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Cohomology rings, Rochlin function, linking pairing and the Goussarov--Habiro theory of three--manifolds

We prove that two closed oriented 3-manifolds have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function) if, and only if, they belong to the same class of a certain surgery equivalence relation introduced by Goussarov and Habiro.


Introduction
Goussarov and Habiro have developed a theory of finite type invariants for compact oriented 3-manifolds [6,7,4]. Their theory is based on a new kind of 3-dimensional topological calculus, called calculus of claspers. In strong connection with their finite type invariants, some equivalence relations have been studied by Goussarov and Habiro. For any integer k ≥ 1, the Y k -equivalence is the equivalence relation among compact oriented 3-manifolds generated by positive diffeomorphisms and surgeries along graph claspers of degree k . The reader will find the precise definition of the Y k -equivalence in Section 2 and, waiting for this, will be enlightened by the following characterization due to Habiro [7]. Two manifolds M and M ′ are Y k -equivalent if, and only if, there exists a compact oriented connected surface Σ in M and an element h of the k -th lower central series subgroup of the Torelli group of Σ such that M ′ is diffeomorphic to the manifold obtained from M by cutting it along Σ and regluing it using h. In particular, we see that the Y k -equivalence becomes finer and finer as k increases.
Thus, the problem of characterizing the Y k -equivalence relation in terms of invariants of the manifolds naturally arises. In the case k = 1, this problem has been solved for manifolds without boundary. Indeed, a result of Matveev [14], anterior to the Goussarov-Habiro theory, can be re-stated as follows: two closed oriented 3-manifolds are Y 1 -equivalent if and only if they have isomorphic pairs (homology, linking pairing). That problem has also been given a solution in the case k = 2 for a certain class of manifolds with boundary [13].
In some situations, spin structures and, more recently, complex spin structures have proved to be of use to low-dimensional topologists. It happens that the Goussarov-Habiro theory can be refined to the settings where the compact oriented 3-manifolds are equipped with those additional structures. So, the problem of characterizing the Y k -equivalence makes sense in those refined contexts as well. In the case k = 1 and for manifolds without boundary, Matveev's theorem has been extended to the realm of spin manifolds and complex spin manifolds in [12] and [3] respectively.
In this paper, we deal with the Y 2 -equivalence for manifolds without boundary. It is known that surgery along a graph clasper of degree 2 preserves triple cup products, as well as Rochlin invariant. Also, according to Habiro [7], two homology 3-spheres are Y 2 -equivalent if and only if they have identical Rochlin invariant. We prove that, in general, two closed oriented 3-manifolds are Y 2equivalent if and only if they have isomorphic quintuplets (homology, space of spin structures, linking pairing, cohomology rings, Rochlin function). We also consider the spin case and, with less emphasis, the complex spin case. In order to give a precise statement of the results, let us fix some notation for those classical invariants.
Let us consider a closed oriented 3-manifold M . A spin structure on M is a trivialization of its oriented tangent bundle, up to homotopy on M deprived of one point. We denote by Spin(M ) the set of spin structures of M which, by obstruction theory, is an affine space over the for any x 1 , x 2 ∈ Tors H 1 (M ; Z). Lannes, Latour, Morgan and Sullivan [10,16] have defined a map which assigns to any spin structure σ a linking quadratic function q M,σ . For any integer n ≥ 0, will denote the skew-symmetric trilinear map given by the evaluation of triple cup products with coefficients in Z n on the fundamental class of M . One can verify, using Poincaré duality, that the cohomology rings of M (with coefficients in Z n , n ≥ 0) are determined by those triple cup product forms and the group H 1 (M ; Z). Finally, if M ′ is another closed oriented 3-manifold and if ψ : (a) For any x 1 , x 2 ∈ Tors H 1 (M ; Z), we have (b) For any integer n ≥ 0 and for any y ′ (d) The bijection Ψ is compatible with the isomorphism ψ in the sense that it is affine over ψ (2) and the following diagram is commutative: For any integer n ≥ 0 and for any y ′ A similar result holds for manifolds equipped with a complex spin structure (see Section 5, Theorem 5.3). Let us now discuss the relationship between Theorem 1.1 and some previously known results.
