An introduction to the abelian Reidemeister torsion of three-dimensional manifolds

These notes accompany some lectures given at the autumn school"Tresses in Pau"in October 2009. The abelian Reidemeister torsion for 3-manifolds, and its refinements by Turaev, are introduced. Some applications, including relations between the Reidemeister torsion and other classical invariants, are surveyed.


Introduction
The Reidemeister torsion and the Alexander polynomial are among the most classical invariants of 3-manifolds. The first one was introduced in 1935 by Kurt Reidemeister [54] in order to classify lens spaces -those closed 3-manifolds which are obtained by gluing two copies of the solid torus along their boundary. The second invariant is even older since it dates back to 1928, when James Alexander defined it for knot and link complements [1]. Those two invariants can be extended to higher dimensions and refer by their very definition to the maximal abelian cover of the 3-manifold.
In 1962, John Milnor interpreted in [39] the Alexander polynomial of a link as a kind of Reidemeister torsion (with coefficients in a field of rational fractions, instead of a cyclotomic field as was the case for lens spaces). This new viewpoint on the Alexander polynomial clarified its properties, among which its symmetry. This approach to study the Alexander polynomial was systematized by Vladimir Turaev in [69,72], where he re-proved most of the known properties of the Alexander polynomial of 3-manifolds and links using general properties of the Reidemeister torsion. Vladimir Turaev also defined a kind of "maximal" torsion, which is universal among Reidemeister torsions with abelian coefficients [70]. If H denotes the first homology group of a compact 3-manifold M , his invariant is an element τ (M ) ∈ Q(Z[H])/ ± H of the ring of fractions of the group ring Z[H], up to some multiplicative indeterminacy. He also explained how the sign ambiguity ±1 and the ambiguity in H can be fixed by endowing M with two extra structures which he introduced on this purpose (a "homological orientation" [72] and an "Euler structure" [73]). These notes are aimed at introducing those invariants of 3-manifolds by Alexander, Reidemeister, Milnor and Turaev, and at presenting some of their properties and applications. This will also give us the opportunity to present a few aspects of 3-dimensional topology. We have focussed on closed oriented 3-manifolds to simplify the exposition, although most of the material extends to compact 3-manifolds with toroidal boundary (which include link complements). Besides, we have restricted to abelian Reidemeister torsions. In particular, geometrical aspects of Reidemeister torsions are not considered (see [52] for hyperbolic 3-manifolds) and we have not dealt with twisted Alexander polynomials (see [23] for a survey). The theme of simple homotopy and Whitehead torsion is no treated neither and, for this, we refer to Milnor's survey [40] and Cohen's book [13]. For further reading on abelian Reidemeister torsions, we recommend Turaev's books [76,78], which we used to prepare these notes, as well as Nicolaescu's book [47].
Throughout the notes, we use the following notation.
1. An abelian group G is denoted additively, except when it is thought of as a subset of the group ring C[G], in which case multiplicative notation is used. Besides, (co)homology groups are taken with Z coefficients if no coefficients are specified.
Moreover, as already mentioned, we are mainly interested in closed oriented connected topological 3-manifolds. So, without further mention, the term 3-manifold will refer to a manifold of this kind.

Three-dimensional manifolds
We start by recalling a few general facts about 3-manifolds. We also introduce an important class of 3-manifolds, namely the lens spaces.
2.1. About triangulations and smooth structures. One way to present a 3-manifold M is as the result of gluing finitely many tetrahedra, face by face, using affine homeomorphisms. For instance, S 3 can be identified with the boundary of a 4-dimensional simplex, and so appears as the union of 5 tetrahedra. In general, any 3-manifold can be presented in that way: The pair (K, ρ) is called a triangulation of the manifold M . The reader is refered to [56] for an introduction to piecewise-linear topology and its terminology.
Furthermore, the triangulation of a 3-manifold is essentially unique.
Those two theorems, which are due to Moise [42], show that piecewise-linear invariants give rise to topological invariants in dimension 3. As shown by Munkres [44] and Whitehead [81], these results also imply that every 3-manifold has an essentially unique smooth structure. The reader is refered to [66, §3.10].

Theorem 2.3 (Smoothing in dimension 3)
. Any 3-manifold M has a smooth structure, and any homeomorphism between smooth 3-manifolds is homotopic to a diffeomorphism.

Heegaard splittings.
A handlebody of genus g is a compact oriented 3-manifold (with boundary) obtained by attaching g handles to a 3-dimensional ball: 1 g · · · (Those handles have "index 1" in the terminology of handle decompositions.) It is easily checked that any two handlebodies are homeomorphic if, and only if, they have the same genus. We fix a "standard" handlebody of genus g, which we denote by H g . We also set Σ g := ∂H g which is our "standard" closed oriented connected surface of genus g.  Proof. Here is the classical argument. Let M be a 3-manifold and let K be a triangulation of M : |K| = M . Then, we notice in M two embedded graphs: one is the 1-skeleton a of K and the other one is the 1-skeleton b of the cell decomposition dual to K. We fatten the graph a to a 3-dimensional submanifold A ⊂ M : each vertex of a is fattened to a ball, while each edge of a is fattened to a handle connecting two such balls. By choosing a maximal subtree of the graph a, we see that A is a handlebody. Similarly, we fatten the graph b to a handlebody B ⊂ M . Moreover, we can choose A and B fat enough so that M = A ∪ B and A ∩ B = ∂A = −∂B.
Up to homeomorphism, any Heegaard splitting of genus g can be written in terms of the "standard" genus g handlebody as where f : Σ g → Σ g is an orientation-preserving homeomorphism. This viewpoint leads us to consider the mapping class group of the surface Σ g , i.e. the group M(Σ g ) := Homeo + (Σ g )/ ∼ = of orientation-preserving homeomorphisms Σ g → Σ g up to isotopy. Proof. For any orientation-preserving homeomorphisms E : H g → H g and f : Σ g → Σ g , we have If e : Σ g → Σ g is a homeomorphism isotopic to the identity, we can use a collar neighborhood of Σ g in H g to construct a homeomorphism E : H g → H g such that E| Σg = e. So, we obtain M f •e ∼ = + M f .
The previous lemma, together with the next proposition, shows that there is only one Heegaard splitting of genus 0, namely S 3 ⊂ R 4 decomposed into two hemispheres. Proposition 2.7 (Alexander). The group M(S 2 ) is trivial.
Proof. We start with the following Claim. Let n ≥ 1 be an integer. Any homeomorphism h : D n → D n which restricts to the identity on the boundary, is isotopic to the identity of D n relatively to ∂D n . Indeed, for all t ∈ [0, 1], we define a homeomorphism h t : D n → D n by Here D n is seen as a subset of R n and x is the euclidian norm of x ∈ R n . Then, the map is an isotopy (relatively to ∂D n ) between h and the identity. This way of proving the claim is known as the "Alexander's trick". Now, let f : S 2 → S 2 be an orientation-preserving homeomorphism. We are asked to prove that f is isotopic to the identity. We choose an oriented simple closed curve γ ⊂ S 2 . It follows from the Jordan-Schönflies theorem that the curves f (γ) and γ are isotopic and, so, they are ambiently isotopic. Therefore we can assume that f (γ) = γ. Since any orientation-preserving homeomorphism S 1 → S 1 is isotopic to the identity (as can be deduced from the claim for n = 1), we can assume that f | γ is the identity of γ. Then, because γ splits S 2 into two disks -say D and D ′ -it is enough to show that f | D : D → D is isotopic to the identity of D relatively to ∂D, and similarly for D ′ . But, this is an application for n = 2 of the above claim.
2.3. Example: lens spaces. In contrast with the genus 0 case, there are infinitely many 3-manifolds with a Heegaard splitting of genus 1. Such manifolds are called lens spaces. In order to enumerate them, we need to determine the mapping class group of a 2-dimensional torus. which sends an [f ] to the matrix of f * : H 1 (S 1 × S 1 ) → H 1 (S 1 × S 1 ) relative to the basis (a, a ♯ ), is a group isomorphism.
Proof. Here is the classical argument. The fact that we have a group homomorphism M(S 1 × S 1 ) → GL(2; Z) is clear. For all [f ] ∈ M(S 1 × S 1 ), the matrix of f * has determinant one for the following reason: f preserves the orientation, so that f * leaves invariant the intersection pairing H 1 (S 1 × S 1 ) × H 1 (S 1 × S 1 ) → Z.
The surjectivity of the homomorphism can be proved as follows. We realize S 1 × S 1 as R 2 /Z 2 , in such a way that the loop S 1 × 1 lifts to [0, 1] × 0 and 1 × S 1 lifts to 0 × [0, 1]. Any matrix T ∈ SL(2; Z) defines a linear homeomorphism R 2 → R 2 , which leaves Z 2 globally invariant and so induces an (orientation-preserving) homeomorphism t : R 2 /Z 2 → R 2 /Z 2 . It is easily checked that the matrix of t * is exactly T .
To prove the injectivity, let f : S 1 × S 1 → S 1 × S 1 be a homeomorphism whose corresponding matrix is trivial. Since π 1 (S 1 × S 1 ) is abelian, this implies that f acts trivially at the level of the fundamental group. The canonical projection R 2 → R 2 /Z 2 gives the universal cover of S 1 × S 1 . Thus, f can be lifted to a unique homeomorphism f : R 2 → R 2 such that f (0) = 0 and, by our assumption on f , f is Z 2 -equivariant. Therefore, the "affine" homotopy between the identity of R 2 and f , descends to a homotopy between Id S 1 ×S 1 and f . An old result of Baer asserts that two self-homeomorphisms of a closed surface are homotopic if and only if they are isotopic [2,3], so we deduce that [f ] = 1 ∈ M(S 1 × S 1 ).
In the sequel, we identify the standard handlebody H 1 with the solid torus D 2 × S 1 , so that Σ 1 is identified with S 1 × S 1 . A Heegaard splitting of genus 1 is thus encoded by 4 parameters p, q, r, s ∈ Z such that matrix of f * = q s p r with qr − ps = 1.
So, 3-manifolds having a genus 1 Heegaard splitting can be indexed by quadruplets (p, q, r, s) of that sort, but many repetitions then occur. Indeed, let E k : We deduce from (2.1) that the homeomorphism class of M f only depends on the pair of parameters (p, q). Furthermore, we have matrix of (E k | S 1 ×S 1 • f ) * = q + pk s + rk p r so that (for a given p) only the residue class of q modulo p does matter. Finally, the selfhomeomorphism C : matrix of (C| S 1 ×S 1 ) * = −1 0 0 −1 so that the 3-manifolds corresponding to the pairs (p, q) and (−p, −q) are orientationpreserving homeomorphic. Thus, we can assume that p ≥ 0. Definition 2.9. Let p ≥ 0 be an integer and let q be an invertible element of Z p . The lens space of parameters (p, q) is the 3-manifold where f : S 1 × S 1 → S 1 × S 1 is an orientation-preserving homeomorphism such that f * (a) = q · a + p · a ♯ (in the notation of Proposition 2.8).
The cases p = 0 and p = 1 are special. For p = 0, q ∈ Z 0 = Z must take the value +1 or −1 but, by the previous discussion, we have L 0,1 ∼ = + L 0,−1 . For p = 1, there is no choice for q ∈ Z 1 = {0}. Observe that L 0,1 ∼ = S 2 × S 1 (by decomposing S 2 into two hemispheres D 2 + and D 2 − ) and that L 1,0 ∼ = S 3 (by identifying S 3 with the boundary of D 2 × D 2 ). Thus, the topological classification of lens spaces is interesting only for p ≥ 2.
Theorem 2.10 (Reidemeister [54]). For any integers p, p ′ ≥ 2 and invertible residue classes q ∈ Z p , q ′ ∈ Z p ′ , we have The direction "⇐" is easily checked by exchanging the two solid tori in the Heegaard splitting of L p,q . The direction "⇒" needs topological invariants and will be proved in §5 by means of the Reidemeister torsion.
Exercice 2.11. The terminology "lens spaces" (which dates back to Seifert and Threlfall [58]) is justified by the following equivalent description. Draw the planar regular p-gone Next, consider the polyhedron B p ⊂ R 3 whose boundary is the bicone with base G p and with vertices n := (0, 0, 1) (the "north pole") and s := (0, 0, −1) (the "south pole"): Let ∼ q be the equivalence relation in B p such that n ∼ q s and v i ∼ q v i+q for all i ∈ Z p , and which identifies linearly the north face (v i , v i+1 , n) of B p with its south face (v i+q , v i+q+1 , s). Show that the quotient space B p / ∼ q is homeomorphic to L p,q and deduce that L 2,1 is homeomorphic to the projective space RP 3 .

