GOLD STANDARD · TOPOLOGY · HOW X WORKS
Find the decision hidden inside the model
A disconnected overlap can make the one-basepoint Van Kampen theorem unusable even when every local piece is simple. The fundamental groupoid keeps one basepoint in each relevant component, so the missing connecting information survives the gluing.
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Chapter contents and reading routes
Choose the route closest to your present task. Every chapter remains expanded, so the article also reads naturally from start to finish.
Orient groupoids and basepoints
Build the groupoid pushout
Calculate the circle and test limits
Transfer the method
Use evidence and next routes
CHAPTER 1 OF 12 · ORIENT THE GROUPOID
1. One basepoint can miss how pieces reconnect
The ordinary two-set Van Kampen theorem asks for a path-connected intersection containing the basepoint. If U and V are individually simple but U∩V has several components, forcing one basepoint discards the paths that record how those components are joined in the union.
The fundamental groupoid fixes this by allowing several objects. It keeps homotopy classes of paths between chosen basepoints, not only loops that begin and end at one point. The extra arrows are exactly the change-of-basepoint information the disconnected overlap requires.
CHAPTER 2 OF 12 · ORIENT THE GROUPOID
2. Define the fundamental groupoid on a set
Choose a set A of points in X. The groupoid π₁(X,A) has the points of A as objects. A morphism a→b is a homotopy class, relative to endpoints, of paths from a to b. Composition concatenates paths when the endpoint of one matches the start of the next.
Every morphism is invertible by reversing the path, so this category is a groupoid. The automorphisms at one object a form the ordinary fundamental group π₁(X,a). Keeping several objects delays the arbitrary choice of connectors until after the gluing is understood.
CHAPTER 3 OF 12 · BUILD THE PUSHOUT
3. State the multiple-basepoint cover condition
Let X = U∪V. Choose A so it meets every path component of U, V and U∩V that matters to the cover. Under the usual openness or suitable cofibration conditions, the diagram of fundamental groupoids from U∩V into U and V is a pushout.
The condition is practical: each disconnected overlap component needs a representative object. If one is omitted, the algebra cannot record the corresponding route between the pieces. More basepoints can make the theorem easier, not harder.
CHAPTER 4 OF 12 · BUILD THE PUSHOUT
4. Cover the circle with two arcs
Cover S¹ by two connected open arcs U and V chosen so their intersection has two components. Put basepoints p and q, one in each overlap component. Both U and V are contractible, so each groupoid has one essential arrow between p and q, represented by the path through that arc.
Call the arrow through U, u:p→q, and the arrow through V, v:p→q. The overlap groupoid has only identity arrows at p and q because its two components are contractible and disconnected from one another. The pushout therefore retains both u and v.
| Region | Objects | Non-identity path data |
|---|---|---|
| U | p, q | u:p→q |
| V | p, q | v:p→q |
| U∩V | p, q | None between components |
| S¹ | p, q | Both u and v |
CHAPTER 5 OF 12 · BUILD THE PUSHOUT
5. Recover the circle generator from two arrows
At object p, compose u with the inverse of v. The loop uv⁻¹:p→p travels from p to q through U and returns through V. Its powers give the winding classes, so the vertex group at p is infinite cyclic.
The ordinary one-basepoint shortcut would see trivial local fundamental groups and could falsely suggest a trivial union. The groupoid calculation preserves the difference between the two connectors and recovers π₁(S¹,p) ≅ Z.
CHAPTER 6 OF 12 · CALCULATE AND TEST
6. Read the construction as a groupoid pushout
A groupoid pushout is the most general groupoid receiving compatible functors from the two piece groupoids. Arrows coming from the overlap are identified, while arrows not related there remain available for composition.
This is the same universal gluing idea as the group pushout in ordinary Van Kampen, but with several objects. The object set prevents premature collapse of path-component structure. Only after forming the pushout do we choose a vertex group or a spanning tree to obtain a group presentation.
CHAPTER 7 OF 12 · CALCULATE AND TEST
7. Reduce a connected groupoid to one vertex group carefully
For a connected groupoid, choose paths forming a tree that connects the objects to a preferred basepoint. These choices transport arrows into loops at that basepoint. Different trees change the presentation but not the isomorphism type of the resulting vertex group.
In the circle example, choosing u as the tree arrow leaves v as the non-tree alternative, producing the generator uv⁻¹. Collapsing both connectors would erase the loop; keeping neither choice explicit makes the final relation hard to audit.
| Step | What is chosen | What must be preserved |
|---|---|---|
| Choose A | One point per relevant component | Disconnected overlap data |
| Form pushout | Compatible piece groupoids | All distinct connecting arrows |
| Choose a tree | Connect objects to one basepoint | Non-tree cycles |
| Take vertex group | Loops at preferred object | Result up to isomorphism |
CHAPTER 8 OF 12 · TRANSFER AND REDUCE
8. Transfer the method to a graph with several overlaps
Consider a graph assembled from two subgraphs whose overlap has several connected components. Place a basepoint in each overlap component and compute the path groupoids of the pieces. Distinct routes between the same objects become candidate cycles after gluing.
The stable mechanism is combinatorial: a tree connects objects, and every independent non-tree route contributes a cycle generator unless a relation removes it. Transfer succeeds by tracking endpoints and composition, not by counting visible loops informally.
CHAPTER 9 OF 12 · TRANSFER AND REDUCE
9. Use multiple basepoints before adding artificial corridors
A common workaround is to add paths that force the overlap to become connected. Those paths introduce choices and conjugations that can make presentations longer than the topology. The groupoid formulation treats the disconnected geometry directly.
Artificial connectors can still be useful for a final group presentation, but they should come after the groupoid pushout as an explicit tree choice. That order exposes which relations are geometric and which are bookkeeping.
CHAPTER 10 OF 12 · TRANSFER AND REDUCE
10. Know the theorem’s scope
The basepoint set must meet the required components, and the cover must satisfy the hypotheses of the chosen theorem. A missing object or poorly behaved subspace can invalidate the pushout claim. Check openness, deformation-retract neighbourhoods or the exact version being used.
The fundamental groupoid is a one-dimensional homotopy invariant. It does not by itself compute higher homotopy or all information about the space. Higher van Kampen theorems use richer structures such as crossed modules and higher groupoids.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions readers often ask
Why not choose one basepoint and change basepoint along paths?
You can when suitable paths are already available, but choosing them too early hides disconnected overlap components inside conjugations. Multiple basepoints keep the geometric routes explicit until the pushout is complete.
Does adding objects change the fundamental group?
No. At any chosen object, its automorphism group is the ordinary fundamental group there. The extra objects organise paths between basepoints and make gluing information visible.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Evidence, connected routes and the next check
Brown’s 1967 Groupoids and Van Kampen’s Theorem introduces π₁(X,A) on a set of basepoints and proves the groupoid form. His groupoid overview and proof resources explain why multiple basepoints extend the theorem to disconnected pieces and intersections.
Continue through Van Kampen pushouts and fundamental-group calculations, covering spaces, path lifting and monodromy, and the Topology subject root. The next check is to mark one basepoint in every overlap component before writing a single group presentation.