Let Σ g,1 be the compact connected oriented surface of genus g with one boundary component. Homology cylinders over Σ g,1 are homology cobordisms with an extra homological triviality condition [7,6]. Homology cylinders form a monoid which contains the Torelli group of Σ g,1 as a submonoid. Moreover, the Johnson homomorphisms and the Birman-Craggs homomorphisms extend naturally to this monoid. An analog of Johnson's result on the Abelianization of the Torelli group of Σ g,1 [8] has been proved by Meilhan and the author for homology cylinders [13]: two homology cylinders over Σ g,1 are Y 2 -equivalent if and only if they are not distinguished by the first Johnson homomorphism nor the Birman-Craggs homomorphisms. On the other hand, there is a canonical construction producing from any homology cylinder h over Σ g,1 (for instance, an element h of the Torelli group) a closed oriented 3-manifold with first homology group isomorphic to Z 2g . More precisely, one glues to the mapping torus of h, which is a 3-manifold with boundary ∂Σ g,1 × S 1 , the solid torus ∂Σ g,1 × D 2 along the boundary. Since Johnson [9], it is known (at least for elements of the Torelli group) that the first Johnson homomorphism and the many Birman-Craggs homomorphisms correspond, through that construction, to the triple cup products form and the Rochlin function respectively. This results in a connection between Theorem 1.1 and that characterization of the Y 2 -equivalence for homology cylinders. As a matter of fact, some constructions and arguments from [13] will be re-used here.
Also, it is worth comparing Theorem 1.1 to a result of Cochran, Gerges and Orr. They have studied in [1] another equivalence relation among closed oriented 3-manifolds, namely the 2-surgery equivalence. A 2-surgery, defined as the surgery along a null-homologous knot with framing number ±1, is the elementary move of the Cochran-Melvin theory of finite type invariants [2]. While the Y 2 -equivalence coincides with the relation "have isomorphic quintuplets (homology, space of spin structures, linking form, cohomology rings, Rochlin function)" between closed oriented 3-manifolds, the 2-surgery equivalence is the relation "have isomorphic triplets (homology, linking form, cohomology rings)". Indeed, it can be verified that the Y 2 -equivalence is finer than the 2-surgery equivalence, but this will not be used here.
Finally, we mention a result of Turaev, to which Theorem 1.1 is complementary. Consider quintuplets H, S, λ, u (n) n≥0 , R formed by a finitely generated Abelian group H , an affine space S over the Z 2 -vector space Hom(H, Z 2 ), a nondegenerate symmetric bilinear pairing λ : Tors H × Tors H → Q/Z, skew-symmetric trilinear forms u (n) : Hom(H; Z n ) 3 → Z n and a function R : S → Z 16 . Turaev has found in [18] necessary and sufficient algebraic conditions on such a quintuplet to be realized, up to isomorphisms, as the quintuplet The paper is organized as follows. In Section 2, we briefly review calculus of claspers and its refinement to spin manifolds. Next, in Section 3, we recall or precise how the classical invariants involved in Theorem 1.1 behave under the surgery along a graph clasper. In Section 4, we fix a closed spin 3-manifold (M, σ) and associate to it a certain set of Y 2 -equivalence classes. We define a surgery map from a certain space of abstract graphs to this quotient set, and we prove this map to be bijective. In Section 5, we derive from that bijectivity Theorem 1.2 and, next, Theorem 1.1. We also give the analogous result for closed oriented 3-manifolds equipped with a complex spin structure. Last section is an appendix containing a few algebraic lemmas needed to obtain the above results.
In the sequel, unless otherwise specified, all manifolds are assumed to be 3dimensional smooth compact and oriented, and the diffeomorphisms are supposed to preserve the orientations.

