TITLE: Compelling evidence that two basepoints are better than one
QUESTION [76 upvotes]: This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his book Topology and Groupoids. The idea of the fundamental groupoid, put forward as a multi-basepoint alternative to the fundamental group, is the highlight of the theory. The headline result seems to be that the van-Kampen Theorem looks more natural in the groupoid context.
I don't know whether I find this headline result compelling- the extra baggage of groupoids and pushouts makes me question whether the payoff is worth the effort, all the more so because I am a geometric topology person, rather than a homotopy theorist.

Do you have examples in geometric topology (3-manifolds, 4-manifolds, tangles, braids, knots and links...) where the concept of the fundamental groupoid has been useful, in the sense that it has led to new theorems or to substantially simplified treatment of known topics?

One place that I can imagine (but, for lack of evidence, only imagine) that fundamental groupoids might be useful (at least to simplify exposition) is in knot theory, where we're constantly switching between (at least) three different "natural" choices of basepoint- on the knot itself, on the boundary of a tubular neighbourhood, and in the knot complement. This change-of-basepoint adds a nasty bit of technical complexity which I have struggled with when writing papers. A recent proof (Proposition 8 of my paper with Kricker) which would have been a few lines if we hadn't had to worry about basepoints, became 3 pages. In another direction, what about fundamental groupoids of braids?
Have the ideas of fundamental groupoids been explored in geometric topological contexts? Conversely, if not, then why not?

REPLY [4 votes]: The most convincing example I have found of "two basepoints being better than one" is the incorrect statement of the main result of the following paper:

Garoufalidis, Stavros, and Andrew Kricker. "A surgery view of boundary links." Mathematische Annalen 327.1 (2003): 103-115. arXiv:math/0205328

and its corrected version here:

Habiro, Kazuo, and Tamara Widmer. "On Kirby calculus for null-homotopic framed links in 3–manifolds." Algebraic & Geometric Topology 14.1 (2014): 115-134. arXiv:1302.0612

The result implies a Kirby theorem for framed links in certain classes of link complements. The condition for the statement to be true is that a certain commutative diagram commutes. The diagram for fundamental groups fails to commute in general, but it does commute for fundamental groupoids, and this implies the desired Kirby Theorem.   
Further details are provided below.
Quantum topology of knots, links, and 3-manifolds is the study of diagrammatically-constructed topological invariants. The hard kernel of any such construction is a theorem that translates from topology to the combinatorics of a class of diagrams. In 3-manifold topology this is the Kirby Theorem. The Kirby Theorem states that two 3-manifolds are homeomorphic if and only if they are obtained from Dehn surgery on links $L$ and $L^\prime$ correspondingly such that $L^\prime$ can be obtained from $L$ by a sequence of so-called Kirby moves: Stabilization and handle-slide.
To obtain a quantum-topological $3$--manifold invariant, the recipe is to define a quantum topological invariant for a framed link diagram, and to mod out by the relations induced by stabilization and and handle-slide. This turns out to be achievable and this procedure has given rise to interesting invariants, such as the LMO invariant. 
For links in general $3$--manifolds,  Fenn and Rourke proved that the analogous result holds if we allow a third move: circumcision. From the quantum-topological perspective, this doesn't help us because circumcision is too violent a move- if we mod out by it, the resulting invariants are usually killed. Fenn and Rourke showed that we could do without circumcision when a certain diagram of fundamental groups of 4-manifolds (cobordisms defined by the respective surgeries) commutes. The Fenn--Rourke results was generalized to $3$--manifolds with boundary by Roberts.
Kricker and Garoufalidis considered a certain restricted class of $3$--manifolds with boundary---- complements of so-called boundary links. They argue that the Fenn-Rourke diagram commutes- but it doesn't unless the boundary link happens to be a knot. As shown by Habiro and Widmer, it commutes only when we place a basepoint on each component of the boundary.