A CW 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]

Definition

CW complex

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 , 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 construction, in words

The CW complex construction is a straightforward generalization of the following process:

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 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 (https://twiki.di.uniroma1.it/pub/Users/SergioDeAgostino/DeAgostino.pdf).

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 extra-block can be treated as a (-1)-dimensional cell in the former definition.[3][4][5]

Examples

0-dimensional CW complexes

Every discrete topological space is a 0-dimensional CW complex.

1-dimensional CW complexes

Some examples of 1-dimensional CW complexes are:[6]

Finite-dimensional CW complexes

Some examples of finite-dimensional CW complexes are:[6]

Infinite-dimensional CW complexes

Non CW-complexes

Properties

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:

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

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 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

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).[13] 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

References

Notes

  1. ^ Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 0-521-79540-0. 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.
  2. ^ a b Whitehead, J. H. C. (1949a). "Combinatorial homotopy. I." Bulletin of the American Mathematical Society. 55 (5): 213–245. doi:10.1090/S0002-9904-1949-09175-9. MR 0030759. (open access)
  3. ^ Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Providence, R.I.: American Mathematical Society.
  4. ^ "CW complex in nLab".
  5. ^ "CW-complex - Encyclopedia of Mathematics".
  6. ^ a b Archived at Ghostarchive and the Wayback Machine: channel, Animated Math (2020). "1.3 Introduction to Algebraic Topology. Examples of CW Complexes". Youtube.
  7. ^ Turaev, V. G. (1994). Quantum invariants of knots and 3-manifolds". De Gruyter Studies in Mathematics. Vol. 18. Berlin: Walter de Gruyter & Co. ISBN 9783110435221.
  8. ^ Milnor, John (February 1959). "On Spaces Having the Homotopy Type of a CW-Complex". Transactions of the American Mathematical Society. 90 (2): 272. doi:10.2307/1993204. ISSN 0002-9947.
  9. ^ Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN 0-521-79540-0. A free electronic version is available on the author's homepage
  10. ^ Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the authors homepage
  11. ^ Milnor, John (1959). "On spaces having the homotopy type of a CW-complex". Trans. Amer. Math. Soc. 90 (2): 272–280. doi:10.1090/s0002-9947-1959-0100267-4. JSTOR 1993204.
  12. ^ "Compactly Generated Spaces" (PDF).
  13. ^ 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], "CW-complex", 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 0-442-04910-2.
  • 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 978-3-03719-083-8. More details on the [1] first author's home page]