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 "closurefinite", and the W for "weak" topology.^{[2]}
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4.01 CW complexes

4.03 CW complexes and the HEP

Algebraic Topology 0: Cell Complexes

5.02 Fundamental group of a CW complex

Lecture 2  Quotient topology and CW Complexes
Transcription
Definition
CW complex
A CW complex is constructed by taking the union of a sequence of topological spaces
Each is called the kskeleton 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
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:
 For each , there exists a continuous surjection from the dimensional closed ball such that
 The restriction to the open ball is a homeomorphism.
 (closurefiniteness) The image of the boundary is covered by a finite number of closed cells, each having cell dimension less than k.
 (weak topology) A subset of X is closed if and only if it meets each closed cell in a closed set.
This partition of X is also called a cellulation.
The construction, in words
The CW complex construction is a straightforward generalization of the following process:
 A 0dimensional CW complex is just a set of zero or more discrete points (with the discrete topology).
 A 1dimensional CW complex is constructed by taking the disjoint union of a 0dimensional CW complex with one or more copies of the unit interval. For each copy, there is a map that "glues" its boundary (its two endpoints) to elements of the 0dimensional complex (the points). The topology of the CW complex is the topology of the quotient space defined by these gluing maps.
 In general, an ndimensional CW complex is constructed by taking the disjoint union of a kdimensional CW complex (for some ) with one or more copies of the ndimensional ball. For each copy, there is a map that "glues" its boundary (the dimensional sphere) to elements of the dimensional complex. The topology of the CW complex is the quotient topology defined by these gluing maps.
 An infinitedimensional CW complex can be constructed by repeating the above process countably many times. Since the topology of the union is indeterminate, one takes the direct limit topology, since the diagram is highly suggestive of a direct limit. This turns out to have great technical benefits.
Regular CW complexes
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 1dimensional CWcomplex. A closed 2cell graph embedding on a surface is a regular 2dimensional CWcomplex. Finally, the 3sphere regular cellulation conjecture claims that every 2connected graph is the 1skeleton of a regular CWcomplex on the 3dimensional sphere.^{[3]}
Relative CW complexes
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 extrablock can be treated as a (1)dimensional cell in the former definition.^{[4]}^{[5]}^{[6]}
Examples
0dimensional CW complexes
Every discrete topological space is a 0dimensional CW complex.
1dimensional CW complexes
Some examples of 1dimensional CW complexes are:^{[7]}
 An interval. It can be constructed from two points (x and y), and the 1dimensional ball B (an interval), such that one endpoint of B is glued to x and the other is glued to y. The two points x and y are the 0cells; the interior of B is the 1cell. Alternatively, it can be constructed just from a single interval, with no 0cells.
 A circle. It can be constructed from a single point x and the 1dimensional ball B, such that both endpoints of B are glued to x. Alternatively, it can be constructed from two points x and y and two 1dimensional balls A and B, such that the endpoints of A are glued to x and y, and the endpoints of B are glued to x and y too.
 A graph. Given a graph, a 1dimensional CW complex can be constructed in which the 0cells are the vertices and the 1cells are the edges of the graph. The endpoints of each edge are identified with the incident vertices to it. This realization of a combinatorial graph as a topological space is sometimes called a topological graph.
 3regular graphs can be considered as generic 1dimensional CW complexes. Specifically, if X is a 1dimensional CW complex, the attaching map for a 1cell is a map from a twopoint space to X, . This map can be perturbed to be disjoint from the 0skeleton of X if and only if and are not 0valence vertices of X.
 The standard CW structure on the real numbers has as 0skeleton the integers and as 1cells the intervals . Similarly, the standard CW structure on has cubical cells that are products of the 0 and 1cells from . This is the standard cubic lattice cell structure on .
Finitedimensional CW complexes
Some examples of finitedimensional CW complexes are:^{[7]}
 An ndimensional sphere. It admits a CW structure with two cells, one 0cell and one ncell. Here the ncell is attached by the constant mapping from its boundary to the single 0cell. An alternative cell decomposition has one (n1)dimensional sphere (the "equator") and two ncells that are attached to it (the "upper hemisphere" and the "lower hemisphere"). Inductively, this gives a CW decomposition with two cells in every dimension k such that .
 The ndimensional real projective space. It admits a CW structure with one cell in each dimension.
 The terminology for a generic 2dimensional CW complex is a shadow.^{[8]}
 A polyhedron is naturally a CW complex.
 Grassmannian manifolds admit a CW structure called Schubert cells.
 Differentiable manifolds, algebraic and projective varieties have the homotopytype of CW complexes.
