Жасушалық кешендер: Топологиялық кеңістіктердің бір түрі

Type of topological space
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was 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 (often with a much smaller complex). The C stands for "closure finite", and the W for "weak" topology.

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.

Every discrete topological space is a 0 dimensional CW complex.

An infinite dimensional Hilbert space is not a CW complex: it is a Baire space and therefore cannot be written as a countable union of n skeletons, each of which being a closed set with empty interior. This argument extends to many other infinite dimensional spaces. The hedgehog space is homotopic to a CW complex (the point) but it does not admit a CW decomposition, since it is not locally contractible. The Hawaiian earring is not homotopic to a CW complex. It has no CW decomposition, because it is not locally contractible at origin. It is not homotopy equivalent to a CW complex, because it has no good open cover.

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. 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. 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). 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.

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:

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 If we let be the corresponding CW complex then there is a homotopy equivalence 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 where the new 3 cell has an attaching map that consists of the new 2 cell and remainder mapping into A similar slide gives a homotopy equivalence
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.

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). 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).