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Arithmetic topology of 4-manifolds

We construct a functor from the smooth 4-dimensional manifolds to the hyper-algebraic number fields, i.e. fields with non-commutative multiplication. It is proved that that the simply connected 4-manifolds correspond to the abelian extensions. We recover the Rokhlin and Donaldson's Theorems from the Galois theory of the non-commutative fields.


Introduction
The arithmetic topology studies an interplay between the 3-dimensional manifolds and the fields of algebraic numbers [Morishita 2012] [6]. The idea dates back to C. F. Gauss. Namely, there exists a map F from the 3-dimensional manifolds M 3 to the algebraic number fields K. Such a map transforms links Z ⊂ M 3 (knots K ⊂ M 3 , resp.) into the ideals (prime ideals, resp.) of the ring of integers O K of K. The map F is a functor on the category of 3-dimensional manifolds M 3 with values in the category of algebraic number fields K [7, Theorem 1.2].
The aim of our note is an extension of F to the smooth 4-dimensional manifolds M 4 . The range of F is no longer the fields of algebraic numbers, but the fields K with non-commutative multiplication, e.g. the quaternions or the cyclic division algebras. We refer to K as a hyper-algebraic number field. The Galois theory of such fields was elaborated by [Cartan 1947] [2] and [Jacobson 1940] [4], see [Harqenko 1996] [14] for a detailed account.
To introduce the map F , let N be a finite index subgroup of the mapping class group M od M 3 , where M 3 is a 3-dimensional manifold. The subgroup N gives rise to a smooth branched cover M 4 N of the 4-sphere S 4 , see [Piergallini 1995] [12] and Section 2.1. On the other hand, we have N / Z ∼ = G K , where N ( Z, resp.) is a profinite completion of N (Z, resp.) and G K is the absolute Galois group of K [8, Theorem 1.2]. Recall that the field K can be recovered up to an isomorphism from the group G K , see [Uchida 1976] [13, Corollary 2] and remark 2.5. We define F as a composition of maps: Let O K be the ring of integers of the field K. Unlike O K , the ring O K is always simple, i.e. all two-sided ideals of O K are trivial. Hence we must use the dynamical ideals (dynamials), i.e. the crossed products O K ⋊ mZ by an endomprphism of O K of degree m ≥ 1, see [9] for the details and motivation. The dynamial O K ⋊ mZ is minimal if and only if m = p is a prime number [9,Theorem 1.7].
To formalize our results, recall that the groups N ∼ = N ′ are a Grothendieck pair, if N ∼ = N ′ . We shall denote by M 4 a category of all smooth 4-dimensional manifolds M 4 N , such that: (i) the set M 4 contains no M 4 N and M 4 N ′ such that N and N ′ are a Grothendieck pair and (ii) the arrows of M 4 are differentiable maps between such manifolds. Denote by K a category of the hyper-algebraic number fields, such that the arrows of K are injective homomorphisms between these fields. The knotted surface is a transverse immersion ι : X g1 ∪ . . . X gn ֒→ M 4 of a collection of the 2-dimensional orientable surfaces X gi of the genera g i ≥ 0. The ι(X g1 ∪ . . . X gn ) is called a surface knot K if n = 1 and a surface link Z otherwise. Our main result can be formulated as follows.
Theorem 1.2. The map F : M 4 → K is a covariant functor, such that: Remark 1.3. The functor F is independent of the choice of manifold M 3 used in the construction of F according to the formula (1.1). Indeed, using the manifolds i described in Section 2.1, one can always take a connected sum #W 4 i , so that M od M 3 ∼ = π 1 (#W 4 i ). This observation is corroborated by the C * -algebra approach, see remark 1.1 Let K be a Galois extension of H, i.e. an extension having the Galois group Gal K and obeying the fundamental correspondence of the Galois theory [Cartan 1947] [2], [Jacobson 1940] [4] and [Harqenko 1996] [14]. Recall that K is said to be an abelian extension, if Gal K is an abelian group. Theorem 1.2 implies the following result. The article is organized as follows. Section 2 contains a brief review of the 4-dimensional topology, the Galois theory of non-commutative fields and the arithmetic topology. Theorem 1.2, corollary 1.4 and remark 1.5 are proved in Section 3. In Section 4 we give a proof of the Rokhlin and Donaldson's Theorems using the Galois theory of non-commutative fields.

