CW complex

A CW complex (also called cellular complex or cell 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]

Definition

CW complex

A CW complex is constructed by taking the union of a sequence of topological spaces

such that each ${\displaystyle X_{k}}$ is obtained from ${\displaystyle X_{k-1}}$ by gluing copies of k-cells ${\displaystyle (e_{\alpha }^{k})_{\alpha }}$, each homeomorphic to the open ${\displaystyle k}$-ball ${\displaystyle D^{k}}$, to ${\displaystyle X_{k-1}}$ by continuous gluing maps ${\displaystyle g_{\alpha }^{k}:\partial e_{\alpha }^{k}\to X_{k-1}}$. The maps are also called .

Each ${\displaystyle X_{k}}$ is called the k-skeleton of the complex.

The topology of ${\displaystyle X=\cup _{k}X_{k}}$ is weak topology: a subset ${\displaystyle U\subset X}$ is open

${\displaystyle U\cap e_{\alpha }^{k}}$ is open for each cell ${\displaystyle e_{\alpha }^{k}}$.

In the language of category theory, the topology on ${\displaystyle X}$ 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 (with the discrete topology).
• A 1-dimensional CW complex is constructed by taking the
  glues" its boundary (its two endpoints) to elements of the 0-dimensional complex (the points). 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 (for some ${\displaystyle k<n}$) with one or more copies of the n-dimensional ball. For each copy, there is a map that "glues" its boundary (the ${\displaystyle (n-1)}$-dimensional sphere) to elements of the ${\displaystyle k}$-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 ${\displaystyle \cup _{k}X_{k}}$ 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.^[3]

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 (-1)-dimensional cell in the former definition.^[4][5][6]

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:^[7]

• An interval. It can be constructed from two points (x and y), and the 1-dimensional ball B (an interval), 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, ${\displaystyle f:\{0,1\}\to X}$. This map can be perturbed to be disjoint from the 0-skeleton of X if and only if ${\displaystyle f(0)}$ and ${\displaystyle f(1)}$ are not 0-valence vertices of X.
• The standard CW structure on the real numbers has as 0-skeleton the integers ${\displaystyle \mathbb {Z} }$ and as 1-cells the intervals ${\displaystyle \{[n,n+1]:n\in \mathbb {Z} \}}$. Similarly, the standard CW structure on ${\displaystyle \mathbb {R} ^{n}}$ has cubical cells that are products of the 0 and 1-cells from ${\displaystyle \mathbb {R} }$. This is the standard cubic lattice cell structure on ${\displaystyle \mathbb {R} ^{n}}$.

Finite-dimensional CW complexes

Some examples of finite-dimensional CW complexes are:^[7]

• An n-dimensional sphere. It admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell ${\displaystyle D^{n}}$ is attached by the constant mapping from its boundary ${\displaystyle S^{n-1}}$ to the single 0-cell. An alternative cell decomposition has one (n-1)-dimensional sphere (the "equator") and two n-cells that are attached to it (the "upper hemi-sphere" and the "lower hemi-sphere"). Inductively, this gives ${\displaystyle S^{n}}$ a CW decomposition with two cells in every dimension k such that ${\displaystyle 0\leq k\leq n}$.
• 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.^[8]
• 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 (the compactification point) 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

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 which being a closed set with empty interior. This argument extends to many other infinite-dimensional spaces.
• The hedgehog space ${\displaystyle \{re^{2\pi i\theta }:0\leq r\leq 1,\theta \in \mathbb {Q} \}\subseteq \mathbb {C} }$ is homotopic to a CW complex (the point) but it does not admit a CW decomposition, since it is not locally contractible.
• The Hawaiian earring is not homotopic to a CW complex. It has no CW decomposition, because it is not locally contractible at origin. It is not homotopy equivalent to a CW complex, because it has no good open cover.

Properties

• CW complexes are locally contractible (Hatcher, prop. A.4).
• If a space is homotopic to a CW complex, then it has a good open cover.^[9] 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.^[10][11]
• 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
  compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compact-open topology; the above statements remain true.^[13]

Homology and cohomology of CW complexes

.

