CW complex

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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] A CW complex can be defined inductively. [3]

Contents

In an n-dimensional CW complex, for every , a k-cell is the interior of a k-dimensional ball added at the k-th step. The k-skeleton of the complex is the union of all its k-cells.

Examples

As mentioned above, every collection of discrete points is a CW complex (of dimension 0).

1-dimensional CW-complexes

Some examples of 1-dimensional CW complexes are: [4]

Multi-dimensional CW-complexes

Some examples of multi-dimensional CW complexes are: [4]

Non CW-complexes

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.

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 (or meets only finitely many other cells [6] ).

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:

The partition of X is also called a cellulation.

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. 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. [7] [8] [9]

Inductive construction 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 by 'attaching' cells of increasing dimension. By an 'attachment' of an n-cell to a topological space X one means an adjunction space where f is a continuous map from the boundary of a closed n-dimensional ball to X. To construct a CW complex, begin with a 0-dimensional CW complex, that is, a discrete space . Attach 1-cells to to obtain a 1-dimensional CW complex . Attach 2-cells to to obtain a 2-dimensional CW complex . Continuing in this way, we obtain a nested sequence of CW complexes of increasing dimension such that if then is the i-skeleton of .

Up to isomorphism every n-dimensional CW complex can be obtained from its (n  1)-skeleton via attaching n-cells, 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.

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). [10] 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

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This is a glossary of properties and concepts in algebraic topology in mathematics.

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. 1 2 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. channel, Animated Math (2020). "1.2 Introduction to Algebraic Topology. CW Complexes". Youtube.
  4. 1 2 channel, Animated Math (2020). "1.3 Introduction to Algebraic Topology. Examples of CW Complexes". Youtube.
  5. Turaev, V. G. (1994). Quantum invariants of knots and 3-manifolds". De Gruyter Studies in Mathematics. 18. Berlin: Walter de Gruyter & Co. ISBN   9783110435221.
  6. Hatcher, Allen, Algebraic topology, p.520, Cambridge University Press (2002). ISBN   0-521-79540-0.
  7. Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology. Providence, R.I.: American Mathematical Society.
  8. https://ncatlab.org/nlab/show/CW+complex
  9. https://www.encyclopediaofmath.org/index.php/CW-complex
  10. 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
  11. Milnor, John (1959). "On spaces having the homotopy type of a CW-complex". Trans. Amer. Math. Soc. 90: 272–280. doi: 10.1090/s0002-9947-1959-0100267-4 . JSTOR   1993204.
  12. "Compactly Generated Spaces" (PDF).
  13. Hatcher, Allen, Algebraic topology, Cambridge University Press (2002). ISBN   0-521-79540-0. A free electronic version is available on the author's homepage
  14. Hatcher, Allen, Vector bundles and K-theory, preliminary version available on the authors homepage

General references