Review of calculus of claspers
We begin by recalling basic concepts from calculus of claspers. The reader is refered to [7,4] for details and complete expositions, or to the monograph [17].

A flash review of calculus of claspers
Graph claspers can be defined as follows. We start with a finite trivalent graph G decomposed as G 1 ∪ G 2 , where G 1 is a unitrivalent subgraph of G and G 2 is a union of looped edges of G. We give G a thickening 1 with the property to be trivial on each looped edge of G 2 , and we consider an embedding G of this thickened graph into the interior of a manifold M . Then, G is said to be a graph clasper in the manifold M . The leaves of G are the framed knots in M corresponding to the thickening of G 2 . The degree of G is the internal degree 2 of G 1 . We assume this degree to be at least 1.
Example 2.1 Using the above notations, if G 1 is Y -shaped (respectively H -shaped), then the graph clasper G is called a Y -graph (respectively a Hgraph). Actually, Y -graphs play a specific role in the theory. A Y -graph and a H -graph have been depicted on Figure 2.1 before embedding in a manifold M . On these diagrams, the bold vertices correspond to the trivalent vertices of G 1 and, as everywhere else in the sequel, the graphs are thickened by the "blackboard framing convention".  2 The internal degree of a unitrivalent graph G is the number of its trivalent vertices if it is connected, or is the minimum of the internal degrees of its connected components otherwise.
A graph clasper carries surgery instructions to modify the manifold where it is embedded. Surgery along a graph clasper is defined in the following way.
First of all, we consider the particular case when G is a Y -graph in a manifold M . Let N(G) be the regular neighborhood of G in M , which is a genus 3 handlebody. The manifold obtained from M by surgery along G is denoted by and defined as   The Y k -equivalence, mentioned in the introduction, is defined to be the equivalence relation among manifolds generated by Y k -moves and diffeomorphisms.
Example 2.2 It follows from the definitions that the Y 1 -equivalence and the Y 2 -equivalence are generated by surgeries along Y -graphs and H -graphs respectively, and diffeomorphisms.
Finally, let us give an idea of what "calculus of claspers" is. Let G 1 and G 2 be graph claspers in a manifold M . They are said to be equivalent, which we denote by G 1 ∼ G 2 , if there exists an embedded handlebody H in M whose interior contains both G 1 and G 2 , and if there exists a diffeomorphismf : Thanks to the canonical identifications The calculus of claspers is a corpus of calculi rules which state equivalence of claspers. Thus, the calculus of claspers allows one to prove diffeomorphisms between manifolds.
Example 2.3 Figure 2.4 illustrates one of the Goussarov-Habiro moves, which says that any H -graph is equivalent in its regular neighborhood (a genus 4 handlebody) to a Y -graph with a null-homologous leaf. In particular, the Y k+1 -equivalence relation is finer than the Y k -equivalence.

Calculus of claspers for spin manifolds
The most important property of a Y k -move M ❀ M G is certainly to preserve homology. There is a canonical isomorphism Φ G : , whose existence follows from the fact that the surgery along a Y -graph can be realized by cutting a genus three handlebody and gluing it back using a diffeomorphism of its boundary which acts trivially in homology [14]. If H is an embedded handlebody in M whose interior contains G, Φ G is the only map making the diagram commute, where the oblique arrows are induced by inclusions and are surjective.
There exists a canonical bijection Θ G : Spin(M ) → Spin(M G ), which we shall denote by σ → σ G . This map has been defined in [12] for G a Y -graph, the general case can be reduced to this special case by definition of a Y k -move. If H is a handlebody as above, Θ G is the only map making the diagram commute, where the oblique arrows are induced by inclusions and are injective. Let us observe, from diagrams (2.1) and (2.2), that the bijection Θ G is affine over the inverse of the isomorphism Φ G If G is a degree k graph clasper in a manifold M and if σ is a spin structure on M , the spin manifold (M G , σ G ) is said to be obtained from the spin manifold (M, σ) by surgery along G, or, by a Y k -move. The Y k -equivalence among spin manifolds is the equivalence relation generated by such Y k -moves and spin diffeomorphisms. Next lemma says that the calculus of claspers extends to the context of manifolds equipped with a spin structure.
Lemma 2.4 Let (M, σ) be a spin manifold. If G 1 and G 2 are equivalent graph claspers in M , then the spin manifolds (M G 1 , σ G 1 ) and (M G 2 , σ G 2 ) are spin diffeomorphic.
Proof Let H be an embedded handlebody in M whose interior contains G 1 and G 2 , and letf : H G 1 → H G 2 be a diffeomorphism which restricts to the identity on the boundaries. Let f : Example 2.5 As in Example 2.3, we observe that the Y k+1 -equivalence is finer than the Y k -equivalence in the context of spin manifolds too.