Exercice 2.12.
Here is yet another description of lens spaces. Consider the 3-sphere Show that the quotient space S 3 /Z p is homeomorphic to L p,q .
Exercice 2.13. Check that, for any integer p ≥ 0 and for any invertible q ∈ Z p , we have L p,−q ∼ = + −L p,q . Deduce from Theorem 2.10 that L p,q has an orientation-reversing self-homeomorphism if, and only if, q 2 = −1 ∈ Z p .

The abelian Reidemeister torsion
We aim at introducing the abelian Reidemeister torsion of 3-manifolds. This is a combinatorial invariant in the sense that it is defined (and can be computed) from triangulations or, more generally, from cell decompositions. Thus, we start by introducing the abelian Reidemeister torsion of CW-complexes.
3.1. The torsion of a chain complex. Let F be a commutative field, and let C be a finite-dimensional chain complex over F: We assume that C is acyclic and based in the sense that we are given a basis c i of C i for each i = 0, . . . , m.
We denote by B i ⊂ C i the image of ∂ i+1 and, for each i, we choose a basis b i of B i . The short exact sequence of F-vector spaces shows that we can obtain a new basis of C i by taking, first, the vectors of b i and, second, some lifts b i−1 of the vectors b i−1 . We denote by b i b i−1 this new basis, and we compare it to c i by computing This scalar does not depend on the choice of the lift b i−1 of b i−1 .
One easily checks that τ (C, c) does not depend on the choice of b 0 , . . . , b m .
Remark 3.2. The torsion can also be defined for an acyclic free chain complex C over an associative ring Λ. Still, we must assume that the rank of free Λ-modules is well defined, i.e. Λ r is not isomorphic to Λ s for r = s. Then, the torsion of C based by c is defined, without taking determinants, as an element of (Let us note that, when Λ is a commutative field, the determinant provides an isomorphism between K 1 (Λ) and Λ \ {0}.) This generalization is for instance needed in the definition of the Whitehead torsion of a homotopy equivalence between finite CWcomplexes, which is the obstruction for it to be simple, i.e. to be homotopic to a finite sequence of elementary "collapses" and "expansions". We refer to Milnor's survey [40] or Cohen's book [13] for an introduction to this important subject.
By its definition, the torsion of a chain complex C can be seen as a multiplicative analogue of its Euler characteristic, namely Keeping in mind this analogy, let us state some of the most important properties of the torsion. We refer to Milnor's survey [40] or to Turaev's book [76] for proofs.
Firstly, the Euler characteristic is additive in the sense that χ(C 1 ⊕ C 2 ) = χ(C 1 ) + χ(C 2 ). Similarly, the torsion is multiplicative. Proposition 3.3 (Multiplicativity). Let C 1 , C 2 be some finite-dimensional acyclic based chain complexes over F. If their direct sum C 1 ⊕ C 2 is based in the usual way, then we have Secondly, the Euler characteristic behaves well with respect to duality in the sense that χ(C * ) = (−1) m · χ(C). Here, C * is the dual chain complex The torsion enjoys a similar property. Proposition 3.4 (Duality). Let C be a finite-dimensional acyclic based chain complex over F. If the dual chain complex C * is equipped with the dual basis, then we have Finally, the Euler characteristic can be computed homologically by the classical formula χ(C) = χ(H * (C)). If F = Q(R) is the field of fractions of a domain R, and if C = Q(R) ⊗ R D is the localization of a chain complex D over R, this formula takes the form There is a multiplicative analogue of this identity for the torsion, where ranks of Rmodules are replaced by their orders.
Theorem 3.5 (Homological computation). Let R be a noetherian unique factorization domain, and let D be a finitely generated free chain complex over R. We assume that D is based and that rank H i (D) = 0 for all i. Then, we have (The definition of the order of a finitely generated module over a unique factorization domain is recalled in §4.1.) This theorem is due to Milnor for a principal ideal domain R [41], and to Turaev in the general case [72].
The torsion can also be defined for a non-acyclic chain complex C. In this case, C should also be equipped with a basis h i of H i (C) for each i = 0, . . . , m. Let Z i ⊂ C i be the kernel of ∂ i . The short exact sequence shows that a basis of Z i is obtained by taking, first, a basis b i of B i and, second, some lifts h i of the vectors h i . The resulting basis is denoted by b i h i and, according to the short exact sequence Definition 3.6. The torsion of C based by c = (c 0 , . . . , c m ) and homologically based by where N (C) is the mod 2 integer The sign (−1) N (C) is here for technical convenience and follows the convention of [72].
3.2. Abelian Reidemeister torsions of a CW-complex. Let X be a finite connected CW-complex, whose first homology group is denoted by H := H 1 (X). Assume that ϕ : Z[H] −→ F is a ring homomorphism with values in a commutative field F. We consider the cellular chain complex of X with ϕ-twisted coefficients where X denotes the maximal abelian covering space of X. This is a finite-dimensional chain complex over F whose homology H ϕ * (X) := H * (C ϕ (X)) may be trivial, or may be not.
Let E be the set of cells of X. For each e ∈ E, we choose a lift e to X, and we denote by E the set of the lifted cells. We also put a total ordering on the finite set E, and we choose an orientation for each cell e ∈ E: this double choice (ordering+orientation) is denoted by oo. The choice of E combined to oo induces a basis E oo of C( X), which defines itself a basis 1 ⊗ E oo of C ϕ (X).
with the convention that τ ϕ (X) := 0 if H ϕ * (X) = 0. Because of the choices that we were forced to make, the quantity τ ϕ (X) has two kinds of indeterminacy: a sign ±1 and the image by ϕ of an element of H. Those two ambiguities are resolved by Turaev [72,73]. The ϕ(H) indeterminacy is the most interesting and it will be discussed in §7. To kill the sign indeterminacy, Turaev defines in [72] a homological orientation ω of X to be an orientation of the R-vector space H * (X; R). Then, the quantity τ ϕ (X, ω) := sgn τ C(X; R), oo, w · τ C ϕ (X), 1 ⊗ E oo ∈ F/ϕ(H), where w is any basis of H * (X; R) representing ω, does not depend on the choice of oo and only depends on ω. Here again, we set τ ϕ (X, ω) := 0 if H ϕ * (X) = 0. This invariant τ ϕ (X, ω) is sometimes called the sign-refined Reidemeister torsion of (X, ω). We have τ ϕ (X, −ω) = −τ ϕ (X, ω). The algebraic properties of the torsion of chain complexes (some of those have been recalled in §3.1) have topological implications. For instance, Proposition 3.3 translates into a kind of Mayer-Vietoris theorem for the Reidemeister torsion of CW-complexes. Also, Theorem 3.5 allows one to compute the Reidemeister torsion of a CW-complex by homological means when F is the field of fractions of a domain.
Remark 3.8. Here, we have restricted ourselves to the abelian version of the Reidemeister torsion, in the sense that coefficients are taken in a commutative field F. Nonetheless, the same construction applies to any ring homomorphism with values in a ring Λ for which the rank of free modules is well-defined. In this situation, we need the torsion of Λ-complexes evoked in Remark 3.2 and we work with the universal cover of X instead of its maximal abelian cover. The Reidemeister torsion is then an element τ ϕ (X) ∈ K 1 (Λ)/ ± ϕ(π 1 (X)) ∪ {0}.  We apply this to H := H 1 (X) to write Q(Z[H 1 (X)]) as a direct sum of commutative fields: We denote by ϕ i : Q(Z[H]) → F i the corresponding projections.
Definition 3.10 (Turaev [70]). The maximal abelian torsion of X is Recall that the construction of τ ϕ i (X) needs some choices (Definition 3.7). Here, we do the same choices for all i = 1, . . . , n so that the indeterminacy is "global" (in ±H) instead of "local" (one for each component i = 1, . . . , n). A sign-refined version τ (X, ω) ∈ Q(Z[H])/H of τ (X) is also defined in the obvious way, for any homological orientation ω of X.
The invariant τ (X) determines the abelian Reidemeister torsion τ ϕ (X) for any ring homomorphism ϕ : Z[H] → F, which justifies the name given to τ (X). Here is the precise statement, where it is assumed that τ ϕ (X) = 0. We consider the subring Proof of Proposition 3.9. We follow Turaev [76, §12]. The unicity of the splitting means that, for two decompositions of Q(Z[H]) as a direct sum of fields we must have n = n ′ and F j = F ′ α(j) for some permutation α ∈ S n . This is an instance of the following general fact, which is easily proved: If a ring splits as a direct sum of finitely many domains, then the splitting is unique in the above sense.
To prove the existence of the splitting, let us assume first that H is finite. Each character σ : H → C * of H extends to a ring homomorphism σ : Q[H] → C by linearity. The subgroup σ(H) of C * is finite and, so, is cyclic: thus, σ (Q[H]) is the cyclotomic field Q e 2iπ/mσ where m σ is the order of σ(H). Two characters σ and σ ′ of H are declared to be equivalent if m σ = m σ ′ and if σ, σ ′ : Q[H] → Q e 2iπ/mσ differ by a Galois automorphism of Q e 2iπ/mσ over Q. Let σ 1 , . . . , σ n be some representatives for the equivalence classes of Hom(H, C * ). Then, the ring homomorphism is injective since any non-trivial element of the ring Q[H] can be detected by a character. (Indeed, by Maschke's theorem, the C-algebra C[H] is semi-simple so that, according to the Artin-Wedderburn theorem, it is isomorphic to End(V 1 ) ⊕ · · · ⊕ End(V r ) for some Cvector spaces V 1 , . . . , V r . But, since the C-algebra C[H] is commutative, each V i should be one-dimensional. Thus, the C-algebra C[H] is isomorphic to a direct sum of r copies of C and each corresponding projection C[H] → C restricts to a character H → C * .) Therefore, the ring homomorphism (3.1) is bijective since its source and target have the same dimension over Q:

