Van Kampen and gluing loops

Try

A loop in a space built from overlapping open pieces can be cut into segments lying in those pieces. The issue is how to turn those segments into loops at one common basepoint and how to record the relations created by the overlap. Van Kampen’s theorem gives a precise answer using a free product and a normal subgroup of overlap relations.

Predict

Predict the group of two circles joined at one point. The two directions should generate freely rather than commute automatically. Now cover one circle by two contractible open arcs. Their intersection has two components. Ignoring that fact would falsely predict a trivial group, so the connected-overlap hypothesis must be checked before using the simplest theorem.

Experiment

For an open cover X=U∪V, organize a subdivided loop.

Piece of route	Connector requirement	Resulting based loop
Segment in U	Paths from x₀ in U∩V	Loop in U
Segment in V	Paths from x₀ in U∩V	Loop in V
Change of piece	Switching point lies in overlap	Connector can be reused

Insert each connector followed by its reverse. Those pairs cancel up to endpoint homotopy. The original route becomes a word of loops from the two pieces.

Understand

Assume U,V are open, X=U∪V, and U,V,U∩V are path connected. Choose x₀ in U∩V. The theorem states that π₁(X,x₀) is the free product of π₁(U,x₀) and π₁(V,x₀), modulo the normal closure of relations identifying the images of every class from π₁(U∩V,x₀).

For generation, compactness and a Lebesgue subdivision split any loop into finitely many subpaths contained in U or V. Merge consecutive subpaths carrying the same label. At each switch the common endpoint belongs to the overlap. Choose a connector from x₀ to that endpoint inside U∩V. Inserting connector-reverse pairs expresses the loop as a product of based loops in the pieces. Hence the natural homomorphism from the free product is surjective.

To identify its kernel, take a null homotopy of a represented loop word. Subdivide the square finely enough that every closed cell maps into U or V. At a grid vertex whose incident cells all have one label, choose a connector from x₀ within that piece. At a vertex incident to both labels, its image lies in U∩V, so choose the connector in the overlap. Each oriented edge, together with its endpoint connectors, defines a based loop in the label of an incident cell. If its two adjacent cells have different labels, the edge and both connectors lie in the overlap; the imposed identification makes their two interpretations equal.

Each cell boundary is null homotopic within its piece, since the cell itself fills it. Multiply these boundary relations with the connector conjugations needed to move their starting vertices to x₀. Internal edges cancel with reversed partners. The surviving outside boundary is the original loop word, up to subdivisions and inserted inverse pairs. Thus every kernel element lies in the normal subgroup generated by overlap identifications. Conversely those identifications hold in X, so the claimed quotient is exact. Normal closure records that a relation may occur inside a longer loop word.

Apply

For a bouquet of two circles, use open neighborhoods that each contain one whole circle and short arms of the other. Their overlap is a contractible tree-like neighborhood of the vertex. Its group is trivial, so no relation identifies the two generators. The result is a free product of two copies of Z. A separate reduced-word argument next makes its noncommutativity explicit.

Continue

Do not apply this formulation to disconnected intersections or to arbitrary closed pieces without constructing suitable open neighborhoods. More general groupoid formulations handle disconnected overlaps, but are not proved here. The supplied grid argument explains both generation and relations under the stated hypotheses.

Continue to Graph loops and free groups.