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I'd like to address another aspect of your questions. My feeling is that PL topology, or smooth topology, are foundational subjects to the low dimensional topologist, in the sense that set theory is a foundational subject to most mathematicians. A large proportion of low dimensional topologists use the foundational theorems in PL topology as black boxes, certainly without understanding or having read the proofs, and in fact they can do good mathematics that way. In the smooth category, the situation is even worse- I'm sure that there are very few people in the world who understand the proof of Kirby's Theorem, which is a difficult result, but it gets used all over low dimensional topology as a black box. Indeed, the fact that a diffeomorphism of $S^2$ extends to the $3$--ball is fundamental, under the hood everywhere, and highly non-trivial.
Open problem: Construct a discrete $3$-dimensional Chern-Simons theory, compatible with gauge symmetry, replacing the path integrals of the smooth picture (which are not mathematically well-defined) with finite dimensional integrals.