Allowing ww^{-1} insertions in defect-cobordism models of the word problem

Determine whether, in modeling group word transformations via defect cobordisms in the category Bord_2^{def}(𝓟_G), it is necessary or appropriate to allow insertion of the trivial relation ww^{-1} during word factorization, or whether one can restrict to progressive pants/co-pants together with cups and caps without loss of generality.

Background

The paper interprets word manipulations in a presentation 𝓟_G as defect-cobordisms in Bord_2^{def}(𝓟_G), translating group-theoretic moves into compositions of defect cups, caps, pants, and co-pants. While reductions usually avoid explicit ww^{-1} insertions, it is unclear whether such insertions are needed in the cobordism model.

Clarifying this would pin down a minimal generating set of elementary defect-cobordisms adequate for representing all word transformations relevant to the word problem.