CW complex

• 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 function spaces Hom(X,Y) (with the compact-open topology) are not CW complexes in general. If X is finite then Hom(X,Y) is homotopy equivalent to a CW complex by a theorem of John Milnor (1959).[12] Note that X and Y are 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]

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.

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.

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 n-connected 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 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).[14] 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).

• 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.