Some invariants and surgery along a graph clasper
From now on, we restrict ourselves to closed manifolds and, in this section, we describe how their invariants that are involved in Theorem 1.1 behave under the surgery along a graph clasper.

Linking pairing and surgery along a graph clasper
A theorem of Matveev says that two closed manifolds are Y 1 -equivalent if and only if they have isomorphic pairs (homology, linking pairing) [14]. In the spin case, we have the following refinement of Matveev's theorem.
More precisely, any isomorphism ψ : can be realized by a sequence of Y 1 -moves and spin diffeomorphisms from Let us comment that characterization. For any graph clasper G in a closed manifold M , we have that This implies the necessary condition in Theorem 3.1. Reciprocally, given an isomorphism ψ : H 1 (M ; Z) → H 1 (M ′ ; Z) satisfying (3.1), there exists a sequence of Y 1 -moves and spin diffeomorphisms This is what the second statement 3 of Theorem 3.1 means.

Triple cup products and surgery along a graph clasper
In contrast with the linking quadratic functions, the cohomology rings can be modified by the surgery along a graph clasper.
Let M be a closed manifold. For any integer n ≥ 0, we consider the bilinear A Y -graph G in M defines an element of Λ 3 H 1 (M ; Z) in the following way.
Order the leaves of G and denote them by L 1 , L 2 , L 3 accordingly: L 1 < L 2 < L 3 . This ordering induces an orientation for each leaf, as shown in Figure 3.1. Let [L i ] be the homology class of the i-th oriented leaf. Clearly, Lemma 3.2 Let G be a Y -graph in a closed manifold M whose leaves are ordered, denoted by L 1 , L 2 , L 3 accordingly and oriented as shown in Figure 3.1. Then, for any integer n ≥ 0 and 3 In fact, this realization property does not appear explicitely in [12] but it can be verified from the proof of [12, Theorem 1]. One of the key ingredients, there, is an algebraic result, due to Durfee and Wall, according to which two even symmetric bilinear lattices A and B produce isomorphic quadratic functions q A and q B if and only if they are stably equivalent. The point is that, as can be verified from [19,Theorem], any given isomorphism between q A and q B can be lifted to a stable equivalence between A and B . The inclusions induce isomorphisms between H 1 (N ; Z n ) and H 1 (M ; Z n ), as well as between H 1 (N ; Z n ) and We conclude from (3.4) that Remark 3.3 Lemma 3.2 essentially appears in [18,Section 4.3] where "Borromean replacements" are performed on surgery presentations of the manifolds in S 3 . Indeed, this operation has been used by Turaev to prove his result, mentioned in the introduction, on realization of skew-symmetric trilinear forms as triple cup products forms of manifolds. Proof

Rochlin invariant and surgery along a graph clasper
As the cohomology rings, the Rochlin invariant can be changed by the surgery along a graph clasper.
Let M be a closed manifold and let FM be its bundle of oriented frames, which is a GL + (3; R)-principal bundle: (For equivalences between the various definitions of a spin structure, the reader is refered to [15].) An element t K ∈ H 1 (E(FM ); Z) can be associated to any oriented framed knot K ⊂ M in the following way: add to K an extra (+1)-twist and, next, consider the homology class of its lift in FM . Some elementary properties of the map K → t K are listed in [13, Lemma 2.7].
Lemma 3.5 (See [13]) Let G be a Y -graph in a closed manifold M whose leaves are ordered, denoted by L 1 , L 2 , L 3 and oriented. Then, for any spin structure σ on M , we have that

6)
where 8· : Z 2 → Z 16 denotes the usual monomorphism of groups. Proof Again, we can suppose that H is connected and, by Example 2.3, H is equivalent to a Y -graph G with a null-homologous leaf. By Lemma 2.4, . It follows from [13, Lemma 2.7] that t K = 0 for any null-homologous oriented knot K with 0-framing. So, by Lemma 3.5, we have that R M G (σ G ) = R M (σ).

A surgery map
In this section, we fix a closed spin manifold (M, σ). We associate to (M, σ) a bijective surgery map from a certain space of abstract graphs to a certain set of Y 2 -equivalence classes. This is a refinement of the surgery map defined in [13, Section 2.3].

