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 Wolfgang Franz (1935) and Georges de Rham (1936). Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger (1977, 1979) and Werner 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). It has also given some important motivation to arithmetic topology; see (Mazur). For more recent work on torsion see the books (Turaev 2002) and (Nicolaescu 2002, 2003).

If M is a Riemannian manifold and E a vector bundle over M, then there is a Laplacian operator acting on the k-forms with values in E. If the eigenvalues on k-forms are λ_j then the zeta function ζ_k is defined to be

for s large, and this is extended to all complex s by analytic continuation. The zeta regularized determinant of the Laplacian acting on k-forms is

which is formally the product of the positive eigenvalues of the laplacian acting on k-forms. The analytic torsion T(M,E) is defined to be

Let ${\displaystyle X}$ be a finite connected CW-complex with fundamental group ${\displaystyle \pi :=\pi _{1}(X)}$ and universal cover ${\displaystyle {\tilde {X}}}$, and let ${\displaystyle U}$ be an orthogonal finite-dimensional ${\displaystyle \pi }$-representation. Suppose that

for all n. If we fix a cellular basis for ${\displaystyle C_{*}({\tilde {X}})}$ and an orthogonal ${\displaystyle \mathbf {R} }$-basis for ${\displaystyle U}$, then ${\displaystyle D_{*}:=U\otimes _{\mathbf {Z} [\pi ]}C_{*}({\tilde {X}})}$ is a contractible finite based free ${\displaystyle \mathbf {R} }$-chain complex. Let ${\displaystyle \gamma _{*}:D_{*}\to D_{*+1}}$ be any chain contraction of D_*, i.e. ${\displaystyle d_{n+1}\circ \gamma _{n}+\gamma _{n-1}\circ d_{n}=id_{D_{n}}}$ for all ${\displaystyle n}$. We obtain an isomorphism ${\displaystyle (d_{*}+\gamma _{*})_{\text{odd}}:D_{\text{odd}}\to D_{\text{even}}}$ with ${\displaystyle D_{\text{odd}}:=\oplus _{n\,{\text{odd}}}\,D_{n}}$, ${\displaystyle D_{\text{even}}:=\oplus _{n\,{\text{even}}}\,D_{n}}$. We define the Reidemeister torsion