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  "body": "\n## Summary\n\nAnalytic torsion, or Ray–Singer torsion, is a spectral invariant of a closed Riemannian manifold with an orthogonal or unitary representation of its fundamental group.\nIt is an alternating product of [[zeta-regularized determinants|YT3ZQ2F7FG]] of the Laplacians on forms with values in the associated flat bundle.\nRay and Singer introduced it as an analytic counterpart of Reidemeister torsion, a combinatorial invariant that can distinguish homotopy equivalent lens spaces.\nThe [[Cheeger–Müller theorem|C9DT4Y63CY]] says that the two torsions agree.\n\n## Explanations\n\n### L1 human\n\nHomology counts holes, and when all homology with twisted coefficients vanishes, a finer invariant called torsion still remains.\nReidemeister torsion computes it from a triangulation.\nAnalytic torsion computes it instead from the vibration frequencies of the manifold, the eigenvalues of Laplacians on forms.\nOn a circle with a twist by the unit complex number $e^{2\\pi i a}$, both give $2\\sin(\\pi a)$.\n",
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