The Alexander polynomial
We introduce the Alexander polynomial of a topological space X of finite type 1 and, following Milnor and Turaev, we explain how to obtain it as a kind of Reidemeister torsion. In a few words, "the Alexander polynomial of X is the order of the first homology group of the free maximal abelian cover of X." Thus, we start by recalling what the order of a module is.
4.1. The order of a module. Let R be a unique factorization domain, whose multiplicative group of invertible elements is denoted by R × . Let also M be a finitely generated R-module.
We choose a presentation of M with, say, n generators and m relations, and we denote by A the corresponding m × n matrix: Here, m may be infinite. Besides, we can assume that m ≥ n with no loss of generality.
The order of M is ∆ 0 (M ) and is denoted by ord(M ).
It is easily checked that the elementary ideals and, a fortiori, their greatest common divisors, do not depend on the choice of the presentation matrix A. We have the following inclusions of ideals: hence the following divisibility relations: Example 4.2. Let R be a principal ideal domain. Then, M can be decomposed as a direct sum of cyclic modules: M = R/n 1 R ⊕ · · · ⊕ R/n k R. The order of M is represented by the product n 1 · · · n k ∈ R.  4.2. The Alexander polynomial of a topological space. Let X be a connected topological space of finite type. Then, the free abelian group is finitely generated. We observe that is the first Betti number of X, so that the ring Z[G] has essentially the same properties as a polynomial ring with integer coefficients. In particular, Z[G] is a unique factorization domain (by Gauss's theorem, since Z is so) and it is noetherian (by Hilbert's basis theorem, since Z is so). Moreover, we have Z[G] × = ±G.
We are interested in the maximal free abelian cover X → X whose group of covering automorphisms is identified with G. More precisely, we are interested in the homology of X as a Z[G]-module. By our assumptions, the Z[G]-module H i (X) is finitely generated for any i ≥ 0 and, sometimes, it is called the i-th Alexander module of X. Here, we are mainly interested in the first Alexander module which only depends on the fundamental group of X. Indeed, the Hurewicz theorem gives a canonical isomorphism where K(X) denotes the kernel of the canonical epimorphism π 1 (X) → G.
Here is a recipe to compute the Alexander polynomial by means of Fox's "free derivatives". (The basics of Fox's free differential calculus are recalled in the appendix.) Theorem 4.6 (Fox [18]). Let X be a connected topological space of finite type. Consider a finite presentation of π 1 (X) (4.1) π 1 (X) = x 1 , . . . , x n |r 1 , . . . , r m and the associated matrix where ∂ ∂x 1 , . . . , ∂ ∂xn denote the free derivatives with respect to (x 1 , . . . , x n ). Then, the Alexander polynomial of X is Proof. We follow Turaev [76, §16]. Let Y be the 2-dimensional cellular realization of the group presentation (4.1). More explicitely, Y has a unique 0-cell, n 1-cells (in bijection with the generators x 1 . . . , x n ) and m 2-cells (in bijection with the relations r 1 , . . . , r m which are interpreted as attaching maps for the 2-cells). Then, π 1 (Y ) has the same presentation (4.1) as the group π 1 (X). Since ∆(X) only depends on π 1 (X), we have where Y denotes the maximal free abelian cover of Y . It follows from the topological interpretation of Fox's free derivatives (see §A.2) that A reduced to Z[π 1 (Y )] is the matrix of the boundary operator of the universal cover Y of Y with respect to some appropriate basis (which are obtained by lifting the cells of Y ). Let Y 0 be the 0-skeleton of Y . Because we have that topological interpretation of the matrix A leads to The following statement is proved by Blanchfield in [7]. See also [26, §3.1].
Fact. Let M be a finitely generated module over a noetherian unique factorization domain. Then, we have ). The conclusion then follows from equations (4.2) and (4.3).

Alexander polynomial and Milnor torsion.
Let X be a finite connected CWcomplex, with maximal free abelian cover X. As before, we set We consider the fraction Observe that the Alexander polynomial ∆(X) appears as a numerator of A(X), which is sometimes called the Alexander function of X.
Definition 4.7. The Milnor torsion of X is the Reidemeister torsion The next result shows that the combinatorial invariant τ µ (X) is in fact a topological invariant, and is more precisely a homotopy invariant.  [39,41], Turaev [72]). For any finite connected CW-complex X, Proof. Assume that τ µ (X) = 0. By our convention, this means that Here, the last identity follows from the universal coefficients theorem (which can be applied since C(X) is Z[G]-free) and the fact that the field of fractions of Z[G] is Z[G]-flat. So, the Z[G]-module H i X is not fully torsion or, equivalently, it has order 0. It follows that A(X) = 0 by convention. If τ µ (X) is not zero, then the identity τ µ (X) = A(X) is an application of Theorem 3.5.

