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]
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 -ball , to by continuous gluing maps . The maps are also called attaching maps.
Each is called the k-skeleton of the complex.
The topology of is weak topology: a subset is open iff is open for each cell .
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:
Theorem — A Hausdorff space X is homeomorphic to a CW complex iff there exists a partition of X into "open cells" , each with a corresponding closure (or "closed cell") that satisfies:
This partition of X is also called a cellulation.
The CW complex construction is a straightforward generalization of the following process:
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]
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]
Every discrete topological space is a 0-dimensional CW complex.
Some examples of 1-dimensional CW complexes are: [7]
Some examples of finite-dimensional CW complexes are: [7]
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:
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 where the equivalence relation is generated by if they are contained in a common tree in the maximal forest F. The quotient map is a homotopy equivalence. Moreover, naturally inherits a CW structure, with cells corresponding to the cells of that are not contained in F. In particular, the 1-skeleton of 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 consists of a single point? The answer is yes. The first step is to observe that and the attaching maps to construct from 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:
If a CW complex X is n-connected one can find a homotopy-equivalent CW complex whose n-skeleton consists of a single point. The argument for is similar to the case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for (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.
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