CW complex

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In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical properties), but still retained a combinatorial nature, so that computational considerations were not ignored. The name itself is unrevealing: CW stands for closure-finite weak topology.

For these purposes a closed cell is a topological space homeomorphic to a simplex, or equally a ball (of which a sphere is the boundary) or cube in n dimensions. Only the topological nature matters: but one does want to keep track of the subspace on the 'surface' (the sphere that bounds the ball), and its complement, the interior points. A general cell complex would be a topological space X that is covered by cells; or to put it another way, we start with a space that is the disjoint union of some collection of cells, and take X as a quotient space, for some equivalence relation.

Contents 1 Attaching cells 1.1 Regular CW-complex 2 CW complexes are defined inductively 3 'The' homotopy category 4 Properties 5 See also 6 References

[edit] Attaching cells

A cell is attached by gluing a closed n-dimensional ball Dⁿ to the (n−1)-skeleton X_n−1, i.e., the union of all lower dimensional cells. The gluing is specified by a continuous function f from ∂Dⁿ = Sⁿ⁻¹ to X_n−1. The points on the new space are exactly the equivalence classes of points in the disjoint union of the old space and the closed cell Dⁿ, the equivalence relation being the transitive closure of x ≡ f(x). The function f plays an essential role in determining the nature of the newly enlarged complex. For example, if D² is glued onto S¹ in the usual way, we get D² itself; if f has winding number 2, we get the real projective plane instead.

[edit] Regular CW-complex

If all attaching maps are homeomorphisms, the structure is called a regular CW-complex.

[edit] CW complexes are defined inductively

Assume that X to be a Hausdorff space: for the purposes of homotopy theory this loses nothing important. Then since closed cells are compact spaces, we can be sure that their images in X are also compact, closed subspaces. From now on, we refer to 'closed cells', and 'open cells', as subspaces of X, the open cell being the image of the distinguished interior.

A 0-cell is just a point; if we only have 0-cells building up a Hausdorff space, it must be a discrete space. The general CW-complex definition can proceed by induction, using this as the base case.

The first restriction is the closure-finite one: each closed cell should be covered by a finite union of open cells.

The other restriction is to do with the possibility of having infinitely many cells, of unbounded dimension. The space X will be presented as a limit of subspaces X_i for i = 0, 1, 2, 3, … . How do we infer a topological structure for X? This is a colimit in category theory terms. From the continuity of each mapping X_i to X, a closed set in X must have a closed inverse image in each X_i, and so must intersect each closed cell in a closed subset. We can turn this round, and require that a subset C ⊂ X is by definition closed precisely when the intersection of C with the closed cells in X is always closed. This yields the weak topology on X.

With all those preliminaries, the definition of CW-complex runs like this: given X₀ a discrete space, and inductively constructed subspaces X_i obtained from X_i−1 by attaching some collection of i-cells, the resulting colimit space X is called a CW-complex provided it is given the weak topology, and the closure-finite condition is satisfied for its closed cells.

[edit] 'The' homotopy category

The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category. Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion, but half a century since Whitehead has left this definition of homotopy category in good shape. One basic result is that the representable functors on the homotopy category have a simple characterisation (the Brown representability theorem).

[edit] Properties

• The product of two CW-complexes is a CW-complex. The weak topology on this product X×Y is the same as the more familiar product topology on most spaces of interest, but can be finer if X×Y has uncountably many cells and neither X nor Y is locally compact.

• The function spaces Hom(X,Y) are not CW-complexes in general but are homotopic to CW-complexes by a theorem of John Milnor (1958). Actual function spaces occur in the somewhat larger category of compactly generated Hausdorff spaces.

[edit] See also

• The manifold analog of attaching a cell is attaching a handle, which leads to Surgery theory

[edit] References

• J. H. C. Whitehead, Combinatorial homotopy. I., Bull. Amer. Math. Soc. 55 (1949), 213–245
• J. H. C. Whitehead, Combinatorial homotopy. II., Bull. Amer. Math. Soc. 55 (1949), 453–496
• Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN 0-521-79540-0. This textbook defines CW complexes in the first chapter and uses them throughout; includes an appendix on the topology of CW complexes. A free electronic version is available on the author's homepage.