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 (in modern language, which 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.

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[edit] Attaching cells

A cell is attached by gluing a closed n-dimensional ball Dn to the (n−1)-skeleton Xn−1, i.e., the union of all lower dimensional cells. The gluing is specified by a continuous function f from ∂Dn = Sn−1 to Xn−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 is 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 Xi 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 Xi to X, a closed set in X must have a closed inverse image in each Xi, 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 Xi obtained from Xi−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] Examples and Computing their Cohomology

Many algebraic and projective varieties are easily seen to be CW-complexes. A more precise statement is that every topological manifold can be represented as a CW-complex by means of CW-approximation.

A particular example are spheres \mathbb{S}^n ={D}^n\cup \mathrm{pt} where pt is the 0-skeleton and the boundary of the only cell in dimension n is glued to it in the unique way, i.e. everything maps to this point.

Another example from algebraic geometry are the projective spaces themselves, \mathbb{P}^n\mathbb{C} and \mathbb{P}^n\mathbb{R}. These are formally defined as the space of all lines in \mathbb{C}^{n+1} or \mathbb{R}^{n+1} through the origin. The real ones can be easily described via compact spaces, namely lines in \mathbb{R}^{n+1} are parametrized by the points of \mathbb{S}^n, but antipodal points give the same line. So \mathbb{P}^n\mathbb{R}=\mathbb{S}^n/\mathbb{Z}_2. To find the CW structure consider the recursion following. \mathbb{P}^n\mathbb{R} is \mathbb{P}^{n-1}\mathbb{R} with an additional n-cell attached. Therefore:

\mathbb{P}^n\mathbb{R}=\mathrm{pt}\cup{D}^1\cup \ldots\cup{D}^n

Analog considerations lead to:

\mathbb{P}^n\mathbb{C}=\mathrm{pt}\cup{D}^2\cup \ldots\cup{D}^{2n}

[edit] Computing cohomology

There is a cohomology theory associated to CW-spaces, the cell cohomology. The main-property is that it coincides with the singular cohomology of the CW-spaces. But moreover it is often easily computable.

For the spheres we get from the above cell-decomposition:

H^k(\mathbb{S}^n)=\begin{cases} \mathbb{Z}, \quad\text{for } k=0\text{ or }n\\ 0, \quad\text{otherwise} \end{cases}

The generators of the cochains Ck are (the identity maps of) the cells. There is no relation between these generators, because the gluing map is trivial.

For \mathbb{P}^n\mathbb{C} we get similarly

H^k(\mathbb{P}^n\mathbb{C}) = \begin{cases} \mathbb{Z} \quad\text{for } 0\le k\le 2n,\text{even}\\ 0 \quad\text{otherwise} \end{cases}

This case is simpler than for the real analog, because relations between the generators would come from the differential \mathrm{d}: C^k(\ldots) \to C^{k+1}(\ldots), but for the complex case one of these 2 spaces always vanishes, therefore the differential is trivial again.

[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.

[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.