Analytic torsion

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 X be a finite connected CW-complex with fundamental group π := π 1 ( X ) and universal cover X ~ , and let U be an orthogonal finite-dimensional π -representation. Suppose that

for all n. If we fix a cellular basis for C ∗ ( X ~ ) and an orthogonal R -basis for U , then D ∗ := U ⊗ Z [ π ] C ∗ ( X ~ ) is a contractible finite based free R -chain complex. Let γ ∗ : D ∗ → D ∗ + 1 be any chain contraction of D_*, i.e. d n + 1 ∘ γ n + γ n − 1 ∘ d n = i d D n for all n. We obtain an isomorphism ( d ∗ + γ ∗ ) o d d : D o d d → D e v e n with D o d d := ⊕ n o d d D n , D e v e n := ⊕ n e v e n D n . We define the Reidemeister torsion

where A is the matrix of ( d ∗ + γ ∗ ) o d d with respect to the given bases. The Reidemeister torsion ρ ( X ; U ) is independent of the choice of the cellular basis for C ∗ ( X ~ ) , the orthogonal basis for U and the chain contraction γ ∗ .

Let M be a compact smooth manifold, and let ρ : π ( M ) → G L ( E ) be a unimodular representation. M has a smooth triangulation. For any choice of a volume μ ∈ d e t H ∗ ( M ) , we get an invariant τ M ( ρ : μ ) ∈ R + . Then we call the positive real number τ M ( ρ : μ ) the Reidemeister torsion of the manifold M with respect to ρ and μ .

A short history of Reidemeister torsion

Reidemeister torsion was first used to combinatorially classify 3-dimensional lens spaces in (Reidemeister 1935) 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 (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 Reidemister torsion of its knot complement in S³. (Milnor 1962) For each q the Poincaré duality P o 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 ( M , g ) be an orientable compact Riemann manifold of dimension n and ρ : π ( M ) → G L ( E ) a representation of the fundamental group of M on a real vector space of dimension N. Then we can define the De Rham complex

and the formal adjoint d p and δ p due to the flatness of E q . As usual, we also obtain the Hodge Laplacian on p-forms

Assuming that ∂ M = 0 , 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 Δ q on Λ q ( E ) by

where P is the projection of L 2 Λ ( E ) onto the kernel space H q ( E ) of the Laplacian Δ q . It was moreover shown by (Seeley 1967) that ζ q ( s ; ρ ) extends to a meromorphic function of s ∈ C which is holomorphic at s = 0 .

As in the case of an orthogonal representation, we define the analytic torsion T M ( ρ ; E ) by

In 1971 D.B. Ray and I.M. Singer conjectured that T M ( ρ ; E ) = τ M ( ρ ; μ ) for any unitary representation ρ . 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. 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.