Questions: CW Complexes

Question 1

The 2-sphere S^2 can be given a CW structure with one 0-cell, zero 1-cells, and one 2-cell. How is the 2-cell attached?

• A. Its boundary circle is mapped homeomorphically to the 1-skeleton
• B. Its boundary circle is mapped to the single 0-cell (the entire boundary is collapsed to a point)
• C. Its boundary circle is divided into two arcs mapping to two 1-cells
• D. This CW structure is impossible — S^2 needs at least one 1-cell

Answer: (B)

The 0-skeleton is a single point. There are no 1-cells, so the 1-skeleton equals the 0-skeleton (still just a point). To form S^2, we attach a 2-cell e^2 with attaching map φ: S^1 → X^1 = {point}. This collapses the entire boundary circle to the point, and the quotient D^2/S^1 is homeomorphic to S^2. This is a minimal CW structure: just one 0-cell and one 2-cell. Compare this to the minimal simplicial triangulation of S^2, which requires 4 vertices, 6 edges, and 4 triangles.

Question 2

Every CW complex is a Hausdorff space.

• T. True
• F. False

Answer: True

This is a theorem, not obvious from the definition. The proof uses the 'weak topology' property (the W in CW): a set is closed if and only if its intersection with every finite subcomplex is closed. Combined with the fact that cells are attached via continuous maps from compact Hausdorff spaces (closed disks), this implies the Hausdorff property. The C in CW stands for 'closure-finite' (each cell touches only finitely many other cells), and together these properties ensure CW complexes are well-behaved topological spaces.

Question 3

Complex projective space CP^n has a CW structure with one cell in each even dimension: one 0-cell, one 2-cell, one 4-cell, ..., one 2n-cell, and no odd-dimensional cells.

• T. True
• F. False

Answer: True

CP^n = {[z_0 : ... : z_n] | z_i ∈ C} has a natural CW decomposition using the standard affine coordinate charts. The cell e^{2k} consists of points [z_0 : ... : z_k : 0 : ... : 0] with z_k = 1, parametrized by (z_0, ..., z_{k-1}) ∈ C^k ≅ R^{2k}. The attaching map sends the boundary of the 2k-disk to CP^{k-1} via the Hopf-like map. The absence of odd-dimensional cells immediately gives H_{2k+1}(CP^n) = 0 and H_{2k}(CP^n) = Z for 0 ≤ k ≤ n by cellular homology.

Question 4

Why are CW complexes preferred over simplicial complexes in modern algebraic topology?

Model answer: CW complexes require far fewer cells than simplicial complexes require simplices to model the same space, because cells can be attached via arbitrary continuous maps (not just face-to-face gluings). S^2 needs 2 cells (CW) vs 14 triangles (simplicial). This economy translates directly to smaller chain complexes in cellular homology, making computations faster. Additionally, many natural constructions in homotopy theory (mapping cones, suspensions, loop spaces) produce CW complexes directly, not simplicial complexes.

The trade-off: simplicial complexes have more rigid combinatorial structure, which is better for computational algorithms (like persistent homology). CW complexes have more flexible attaching maps, which is better for theoretical work and hand computation. For homotopy theory specifically, CW complexes are essential: Whitehead's theorem (a weak homotopy equivalence between CW complexes is a homotopy equivalence) and the CW approximation theorem (every space is weakly equivalent to a CW complex) make CW complexes the canonical representatives of homotopy types.