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  "body": "\n## Statement\n\nLet $M$ be a closed oriented Riemannian manifold, and let $\\rho$ be an acyclic orthogonal or unitary representation of $\\pi_1(M)$.\nThen the [[analytic torsion|MZT9D34B0A]] of $(M, \\rho)$ equals the Reidemeister torsion of $(M, \\rho)$, computed from any smooth triangulation of $M$.\nHere both torsions are normalized with the same convention.\n\n## Remarks\n\nRay and Singer conjectured the equality, and Cheeger and Müller proved it independently.\nCheeger's proof compares the two sides under surgery; Müller's approximates the de Rham complex by the combinatorial cochain complex of finer and finer triangulations.\nMüller extended it to unimodular representations, and Bismut and Zhang gave a proof by Witten deformation that covers all flat bundles, with an explicit correction term.\nExample: on the circle with the character $\\rho(1) = e^{2\\pi i a}$, $0 \u003c a \u003c 1$, the Reidemeister torsion is $\\abs{1 - e^{2\\pi i a}}$, and the analytic torsion is $2\\sin(\\pi a)$, the same number.\nFor lens spaces the theorem gives an analytic way to compute the Reidemeister torsions that distinguish non-homeomorphic homotopy equivalent lens spaces.\n\n## Explanations\n\n### L1 human\n\nReidemeister torsion is computed by cutting a manifold into simplices and doing linear algebra.\nAnalytic torsion is computed from the eigenvalues of Laplacians, with no triangulation at all.\nThe theorem says that these two very different recipes give the same number.\n\n### L2 human\n\nThe theorem identifies a spectral invariant with a combinatorial one, in the spirit of index theory but for a secondary invariant.\nIt motivated the study of torsion in families (the Bismut–Lott torsion forms) and the use of analytic torsion to study torsion in the cohomology of arithmetic groups.\n",
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