Analytic torsion
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister (Reidemeister (1935)) for 3-manifolds and generalized to higher dimensions by Franz (1935) and de Rham (1936). Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Ray and Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Cheeger (1977, 1979) and Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.
Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces.
Reidemeister torsion is closely related to Whitehead torsion; see (Milnor 1966). For later work on torsion see the books (Turaev 2002), (Nicolaescu 2002, 2003). And it had given one of important motivation to arithmetic topology. (Mazur)
If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the i-forms with values in E. If the eigenvalues on i-forms are λj then the zeta function ζi is defined to be
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