CW complex

A CW complex is a kind of a topological space that is particularly important in algebraic topology.[1] It was introduced by J. H. C. Whitehead[2] to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.[2] A CW complex can be defined inductively.[3]

As mentioned above, every collection of discrete points is a CW complex (of dimension 0).

Roughly speaking, a CW complex is made of basic building blocks called cells. The precise definition prescribes how the cells may be topologically glued together.

An n-dimensional closed cell is the image of an n-dimensional closed ball under an attaching map. 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 n-dimensional open ball. A 0-dimensional open (and closed) cell is a singleton space. Closure-finite means that each closed cell is covered by a finite union of open cells (or meets only finitely many other cells[6]).

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:

A CW complex is called regular if for each n-dimensional open cell C in the partition of X, the continuous map f from the n-dimensional closed ball to X is a homeomorphism onto the closure of the cell C. Accordingly, the partition of X is also called a regular cellulation. A loopless graph is a regular 1-dimensional CW-complex. A closed 2-cell graph embedding on a surface is a regular 2-dimensional CW-complex. Finally, the 3-sphere regular cellulation conjecture claims that every 2-connected graph is the 1-skeleton of a regular CW-complex on the 3-dimensional sphere ().

Roughly speaking, a relative CW complex differs from a CW complex in that we allow it to have one extra building block that does not necessarily possess a cellular structure. This extra-block can be treated as a (-1)-dimensional cell in the former definition.[7][8][9]

If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The n-skeleton of a CW complex is the union of the cells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex, called a subcomplex. Thus the n-skeleton is the largest subcomplex of dimension n or less.

Up to isomorphism every n-dimensional CW complex can be obtained from its (n − 1)-skeleton via attaching n-cells, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the direct limit of the skeleta: a set is closed in X if and only if it meets each skeleton in a closed set.

Singular homology and cohomology of CW complexes is readily computable via cellular homology. Moreover, in the category of CW complexes and cellular maps, cellular homology can be interpreted as a homology theory. To compute an extraordinary (co)homology theory for a CW complex, the Atiyah–Hirzebruch spectral sequence is the analogue of cellular homology.

Both of the above examples are particularly simple because the homology is determined by the number of cells—i.e.: the cellular attaching maps have no role in these computations. This is a very special phenomenon and is not indicative of the general case.

There is a technique, developed by Whitehead, for replacing a CW complex with a homotopy-equivalent CW complex that has a simpler CW decomposition.

Another way of stating the above is that a connected CW complex can be replaced by a homotopy-equivalent CW complex whose 0-skeleton consists of a single point.

The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category (for technical reasons the version for pointed spaces is actually used).[10] Auxiliary constructions that yield spaces that are not CW complexes must be used on occasion. One basic result is that the representable functors on the homotopy category have a simple characterisation (the Brown representability theorem).