In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister  for 3-manifolds and generalized to higher dimensions by  and .
Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by  as an analytic analogue of Reidemeister torsion.  and  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 . It has also given some important motivation to arithmetic topology; see . For more recent work on torsion see the books  and .

Definition of analytic torsion
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.
The analytic torsion T(M,E) is defined to be

Definition of Reidemeister torsion

Let  be a finite connected CW-complex with fundamental group 
and universal cover , and let  be an orthogonal finite-dimensional -representation. Suppose that

for all n. If we fix a cellular basis for  and an orthogonal -basis for , then  is a contractible finite based free -chain complex. Let  be any chain contraction of D*, i.e.  for all . We obtain an isomorphism  with , . We define the Reidemeister torsion

where A is the matrix of  with respect to the given bases. The Reidemeister torsion  is independent of the choice of the cellular basis for , the orthogonal basis for  and the chain contraction .

Let  be a compact smooth manifold, and let  be a unimodular representation.  has a smooth triangulation. For any choice of a volume , we get an invariant . Then we call the positive real number  the Reidemeister torsion of the manifold  with respect to  and .

A short history of Reidemeister torsion

Reidemeister torsion was first used to combinatorially classify 3-dimensional lens spaces in  by Reidemeister, and in higher-dimensional spaces by Franz. 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  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 

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 .  For each q the Poincaré duality  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.

Cheeger–Müller theorem

Let  be an orientable compact Riemann manifold of dimension n and  a representation of the fundamental group of  on a real vector space of dimension N. Then we can define the de Rham complex

and the formal adjoint  and  due to the flatness of . 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  on  by

where  is the projection of  onto the kernel space  of the Laplacian .  It was moreover shown by  that  extends to a meromorphic function of  which is holomorphic at .

As in the case of an orthogonal representation, we define the analytic torsion  by

In 1971 D.B. Ray and I.M. Singer conjectured that  for any unitary representation . This Ray–Singer conjecture was eventually proved, independently,  by    and  . 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. This Cheeger–Müller theorem (that the two notions of torsion are equivalent), along with Atiyah–Patodi–Singer theorem, later provided the basis for Chern–Simons perturbation theory.

A proof of the Cheeger-Müller theorem for arbitrary representations was later given by J. M. Bismut and Weiping Zhang. Their proof uses the Witten deformation.

References
 
 

 Online book

Differential geometry
3-manifolds
Surgery theory