Preliminaries
This section contains a brief review of the smooth 4-dimensional manifolds, the Galois theory of non-commutative fields and the arithmetic topology. We refer the reader to [Cartan 1947] [2], [Morishita 2012] [6] and [Piergallini 1995] [12] for a detailed account.
2.1. Topology of 4-dimensional manifolds. Let M 3 be a closed 3-dimensional manifold. Denote by M od M 3 the mapping class group of M 3 , i.e. a group of isotopy classes of the orientation-preserving diffeomorphisms of M 3 . The M od M 3 is a finitely presented group [Hatcher & McCullough 1990] [3].
Since each finitely presented group can be realized as the fundamental group of a smooth 4-manifold, we denote by W 4 ∈ M 4 a 4-manifold with π 1 (W 4 ) ∼ = M od M 3 . It will be useful for us to represent W 4 as a PL (and, therefore, smooth) 4-fold (or 5-fold, if necessary) branched cover of the sphere S 4 . Let X g be a closed surface, which we assume for the sake of brevity to be orientable of genus g ≥ 0. Recall that there exists a transverse immersion ι : X g ֒→ S 4 , such that W 4 is the 4-fold PL cover of S 4 branched at the points of X g [Piergallini 1995] [12]. In other words, the M od M 3 is a normal subgroup of index 4 of the fundamental group π 1 (S 4 − X g ).
Remark 2.1. The immersion ι and surface X g need not be unique for given W 4 . Yet one can always restrict to a canonical choice of ι and X g , which we always assume to be the case.
N we understand a cover of the manifold W 4 corresponding to the finite index subgroup N of the fundamental group π 1 (W 4 ).

2.2.
Galois theory for non-commutative fields. Denote by K a division ring, i.e. a ring such that the set K × := K − {0} is a group under multiplication. If the group K × is commutative, then K is a field. For otherwise, we refer to K as a non-commutative field.
Roughly speaking, the Galois theory for K is a correspondence between the subfields of K and the subgroups of a Galois group Gal K of the field K. Namely, let G be a group of automorphisms of the field K. It is easy to see, that the set is a subfield of the field K.
Definition 2.4. An extension K of the field L is called Galois, if there exists a group G of automorphisms of the field K, such that L ∼ = I G . The Galois group of K with respect to L is defined as Gal K ∼ = G. The absolute Galois group G K is defined as a Galois group of the algebraic closure of K. The G K is a profinite group.
Remark 2.5. By an adaption of the argument for the case of fields [Uchida 1976] [13, Corollary 2], one can show that the group G K defines the underlying field K up to an isomorphism, see lemma 3.3.
Unlike the case of fields, the inner automorphisms of K are a non-trivial group. Indeed, consider an automorphism h g : K → K given by the formula: By Inn (K) we denote a group of such automorphisms under the composition Let G be a finite group of automorphisms of the field K. A normal subgroup Γ := G ∩ Inn (K) of G consists of the inner automorphisms of the field K. Consider a group ring It is easy to see, that B(Γ) is a finite-dimensional algebra over its center C. The following result has been established in [Cartan 1947 where ι is an inclusion and |G/Γ| = dim C B(Γ).
Recall that B(Γ) is a Frobenius algebra [Harqenko 1996] [14, Theorem 3.5.1]. In other words, there exists a non-degenerate bilinear form The following construction of F is based on [Uchida 1976] [13]. Let X g,n be an orientable surface of genus g ≥ 0 with n ≥ 0 boundary components. Denote by M od X g,n the mapping class group of X g,n , i.e. a group of isotopy classes of the orientation and boundary-preserving diffeomorphisms of the surface X g,n . Let N be a finite index subgroup of M od X g,n . We omit the construction of a 3-manifold M 3 N ∈ M 3 from N referring the reader to [7, Remark 1.1]. As usual, let G K be the absolute Galois group of K ∈ K, while N and Z denote the profinite completion of the groups N and Z, respectively. Let G K → K be an injective map defined in [Uchida 1976] [13,Corollary 2].
N is homeomorphic to M 4 N ′ , one gets an isomorphism of the fundamental groups . Because the maps p * and p ′ * are injective, we conclude that N and N ′ are isomorphic subgroups of π 1 (S 4 − X g ) and, therefore, of the group M od M 3 . In view of the above and remark 2.1, the necessary condition of lemma 3.1 is proved.
(ii) Let N ∼ = N ′ be isomorphic subgroups of M od M 3 . As explained in Section 2.1, the groups N and N ′ define a pair of branched covers M 4 N and M 4 N ′ of S 4 . In view of the diagram in Figure 1, one obtains an inclusion of groups N, N ′ ⊂ π 1 (S 4 − X g ). Since N ∼ = N ′ , the subgroups N and N ′ are conjugate in the group π 1 (S 4 − X g ). In other words, there exists an element g ∈ π 1 (S 4 − X g ), such that N ′ = g −1 N g. It is well known, that the conjugate subgroups of π 1 (S 4 − The necessary condition of lemma 3.2 is proved. (ii) Suppose that G K ∼ = N / Z and G K ′ ∼ = N ′ / Z are isomorphic groups. Let us show, that N ∼ = N ′ are isomorphic subgroups of M od M 3 . To the contrary, let N ∼ = N ′ . Since N and N ′ cannot be a Grothendieck pair, we conclude that N ∼ = N ′ . But G K ∼ = N / Z and G K ′ ∼ = N ′ / Z and, therefore, G K ∼ = G K ′ . One gets a contradiction proving the sufficient condition of lemma 3.2. Proof. The proof is an adaption of the argument of [Uchida 1976] [13,Corollary 2] to the case of non-commutative fields. Namely, we introduce a topology on G K according to the formula (3.1). Let G K and G K ′ be open subgroups of an absolute Galois group G, and let σ : G K → G K ′ be a topological isomorphism. To prove lemma 3.3, it is enough to show that σ can be extended to an inner automorphism of G, which corresponds an isomorphism K ∼ = K ′ . Indeed, one takes an open normal subgroup N of G contained in G K and G K ′ . The group N induces an isomorphism σ N : G K /N → G K ′ /N . It can be shown that σ N extends to an inner automorphism of the group G/N . We repeat the construction over all open normal subgroups of G and obtain an explicit formula for the required inner automorphism of G. Lemma 3.3 follows.

