Analytic torsion

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 "i"-forms with values in "E". If the eigenvalues on "i"-forms are λ"j" then the zeta function ζ"i" 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 "i"-forms is
which is formally the product of the positive eigenvalues of the laplacian acting on "i"-forms.
The analytic torsion "T"("M","E") is defined to be
Definition of Reidemeister torsion.
Let formula_4 be a finite connected CW-complex with fundamental group formula_5
and universal cover formula_6, and let formula_7 be an orthogonal finite-dimensional formula_8-representation. Suppose that
for all n. If we fix a cellular basis for formula_10 and an orthogonal formula_11-basis for formula_7, then formula_13 is a contractible finite based free formula_11-chain complex. Let formula_15 be any chain contraction of D*, i.e. formula_16 for all formula_17. We obtain an isomorphism formula_18 with formula_19, formula_20. We define the Reidemeister torsion
where A is the matrix of formula_22 with respect to the given bases. The Reidemeister torsion formula_23 is independent of the choice of the cellular basis for formula_10, the orthogonal basis for formula_7 and the chain contraction formula_26.
Let formula_27 be a compact smooth manifold, and let formula_28 be a unimodular representation. formula_27 has a smooth triangulation. For any choice of a volume formula_30, we get an invariant formula_31. Then we call the positive real number formula_32 the Reidemeister torsion of the manifold formula_27 with respect to formula_34 and formula_35.
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 formula_36. For each "q" the Poincaré duality formula_37 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 formula_40 be an orientable compact Riemann manifold of dimension n and formula_41 a representation of the fundamental group of formula_27 on a real vector space of dimension N. Then we can define the de Rham complex
and the formal adjoint formula_44 and formula_45 due to the flatness of formula_46. As usual, we also obtain the Hodge Laplacian on p-forms 
Assuming that formula_48, 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 formula_50 on formula_51 by
where formula_53 is the projection of formula_54 onto the kernel space formula_55 of the Laplacian formula_50. It was moreover shown by that formula_57 extends to a meromorphic function of formula_58 which is holomorphic at formula_59.
As in the case of an orthogonal representation, we define the analytic torsion formula_60 by
In 1971 D.B. Ray and I.M. Singer conjectured that formula_62 for any unitary representation formula_34. 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.