CW Complexes

Spaces built one cell at a time: skeleta, attaching maps, and the operations that glue them together.

The idea

A CW complex is a presentation of a space as disks glued together one dimension at a time, and it reduces questions about the space to questions about a finite list.

The construction proceeds by dimension. Start with a discrete set of points. Glue on arcs, each by its two endpoints; then disks, each by its boundary circle; and in general, glue each $n$-disk to the part already built by a continuous attaching map $\varphi_{\alpha} \colon S^{n-1} \to X^{n-1}$ defined on its boundary sphere. Nothing is ever attached to material built later, so the space grows through an increasing sequence of skeleta $X^{0} \subseteq X^{1} \subseteq X^{2} \subseteq \cdots$ whose union is the space.

The data of the presentation are the cells — the open disks, one for each piece used — together with the attaching map of each. The cells record how much material the space is made of; the attaching maps record how the pieces meet.

Spheres, surfaces and projective spaces all admit such presentations with a handful of cells each. Once a space has one, questions about it become questions about the list: how many cells of each dimension there are, and where each boundary sphere was sent.

Ways to work on it

• Walkthrough. Attaching cells and skeleta, the torus and Klein bottle squares, and how wedge sums and products count cells.
• Practice. Count cells and Euler characteristics, read attaching words off a square, and run the suspension, cone and quotient rules.
• Hardest. Real projective space, built one cell per dimension, and what its quotients and products count.

Not sure where to start? Take the ten-question placement test.