In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called cells) of different dimensions in specific ways.
It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology.
[1] It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory.
The C in CW 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
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"
that satisfies: This partition of X is also called a 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.
[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.
: the cellular attaching maps have no role in these computations.
This is a very special phenomenon and is not indicative of the general case.
Its 1-skeleton can be fairly complicated, being an arbitrary graph.
if they are contained in a common tree in the maximal forest F. The quotient map
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
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.
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.
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).
[16] 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).