On the Reidemeister torsion of rational homology spheres

We prove that the mod Z reduction of the torsion of a rational homology 3-sphere is completely determined by three data: a certain canonical spin^c structure, the linking form and a Q/Z-valued constant c. This constant is a new topological invariant of the rational homology sphere. Experimentations with lens spaces suggest this constant may be as powerful an invariant as the torsion itself.


Introduction
In the paper [3] V. Turaev has proved a certain identity involving the Reidemeister torsion of a rational homology sphere. In this very short note we will suitably interpret this identity as a second order finite difference equation satisfied by the torsion which will allow us to prove a general structure result for the mod Z reduction of the torsion. More precisely we prove that the mod Z reduction of the torsion is completely determined by three data.
• a certain canonical spin c structure, • the linking form of the rational homology sphere and • a constant c ∈ Q/Z.
As a consequence, the constant c is a Q/Z-valued invariant of the rational homology sphere. Experimentations with lens spaces suggest this invariant is as powerful as the torsion itself. 1 The Reidemeister torsion

Contents
We review briefly a few basic facts about the Reidemeister torsion a rational homology 3-sphere. For more details and examples we refer to [1,3]. Suppose M is a rational homology sphere. We set H := H 1 (M, Z) and use the multiplicative notation to denote the group operation on H. Denote Spin c (M ) the H-torsor of isomorphism classes of spin c structure on M . We denote by F the space of functions The group H acts on F M by We denote by H the augmentation map According to [3] Reidemeister torsion is a H-equivariant map V. Turaev has proved in [3] that τ σ satisfies the identity 2 A second order "differential equation" The identity (1.1) admits a more suggestive interpretation. To describe it we need a few more notation. Denote by S the space of functions H → Q/Z. Each g ∈ H defines a first order differential operator If Ξ = Ξ σ denotes the mod Z reduction of τ σ then we can rewrite (1.1) as We will prove uniqueness and existence results for this equation. We begin with the (almost) uniqueness part.
Lemma 2.1. The second order linear differential equation (2.1) determines Ξ up to an "affine" function.
Proof Suppose Ξ 1 , Ξ 2 are two solutions of the above equation. Set Ψ := Ξ 1 − Ξ 2 . Ψ satisfies the equation Now observe that any function F ∈ S satisfying the second order equation The function G = F − F (1) satisfies the same differential equation and the additional condition G(1) = 0. If we set h = 1 in the above equation we deduce so that G is a character and F = F (1) + G. Thus, the differential equation (2.1) determines Ξ up to a constant and a character.
Then there exists a quadratic form q : H → Q/Z such that Proof 1 Let us briefly recall the terminology in this lemma. b is nonsingular if the induced map G → G ♯ is an isomorphism. A quadratic map form is a function q : H → Q/Z such that q(0) = 0, q(u k ) = k 2 q(u), ∀u ∈ H, k ∈ Z and ∆q is a bilinear form.
Suppose b is a nonsingular, symmetric , bilinear form H × H → Q/Z. Then, according to [4, §7], b admits a resolution. This is a nondegenerate, symmetric, bilinear form It is clear that this quantity is well defined i.e. We deduce that there exists a constant c, a character λ : H → Q/Z and a quadratic form q such that In the above discussion the choice of the spin c structure σ is tantamount to a choice of an origin of H which allowed us to identify the torsion of M as a function H → Q. Once we make such a non-canonical choice, we have to replace Ξ with the family of translates {Ξ g (•) := Ξ(g•); g ∈ H} In particular Ξ g (h) := Ξ(gh) = c + λ(gh) + q(gh) = c + λ(g) + q(g) where λ g (•) = λ(•) + lk M (g, •). Since the linking from is nondegenerate we can find an unique g such that λ g = 0.
We have proved the following result.
where c ∈ Q/Z is a constant while q(u) is the unique quadratic form such that In particular, and the constant c ∈ Q/Z is a topological invariant of M .

Examples
We want to show on some simple examples that the invariant c is nontrivial.
(a) Suppose M = L(8, 3). Then its torsion is (see [2]) x 7 − 3 32 x 6 − 9 32 x 5 + 5 32 x 4 + 7 32 x 3 − 3 32 x 2 + 7 32 x + 5 32 where x 8 = 1 is a generator of Z 8 . Then The set of possible values −3m 2 16 mod Z is The set possible values of Ξ(h) is We need to find a constant c ∈ Q/Z such that Equivalently, we need to figure out orderings {a 1 , a 2 , a 3 , a 4 } and {b 1 , b 2 , b 3 , b 4 } of A and B such that b i − a i mod Z is a constant independent of i. A little trial and error shows that and the constant is c = − 3 32 . This is the coefficient of x 2 . We deduce that (modulo Z) The translation of F by x −2 is x.
It would be very interesting to know whether the invariant c satisfies any surgery properties. This is not a trivial issue because we cannot relate the potential surgery properties of c to the surgery properties of the torsion. In the case of torsion the surgery formula involve finite difference operators which kill the constants so c will not appear in any of them.