Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions

Understand

We prove that the quivers with potentials associated to triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly 5 punctures.

• This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related by a flip, their associated QPs are related by the corresponding QP-mutation.
• As a byproduct, for (arbitrarily punctured) surfaces with non-empty boundary we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series.
• The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called \emph{Popping Theorem}, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence.