A CW-complex is a nice topological space which is or can be built up inductively, by a process of attaching -dimensional disks along their boundary spheres for all : a cell complex built from the basic topological cells .
Being, therefore, essentially combinatorial objects, CW complexes are the principal objects of interest in algebraic topology; in fact, most spaces of interest to algebraic topologists are homotopy equivalent to CW-complexes. Notably the geometric realization of every simplicial set, hence also of every groupoid, 2-groupoid, etc., is a CW complex.
Also, CW complexes are the cofibrant objects in the standard model structure on topological spaces. This means in particular that every (Hausdorff) topological space is weakly homotopy equivalent to a CW-complex (but need not be strongly homotopy equivalent to one). Since every topological space is a fibrant object in this model category structure, this means that the full subcategory of Top on the CW-complexes is a category of “homotopically very good representatives” of homotopy types. See at homotopy theory and homotopy hypothesis for more on this.
A CW-complex is a topological space equipped with a sequence of spaces and continuous maps
and a cocone making into its colimit (in Top, or else in the category of compactly generated spaces or one of many other nice categories of spaces) where each space (called the -skeleton of ) is the result of attaching copies of the -disk along their boundaries to . Specifically, is the empty space, and each is a pushout in Top of a diagram of the form
where is some index set, each is the boundary inclusion of a copy of , and is a continuous map, often called an attaching map. The coprojections of these pushouts give the arrows on which diagram (1) is based.
A relative CW-complex is defined as above, except is allowed to be any space.
A finite CW-complex is one which admits a presentation in which there are only finitely many attaching maps, and similarly a countable CW-complex is one which admits a presentation with countably many attaching maps.
A CW-complex is a locally contractible topological space.
For instance (Hatcher, prop. A.4).
For instance (Hatcher, prop. A.5).
Let be a CW-complex and write
for its filtered topological space-structure with the topological space obtained from by gluing on -cells. For write for the set of -cells of .
The relative singular homology of the filtering degrees is
where denotes the free abelian group on the set of -cells.
The proof is spelled out at Relative singular homology - Of CW complexes.
With we have
In particular if is a CW-complex of finite dimension (the maximum degree of cells), then
Moreover, for the inclusion
is an isomorphism and for we have an isomorphism
This is mostly for instance in (Hatcher, lemma 2.34 b),c)).
Now by prop. 3 the leftmost and rightmost homology groups here vanish when and and hence exactness implies that
Finally for the last claim use that the above exact sequence gives
and hence that with the above the map is surjective.
Basic textbook accounts include
Original articles include
John Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280.