 The onepoint compactification of a cusped hyperbolic manifold has a canonical CW decomposition with only one 0cell (the compactification point) called the Epstein–Penner Decomposition. Such cell decompositions are frequently called ideal polyhedral decompositions and are used in popular computer software, such as SnapPea.
Infinitedimensional CW complexes
Non CWcomplexes
 An infinitedimensional Hilbert space is not a CW complex: it is a Baire space and therefore cannot be written as a countable union of nskeletons, each of which being a closed set with empty interior. This argument extends to many other infinitedimensional 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.
Properties
 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.^{[9]} 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.^{[10]}^{[11]}
 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 compactopen 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).^{[12]} Note that X and Y are compactly generated Hausdorff spaces, so Hom(X,Y) is often taken with the compactly generated variant of the compactopen topology; the above statements remain true.^{[13]}
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 sphere, take the cell decomposition with two cells: a single 0cell and a single ncell. The cellular homology chain complex and homology are given by:
 since all the differentials are zero.
 Alternatively, if we use the equatorial decomposition with two cells in every dimension
 and the differentials are matrices of the form This gives the same homology computation above, as the chain complex is exact at all terms except and
 For we get similarly
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.
Modification of CW structures
There is a technique, developed by Whitehead, for replacing a CW complex with a homotopyequivalent CW complex that has a simpler CW decomposition.
Consider, for example, an arbitrary CW complex. Its 1skeleton 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 1skeleton 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 homotopyequivalent CW complex whose 0skeleton consists of a single point.
Consider climbing up the connectivity ladder—assume X is a simplyconnected CW complex whose 0skeleton consists of a point. Can we, through suitable modifications, replace X by a homotopyequivalent 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 1cell and a 2cell whose attaching map consists of the new 1cell 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 2cell into X.
 2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing X by where the new 3cell has an attaching map that consists of the new 2cell and remainder mapping into . A similar slide gives a homotopyequivalence .
If a CW complex X is nconnected one can find a homotopyequivalent CW complex whose nskeleton 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
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).^{[14]} 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).
See also
 Abstract cell complex
 The notion of CW complex has an adaptation to smooth manifolds called a handle decomposition, which is closely related to surgery theory.
References
Notes
 ^ Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 0521795400. 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} ^{b} Whitehead, J. H. C. (1949a). "Combinatorial homotopy. I." (PDF). Bulletin of the American Mathematical Society. 55 (5): 213–245. doi:10.1090/S000299041949091759. MR 0030759. (open access)
 ^ De Agostino, Sergio (2016). The 3Sphere Regular Cellulation Conjecture (PDF). International Workshop on Combinatorial Algorithms.
 ^ Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Providence, R.I.: American Mathematical Society.
 ^ "CW complex in nLab".
 ^ "CWcomplex  Encyclopedia of Mathematics".
 ^ ^{a} ^{b} Archived at Ghostarchive and the Wayback Machine: channel, Animated Math (2020). "1.3 Introduction to Algebraic Topology. Examples of CW Complexes". Youtube.
 ^ Turaev, V. G. (1994). Quantum invariants of knots and 3manifolds. De Gruyter Studies in Mathematics. Vol. 18. Berlin: Walter de Gruyter & Co. ISBN 9783110435221.
 ^ Milnor, John (February 1959). "On Spaces Having the Homotopy Type of a CWComplex". Transactions of the American Mathematical Society. 90 (2): 272–280. doi:10.2307/1993204. ISSN 00029947. JSTOR 1993204.
 ^ Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN 0521795400. A free electronic version is available on the author's homepage
 ^ Hatcher, Allen, Vector bundles and Ktheory, preliminary version available on the authors homepage
 ^ Milnor, John (1959). "On spaces having the homotopy type of a CWcomplex". Trans. Amer. Math. Soc. 90 (2): 272–280. doi:10.1090/s00029947195901002674. JSTOR 1993204.
 ^ "Compactly Generated Spaces" (PDF). Archived from the original (PDF) on 20160303. Retrieved 20120826.
 ^ 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 Baladze, D.O. (2001) [1994], "CWcomplex", Encyclopedia of Mathematics, EMS Press
General references
 Lundell, A. T.; Weingram, S. (1970). The topology of CW complexes. Van Nostrand University Series in Higher Mathematics. ISBN 0442049102.
 Brown, R.; Higgins, P.J.; Sivera, R. (2011). Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical homotopy groupoids. European Mathematical Society Tracts in Mathematics Vol 15. ISBN 9783037190838. More details on the [1] first author's home page]