The abelian Reidemeister torsion for three-dimensional manifolds
We introduced abelian Reidemeister torsions of CW-complexes in §3, and we saw in §4 that the Alexander polynomial fits into this framework. In this section, we apply the theory of abelian Reidemeister torsions to 3-manifolds and we compute them for two important classes of 3-manifolds: lens spaces (which are classified in this way) and surface bundles. 5.1. Abelian Reidemeister torsions of a 3-manifold. A theorem of Chapman asserts that the Reidemeister torsion of CW-complexes is invariant under homeomorphisms, so that it defines a topological invariant of those topological spaces which admit cell decompositions [10]. We will not need this deep result here. It is not too difficult (although technical) to prove that the Reidemeister torsion is invariant under cellular subdivisions, so that the Reidemeister torsion defines a piecewise-linear invariant of polyhedra [40]. It follows from Theorem 2.1 and Theorem 2.2 that the Reidemeister torsion induces a topological invariant of 3-manifolds.
In more details, let M be a 3-manifold and let ϕ : is a triangulation of M and the cellular homology of K is identified with its singular homology.
Theorem 2.1 ensures the existence of the triangulation (K, ρ) of M . Theorem 2.2 and the invariance of the Reidemeister torsion under subdivisions imply that the above definition does not depend on the choice of (K, ρ). Thus, for any cell decomposition X of M (which can be subdivided to a triangulation), we also have where the cellular homology of X and the singular homology of M are identified.
We mentioned in §3.2 two properties of the Reidemeister torsion of CW-complexes, which were inherited from algebraic properties of the torsion of chain complexes. Here is a third important property, which is specific to manifolds. [20], Milnor [39]). Let M be a 3-manifold and let F be a commutative field with an involution f → f . Consider a ring homomorphism ϕ : Then, we have the following symmetry property: Sketch of the proof. Let K be a triangulation of M : |K| = M . It can be lifted to a triangulation K of the maximal abelian cover M of M . We fix a total ordering on the set E of simplices of K, we choose an orientation for each simplex e ∈ E and we choose a lift e of e to K. Thus, we obtain a basis E oo of C( K) and, by definition, we have with the convention that τ ϕ (M ) := 0 if H ϕ * (M ) = 0. Besides, we can consider the cell decomposition K * dual to the triangulation K, which can be lifted to a cell decomposition K * of M . Each cell e * of K * is dual to a unique simplex e of K, so that it has a preferred orientation and a preferred lift e * which are determined by the choices we did for e. Moreover, the total ordering on E induces a total ordering on the set E * of the cells of K * . Thus, we obtain a basis E * oo of C( K * ) and, by definition, we have Here we have denoted by ϕ the ring homomorphism ϕ composed with the involution of F.
As in the proof of the Poincaré duality theorem (relating the ϕ-twisted cohomology of K to the ϕ-twisted homology of K * ), one can prove that the dual of the based chain We conclude with the duality property for torsions of chain complexes (Proposition 3.4) that τ ϕ (M ) = ±τ ϕ (M ). We refer to [39] or to [76] for the details of proof. See [72] for the computation of signs. We consider the 2g oriented simple closed curves α 1 , . . . , α g , α ♯ 1 , . . . , α ♯ g on the genus g surface ∂A shown on Figure 5.1, as well as the curves β 1 , . . . , β g , β ♯ 1 , . . . , β ♯ g on ∂B. If we focus on A, the 3-manifold M is determined by how the curves β 1 , . . . , β g read in the surface ∂A when ∂B is identified to ∂A. So, we are aiming at a formula expressing τ ϕ (M ) in terms of the curves β 1 , . . . , β g ⊂ ∂A = ∂B.
We choose a small disk D ⊂ ∂A and a base point ⋆ ∈ ∂D. We base the oriented simple closed curves α, α ♯ at ⋆ to get a basis of the free group π 1 (∂A \ D, ⋆). We also denote by s free derivatives with respect to this basis. (See the appendix.) We also base the simple oriented closed curves β at ⋆, so that they define elements of π 1 (∂A \ D, ⋆). Then, we can consider the following g × g matrix with coefficients in Z[H 1 (M )]: The curves α, α ♯ on ∂A and the curves β, β ♯ on ∂B.
Lemma 5.3. With the above notation and for any indices i, j ∈ {1, . . . , g}, we have Sketch of the proof. The lemma is an application of a formula by Turaev which computes τ ϕ (M ) from a cell decomposition of M with a single 0-cell and a single 3-cell [78, §II.1]. In more details, we consider the cell decomposition X defined as follows by the Heegaard splitting. There is only one 0-cell e 0 , the center of the ball to which handles have been added to form A; the 1-cells are e 1 1 , . . . , e 1 g where e 1 i is obtained from the core of the i-th handle of A, which is bounded by two points, by adding the trace of those two points when the previous ball is "squeezed" to e 0 ; the 2-cells are e 2 1 , . . . , e 2 g where e 2 j is obtained from the co-core of the j-th handle of B (with boundary β j ) by adding the trace of β j ⊂ ∂A when A is "squeezed" to e 0 ∪ e 1 1 ∪ · · · ∪ e 1 g ; there is only one 3-cell e 3 , namely the complement in M of the cells of smaller dimension. See Figure 5.2. The cell decomposition X is related to the curves (α, α ♯ ) and (β, β ♯ ) in the following way: each curve α ♯ i is isotopic to e 1 i in A, and each curve β j is the boundary of e 2 j ∩ B. The cells of X can be oriented as follows: e 0 is given the + sign, the 1-cell e 1 i is oriented coherently with α ♯ i , the 2-cell e 2 j is oriented so that the intersection number e 2 j β ♯ j is +1 and e 3 inherits the orientation from M . Let X be the maximal abelian cover of X, and choose some lifts e 0 , e 1 1 , . . . , e 1 g , e 2 1 , . . . , e 2 g , e 3 of the cells to X. The computation of τ ϕ (M ) in terms of the cell decomposition X involves the sign where oo refers to the above choice of order and orientations for the cells of X. It also involves the cell chain complex of X. For some appropriate choices of the lifts, the boundary operators ∂ 3 and ∂ 1 are given by respectively. Thus, the main indeterminate is the boundary operator ∂ 2 and we denote Then, a computation of τ ϕ (X) from its definition gives ) See the proof of Theorem II.1.2 in [78] for the details of computation. The topological interpretation of free differential calculus (see §A.2) shows that, for appropriate choices of the lifted cells, the matrix A ′ is equal to A. The conclusion follows.
Proof. This can be deduced from Theorem 4.8. Alternatively, we can use the following argument which is more direct. The torsion τ µ (M ) can be computed from a Heegaard splitting as explained by Lemma 5.3. Using the same notations, we get in the basis e 2 and e 1 .
Let Y be the 2-skeleton of X and, as before, we denote by Y the maximal free abelian cover of Y . The proof of Theorem 4.6 tells us that . We can conclude thanks to (5.2) and (5.3).
Its kernel is called the augmentation ideal of Q[H]. The Reidemeister torsion with ε-twisted coefficients is zero since H * (M ; Q) = 0. We conclude that for any 3-manifold M with β 1 (M ) = 0. Assume that β 1 (M ) = 1. We proceed as in the previous case, and we denote by t a generator of G = H/ Tors H. We know from Theorem 5.4 that µ (τ (M )) coincides with ∆(M )/(t − 1) 2 , so that it belongs to Z[t ± ]/(t − 1) 2 . More generally, Turaev "extracts" from τ (M ) an integral part denoted by where t ∈ H projects to a generator of G ≃ Z. We refer to [78,§II] for further details.

5.4.
Example: torsion of lens spaces. Recall that the lens space L p,q is defined by the Heegaard splitting Let T be the preferred generator of H 1 (L p,q ) defined by the core 0 × S 1 of the left-hand solid torus in the decomposition (5.4).
Lemma 5.7. Let ϕ : Z[H 1 (L p,q )] → F be a non-trivial ring homomorphism with values in a commutative field. Then, we have where ε p is a sign which does not depend on q ∈ Z p and r := q −1 ∈ Z p .
. Just like the cellular chain complex for the cell decomposition induced by the Heegaard splitting, the sign τ 0 in (5.3) does not depend on q.
The topological classification of lens spaces can now be completed and, for this, we need a number-theoretic result. This is for instance proved in [14, §I] using elementary properties of Gauss sums and a non-vanishing property of Dirichlet series.
Lemma 5.8 (Franz [19]). Let p ≥ 1 be an integer and let Z × p be the multiplicative group of invertible elements of Z p . Let a : Z × p → Z be a map such that j∈Z × p (ζ j − 1) a(j) = 1 for any p-th root of unity ζ = 1.
Then, a(j) = 0 for all j ∈ Z × p . Proof of Theorem 2.10. It remains to prove the necessary condition for two lens spaces L p,q and L p ′ ,q ′ to be homeomorphic with their orientations preserved. The interesting case is when p ≥ 3, which we assume.
Let f : L p,q → L p ′ ,q ′ be an orientation-preserving homeomorphism. Because f induces an isomorphism in homology and because H 1 (L p,q ) ≃ Z p , we must have p = p ′ . Let T ∈ H 1 (L p,q ) and T ′ ∈ H 1 (L p,q ′ ) be the preferred generators defined by the Heegaard splittings. There is a k ∈ Z × p such that T ′ = k · f * (T ). For any p-th root of unity ζ = 1, we consider the ring homomorphism ϕ : Since the Reidemeister torsion is a topological invariant, we have τ ϕ (L p,q ) = τ ϕ ′ (L p,q ′ ) and we deduce from Lemma 5.7 that where r := q −1 , r ′ := (q ′ ) −1 and u ∈ Z p is unknown. If we multiply (5.5) by its conjugate, we get the formula For all j ∈ Z × p , let m(j) be the number of times j appears in the sequence (1, −1, r, −r) and let m ′ (j) be the number of times it appears in the sequence (k, −k, kr ′ , −kr ′ ). We have m(j) = m(−j) and j∈Z × p m(j) = 4, and similarly for m ′ . We deduce from this and from (5.6) that the map a := m − m ′ satisfies the hypothesis of Lemma 5.8. Thus, we have m = m ′ so that the two sequences (1, −1, r, −r) and (k, −k, kr ′ , −kr ′ ) coincide up to some permutation, which we write Moreover, let P ∈ C[X] be a polynomial such that the identity holds in the group ring C[H] where, following our convention, the group H := H 1 (L p,q ) is written multiplicatively. Since T p = 1 ∈ C[H], we can assume that P has degree at most p − 1. But, since the identity (5.5) holds for any p-th root of unity ζ (including ζ = 1), this polynomial has at least p roots. Therefore, P is zero and Equation (5.7) leaves out 8 possible cases which must be analized separately, possibly using (5.8). For instance, if k = −r and kr ′ = −1, then we have rr ′ = 1 and we are done. As another example, let us consider the case when k = 1 and kr ′ = −r. Then, (5.8) gives To pursue, we point out the remarkable identity where, again, H is written multiplicatively and Σ H := h∈H h. Using that identity for h = T and h = T r , we get So, the case (k = 1, kr ′ = −r) must be excluded. The 6 remaining cases are treated in a similar way.
Remark 5.9. Bonahon proved by purely topological methods that a lens space has, up to isotopy, only one Heegaard splitting of genus one [8]. The topological classification of lens spaces is easily deduced from this result, without using the Reidemeister torsion. The proof of Theorem 2.10 that we have presented here, which dates back to Reidemeister [54] and Franz [19], has the advantage to extend to higher dimensions. (Lens spaces can be defined in any odd dimension by generalizing the description proposed in Exercice 2.12: see [13, §V] for instance.) To conclude with lens spaces, we compute the maximal abelian torsion of L p,q . We deduce from Lemma 5.7 that Thus, we obtain the formula

