CW complex

In topology, a CW complex is a type of topological space 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).

Formulation

Roughly speaking, a CW complex is made of basic building blocks called cells. The precise definition prescribes how the cells may be topologically glued together. The C stands for "closure-finite", and the W for "weak topology".

An n-dimensional closed cell is the image of an n-dimensional closed ball under an attaching map. For example, a simplex is a closed cell, and more generally, a convex polytope is a closed cell. An n-dimensional open cell is a topological space that is homeomorphic to the n-dimensional open ball. A 0-dimensional open (and closed) cell is a singleton space. Closure-finite means that each closed cell is covered by a finite union of open cells.

A CW complex is a Hausdorff space X together with a partition of X into open cells (of perhaps varying dimension) that satisfies two additional properties:

Regular CW complexes

A CW complex is called regular if for each n-dimensional open cell C in the partition of X, the continuous map f from the n-dimensional closed ball to X is a homeomorphism onto the closure of the cell C.

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 which does not necessarily possess a cellular structure. This extra-block can be treated as a (-1)-dimensional cell in the former definition.[1][2][3]

Inductive definition of CW complexes

If the largest dimension of any of the cells is n, then the CW complex is said to have dimension n. If there is no bound to the cell dimensions then it is said to be infinite-dimensional. The n-skeleton of a CW complex is the union of the cells whose dimension is at most n. If the union of a set of cells is closed, then this union is itself a CW complex, called a subcomplex. Thus the n-skeleton is the largest subcomplex of dimension n or less.

A CW complex is often constructed by defining its skeleta inductively. Begin by taking the 0-skeleton to be a discrete space. Next, attach 1-cells to the 0-skeleton. Here, each 1-cell begins as a closed 1-ball and is attached to the 0-skeleton via some (continuous) map from the boundary of the 1-ball, that is, from the 0-sphere . Each point of can be identified with its image in the 0-skeleton under the aforementioned map; this is an equivalence relation. The 1-skeleton is then defined to be the identification space obtained from the union of the 0-skeleton and 1-cells under this equivalence relation.

In general, given the (n  1)-skeleton, the n-skeleton is formed by attaching n-cells to it. Each n-cell begins as a closed n-ball and is attached to the (n  1)-skeleton via some continuous map from the boundary of the n-ball, that is, from the (n  1)-sphere . Each point of can be identified with its image in the (n  1)-skeleton under the aforementioned map; this is again an equivalence relation. The restriction of the attaching map to the inside of the ball, i.e. to the open n-disk, is required to be a homeomorphism onto its image. The n-skeleton is then defined to be the identification space obtained from the union of the (n  1)-skeleton and n-cells under this equivalence relation.

Up to isomorphism every n-dimensional complex can be obtained from its (n  1)-skeleton in this sense, and thus every finite-dimensional CW complex can be built up by the process above. This is true even for infinite-dimensional complexes, with the understanding that the result of the infinite process is the direct limit of the skeleta: a set is closed in X if and only if it meets each skeleton in a closed set.

Examples

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 0-cell and a single n-cell. 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 homotopy-equivalent CW complex which 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 which 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).[5] 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).

Properties

See also

References

Notes

  1. Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Providence, R.I.: American Mathematical Society.
  2. https://ncatlab.org/nlab/show/CW+complex
  3. https://www.encyclopediaofmath.org/index.php/CW-complex
  4. Turaev, V. G. (1994), "Quantum invariants of knots and 3-manifolds", De Gruyter Studies in Mathematics (Berlin: Walter de Gruyter & Co.) 18
  5. 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", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  6. Milnor, John, "On spaces having the homotopy type of a CW-complex" Trans. Amer. Math. Soc. 90 (1959), 272–280.
  7. "Compactly Generated Spaces" (PDF).
  8. Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN 0-521-79540-0. A free electronic version is available on the author's homepage
  9. Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the authors homepage

General references

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