CW complex: Information from Answers.com

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 some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation.

1 Formulation
2 Inductive definition of CW-complexes
3 Examples
4 Homology and cohomology of CW-complexes
5 Modification of CW-structures
6 'The' homotopy category
7 Properties
8 See also
9 References

Formulation

Roughly speaking, a CW-complex is built by gluing together certain basic building blocks called cells. The definition speaks to a precise definition of the topological structure on the result. The C stands for "closure-finite", and the W for "weak topology".

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 continuous map f from the n-dimensional closed ball to 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 contained in a finite union of elements of the partition whose cell dimension is less than n.
A subset of X is closed if and only if it meets the closure of each cell in a closed set.

Inductive definition of CW-complexes

The n-skeleton of a CW complex is the union of the cells whose dimension is at most n. 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 closed in X if and only if it meets each n-skeleton in a closed set.

Examples

The space $\{re^{2\pi i \theta} : 0 \leq r \leq 1, \theta \in \mathbb Q\} \subset \mathbb R^2$ has the homotopy-type of a CW-complex (it is contractible) but it does not admit a CW-decomposition, since it is not locally contractible.

The Hawaiian earring is an example of a topological space that does not have the homotopy-type of a CW-complex.

The standard CW-structure on the real numbers has 0-skeleton the integers $\mathbb Z$ and 1-cells the intervals $\{ [n,n+1] : n \in \mathbb Z\}$ . Similarly, the standard CW-structure on $\mathbb R^n$ has cubical cells that are products of the 0 and 1-cells from $\mathbb R$ . This is the standard cubical lattice cell structure on $\mathbb R^n$ .

A polyhedron is naturally a CW-complex.

A graph is a 1-dimensional CW-complex. Trivalent 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, $f : \{0,1\} \to X$ . This map can be perturbed to be disjoint from the 0-skeleton of X if and only if $f (0)$ and $f (1)$ are not 0-valence vertices of X.

The terminology for a generic 2-dimensional CW-complex is a shadow ^[1].

The n-dimensional sphere admits a CW-structure with two cells.

The n-dimensional real projective space admits a CW-structure with one cell in each dimension.

Grassmannian manifolds admit a CW-structure called Schubert cells.

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, SnapPea.

Homology and cohomology of CW-complexes

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 spheres we get from the above cell-decomposition:

$H^k(\mathbb{S}^n)=\begin{cases} \mathbb{Z}, \quad\text{for } k=0\text{ or }n\\ 0, \quad\text{otherwise}. \end{cases}$

The generators of the cochains $C k$ are (the identity maps of) the cells. There is no relation between these generators, because the gluing map is trivial.

For $\mathbb{P}^n\mathbb{C}$ we get similarly

$H^k(\mathbb{P}^n\mathbb{C}) = \begin{cases} \mathbb{Z} \quad\text{for } 0\le k\le 2n,\text{even}\\ 0 \quad\text{otherwise}. \end{cases}$

Both of the above examples are particularly simple because the homology is determined by the number of cells -- ie: 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 which has a simpler CW-decomposition.

Consider, for example, an arbitrary CW-complex. It's 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 $X/\sim$ where the equivalence relation is generated by $x \sim y$ if they are contained in a common tree in the maximal forest F. The quotient map $X \to X/\sim$ is a homotopy equivalence. Moreover, $X/\sim$ naturally inherits a CW-structure, with cells corresponding to the cells of $X$ which are not contained in F. In particular, the 1-skeleton of $X/\sim$ 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 $X 1$ consists of a single point? The answer is yes. The first step is to observe that $X 1$ and the attaching maps to construct $X 2$ from $X 1$ 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

X 1

. If we let $\tilde X$ be the corresponding CW-complex $\tilde X = X \cup e^1 \cup e^2$ then there is a homotopy-equivalence $\tilde X \to X$ 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 $\tilde X = X \cup e^2 \cup e^3$ where the new 3-cell has an attaching map that consists of the new 2-cell and remainder mapping into

X 2

. A similar slide gives a homotopy-equivalence $\tilde X \to X$ .

If a CW-complex X is n-connected one can find a homotopy-equivalent CW-complex $\tilde X$ whose n-skeleton $X n$ consists of a single point. The argument for $n \geq 2$ is similar to the $n = 1$ case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for $H_n(X;\mathbb Z)$ (using the presentation matrices coming from cellular homology. ie: 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 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).^[2] 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).

Properties

CW-complexes are locally contractible.

CW-complexes satisfy the Whitehead theorem -- ie: a map between CW-complexes is a homotopy-equivalence if and only if it induces an isomorphism on all homotopy groups.

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 homotopy equivalent to CW-complexes by a theorem of John Milnor (1958). Actual function spaces occur in the somewhat larger category of compactly generated Hausdorff spaces.

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.
A. T. Lundell and S. Weingram, The topology of CW complexes, Van Nostrand University Series in Higher Mathematics (1970), ISBN 0-442-04910-2

^ Turaev, V. G. (1994), "Quantum invariants of knots and 3-manifolds", De Gruyter Studies in Mathematics (Berlin: Walter de Gruyter & Co.) 18
^ For example, the opinion "The class of CW-complexes (or the class of spaces of the same homotopy type as a CW-complex) is the most suitable class of topological spaces in relation to homotopy theory" appears in the article CW-complex of the Springer Encyclopaedia of Mathematics.

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