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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.

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It is kind of awkward to answer my own question: It seems the answer is related to partition function of certain quadratic functional under gauge invariance. The relevant literature is:

The Partition Function of a Degenerate Functional

and

The partition function of degenerate quadratic functional and Ray-Singer invariants

by A. S. Schwarz. I found pointer to this reference via reading John Lott's lecture notes and personal communication with Pavel Mnev. The paper is not hard to read but skipped a lot detail in the proofs. So if someone can give a sketch of the main argument I will be grateful.

There has also been some recent important work by Phillip Andreae, who pointed out in his thesis that it is impossible to have a "Mckean-Singer" type of formula for analytic torsion. So in particular this rules out possible construction of arithemetic cohomology groups in Arakelov theory via integration of local terms arising from heat kernel asymptotics.

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