Domain and codomain of the surgery map
We are going to consider the triplets where (M ′ , σ ′ ) is a spin manifold and ψ : H 1 (M ; Z) → H 1 (M ′ ; Z) is an isomorphism such that q M ′ ,σ ′ (ψ (x)) = q M,σ (x), for any x ∈ Tors H 1 (M ; Z). The set of such triplets is denoted by C(M, σ).
A diffeomorphism between two triplets (M ′ There is a notion of Y k -move, too, for such triplets: given (M ′ , σ ′ , ψ) ∈ C(M, σ) and a graph clasper G of degree k in M ′ , equation (3.1) allows us to set A is an Abelian group and s ∈ A is of order at most 2. We denote by Ab s the category of Abelian groups with special element whose morphisms are group homomorphisms respecting the special elements. We define a functor Y : Ab s −→ Ab in the following way. For an object (A, s) of Ab s , Y(A, s) is defined to be the free Abelian group generated by abstract Y-shaped graphs, whose edges are given a cyclic order and whose univalent vertices are labelled by A. The notation Y[a 1 , a 2 , a 3 ] will stand for the Y-shaped graph whose univalent vertices are colored by a 1 , a 2 and a 3 ∈ A in accordance with the cyclic order, so that our notation is invariant under cyclic permutation of the a i 's. The Abelian group Y(A, s) is the quotient of Y(A, s) by the following relations 4 : Here, as in Section 3.3, s is the image of the generator of H 1 (GL + (3; R); Z).

A surgery map from Y (P M ) to C(M, σ)
Let us consider an arbitrary element X of Y (P M ) written as where ε (j) = ±1 and x (Here, the class t K ∈ H 1 (E(FM ); Z) associated to an oriented framed knot K in M has been defined in Section 3.3.) Lastly, take G X to be a disjoint union of such Y -graphs G

Bijectivity of the surgery map S
According to the second statement of Theorem 3.1, any element of C(M, σ) is Y 1 -equivalent to (M, σ, Id). Consequently, the surgery map S is surjective. In order to prove that S is injective too, we are going to insert it into a commutative square and, for this, we need to define three other maps. It will be convenient to simplify the notation as follows:   to be the product of the above maps.
Secondly, we come back to the Abelian group with special element P . We denote by A (S, Z 2 ) the space of Z 2 -valued affine functions on S . Let e : H 1 (E(FM ); Z) → A (S, Z 2 ) be the homomorphism sending a homology class x to the map e(x) defined by α → α, x . (The function e(x) is affine because of (3.5).) There exists also a unique homomorphism κ : A(S, Z 2 ) → H (2) such that f (σ + y) = f (σ) + y, κ(f ) for any affine function f : S → Z 2 and cohomology class y ∈ H (2) . Consider the diagram in the category of Abelian groups with special element, where 1 is the function defined by α → 1 and p * is the homomorphism in homology induced by the bundle projection p : E(FM ) → M . By (3.5), that diagram is commutative: in fact, according to [13,Lemma 2.7], this is a pull-back square. In particular, by functoriality, there is a canonical homomorphism whose codomain is the pull-back of Abelian groups obtained from the homomorphisms Y(− ⊗ Z 2 ) and Y(κ). Observe that the groups Y(H, 0) and Y(H (2) , 0) are respectively isomorphic to Λ 3 H and Λ 3 H (2) via the maps de- On the other hand, Y A(S, Z 2 ), 1 is isomorphic to the space of Z 2 -valued cubic functions on S , denoted by C(S, Z 2 ), via the map defined by Y[f 1 , f 2 , f 3 ] → f 1 f 2 f 3 . This is proved in the Appendix (Lemma 6.3). Consequently, there is a canonical homomorphism whose codomain is the pull-back of Abelian groups obtained from the appropriate homomorphisms Λ 3 H → Λ 3 H (2) and C(S, Z 2 ) → Λ 3 H (2) . The homomorphism W is proved to be bijective in the Appendix (Lemma 6.4).
Thirdly, there is a homomorphism where −, − (n) : Λ 3 H (n) × Λ 3 H → Z n is the pairing defined at (3.3). By Lemma 6.1 from the Appendix, an element X of Λ 3 H such that −, X (n) = 0 for all n > 0 must vanish. Consequently, the homomorphism N is injective.
The above discussion can be summed up into the square The commutativity of that diagram follows from Lemma 3.2 and Lemma 3.5. We deduce next lemma, which concludes this section on the surgery map S.

