Mathematics.

algebraic topology

CW Complexes

Topology80 minDifficulty7 out of 10

You should know: topological space

Overview

A CW complex (Closure-finite, Weak-topology complex) is a type of topological space built by attaching cells of increasing dimension. Introduced by J.H.C. Whitehead, CW complexes provide a flexible and computable framework for algebraic topology: every manifold is homotopy equivalent to a CW complex, and the cellular chain complex computes homology efficiently. The 'C' stands for closure-finite (each cell meets only finitely many other cells) and 'W' for weak topology.

Intuition

Build a space in stages: start with a discrete set of points (0-cells), attach intervals (1-cells) to form edges, glue disks (2-cells) along their boundary circles to form faces, and continue in higher dimensions. A sphere S^n has a minimal CW structure with just one 0-cell and one n-cell. The attaching maps determine the topology, and the cellular chain complex encodes how cells attach to compute homology.

Formal Definition

Definition

A CW complex X is constructed inductively: X^0 is a discrete set of 0-cells. The n-skeleton X^n is obtained from X^{n-1} by attaching n-cells {e^n_alpha} via attaching maps phi_alpha: S^{n-1} -> X^{n-1}. Formally, X^n = X^{n-1} ⊔ (D^n_alpha) / (x ~ phi_alpha(x) for x in S^{n-1}). The space X = union of all X^n with the weak topology: A ⊂ X is closed iff A ∩ X^n is closed for all n.

Xn=Xn−1∪φαeαnX^n = X^{n-1} \cup_{\varphi_\alpha} e^n_\alpha
n-skeleton construction
Xn=(Xn−1⊔⨆αDαn)/(x∼φα(x), x∈Sn−1)X^n = \left(X^{n-1}\sqcup \bigsqcup_\alpha D^n_\alpha\right) \big/ \left(x \sim \varphi_\alpha(x),\, x\in S^{n-1}\right)
Pushout diagram
Cncell(X)=Hn(Xn,Xn−1)≅⨁αZC_n^{\mathrm{cell}}(X) = H_n(X^n, X^{n-1}) \cong \bigoplus_{\alpha} \mathbb{Z}
Cellular chain group (one Z per n-cell)
dn:Cncell→Cn−1cell,dn(eαn)=∑βdαβ eβn−1d_n: C_n^{\mathrm{cell}} \to C_{n-1}^{\mathrm{cell}}, \quad d_n(e^n_\alpha) = \sum_\beta d_{\alpha\beta}\, e^{n-1}_\beta
Cellular boundary map

Properties

Cellular Homology Theorem

Thecellularchaincomplex(C∗cell(X),d∗)computesthesamehomologyassingularhomology:Hn(C∗cell(X))≅Hn(X).The cellular chain complex (C_*^{\mathrm{cell}}(X), d_*) computes the same homology as singular homology: H_n(C_*^{\mathrm{cell}}(X)) \cong H_n(X).

Whitehead's Theorem

Amapf:X→YbetweenCWcomplexesthatinducesisomorphismsonallhomotopygroupsisahomotopyequivalence.A map f: X \to Y between CW complexes that induces isomorphisms on all homotopy groups is a homotopy equivalence.

Theorems

Theorem 1: CW Approximation
ForeverytopologicalspaceX,thereexistsaCWcomplexZandaweakhomotopyequivalenceZ→X(inducingisomorphismsonallhomotopygroups).For every topological space X, there exists a CW complex Z and a weak homotopy equivalence Z \to X (inducing isomorphisms on all homotopy groups).
Theorem 2: Euler Characteristic Formula
ForafiniteCWcomplexXwithcncellsindimensionn,theEulercharacteristicisχ(X)=∑n(−1)ncn=∑n(−1)nrank⁡Hn(X).For a finite CW complex X with c_n cells in dimension n, the Euler characteristic is \chi(X) = \sum_n (-1)^n c_n = \sum_n (-1)^n \operatorname{rank} H_n(X).
Theorem 3: Cellular Boundary Formula
Thedegreeofthecellularboundarymapfromeαntoeβn−1isthedegreeofthecompositionSn−1→φαXn−1→Xn−1/Xn−2→Sβn−1.The degree of the cellular boundary map from e^n_\alpha to e^{n-1}_\beta is the degree of the composition S^{n-1} \xrightarrow{\varphi_\alpha} X^{n-1} \to X^{n-1}/X^{n-2} \to S^{n-1}_\beta.

Worked Examples

  1. 1

    Minimal CW structure: one 0-cell e^0 and one n-cell e^n attached by the unique map S^{n-1} -> {e^0} (a constant map).

  2. 2

    The cellular chain complex: C_0 = Z (one 0-cell), C_n = Z (one n-cell), C_k = 0 otherwise.

  3. 3

    All boundary maps are zero: d_n: C_n -> C_{n-1} = 0 (since for n >= 2 there are no (n-1)-cells; for n = 1 the degree is 1-1 = 0 by the two endpoint cancellation).

  4. 4

    Homology = kernel/image: H_0 = Z, H_n = Z, H_k = 0 otherwise.

    Hk(Sn)={Zk=0,n0otherwise.H_k(S^n) = \begin{cases} \mathbb{Z} & k = 0,n \\ 0 & \text{otherwise.}\end{cases}

✓ Answer

H_0(S^n) = Z, H_n(S^n) = Z, H_k(S^n) = 0 for all other k.

Practice Problems

Mediumfree response

Give a CW structure for RP^2 (the real projective plane) and state how many cells in each dimension.

Mediumfree response

Compute the cellular homology of RP^2 using the CW structure with one cell in each dimension 0, 1, 2.

Mediumfree response

Compute chi(CP^n) using the CW structure on CP^n with one cell in each even dimension 0, 2, 4, ..., 2n.

Common Mistakes

Common Mistake

The number of n-cells equals rank H_n(X).

The number of n-cells equals rank C_n, not rank H_n. Boundary maps between cells cancel some contributions, and torsion can appear. For example RP^2 has one 2-cell but H_2(RP^2) = 0.

Common Mistake

Every topological space is a CW complex.

Not every space is a CW complex, but the CW approximation theorem guarantees that every space is weakly homotopy equivalent to one. For most spaces in algebraic topology, working with a CW model is sufficient.

Quiz

What does the 'W' in CW complex stand for?
How many cells does the minimal CW structure of S^n (for n >= 1) have?
The cellular chain group C_n(X) is a free abelian group generated by:

Summary

  • A CW complex is built by inductively attaching n-cells (copies of D^n) to a previously constructed (n-1)-skeleton via attaching maps from S^{n-1}.
  • The cellular chain complex C_*^{cell}(X) with one Z per n-cell computes the singular homology of X.
  • All manifolds, projective spaces, Grassmannians, and classifying spaces admit CW structures, making this a universal construction.
  • The Euler characteristic chi(X) = sum (-1)^n c_n counts cells with signs; it equals sum (-1)^n rank H_n(X) by the cellular homology theorem.
  • Whitehead's theorem: a map between CW complexes inducing isomorphisms on all homotopy groups is a homotopy equivalence.

References

  1. BookHatcher, A. — Algebraic Topology. Cambridge University Press, 2002.
  2. WebsiteWikipedia — CW complex