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 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 λj then the zeta function ζk 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.
be a finite connected CW-complex with fundamental group
Suppose that for all n. If we fix a cellular basis for
is a contractible finite based free
We define the Reidemeister torsion where A is the matrix of
is independent of the choice of the cellular basis for
Then we call the positive real number
The classification includes examples of homotopy equivalent 3-dimensional manifolds which are not homeomorphic — at the time (1935) the classification was only up to PL homeomorphism, but later E.J.
Brody (1960) showed that this was in fact a classification up to homeomorphism.
J. H. C. Whitehead defined the "torsion" of a homotopy equivalence between finite complexes.
This is a direct generalization of the Reidemeister, Franz, and de Rham concept; but is a more delicate invariant.
Whitehead torsion provides a key tool for the study of combinatorial or differentiable manifolds with nontrivial fundamental group and is closely related to the concept of "simple homotopy type", see (Milnor 1966) In 1960 Milnor discovered the duality relation of torsion invariants of manifolds and show that the (twisted) Alexander polynomial of knots is the Reidemeister torsion of its knot complement in
induces and then we obtain The representation of the fundamental group of knot complement plays a central role in them.
It gives the relation between knot theory and torsion invariants.
be an orientable compact Riemann manifold of dimension n and
on a real vector space of dimension N. Then we can define the de Rham complex and the formal adjoint
As usual, we also obtain the Hodge Laplacian on p-forms Assuming that
, the Laplacian is then a symmetric positive semi-positive elliptic operator with pure point spectrum As before, we can therefore define a zeta function associated with the Laplacian
As in the case of an orthogonal representation, we define the analytic torsion
This Ray–Singer conjecture was eventually proved, independently, by Cheeger (1977, 1979) and Müller (1978).
Both approaches focus on the logarithm of torsions and their traces.
This is easier for odd-dimensional manifolds than in the even-dimensional case, which involves additional technical difficulties.
A proof of the Cheeger-Müller theorem for arbitrary representations was later given by J. M. Bismut and Weiping Zhang.