Characterization of the Y 2 -equivalence relation
In this section, we prove the characterization of the Y 2 -equivalence relation for closed manifolds, with or without structure, as announced in the introduction.  Again, the necessary condition is easily verified from previous results. We prove the sufficient condition and we consider, for this, closed manifolds M and M ′ together with an isomorphism ψ : H 1 (M ; Z) → H 1 (M ′ ; Z) and a bijection Ψ : Spin(M ′ ) → Spin(M ) satisfying conditions (a) to (d). We choose a spin structure σ ′ on M ′ and we set σ := Ψ(σ ′ ). By condition (d), we have q M,σ (x) = q M ′ ,σ ′ (ψ(x)) for any x ∈ Tors H 1 (M ; Z). From (d) and (c), we deduce that Thus, Theorem 1.2 applies: the spin manifolds (M, σ) and (M ′ , σ ′ ) are Y 2equivalent and, a fortiori, the manifolds M and M ′ are Y 2 -equivalent.

In the setting of complex spin manifolds
We have seen in Section 2.2 how calculus of claspers makes sense in the context of spin manifolds. The same happens for manifolds equipped with a complex spin structure. In this paragraph, we give a characterization of the Y 2equivalence for complex spin manifolds without boundary. Before that, it is worth recalling the characterization of the Y 1 -equivalence in this context.  [11,5] in case when the Chern class of α is torsion and [3] in the general case.) For instance, if α comes from a spin structure σ , then the quadratic function φ M,α is essentially equivalent to the linking quadratic function q M,σ .
According to [3], For any integer n ≥ 0 and for any y ′ (d) The bijection Ψ is compatible with the isomorphism ψ in the sense that it is affine over ψ (2) and the following diagram is commutative: The necessary condition is proved from previous results (Corollary 3.4, Corollary 3.6, equation (3.2)) and from the following fact: if G is a graph clasper in a closed manifold M and if α is a complex spin structure on M , then we have that . We conclude that the complex spin manifolds (M, α) and (M ′ , α ′ ) are Y 2 -equivalent.

Appendix
This section contains the proofs of the algebraic lemmas that have been used in Section 4. Here, we shall use the following convention for any finitely generated Abelian group A and any integer n > 0. We denote A (n) = A ⊗ Z n , A (n) = Hom (A, Z n ) and −, − (n) : Λ 3 A (n) × Λ 3 A → Z n the pairing defined by A basis of A is a family of pairs {(e i , n i ) : i ∈ I} indexed by a finite set I , such that e i is an element of A of order 5 n i ≥ 0 and A is the direct sum of the cyclic subgroups generated by the e i 's. The dual basis of A (n) is the basis {(e * i , gcd(n, n i )) : i ∈ I} of A (n) , defined by e * i , e j = δ i,j n/ gcd(n, n i ) ∈ Z n .

