Math 2.0

Global analysis, analysis on manifolds Heat kernels, zeta functions and determinants Theorem

Cheeger–Müller theorem

Builds on Analytic torsionRiemannian manifoldUnitary representationFundamental group

Statement

Let MM be a closed oriented Riemannian manifold, and let ρ\rho be an acyclic orthogonal or unitary representation of π1(M)\pi_1(M). Then the analytic torsion of (M,ρ)(M, \rho) equals the Reidemeister torsion of (M,ρ)(M, \rho), computed from any smooth triangulation of MM. Here both torsions are normalized with the same convention.

Remarks

Ray and Singer conjectured the equality, and Cheeger and Müller proved it independently. Cheeger'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. Müller 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. Example: on the circle with the character ρ(1)=e2πia\rho(1) = e^{2\pi i a}, 0<a<10 < a < 1, the Reidemeister torsion is ∣1−e2πia∣\abs{1 - e^{2\pi i a}}, and the analytic torsion is 2sin⁡(πa)2\sin(\pi a), the same number. For lens spaces the theorem gives an analytic way to compute the Reidemeister torsions that distinguish non-homeomorphic homotopy equivalent lens spaces.

Going deeper

For a student

The theorem identifies a spectral invariant with a combinatorial one, in the spirit of index theory but for a secondary invariant. It 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.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Analytic torsion, concept, no evidence levelAnalytic torsionconceptFundamental group, concept, no evidence levelFundamental groupconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptUnitary representation, concept, no evidence levelUnitary representationconceptCheeger–Müller theorem, statement, no evidence levelCheeger–Müller theoremstatement
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  1. Analytic torsion concept
  2. Fundamental group concept
  3. Riemannian manifold concept
  4. Unitary representation concept
  5. Cheeger–Müller theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Analytic_torsion
About this page
TagC9DT4Y63CY
URIm2://C9DT4Y63CY
Kindstatement, theorem
WritingW0 stub
MSC58J52, 57Q10
StatusThis copy of the site has no build data. Run m2 status C9DT4Y63CY.
AboutAnalytic torsion
UsesAnalytic torsion, Riemannian manifold, Unitary representation, Fundamental group

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