Cell attachments and surface groups

Try

Start with two circles joined at one vertex, naming their directed loops a and b. The preceding graph calculation gives a free group: words can cancel adjacent inverse letters, but a and b need not commute. Now glue the boundary of a disk along the word aba⁻¹b⁻¹. The disk provides a contraction of that boundary loop. Predict which new equality becomes available and which loops still need not contract.

We multiply paths in travel order: αβ means first α, then β. Keep that convention when reading a polygon boundary. A two-cell is an attached disk; its attaching map tells us how the boundary travels through the existing space.

Predict

Does filling one boundary word make only that particular written word trivial? Move the basepoint to the boundary along a path g, go around the filled boundary, and return along g⁻¹. This conjugate also contracts. Predict why all products of conjugates and their inverses must disappear, not merely powers of the original relation.

Experiment

Read these finite cell constructions as group calculations. The angle brackets list generators and relations; they do not assert that the group is finite.

Initial graph	Attached boundary word	Resulting fundamental group
One loop a	a³	Cyclic group of order 3
Two loops a,b	aba⁻¹b⁻¹	Z²
One loop a	a²	Cyclic group of order 2

In the middle row, use aba⁻¹b⁻¹=1 to obtain ab=ba. Reduce the word a²b⁻¹ab³ to a³b². Then explain why the exponent pair (3,2) proves this loop is nontrivial. In the last row, draw two traversals of the boundary loop and distinguish a²=1 from a=1.

Understand

For a path-connected CW complex X, attaching a two-cell along a loop r gives the group π₁(X) modulo the normal closure of r. With several attached two-cells, use the normal closure of all their boundary classes. If an attaching loop starts away from the chosen basepoint, use a connecting path; changing that path conjugates the relation and therefore does not change the normal closure.

Here is the van Kampen mechanism for a single attachment. Use an open neighborhood of the old complex together with a collar of the attached disk, and an open disk neighborhood covering the remaining interior. The first retracts to X, the second is simply connected, and their overlap retracts to a circle. Its generator maps to the attaching class on the X side and to the identity on the disk side. Van Kampen’s quotient therefore forces r to be trivial. The kernel is exactly the required normal closure because the pushout has the universal property of a group receiving π₁(X) and killing r. This standard collar construction can be implemented with the mapping-cylinder description of attachment; it does not require the attaching map to be injective.

The square presentation of the torus identifies opposite edges with compatible directions. Its one-skeleton is the two-loop graph, and its face boundary is aba⁻¹b⁻¹. The resulting group is Z²: every word reduces using commutativity to aᵐbⁿ, and the homomorphism to integer exponent pairs proves that distinct pairs remain distinct. This proves more than observing that the generators commute.

The projective plane is a disk with antipodal boundary points identified. Its boundary traverses its one-skeleton twice, yielding the presentation with generator a and relation a²=1. The map a↦1 modulo 2 shows the remaining generator is genuinely nontrivial. This cell description is different from identifying every boundary point to one point, which produces a sphere.

Apply

Attach two disks to a circle with boundary degrees 6 and 10. The quotient of Z by the subgroup generated by 6 and 10 is Z/2Z, using the greatest common divisor. Explain geometrically why both relations hold and algebraically why they do not force a=1. A presentation computes the fundamental group of this cell complex; it does not by itself identify the space up to homeomorphism.

Continue

Return to a polygon you can label consistently and produce its attaching word before manipulating symbols. Higher homotopy groups and homology require further tools: a trivial fundamental group does not mean a space is contractible. These computations use the cell-attachment application of van Kampen developed in Hatcher, Section 1.2.

Continue to The covering lifting criterion.