Embedding of trivectors
This paragraph is aimed at proving the following lemma.
Lemma 6.1 If H is a finitely generated Abelian group, the homomorphism for all integers m > 0. To show that X must vanish, it suffices to prove that for any integer m > 0. Assertion (A m ) trivially holds for m = 1 so that it suffices to prove the following inductive statement.
Claim 6.2 Let n > 0 be an integer. If assertion (A n ) holds, then assertion (A np ) holds too for any prime number p.
To prove Claim 6.2, we need a few preliminaries. Choose a basis of H , and let m be an arbitrary positive integer. Then, {(e i ∧ e j ∧ e k , gcd(n i , n j , n k )) : 1 ≤ i < j < k ≤ r} is a distinguished basis of Λ 3 H , while {((e i ∧ e j ∧ e k ) ⊗ 1, gcd(m, n i , n j , n k )) : 1 ≤ i < j < k ≤ r} is a preferred basis of Λ 3 H ⊗ Z m . Furthermore, a distinguished basis of Λ 3 H (m) is e * i ∧ e * j ∧ e * k , gcd(m, n i , n j , n k ) : 1 ≤ i < j < k ≤ r and Hom Λ 3 H (m) , Z m has the basis e * i ∧ e * j ∧ e * k * , gcd(m, n i , n j , n k ) : 1 ≤ i < j < k ≤ r .
sends the basis element e i ∧ e j ∧ e k to m 2 gcd(m, n i , n j , n k ) gcd(m, n i ) gcd(m, n j ) gcd(m, n k ) We suppose that (A n ) holds, we consider a prime number p and we want to show that (A np ) holds. Writing X in the preferred basis of Λ 3 H , say this amounts to prove that x ijk ≡ 0 mod gcd(np, n i , n j , n k ).
But, from (A n ), we know that x ijk ≡ 0 mod gcd(n, n i , n j , n k ) and, from (H np ) together with (6.2) applied to m = np, we know that x ijk n 2 p 2 gcd(np, n i , n j , n k ) gcd(np, n i ) gcd(np, n j ) gcd(np, n k ) ≡ 0 mod gcd(np, n i , n j , n k ).
Therefore, it is enough to prove that the conditions z ≡ 0 mod gcd(n, n i , n j , n k ) zn 2 p 2 ≡ 0 mod gcd(np, n i ) gcd(np, n j ) gcd(np, n k ) imply that z ≡ 0 mod gcd(np, n i , n j , n k ) for any integer z . But, this can be verified working with the p-valuations of n, n i , n j , n k and z .

Cubic functions and trivectors
Let H be a finitely generated Abelian group and let S be a Z 2 -affine space over H (2) . We denote by A(S, Z 2 ) the space of affine functions S → Z 2 and by 1 ∈ A(S, Z 2 ) the constant function σ → 1. Then, A(S, Z 2 ), 1 is an Abelian group with special element (in the sense of Section 4.1). The space of cubic functions S → Z 2 , ie, functions which are finite sums of triple products of affine functions, is denoted by C(S, Z 2 ).
Proof The demonstration is similar to that of [13,Lemma 4.21]. It is enough to construct an epimorphism ǫ : C(S, Z 2 ) → Y A(S, Z 2 ), 1 such that γ • ǫ is the identity.
We define a homomorphism ǫ : C(S, Z 2 ) → Y A(S, Z 2 ), 1 by setting The homomorphism ǫ is surjective by the slide and multilinearity relations and, clearly, the identity γ • ǫ = Id is satisfied on the basis elements.
It can be verified that d 3 f is multilinear, does not depend on σ (because f is cubic) and is alternate (because 2 · H (2) = 0), hence a homomorphism d 3 : C(S, Z 2 ) → Hom Λ 3 H (2) , Z 2 . By the duality pairing between Λ 3 H (2) and Λ 3 H (2) defined by equation (6.1), this homomorphism can be regarded as taking its values in Λ 3 H (2) . In the sequel, we consider the pull-back of Abelian groups Λ 3 H × Λ 3 H (2) C(S, Z 2 ) (6.3) Proof As in the proof of Lemma 6.3, it suffices to construct an epimorphism ǫ : Λ 3 H × Λ 3 H (2) C(S, Z 2 ) → Y(P ) such that W • ǫ is the identity. Again, we fix a basis {(e i , n i ) : 1 ≤ i ≤ r} of H together with a base point σ 0 ∈ S , and e i : S → Z 2 designates the affine function defined by e i (σ 0 + y) := y, e i for any y ∈ H (2) . From the basis of A(S, Z 2 ) given in the proof of Lemma 6.3, we obtain that ((0, 1), 2) ∪ {((e i , e i ), n i ) : 1 ≤ i ≤ r} is a basis of P . From the basis of C(S, Z 2 ) given in the proof of Lemma 6.3 and the basis {(e i ∧ e j ∧ e k , gcd(n i , n j , n k )) : 1 ≤ i < j < k ≤ r} of Λ 3 H , we construct the following basis of Λ 3 H × Λ 3 H By the slide and multilinearity relations, this homomorphism ǫ is surjective, and it can be readily verified that W • ǫ(z) = z for any of the above basis elements z .