retract.

In practice, CW-complexes are constructed by an inductive procedure: Each stage consists in glueing cells of given dimension to the result of the previous stage. The cellular structure of such a complex is directly related to its homotopy properties. Even for such "good" spaces as polyhedra it is helpful to consider their representation as CW-complexes: There are usually fewer in such a representation than in a simplicial triangulation. If $ X $ is obtained by attaching $ n $- dimensional cells to the space $ A $, then the subset $ X \times 0 \cup A \times I $, where $ I = [ 0 , 1 ] $, is a strong deformation retract of $ X \times I $.

A relative CW-complex is a pair $ ( X , A ) $ consisting of a topological space $ X $ and a closed subset $ A $, together with a sequence of closed subspaces $ ( X , A ) ^ {k} $, $ k \geq 0 $, satisfying the following conditions: a) the space $ ( X , A ) ^ {0} $ is obtained from $ A $ by adjoining $ 0 $- cells; b) for $ k \geq 1 $, $ ( X , A ) ^ {k} $ is obtained from $ ( X , A ) ^ {k-1} $ by adjoining $ k $- dimensional cells; c) $ X = \cup ( X , A ) ^ {k} $; d) the topology of $ X $ is compatible with the family $ \{ ( X , A ) ^ {k} \} $. The space $ ( X , A ) ^ {k} $ is called the $ k $- dimensional skeleton of $ X $ relative to $ A $. When $ A = \emptyset $, the relative CW-complex $ ( X , \emptyset ) = X $ is a CW-complex in the previous sense and its $ k $- dimensional skeleton is $ X ^ {k} $.

Examples. 1) The pair $ ( K , L ) $ of simplicial complexes $ K , L $, with $ L \subset K $, defines a relative CW-complex $ ( | K | , | L | ) $, where $ ( | K | , | L | ) ^ {k} = ( K ^ {k} \cup L ) $. 2) The ball $ V ^ {n} $ is a CW-complex: $ ( V ^ {n} ) ^ {k} = p _ {0} $ for $ k < n - 1 $, $ ( V ^ {n} ) ^ {n-1} = S ^ {n-1} $ and $ ( V ^ {n} ) ^ {k} = V ^ {n} $ for $ k \geq n $. The sphere $ S ^ {n-1} $ is a subcomplex of the CW-complex $ V ^ {n} $. 3) If the pair $ ( X , A ) $ is a relative CW-complex, then so is $ ( X \times I , A \times I ) $, and $ ( X \times I , A \times I ) ^ {k} = (( X , A ) ^ {k} \times \{ 0 , 1 \} ) \cup ( ( X , A ) ^ {k-1} \times I ) $( when $ k = 0 $, $ ( X , A ) ^ {-1} $ is, by definition, $ A $). 4) If $ ( X , A ) $ is a relative CW-complex, then $ X / A $ is a CW-complex and $ ( X , A ) ^ {k} = ( X / A ) ^ {k} $, where $ X / A $ is the quotient space of $ X $ obtained by identifying all points of $ A $ with a single point.

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CW-complexes have been introduced by J.H.C. Whitehead [a4] as a generalization of simplicial complexes (cf. Simplicial complex). An obvious advantage is that the number of cells needed in a decomposition is usually much smaller than the number of simplices in a triangulation. This is particularly profitable when computing homology and cohomology, and fundamental groups (cf. Fundamental group; [a1]). CW-complexes have proved useful in the context of classifying spaces for homotopy functors, and occur as Eilenberg–MacLane spaces (cf. Eilenberg–MacLane space).

Two textbooks specialized on CW-complexes are [a2] and [a3].

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