5.5.
Example: torsion of surface bundles. Let Σ g be a closed connected oriented surface of genus g. The mapping torus of an orientation-preserving homeomorphism f : Σ g → Σ g is the 3-manifold which makes T f a bundle with base S 1 and fiber Σ g . Conversely, any surface bundle over S 1 can be obtained in that way. We wish to compute the maximal abelian torsion of T f . For this, we can assume that g ≥ 1 since T f only depends on the isotopy class of f (up to homeomorphism) and we have seen that M(S 2 ) is trivial. We restrict ourselves to the case when f acts trivially in homology. Then, H := H 1 (T f ) is free abelian (isomorphic to H 1 (Σ g )⊕Z) and Theorem 5.4 says that τ (T f ) = ∆(T f ). Thus, we are reduced in this case to compute the Alexander polynomial of T f . (See Remark 5.11 for the general case.) Thanks to an isotopy of f , we can further assume that f is the identity on a 2-disk D. The bordered surface Σ g \ int(D) is denoted by Σ g,1 , and the restriction of the homeomorphism f to Σ g,1 is denoted by f |. We pick a base point ⋆ ∈ ∂Σ g,1 . Then, we have where t is the homology class of the circle (⋆× [−1, 1])/ ∼. The group π 1 (Σ g,1 , ⋆) is freely generated by the based loops α 1 , . . . , α g , α ♯ 1 , . . . , α ♯ g shown on Figure 5.3. We denote by ∂ ∂α 1 , . . . , ∂ ∂αg , ∂ ∂α ♯ 1 , . . . , ∂ ∂α ♯ g the free derivatives with respect to this basis, and we consider the "Jacobian matrix" of f | * : π 1 (Σ g,1 , ⋆) → π 1 (Σ g,1 , ⋆) defined by It turns out that τ (T f ) is essentially given by the characteristic polynomial with indeterminate t of (a reduction of) the matrix J(f | * ). Figure 5.3. The surface Σ g,1 and a system of meridians and parallels (α, α ♯ ).
Proposition 5.10. With the above notation and assumption, we have Proof. We are asked to compute where T f denotes the maximal (free) abelian cover of T f . If Σ g is the maximal abelian cover of Σ g , then T f can be realized as Σ g × R, so that it decomposes into "slices": The covering transformation corresponding to t ∈ H shifts the slices from left to right.
We set S := H 1 (Σ g ) so that H = S ⊕ (Z · t where f : Σ g → Σ g is the lift of f that fixes a preferred lift ⋆ of ⋆. (The map f * is Z[S]-linear by our assumption that f acts trivially on S.) The maximal abelian cover Σ g,1 of Σ g,1 is a surface with boundary, and the group of covering transformations H 1 (Σ g,1 ) ≃ S acts freely and transitively on the set of its boundary components. Therefore, Σ g is obtained from Σ g,1 by gluing 2-disks D s indexed by s ∈ S. We deduce the short exact sequence The following is a commutative diagram in the category of Z[H]-modules: An application of the "snake" lemma relates the cokernels of the three vertical maps by a short exact sequence. Besides, the order of a module is multiplicative in short exact sequences: this fact generalizes Exercice 4.4 and can be found in [26, §3.3]. We deduce that The surface Σ g,1 deformation retracts to the union of based loops X 2g := α 1 ∪ α ♯ 1 ∪ · · ·∪α g ∪α ♯ g , which becomes a bouquet of 2g circles when all the basing arcs are collapsed to ⋆. Let p : Σ g,1 → Σ g,1 be the projection of the maximal abelian cover of Σ g,1 . Then, X 2g := p −1 (X 2g ) is the maximal abelian cover of X 2g . There is a map φ : Σ g,1 → Σ g,1 homotopic to f | such that φ(X 2g ) = X 2g and φ(⋆) = ⋆, and we have the following commutative square in the category of Z[S]-modules: It follows that (5.13) ord Finally, the long exact sequence for the pair ( X 2g , p −1 (⋆)) gives the short exact sequence where φ r denotes the relative version of φ. Again, by applying the "snake" lemma and the multiplicativity of orders, we obtain is freely generated by the lifts α 1 , . . . , α g , α ♯ 1 , . . . , α ♯ g of the loops α 1 , . . . , α g , α ♯ 1 , . . . , α ♯ g starting at ⋆. Moreover, Proposition A.6 tells us that the matrix of φ r * in that basis is the reduction to Z[H 1 (X 2g )] of the jacobian matrix . We conclude thanks to (5.11), (5.12), (5.13) and (5.14).
Remark 5.11. Proposition 5.10 is proved by Turaev in [78] with no restriction on f ∈ M(Σ g ) using other kinds of arguments. Observing that T f has a preferred Heegaard splitting of genus 2g + 1, another proof would be to apply Lemma 5.3. Proposition 5.10 is the analogue for 3-manifolds of a classical formula expressing the Alexander polynomial of a closed braid (respectively, of a pure braid) from its Burau representation (respectively, from its Gassner representation) -see Birman's book [4].
The natural action on the free group π 1 (Σ g,1 , ⋆) = F(α, α ♯ ) allows us to regard M(Σ g,1 ) as a subgroup of Aut F(α, α ♯ ) . In particular, the Magnus representation defined in §A.3 applies to M(Σ g,1 ): An important subgroup of M(Σ g,1 ) is the Torelli group of Σ g,1 , namely 1 ). By Corollary A.9, we obtain a group homomorphism with values in a group of matrices over a commutative ring. As illustrated by Proposition 5.10, this representation appears naturally in the context of abelian Reidemeister torsions, but it is unfortunately not faithfull [61]. Its kernel consists of those elements of I(Σ g,1 ) which act trivially at the level of the second solvable quotient π 1 (Σ g,1 , ⋆)/π 1 (Σ g,1 , ⋆) ′′ of the fundamental group. Explicit elements of the kernel are described by Suzuki in [62].

Homotopy invariants derived from the abelian Reidemeister torsion
In this section, we relate the maximal abelian torsion of 3-manifolds to some invariants which are defined with just a little bit of algebraic topology, and which only depend on the oriented homotopy type 2 .
6.1. The triple-cup product forms. Let M be a 3-manifold. Apart from the fundamental group, the first homotopy invariant of M which comes to one's mind is probably the cohomology ring. Poincaré duality shows that all the (co)homology groups of M are determined by H 1 (M ; Z) and that, for any integer r ≥ 0, the cohomology ring H * (M ; Z r ) is determined by the triple-cup product form u (r) For instance, u M is a trilinear alternate form on the free abelian group H 1 (M ; Z). It turns out that any algebraic object of this kind can be realized by a 3-manifold. This realization result has first been proved for Z 2 coefficients by Postnikov [53], and it has been generalized to the coefficient ring Z r for any integer r ≥ 0 by Turaev [71].
To prove Theorem 6.1, we need to introduce a way of modifying 3-manifolds. For this, we consider the torus T 3 := S 1 × S 1 × S 1 and its Heegaard splitting of genus 3 (6.1) where A is a closed regular neighborhood of the graph (S 1 ×1×1)∪(1×S 1 ×1)∪(1×1×S 1 ) and B is the complement of int(A). Given a 3-manifold M and an embedding a : A ֒→ M , we can "cut" A out of M and "replace" it by B. More formally, we define the 3-manifold where a ∂ : ∂B = ∂A ֒→ M is the restriction of a to the boundary. This terminology is justified by the facts that (6.1) is the unique Heegaard splitting of T 3 of genus 3 (according to Frohman & Hass [24]) and there is no Heegaard splitting of T 3 of lower genus (for obvious homological reasons). In this sense, the decomposition (6.1) of T 3 is canonical. The T 3 -surgery is implicitly defined in Sullivan's paper [60]. Rediscovered by Matveev under the name of "Borromean surgery" [36], this kind of modification has become fundamental in the theory of finite-type invariants. (See §7.5 in this connection.) Proof of Theorem 6.1. We reformulate Sullivan's proof [60] Let us now compute how the triple-cup product form changes under a T 3 -surgery M ; M a . An application of the Mayer-Vietoris theorem shows that there exists a unique isomorphism Φ a : H 1 (M ) → H 1 (M a ) such that the following diagram is commutative: We also denote by Φ * a : .
We conclude thanks to the claim in the following way. Let b ≥ 0 be an integer and let H := H 1 (♯ b S 1 × S 2 ). Then, H * := H 1 (♯ b S 1 × S 2 ) can be identified with Z b . There is a non-singular bilinear pairing which allows us to identify Hom(Λ 3 H * , Z) with Λ 3 H. Thus, we can write the trilinear alternate form u : For each j = 1, . . . , n, we consider an embedding a (j) : for i = 1, 2, 3, and we assume that the handlebodies a (1) (A), . . . , a (n) (A) are pairwise disjoint. Thus, we can perform simultaneously the n T 3 -surgeries along a (1) , . . . , a (n) and we denote by M ′ the resulting 3-manifold. It follows from the above claim that The triple-cup product form of S 1 × S 2 being trivial (since the rank of H 1 (S 1 × S 2 ) is less than 3), we have u  We shall now state a formula due to Turaev, which relates the torsion τ (M ) to the cohomology ring of M . To simplify the exposition, we will only consider 3-manifolds M with β 1 (M ) ≥ 3, and we will restrict ourselves to the triple-cup product form with Z coefficients. The reader is refered to [78,§III & §IX] for the other cases.
Let L be a free abelian group of finite rank b, and let u : L × L × L → Z be a skew-symmetric trilinear form. We consider the adjoint of u u : L × L −→ L * , (y 1 , y 2 ) −→ u(y 1 , y 2 , −) where L * denotes Hom(L, Z). Let e = (e 1 , . . . , e b ) be a basis of L, and let ( u/e) be the matrix of u in the basis e. This is a b × b matrix with coefficients in the symmetric algebra S(L * ) ⊃ L * . Because of the skew-symmetry of u, its determinant happens to be zero, and we consider instead its (i, j)-th minor ( u/e) i,j . Turaev proves that and one can wonder what is the "leading term" of τ (M ) with respect to that filtration. The triple-cup product form with Z coefficients gives a partial answer to that question.
Then, we have where g i ∈ H denotes a lift of g i ∈ G.
Theorem 6.5 can be proved starting from Lemma 5.3: see [78,§III.2]. It is the analogue for 3-manifolds of a knot-theoretic result by Traldi [67,68]: the leading term of the Alexander polynomial of a link in S 3 (whose linking matrix is assumed to be trivial) is determined by its Milnor's triple linking numbers.
Exercice 6.6. Check Theorem 6.5 for the trivial surface bundle Σ g × S 1 . Here, X and Y are disjoint oriented knots which represent x and y respectively, Σ is a compact connected oriented surface which is transverse to Y and whose boundary goes m ≥ 1 times around X. We have denoted by Σ Y ∈ Z the intersection number. To check the symmetry of λ M , i.e. λ M (x, y) = λ M (y, x), we choose a compact connected oriented surface Θ whose boundary goes n ≥ 1 times around Y . We also assume that Θ is transverse to Σ, so that Σ∩Θ is an oriented 1-dimensional manifold. Thus, the boundary points of Σ ∩ Θ come by pairs. From this, we obtain ∂Σ Θ − Σ ∂Θ = 0 ∈ Z or, equivalently, m · X Θ − n · Σ Y = 0 ∈ Z. We deduce that 1 To prove that λ M is non-degenerate, we use the commutative diagram Here λ M : x → λ M (x, −) denotes the adjoint of λ M . We deduce that λ M is surjective and, so, bijective. The diagram also shows that the isomorphism class of λ M only depends on the oriented homotopy type of M .
Any symmetric non-degenerate bilinear pairing on a finite abelian group can be realized by a 3-manifold. It can be checked that λ f is non-degenerate. Moreover, Wall proved in [79] that any symmetric non-degenerate bilinear pairing on a finite abelian group arises in this way. Thus, there exists a symmetric bilinear form f : H × H → Z on a finitely generated free abelian group H such that (G, λ) ≃ (G f , λ f ). Next, by attaching handles of index 2 to a 4-dimensional ball, one can construct a compact oriented 4-manifold W such that H 2 (W ) ≃ H and whose homological intersection pairing on H 2 (W ) corresponds to −f . Then, one can prove that the 3-manifold M := ∂W is such that H 1 (M ) ≃ Coker( f ) and that its linking pairing corresponds to the bilinear form λ f . Remark 6.9. The above proof gives a way to compute the linking pairing λ M of a 3-manifold M , if M comes as the boundary of a 4-manifold W obtained by attaching handles of index 2 to a 4-ball. This amounts to obtain M by surgery in S 3 along an embedded framed link L ⊂ S 3 . A theorem by Lickorish [33] and Wallace [80] asserts that any 3-manifold M has such a "surgery presentation". Remark 6.10. The set of isomorphism classes of non-degenerate symmetric bilinear pairings on finite abelian groups, equipped with the direct sum ⊕, is an abelian monoid. Generators and relations for this monoid are known by works of Wall [79] and Kawauchi-Kojima [29]. Thus, in contrast with cohomology rings, linking pairings are very wellunderstood from an algebraic viewpoint.
In the case of rational homology 3-spheres, the linking pairing is determined by the maximal abelian torsion. Theorem 6.11 (Turaev [74]). Let M be a 3-manifold with β 1 (M ) = 0, and let H := H 1 (M ). Then, we have where we denote Σ H := h∈H h.
where r is the inverse of q mod p. We deduce that λ M (T, rT ) = ε p /p or, equivalently, that λ M (T, T ) = ε p q/p ∈ Q/Z. Exercice 6.13. Compute the linking pairing of L p,q from (6.3), and prove the easy part ("⇒") of Whitehead's homotopy classification 3 : Two lens spaces L p,q and L p ′ ,q ′ have the same oriented homotopy type if, and only if, p = p ′ and q ′ q ∈ Z p is the square of an invertible element.
Exercice 6.14. Using the homotopy classification of lens spaces (Exercice 6.13) and their topological classification (Theorem 2.10), show that: (a) 3-manifolds with the same π 1 do not have necessarily the same homotopy type, (b) 3-manifolds with the same homotopy type are not necessarily homeomorphic.
6.3. The abelian homotopy type of a 3-manifold. We conclude this section by telling "how much" of the homotopy type the linking pairing and the triple-cup product forms do detect. According to Exercice 6.14, the fundamental group of a 3-manifold M is not enough to determine its homotopy type. Another oriented-homotopy invariant is where f : M → K(π 1 (M ), 1) is a map to the Eilenberg-MacLane space of π 1 (M ) which induces an isomorphism at the level of π 1 . By the general theory of Eilenberg-MacLane spaces (see [9] for instance), the map f is unique up to homotopy, so that the homology class µ(M ) is well-defined. The implication "⇒" is easily checked. See [63] for the proof of "⇐". Theorem 6.15 suggests to approximate the oriented homotopy type of a 3-manifold in the following way. It turns out that the abelian oriented homotopy type is characterized by the two invariants that have been presented in this section, namely the cohomology rings and the linking pairing.  M ′ for all r ≥ 0. The implication "⇒" is easily checked from the fact that the linking pairing and the triple-cup product forms are defined by (co)homology operations, which also exist in the category of groups. The converse "⇐" is proved after a careful analysis of the third homology group of a finitely generated abelian group -see [12].

