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. This class of spaces is broader and has better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation.
Roughly speaking, a CW-complex is built by gluing together certain basic building blocks called cells.
An n-dimensional closed cell is a topological space that is homeomorphic to an n-dimensional closed ball. For example, a simplex is a closed cell, and more generally, a convex polytope is a closed cell. An n-dimensional open cell is a topological space that is homeomorphic to the open ball. A 0-dimensional open (and closed) cell is homeomorphic to the singleton space.
A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties:
- For each n-dimensional open cell C in the partition of X, there exists a homeomorphism f from the n-dimensional closed ball into X such that
- the restriction of f to the interior of the closed ball is a homeomorphism onto the cell C, and
- the image of the boundary of the closed ball is a finite union of elements of the partition whose cell dimension is less than n.
- A subset of X is closed if and only it meets the closure of each cell in a closed set.
If this is the reader's first encounter with the notion of CW complex, then she is encouraged not to dwell too much on the preceding two properties. Instead, skip ahead to the section Inductive definition.
The Hawaiian earring is an example that underlines some subtleties in the definition of CW complex. The Hawaiian earring admits a natural partition consisting of one 0-cell and countably many open 1-cells. But with respect to the usual topology of the Hawaiian earring, this partition does not satisfy the second property of the definition.
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[edit] The n-skeleton
The n-skeleton of a CW complex is the union of the cells whose dimension is at most n.
[edit] Inductive definition
A CW complex may be obtained by defining the n-skeleton inductively. This is the way that one usually encounters CW complexes in practice.
Begin by taking the 0-skeleton to be a discrete space. Next, attach 1-cells to the 0-skeleton. Namely, take a collection of (abstract) closed 1-cells and define maps from the boundary of each closed 1-cell into the 0-skeleton. The 1-skeleton is defined to be the identification space obtained from the union of the 0-skeleton and the closed 1-cells by identifying each point in the boundary of a 1-cell with its image. More generally, given the n-1-skeleton and a collection of (abstract) closed n-cells, define maps from the boundary of each n-cell into the n-1-skeleton. Define the n-skeleton to be the identification space obtained from the union of the n-1-skeleton and the closed n-cells by identifying each point in the boundary of an n-cell with its image.
Note that the process need not stop after finitely many steps. In general, the CW complex X is the direct limit of the n-skeletons with respect to the natural sequence of inclusions. A set is close in X if and only if it meets each n-skeleton in a closed set.
[edit] Examples
Many algebraic and projective varieties are easily seen to be CW-complexes. Every topological manifold can be represented as a CW-complex by means of CW-approximation.
The n-dimensional sphere is perhaps the simplest example. Let x be a point in the sphere. The complement is an open n-cell. For k < n the k-skeleton is {x}. The sphere is constructed by mapping the entire boundary of the closed n-ball into the n-1-skeleton {x}.
Another example is the n-dimensional real projective space. The k-skeleton is homeomorphic to the k-dimensional real projective space. In particular, the n-dimensional real projective space is a union of cells, exactly one of each dimension less than or equal to n.
[edit] Computing cohomology
There is a cohomology theory associated to CW-spaces, the cell cohomology, dual of the Cellular homology. 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:
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 
we get similarly
This case is simpler than for the real analog, because relations between the generators would come from the differential 
, 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.
- 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.
