Ray and Singer proved in their paper that analytic torsion is independent of metric (some details may still need to be checked), and together with Cheeger-Muller theorem this implies analytic torsion is well defined, coincide with Reidemeister torsion.
While it is not difficult to see the formal similarities between analytic torsion and Reidemeister-Franz torsion from the view point of determinants, it is not entirely clear to me why intuitively it is an invariant up to homeomorphism (instead of differeomorphism), as the definition itself clearly depends on the Laplacian. I am sure this is a well-studied topic among experts, but I could not find anything in the literature either. Except results related to index theorem, I also do not know if there are other analytic invariants defined based on $\Psi DO$s but are topological in nature. So here I ask.