Refinement of the abelian Reidemeister torsion
The maximal abelian torsion τ (M ) of a 3-manifold M has been defined in §5 up to multiplication by some element of H 1 (M ). Following Turaev [73], we explain in this section how this indeterminacy can be removed by choosing an "Euler structure" on M . We also give an algebraic description of Euler structures, and we use them to state some polynomial properties of the maximal abelian torsion. 7.1. Combinatorial Euler structures. Let X be a finite connected CW-complex whose Euler characteristic χ(X) is zero. We denote by E the set of cells of X. Definition 7.1 (Turaev [73]). An Euler chain in X is a singular 1-chain p on X with boundary where c e denotes the center of the cell e. Two Euler chains p and p ′ are homologous if p − p ′ is the boundary of a singular 2-chain. An Euler structure of X is a homology class of Euler chains.
The set of Euler structures on X is denoted by This is an H 1 (X)-affine space. In other words, the abelian group H 1 (X) acts freely and transitively on the set Eul(X): Euler structures are used as "instructions to lift cells", as we shall now see. As before, we denote by X the maximal abelian cover of X, whose group of covering transformations is identified with H 1 (X). The cell decomposition of X lifts to a cell decomposition of the space X. vanishes. Here − → e e ′ ∈ H 1 (X) denotes the covering transformation of X needed to move the cell e to e ′ .
If considered up to equivalence, fundamental families of cells form a set Eul ∧ (X) which, again, is an H 1 (X)-affine space.
There is an H 1 (X)-equivariant bijection between Eul ∧ (X) and Eul(X), defined as follows. Given a fundamental family of cells E, connect by an oriented path the center of each cell e ∈ E to a single point in X: this path goes from e to the single point if dim(σ) is odd, and vice-versa if dim(σ) is even. The image of this 1-chain in X is an Euler chain (shaped like a spider). In the sequel, this identification between Eul ∧ (X) and Eul(X) will be implicit.
Remark 7.3. The notion of Euler structure, which we have defined for CW-complexes, is combinatorial by essence. Nonetheless we shall see in the next subsections that, for 3-manifolds, Euler structures also have a topological existence.
Here, w is a basis of H * (X; R) representing ω and E is a fundamental family of cells representing ξ. Again, we agree that τ ϕ (X, ξ, ω) := 0 when H ϕ * (X) = 0. This refined torsion behaves well with respect to the affine action of H 1 (X) on Eul(X): The maximal abelian torsion introduced in §3.3 can also be refined to This refined maximal abelian torsion is H 1 (X)-equivariant: Assume now that X ′ ≤ X is a cellular subdivision. Then there is a "subdivision operator" where γ e ′ is a path contained in the unique open cell e of X in which e ′ sits, and γ e ′ connects the center of e to the center of e ′ . This operator respects the hierarchy of CW-complexes with respect to subdivisions, in the sense that ∀X ′′ ≤ X ′ ≤ X, Ω X ′ ,X ′′ • Ω X,X ′ = Ω X,X ′′ , and it preserves the refined abelian Reidemeister torsion, in the sense that ∀ξ ∈ Eul(X), τ ϕ (X ′ , Ω X,X ′ (ξ), ω) = τ ϕ (X, ξ, ω) ∈ F.
Thus, using triangulations, Turaev proves that the notions of Euler structure and refined abelian Reidemeister torsion extend to polyhedra [73].
We now come back to 3-manifolds. We deduce that the refined abelian Reidemeister torsion defines a topological invariant of pairs (3-manifold, Euler structure). To justify the topological invariance of the Reidemeister torsion, we proceed as in §5.1, i.e. we use triangulations to present 3-manifolds (Theorem 2.1) and we appeal to the Hauptvermutung (Theorem 2.2). To justify that the set of Euler structures is a topological invariant in dimension 3, we use the fact that any two piecewise-linear homeomorphisms between polyhedra act in the same way on Euler structures if they are homotopic as continuous maps [73] and, again, we appeal to Theorem 2.2. Thus, we obtain the following refinement of Definition 5.6.
where the identification between the singular homology of M and the cellular homology of X is implicit.