Some examples:

• For the sphere, ${\displaystyle S^{n},}$ take the cell decomposition with two cells: a single 0-cell and a single n-cell. The cellular homology chain complex ${\displaystyle C_{*}}$ and homology are given by:

${\displaystyle C_{k}={\begin{cases}\mathbb {Z} &k\in \{0,n\}\\0&k\notin \{0,n\}\end{cases}}\quad H_{k}={\begin{cases}\mathbb {Z} &k\in \{0,n\}\\0&k\notin \{0,n\}\end{cases}}}$

since all the differentials are zero.

Alternatively, if we use the equatorial decomposition with two cells in every dimension

${\displaystyle C_{k}={\begin{cases}\mathbb {Z} ^{2}&0\leqslant k\leqslant n\\0&{\text{otherwise}}\end{cases}}}$

and the differentials are matrices of the form ${\displaystyle \left({\begin{smallmatrix}1&-1\\1&-1\end{smallmatrix}}\right).}$ This gives the same homology computation above, as the chain complex is exact at all terms except ${\displaystyle C_{0}}$ and ${\displaystyle C_{n}.}$

• For ${\displaystyle \mathbb {P} ^{n}(\mathbb {C} )}$ we get similarly

${\displaystyle H^{k}\left(\mathbb {P} ^{n}(\mathbb {C} )\right)={\begin{cases}\mathbb {Z} &0\leqslant k\leqslant 2n,{\text{ even}}\\0&{\text{otherwise}}\end{cases}}}$

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.

Modification of CW structures

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

Consider, for example, an arbitrary CW complex. Its 1-skeleton can be fairly complicated, being an arbitrary graph. Now consider a maximal forest F in this graph. Since it is a collection of trees, and trees are contractible, consider the space ${\displaystyle X/{\sim }}$ where the equivalence relation is generated by ${\displaystyle x\sim y}$ if they are contained in a common tree in the maximal forest F. The quotient map ${\displaystyle X\to X/{\sim }}$ is a homotopy equivalence. Moreover, ${\displaystyle X/{\sim }}$ naturally inherits a CW structure, with cells corresponding to the cells of ${\displaystyle X}$ that are not contained in F. In particular, the 1-skeleton of ${\displaystyle X/{\sim }}$ is a disjoint union of wedges of circles.

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.

Consider climbing up the connectivity ladder—assume X is a simply-connected CW complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replace X by a homotopy-equivalent CW complex where ${\displaystyle X^{1}}$ consists of a single point? The answer is yes. The first step is to observe that ${\displaystyle X^{1}}$ and the attaching maps to construct ${\displaystyle X^{2}}$ from ${\displaystyle X^{1}}$ form a group presentation. The Tietze theorem for group presentations states that there is a sequence of moves we can perform to reduce this group presentation to the trivial presentation of the trivial group. There are two Tietze moves:

1) Adding/removing a generator. Adding a generator, from the perspective of the CW decomposition consists of adding a 1-cell and a 2-cell whose attaching map consists of the new 1-cell and the remainder of the attaching map is in ${\displaystyle X^{1}}$. If we let ${\displaystyle {\tilde {X}}}$ be the corresponding CW complex ${\displaystyle {\tilde {X}}=X\cup e^{1}\cup e^{2}}$ then there is a homotopy equivalence ${\displaystyle {\tilde {X}}\to X}$ given by sliding the new 2-cell into X.

2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing X by ${\displaystyle {\tilde {X}}=X\cup e^{2}\cup e^{3}}$ where the new 3-cell has an attaching map that consists of the new 2-cell and remainder mapping into ${\displaystyle X^{2}}$. A similar slide gives a homotopy-equivalence ${\displaystyle {\tilde {X}}\to X}$.

If a CW complex X is

one can find a homotopy-equivalent CW complex ${\displaystyle {\tilde {X}}}$ whose n-skeleton ${\displaystyle X^{n}}$ consists of a single point. The argument for ${\displaystyle n\geq 2}$ is similar to the ${\displaystyle n=1}$ case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for ${\displaystyle H_{n}(X;\mathbb {Z} )}$ (using the presentation matrices coming from cellular homology. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.

'The' homotopy category

The

).

See also

• Abstract cell complex
• The notion of CW complex has an adaptation to smooth manifolds called a handle decomposition, which is closely related to surgery theory.

References

Notes

General references

• Lundell, A. T.; Weingram, S. (1970). The topology of CW complexes.
  ISBN 0-442-04910-2
  .
• Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids.
  ISBN 978-3-03719-083-8. More details on the [1]
  first author's home page]