CW complex


In mathematics, and specifically in topology, a CW complex is a topological space that is built by gluing together topological balls of different dimensions in specific ways. The notion generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory.
CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation.
The C in CW stands for "closure-finite", and the W for "weak" topology.

Definition

CW complex

A CW complex is constructed by taking the union of a sequence of topological spaces such that each is obtained from by gluing copies of k-cells, each homeomorphic to the open unit ball in -dimensional Euclidean space, to by continuous gluing maps. The maps are also called attaching maps. Thus as a set,.
Each is called the k-skeleton of the complex.
The topology of is a weak topology: a subset is open iff is open for each k-skeleton.
In the language of category theory, the topology on is the direct limit of the diagram The name "CW" stands for "closure-finite weak topology", which is explained by the following theorem:
This partition of X is also called a cellulation.

The construction, in words

The CW complex construction is a straightforward generalization of the following process:
  • A 0-dimensional CW complex is just a set of zero or more discrete points.
  • A 1-dimensional CW complex is constructed by taking the disjoint union of a 0-dimensional CW complex with one or more copies of the unit interval. For each copy, there is a map that "glues" its boundary to elements of the 0-dimensional complex. The topology of the CW complex is the topology of the quotient space defined by these gluing maps.
  • In general, an n-dimensional CW complex is constructed by taking the disjoint union of a k-dimensional CW complex with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary to elements of the -dimensional complex. The topology of the CW complex is the quotient topology defined by these gluing maps.
  • An infinite-dimensional CW complex can be constructed by repeating the above process countably many times. Since the topology of the union is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits.

    Regular CW complexes

A regular CW complex is a CW complex whose gluing maps are homeomorphisms. Accordingly, the partition of X is also called a regular cellulation.
A loopless graph is represented by 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.

Relative CW complexes

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 -dimensional cell in the former definition.

Examples

0-dimensional CW complexes

Every discrete topological space is a 0-dimensional CW complex.

1-dimensional CW complexes

Some examples of 1-dimensional CW complexes are:
  • An interval. It can be constructed from two points, and the 1-dimensional ball B, such that one endpoint of B is glued to x and the other is glued to y. The two points x and y are the 0-cells; the interior of B is the 1-cell. Alternatively, it can be constructed just from a single interval, with no 0-cells.
  • A circle. It can be constructed from a single point x and the 1-dimensional ball B, such that both endpoints of B are glued to x. Alternatively, it can be constructed from two points x and y and two 1-dimensional balls A and B, such that the endpoints of A are glued to x and y, and the endpoints of B are glued to x and y too.
  • A graph. Given a graph, a 1-dimensional CW complex can be constructed in which the 0-cells are the vertices and the 1-cells are the edges of the graph. The endpoints of each edge are identified with the incident vertices to it. This realization of a combinatorial graph as a topological space is sometimes called a topological graph.
  • *3-regular graphs can be considered as generic 1-dimensional CW complexes. Specifically, if X is a 1-dimensional CW complex, the attaching map for a 1-cell is a map from a two-point space to X,. This map can be perturbed to be disjoint from the 0-skeleton of X if and only if and are not 0-valence vertices of X.
  • The standard CW structure on the real numbers has as 0-skeleton the integers and as 1-cells the intervals. Similarly, the standard CW structure on has cubical cells that are products of the 0 and 1-cells from. This is the standard cubic lattice cell structure on.

    Finite-dimensional CW complexes

Some examples of finite-dimensional CW complexes are:
  • An n-dimensional sphere. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from its boundary to the single 0-cell. An alternative cell decomposition has one -dimensional sphere and two n-cells that are attached to it. Inductively, this gives a CW decomposition with two cells in every dimension k such that.
  • The n-dimensional real projective space. It admits a CW structure with one cell in each dimension.
  • The terminology for a generic 2-dimensional CW complex is a shadow.
  • A polyhedron is naturally a CW complex.
  • Grassmannian manifolds admit a CW structure called Schubert cells.
  • Differentiable manifolds, algebraic and projective varieties have the homotopy type of CW complexes.
  • The one-point compactification of a cusped hyperbolic manifold has a canonical CW decomposition with only one 0-cell called the Epstein–Penner Decomposition. Such cell decompositions are frequently called ideal polyhedral decompositions and are used in popular computer software, such as SnapPea.

    Infinite-dimensional CW complexes

  • The infinite-dimensional sphere. It admits a CW-structure with 2 cells in each dimension which are assembled in a way such that the -skeleton is precisely given by the -sphere.
  • The infinite-dimensional projective spaces, and. has one cell in every dimension,, has one cell in every even dimension and has one cell in every dimension divisible by 4. The respective skeletons are then given by, and .

    Non CW-complexes

  • An infinite-dimensional Hilbert space is not a CW complex: it is a Baire space and therefore cannot be written as a countable union of n-skeletons, each of them being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
  • The hedgehog space is homotopy equivalent to a CW complex but it does not admit a CW decomposition, since it is not locally contractible.
  • The Hawaiian earring has no CW decomposition, because it is not locally contractible at origin. It is also not homotopy equivalent to a CW complex, because it has no good open cover.

    Properties

  • CW complexes are locally contractible.
  • If a space is homotopy equivalent to a CW complex, then it has a good open cover. A good open cover is an open cover, such that every nonempty finite intersection is contractible.
  • CW complexes are paracompact. Finite CW complexes are compact. A compact subspace of a CW complex is always contained in a finite subcomplex.
  • CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only if it induces an isomorphism on all homotopy groups.
  • A covering space of a CW complex is also a CW complex.
  • The product of two CW complexes can be made into a CW complex. Specifically, if X and Y are CW complexes, then one can form a CW complex X × Y in which each cell is a product of a cell in X and a cell in Y, endowed with the weak topology. The underlying set of X × Y is then the Cartesian product of X and Y, as expected. In addition, the weak topology on this set often agrees with the more familiar product topology on X × Y, for example if either X or Y is finite. However, the weak topology can be finer than the product topology, for example if neither X nor Y is locally compact. In this unfavorable case, the product X × Y in the product topology is not a CW complex. On the other hand, the product of X and Y in the category of compactly generated spaces agrees with the weak topology and therefore defines a CW complex.
  • Let X and Y be CW complexes. Then the function spaces Hom are not CW complexes in general. If X is finite then Hom is homotopy equivalent to a CW complex by a theorem of John Milnor. Note that X and Y are compactly generated Hausdorff spaces, so Hom is often taken with the compactly generated variant of the compact-open topology; the above statements remain true.
  • Cellular approximation theorem

    Homology and cohomology of CW complexes

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 homology theory for a CW complex, the Atiyah–Hirzebruch spectral sequence is the analogue of cellular homology.
Some examples:
  • For the sphere, take the cell decomposition with two cells: a single 0-cell and a single n-cell. The cellular homology chain complex and homology are given by:
  • For we get similarly
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.