Geometric Euler structures.
Let M be a 3-manifold. The Euler structures that we have defined so far for M have been called "combinatorial" because their definition makes reference to cell decompositions of M . We shall now give a more geometric description of the set Eul c (M ). For this, we need to choose a smooth structure on M rather than a cell decomposition. The Poincaré-Hopf theorem shows that non-singular tangent vector fields on M do exist since χ(M ) = 0. We denote by Eul g (M ) the set of geometric Euler structures. Their parameterization is easily obtained from obstruction theory (see, for instance, [59]). Indeed, a non-singular tangent vector field on M is a section of T =0 M , the non-zero tangent bundle of M . This is a fiber bundle with fiber R 3 \ {0} ≃ S 2 , whose first non-trivial homotopy group is π 2 (S 2 ) ≃ Z. Thus, the primary obstruction to find a homotopy between two sections of T =0 M lives 4 in This is the obstruction to construct a homotopy on the 2-skeleton (for any cell decomposition of M ) or, equivalently, on M deprived of an open ball. We deduce that Eul g (M ) is an H 1 (M )-affine space.
Turaev proved that two diffeomorphisms between smooth 3-manifolds act in the same way on geometric Euler structures if they are homotopic as continuous maps. This fact and Theorem 2.3 imply that the set Eul g (M ) is a topological invariant of M . In fact, we have the following. Theorem 7.6 (Turaev [73]). For any smooth 3-manifold M , there is a canonical and H 1 (M )-equivariant bijection between Eul c (M ) and Eul g (M ).
By virtue of this result, we shall now identify the sets Eul c (M ) and Eul g (M ), which we simply denote by Eul(M ).
Sketch of the proof. Turaev defines in [73] a map Eul c (M ) → Eul g (M ) by working with a smooth triangulation of M , and he shows it to be H 1 (M )-equivariant. Here, we shall describe Turaev's map following the Morse-theoretic approach of Hutchings & Lee [27].
Thus, we consider a Morse function f : M → R as well as a riemannian metric on M . If this metric is appropriately choosen with respect to f (i.e. if f satisfies the "Smale condition"), then f defines a cell decomposition X f of M (namely the "Thom-Smale cell decomposition"). The i-dimensional cells of X f are the descending manifolds from index i critical points of f . Let ξ ∈ Eul c (M ) = Eul(X f ) be represented by a Euler chain p, which (without loss of generality) we assume to be contained in a 3-ball B p ⊂ M . By definition of Eul(X f ), we have Let also ∇f be the gradient field of f with respect to the riemannian metric. It is non-singular except at each critical point c of f , where its index is (−1) index of c . Thus, all critical points of ∇f are contained in B p and since the 0-chain (7.1) augments to χ(M ) = 0, there is a non-singular vector field v p on M which coincides with ∇f outside B p . Then, we associate to ξ = [p] the geometric Euler structure represented by v p .
Remark 7.7. Let X be a cell decomposition of M which comes from a Heegaard splitting (as explained during the proof of Lemma 5.3). We can find a Morse function f : M → R (and a riemannian metric on M with the Smale condition satisfied) such that the Thom-Smale cell decomposition X f coincides with X. Then, formula (5.1) can be refined to take into account a geometric Euler structure obtained from the desingularization of ∇f in a ball B that contains all the critical points of f . We refer to [35].
Given a smooth 3-manifold M , one can wonder how the group Diff + (M ) of orientationpreserving self-diffeomorphisms of M acts on the set Eul g (M ) and, in particular, one can ask for the number of orbits. For lens spaces, the Reidemeister-Turaev torsion gives the answer. Theorem 7.8 (Turaev [73]). The number of orbits for the action of Diff + (L p,q ) on Eul g (L p,q ) is [p/2] + 1, if q 2 = 1 or q = ±1, p/2 − b(p, q)/4 + c(p, q)/2, if q 2 = 1 and q = ±1. Here, for x ∈ Q, [x] denotes the greatest integer less or equal than x, b(p, q) is the number of i ∈ Z p for which i, q + 1 − i and qi are pairwise different, and c(p, q) is the number of i ∈ Z p such that i = q + 1 − i = qi.
Finally, the value k = r is realizable if and only k = −r is realizable. Assume that k = r is realizable. Then, (7.3) implies that r 2 = ±1. Working with any p-th root of unity and passing to the group ring C[H 1 (L p,q )] as we did in the proof of Theorem 2.10 ( §5.4), we see that the value r 2 = −1 is impossible for p > 2. So, we must have r 2 = 1 or, equivalently, q 2 = 1 and, indeed, there exists in this situation an orientationpreserving self-diffeomorphism h of L p,q for which k = r (h exchanges the two solid tori in the Heegaard splitting). Then, (7.2) implies that r · i(h(ξ)) = i(ξ) or, equivalently, i(h(ξ)) = q · i(ξ).
The conclusion easily follows from the above analysis.
The set of quadratic functions of polar form L M is denoted by Quad(L M ). A quadratic function q is said to be homogeneous if we have q(nx) = n 2 q(x) for all x ∈ H 2 (M ; Q/Z) and n ∈ Z. It is easily checked that q is homogeneous if, and only if, the map d q : Theorem 7.11 (See [16]). For any 3-manifold M , there is a canonical injection Moreover, the homogeneity defect of φ M,ξ is given by Sketch of the proof of Theorem 7.11. Let us define the quadratic function φ M,ξ in the general case. Our first observation is that, for every x ∈ H 2 (M ; Q/Z), we must have 2φ M,ξ (x) = L M (x, x) + d φ M,ξ (x). Thus, we shall give a formula of the form where c x ∈ Q/Z (which should be selected correlatively to l x ) satisfies 2c x = c(ξ), x . To obtain such a formula, represent the homology class x in the form where n ≥ 1 is an integer, S ⊂ M is an oriented compact immersed surface whose boundary goes n times around an oriented knot K ⊂ M . We can find a non-singular vector field v which represents ξ and is transverse to K. Let V be a sufficiently small regular neighborhood of K in M , and let K v be the parallel of K sitting on ∂V and obtained by pushing K along the trajectories of v. By an isotopy of S (fixed on the boundary), we can ensure that S is transverse to K v . Finally, let w be a non-singular vector field on V which is tangent to K and is linearly independent with v, and let denote the obstruction to extend w| ∂V to a non-singular vector field on M \ int(V ) linearly independent with v| M \int(V ) . With these notations, we set Recall from Remark 6.9 that there is a formula to compute λ M when M is presented as the boundary of a 4-manifold obtained by attaching handles of index 2 to a 4-ball. This can be refined to a 4-dimensional formula for φ M,ξ , which can be more convenient in practice than the 3-dimensional formula (7.5). In particular, it can be used to prove that the map φ M is affine over the homomorphism It can also be checked that Using the fact that M (like any 3-manifold) has a parallelization, one can see that c(ξ) is even. Thus, the restriction of φ M,ξ to H 2 (M ) ⊗ Q/Z comes from Hom(H 2 (M ), Z). The converse is deduced from the fact that the map φ M is affine.
We now assume that M is a rational homology 3-sphere, i.e.
is a quadratic function with polar form λ M . Nicolaescu proves this in [46] (see also [47]) using the relation between the Reidemeister-Turaev torsion and the Seiberg-Witten invariant. The proof given in [15] is purely topological: it uses surgery presentations of 3-manifolds and the 4-dimensional formula for φ M,ξ . 7.5. Some polynomial properties of the Reidemeister-Turaev torsion. Let M be the set of 3-manifolds, up to orientation-preserving homeomorphisms. Recall from Definition 6.2 that the T 3 -surgery is a way of modifying 3-manifolds, which is modelled on the genus 3 Heegaard splitting Definition 7.14. Let c : M → C be an invariant of 3-manifolds with values in an abelian group C. We say that c is a finite-type invariant of degree at most d if, for any 3-manifold M ∈ M and for any embeddings a 0 : A ֒→ M, . . . , a d : A ֒→ M whose images are pairwise disjoint, we have Here, M a P denotes the 3-manifold obtained from M by simultaneous T 3 -surgeries along those embeddings a i : A ֒→ M for which i ∈ P .
The notion of finite-type invariant has been introduced for homology 3-spheres (in an equivalent way) by Ohtsuki [48], and it is similar to the notion of Vassiliev invariant for knots [5]: the T 3 -surgery plays for 3-manifolds the role that the crossing-change move plays for knots. This notion is motivated by the study of quantum invariants. The reader may consult the survey [32] for an introduction to the subject of finite-type invariants.
By analogy with a well-known characterization of polynomial functions (recalled in Exercice 7.16), one can think of finite-type invariants as those maps defined on M which behave "polynomially" with respect to T 3 -surgeries. We shall now see that the Reidemeister-Turaev torsion do have such polynomial properties. But, since the Reidemeister-Turaev torsion depends both on the homology and on the Euler structures of 3-manifolds, we shall first refine the notion of finite-type invariant. Thus, we fix a finitely generated abelian group G and we consider triples of the form where M is a 3-manifold, ξ ∈ Eul(M ) and ψ : G → H 1 (M ) is an isomorphism. Of course, two such triples (M 1 , ξ 1 , ψ 1 ) and (M 2 , ξ 2 , ψ 2 ) are considered to be equivalent if there is an orientation-preserving homeomorphism f : M 1 → M 2 which carries ξ 1 to ξ 2 and satisfies f * • ψ 1 = ψ 2 . We denote by ME(G) the set of equivalence classes of such triples. Then, the Reidemeister-Turaev torsion can be seen as a map We have seen during the proof of Theorem 6.1 that a T 3 -surgery M ; M a induces a canonical isomorphism in homology: Similarly, the move M ; M a induces a canonical correspondence between Euler structures, which is affine over Φ a . We refer to [16,35] for the definition of the map Ω a by cutting and pasting vector fields. Thus, the notion of T 3 -surgery exists also for 3-manifolds with Euler structure and parameterized homology: where ξ a := Ω a (ξ) and ψ a := Φ a • ψ.
Therefore, we get a notion of finite-type invariant for 3-manifolds with Euler structure and parameterized homology: the set M is replaced by ME(G) in Definition 7.14. The proof uses a refinement of Lemma 5.3 which takes into account Euler structures (see Remark 7.7). Theorem 7.15 generalizes the fact that the coefficients of the Alexander polynomial of knots in S 3 are Vassiliev invariants [5]. See [45] in the case of links.
Exercice 7.16. Prove that a function c : Q n → Q is polynomial of degree at most d if, and only if, we have for any m ∈ Q n and any a 0 , . . . , a d ∈ Q n P ⊂{0,...,d} where a P is the sum of those vectors a i for which i ∈ P .

Abelian Reidemeister torsion and the Thurston norm
To conclude, we give a topological application of the abelian Reidemeister torsions: we shall see that they can be used to define a lower bound for the "Thurston norm." Here, the minimum runs over all closed oriented surfaces S ⊂ M which are Poincaré dual to s and which may be disconnected (S = S 1 ⊔ · · · ⊔ S k ), and we denote Example 8.1. Let T f be the mapping torus of an orientation-preserving homeomorphism f : Σ g → Σ g (see §5.5) with g ≥ 1. We consider the cohomology class that is Poincaré dual to the fiber Σ g ⊂ T f . The norm of this class is The inequality "≤" is obvious since the Euler characteristic of the fiber is 2−2g ≤ 0. We shall prove the inequality "≥" with a little bit of 3-dimensional topology. Let S ⊂ T f be a closed oriented surface which minimizes s f T . It can be checked -and this is a general fact (Exercice 8.5) -that the surface S must be incompressible. Consider the cover of T f defined by s f ∈ H 1 (T f ; Z) ≃ Hom(π 1 (T f ), Z), which is Σ g × R with the obvious projection. Let S ′ ⊂ T f be the 2-complex obtained by connecting each component of S by a path to a single point. The inclusion S ′ ֒→ T f lifts to this cover, and by composing with the cartesian projection Σ g × R → Σ g , we get a map h : S ′ → Σ g . The inclusion S ′ ֒→ T f induces an injection at the level of π 1 (−) since S is incompressible. (This is an application of the loop theorem, see [28] for instance.) It follows that h * : π 1 (S ′ ) → π 1 (Σ g ) is injective and, h being of degree 1, it is also surjective. Therefore, π 1 (S ′ ) is isomorphic to π 1 (Σ g ), which implies that S was connected and its genus is equal to g. Thus, χ − (S) = 2g − 2 and we are done.
By a "norm" on a real vector space V , we mean a map − : V → R which satisfies the triangle inequality, i.e.
and which is homogeneous, i.e.
We do not assume it to be non-degenerate: we may have v = 0 for a vector v = 0. Proof. We follow Thurston [65] and start with the following fact.
Claim. Any element s ∈ H 2 (M ; Z) can be realized by a closed oriented surface S ⊂ M . Moreover, if s = k · s ′ is divisible by k ∈ N, then the surface S is the union of k sub-surfaces, each realizing s ′ . Indeed (assuming that M is smooth), s ∈ H 2 (M ) ≃ Hom(H 1 (M ), Z) can be realized by a smooth map f : M → S 1 = K(Z, 1). Then, for any regular value y of f , the surface S := f −1 (y) with the appropriate orientation is such that [S] = s. If now s = k · s ′ , we can find a smooth map f ′ : M → S 1 which realizes s ′ and satisfies π k • f ′ = f , where π k : S 1 → S 1 is the k-fold cover. Let y ′ 1 , . . . , y ′ k be the pre-images of y by π k . They are regular values of f ′ so that S is the union of the subsurfaces f ′−1 (y ′ 1 ), . . . , f ′−1 (y ′ k ). This fact implies that the map − T : H 1 (M ; Z) → Z defined by (8.1) is homogeneous. Let us check that it also satisfies the triangle inequality. Let s, s ′ ∈ H 2 (M ; Z) and let S, S ′ ⊂ M be closed oriented surfaces which are Poincaré dual to s, s ′ respectively and satisfy χ − (S) = s T , χ − (S ′ ) = s ′ T . We put S and S ′ in transverse positions, and we consider their intersection which consists of circles.
Claim. We can assume that none of the components of S ∩ S ′ bounds a disk in S or in S ′ .
Otherwise, one of these circles bounds a disk D, say in S. We can assume that S ′ does not meet the interior of D. (If this is  Since S ′ minimizes s ′ T , the component of S ′ which has been "decompressed" was either a torus (in which case, it has been transformed into a sphere) or was a sphere (in which case, it has been transformed into two spheres). Thus, we have χ − (S ′ D ) = χ − (S ′ ) and we can replace S ′ by S ′ D . This proves the claim by induction.
Next, we consider the closed oriented surface S ′′ obtained from the desingularization of S ∪ S ′ . Here, by "desingularization" we mean to replace S 1 ×(the graph of xy = 0) by S 1 ×(the graph of xy = 1) in a way compatible with the orientations of S and S ′ . The class [S ′′ ] = [S ∪ S ′ ] ∈ H 2 (M ) is Poincaré dual to s + s ′ ∈ H 1 (M ), and we have χ(S ′′ ) = χ(S) + χ(S ′ ). By the above claim, a sphere component of S (respectively S ′ ) can not meet S ′ (respectively S) and, so, remains unchanged in S ′′ . The claim also shows that, conversely, any sphere component of S ′′ comes from a sphere component of S or of S ′ . Thus, we have χ − (S ′′ ) = χ − (S) + χ − (S ′ ) and the triangle inequality is proved: . Again, P τ (M ) is compact and symmetric in 0. It is easily checked that P τ (M ) does not depend on the choice of ξ (and neither on the orientation of M ).
Proof. We mainly follow McMullen's [37] and Turaev's [78] arguments. Let us for instance assume that β 1 (M ) ≥ 2. We choose a representative of the Alexander polynomial of M and we assume that it is non-trivial. (If ∆(M ) = 0, then s A = 0 and there is nothing to prove.) An element s ∈ H 1 (M ; R) is said to be "generic" if we have An is a principal ideal domain, we can compute the order of H s 1 (M ) from a decomposition into cyclic modules (see Example 4.2). We deduce that dim Q H s 1 (M ) = span ord (H s 1 (M )) where the span of a Laurent polynomial i∈Z q i · t i ∈ Q[t ± ] is the difference between max{i ∈ Z : q i = 0} and min{i ∈ Z : q i = 0}. It then follows from (8.4) and the genericity of s that dim Q H s 1 (M ) = span s(∆(M )) + 2 = s A + 2. This proves the previous claim.
Another claim. We can find a connected closed oriented surface S Poincaré dual to s, such that χ − (S) = s T and β 1 (S) ≥ dim Q H s 1 (M ). This will be enough to conclude. Indeed, if that surface S was a sphere, then the Q-vector space H s 1 (M ) would be trivial, which would contradict the first claim. So, To prove the second claim, we consider a closed oriented surface S ⊂ M , Poincaré dual to s, which satisfies χ − (S) = s T and with minimal β 0 (S).
We shall first prove that β 0 (S) = 1. We . Let C w → C and M s → M be the infinite cyclic covers defined by those homomorphisms. The map π : M → C lifts to π s : M s → C w , and the inclusion C → M does to. So, π s induces a surjection at the level of homology, and we deduce that dim Q H s 1 (M ) = rank H 1 (M s ) ≥ rank H 1 (C w ). If C had more than one loop, then β 1 (C w ) would be infinite, which would contradict the first claim. Thus, β 1 (C) = 1 and C is a circle to which trees may be grafted. Using the minimality of β 0 (S), one easily sees that C does not have univalent vertices, nor bivalent vertices with two incident edges pointing in opposite directions. Therefore, C is an oriented circle, and because s = w • π * is assumed to be primitive, it can have only one edge. We conclude that S is connected.
The surface S being connected and s being not trivial, M \ S must be connected. Thus, the infinite cyclic cover M s is an infinite chain of copies of M \ N(S) which are glued "top to bottom": M s = · · · ∪ (M \ N(S)) −1 ∪ (M \ N(S)) 0 ∪ (M \ N(S)) 1 ∪ · · · Since H 1 (M s ; Q) is finite-dimensional, an application of the Mayer-Vietoris theorem shows that it must be generated by the image of one copy of H 1 (S; Q). Thus, we have dim Q H 1 (M s ; Q) ≤ β 1 (S) and the second claim is proved. Example 8. 9. Let T f be the mapping torus of an orientation-preserving homeomorphism f : Σ g → Σ g (as defined in §5.5 with g ≥ 1). Let s f ∈ H 1 (T f ; Z) be the class Poincaré dual to the fiber Σ g ⊂ T f . The above proof shows that The value of s f A is easily deduced from Proposition 5.10: which agrees with the discussion of Example 8.1.
Remark 8.10. Theorem 8.8 has an analogue for 3-manifolds with boundary [37,78]. In this sense, it generalizes a classical fact about the Alexander polynomial ∆(K) of a knot K ⊂ S 3 : twice the genus of K is bounded below by the span of ∆(K) -see [55] for instance. Remark 8.11. Theorem 8.8 has known several recent developments. Friedl and Kim generalize inequality (8.2) to twisted Reidemeister torsions defined by group homomorphisms π 1 (M ) → GL(C; d) [22]. Cochran [11] and Harvey [25] prove generalizations of Theorem 8.8 for non-commutative analogues of the Alexander polynomial. (Their results are further generalized in [77] and [21].) Another spectacular development is the work by Ozsváth and Szabó, who proved that the Heegaard Floer homology determines the Thurston norm [49,51]. This is an analogue of a property proved in 1997 by Kronheimer and Mrowka for the Seiberg-Witten monopole homology [30].
Appendix A. Fox's free differential calculus Fox introduced in the late 40's a kind of differential calculus for free groups [17]. This calculus is an efficient tool in combinatorial group theory as well as in low-dimensional topology. We have used it in these notes for computations related to the Alexander polynomial and to the Reidemeister torsion. We introduce in this appendix the basics of Fox's calculus. If follows easily from this definition that d(1)=0 and that ∀g ∈ G ⊂ Z[G], d(g −1 ) = −g −1 · d(g). is easily checked to be a derivative. We now restrict ourselves to G := F(x), the group freely generated by the set x = {x 1 , . . . , x n }. An element a of the group ring Z[F(x)] can be regarded as a Laurent polynomial with integer coefficients in n indeterminates x 1 , . . . , x n that do not commute. In order to emphasize this interpretation, we will sometimes denote a ∈ Z[F(x)] by a(x) and the augmentation ε(a) ∈ Z by a(1).
By linear extension, we get a map ∂ ∂x i : Z[F(x)] → Z[F(x)] which is easily checked to satisfy (A.1). Thus, ∂ ∂x i is a derivative that satisfies ∂x k ∂x i = δ i,k . Since the ring Z[F(x)] is generated by x 1 , . . . , x n , there is at most one derivative d such that d(x k ) = h k for all k = 1, . . . , n. It is easily checked that, if d 1 and d 2 are two derivatives of Z[F(x)], then d 1 · a 1 + d 2 · a 2 is also a derivative for all a 1 , a 2 ∈ Z[F(x)]. Therefore, the right-hand side of formula (A.3) defines a derivative. Since this derivative takes the values h 1 , . . . , h n on x 1 , . . . , x n respectively, the conclusion follows. This is the fundamental formula of Fox's free differential calculus, which can be regarded as a kind of order 1 Taylor formula with remainder. Consider now a second free group, say the group F(y) freely generated by the set y := {y 1 , . . . , y p }. For all b 1 , . . . , b n ∈ F(y), there is a unique group homomorphism F(x) → F(y) defined by x i → b i . The image of an a ∈ Z[F(x)] by this homomorphism is denoted by a(b 1 , . . . , b n ). This is compatible with our convention to denote a ∈ Z[F(x)] also by a(x). Next, we apply the fundamental formula to each b j ∈ F(y) to get a(b 1 , . . . , b n ) = 1 + j=1...,n k=1,...,p ∂a ∂x j (b 1 , . . . , b n ) · ∂b j ∂y k · (y k − 1) or, equivalently, a(b 1 , . . . , b n ) = 1 + p k=1   n j=1 ∂a ∂x j (b 1 , . . . , b n ) · ∂b j ∂y k   · (y k − 1).
By comparing this identity with the fundamental formula for a(b 1 , . . . , b n ) ∈ Z[F(y)], we exactly obtain identity (A.5).
A.2. The topology behind the free derivatives. The theory of covering spaces gives a topological interpretation to free derivatives. We consider the bouquet X n of n circles x 1 x 2 x n . . .

⋆
Note that π 1 (X n , ⋆) is the free group F(x) on x = {x 1 , . . . , x n }. Let also X n be the infinite oriented graph with vertices indexed by F(x) with edges indexed by n copies of the set F(x) and with incidence map i = (i 0 , i 1 ) : E −→ V × V defined by ∀f ∈ F(x), ∀j ∈ {1, . . . , n}, i 0 (f · x j ) = f · ⋆ and i 1 (f · x j ) = (f x j ) · ⋆.
There is a canonical map π : X n → X n which sends each vertex f · ⋆ to ⋆ and the interior of each edge f · x j homeomorphically onto the interior of x j . It is easily checked that π is the universal covering map of X n . The canonical action of F(x) = π 1 (X n , ⋆) ≃ Aut(π) on X n is compatible with our notation for the vertices and edges of X n . See Figure A.1.

⋆
x 1 x 2 The infinite tree X 2 , where x 1 acts by "horizontal translation" to the right and x 2 acts by "vertical translation" to the top.
For all f ∈ F(x) = π 1 (X n , ⋆), let γ be a closed path ⋆ ; ⋆ representing f and let γ be the unique lift of γ starting at ⋆. This is a path ⋆ ; f · ⋆ which defines an element of H 1 ( X n , π −1 (⋆)). Since the homotopy class of the path γ is determined by that of γ, this element only depends on f and we denote it by f ∈ H 1 X n , π −1 (⋆) .
The left action of F(x) on X n makes H 1 ( X n , π −1 (⋆)) a Z[F(x)]-module.