3.2.
Proof of corollary 1.4. If M 4 ∈ M 4 is a simply connected manifold, then π 1 (M 4 ) is a trivial group. Since M 4 is a cover of S 4 branched over a surface X g ֒→ S 4 , we conclude that π 1 (S 4 − X g ) is a finite group. By the Hurewicz Theorem, the homology group is also a finite abelian group.
Consider a commutative diagram in Figure 2. An automorphism of H 1 (S 4 − X g ; Z) comes from a homeomorphism h : M 4 → M 4 of the manifold M 4 . Theorem 1.2 says that h must correspond to an automorphism σ h of the field K, such that σ h (L) = L for a subfield L ⊆ K. (For simplicity, the reader can think L ∼ = H is the field of quaternions.) Since the automorphisms σ h generate the Galois group of the extension K | L, we conclude that: Recall that the group of automorphisms of a finite abelian group is always an abelian group. Indeed, a finite abelian groups can be written in the form Z/p 1 Z ⊕ · · · ⊕ Z/p k Z for some distinct primes p i . On the other hand, Aut (G ⊕ H) ∼ = Aut (G) ⊕ Aut (H), where G and H are finite abelian groups of the coprime order. Thus Since the group of automorphism of a cyclic group is a cyclic group, using (3.4) we conclude, that the group Aut (H 1 (S 4 − X g ; Z)) is abelian. In view of (3.3), the group Gal K is also abelian. Corollary 1.4 is proved.
3.3. Proof of remark 1.5. Let K be an abelian extension. Then Gal K is a finite abelian group. Let us show, that the group H 1 (S 4 − X g ; Z) can be infinite. Indeed, one can always write H 1 (S 4 − X g ; Z) ∼ = Z k ⊕ T ors, where k ≥ 0 and T ors is a finite abelian group. As explained, we have Aut (H 1 (S 4 − X g ; Z)) ∼ = Aut (Z k ) ⊕ Aut (T ors). Recall that Aut (Z k ) ∼ = GL k (Z). If k = 1, then the group Aut (Z) ∼ = Z/2Z is abelian and finite. Yet the group H 1 (S 4 − X g ; Z) ∼ = Z ⊕ T ors is infinite. In other words, the corresponding 4-manifold M 4 ∈ M 4 cannot be simply connected. Lemma 4.1. The group ring of a finite abelian group Γ ∼ = Z/p 1 Z ⊕ · · · ⊕ Z/p k Z is isomorphic to a direct sum of the cyclotomic fields, i.e.

Rokhlin and Donaldson's Theorems revisited
where ζ pi is the p i -th root of unity.
Proof. An elegant proof of this fact can be found in [Ayoub & Ayoub 1969] [1].
Thus to calculate Q, we can restrict to the cyclotomic fields Q(ζ pi ) and take the tensor product over p i . Recall that the trace form on Q(ζ pi ) is a symmetric bilinear form: T r Q(ζp i ) : Q(ζ pi ) × Q(ζ pi ) → Q, such that (x, y) → tr(xy), where tr is the trace of an algebraic number. The trace form (4.3) is equivalent to the form T r Q(ζp i ) (x, x) := T r Q(ζp i ) (x 2 ) via the formula: The following classification is true: where we use the Witt ring notation p 1 for the diagonal quadratic form of dimension p, 2 n 1 for the Pfister form of dimension 2 n and H for the hyperbolic (split) plane, see [Lam 2005] [   Consider a regular representation of Γ by the automorphisms of H 2 (M 4 ; Z) ∼ = Z k . It is easy to see, that such a representation coincides with ρ and, therefore, we conclude that k = |Γ|. Moreover, since B(Γ) is the group ring of Γ, one gets an isomorphism B(Γ) ∼ = H 2 (M 4 ; Z) between the corresponding Z-modules. Lemma 4.4 is proved. (ii) Signature of the intersection form of a simply connected smooth 4-manifold is divisible by 16.
Proof. As explained, we identify the intersection form: with the trace form T r Q(ζp) (x, y) given by the formulas (4.3)-(4.5). Accordingly, we have to consider the following two cases.
(i) Let us consider the case T r Q(ζp) (x 2 ) = p 1 , where p is an odd prime. Let {1, ζ p , . . . , ζ p−1 p } be the standard basis in the ring of integers Z[ζ p ] of the cyclotomic field Q(ζ p ). The corresponding formula (4.5) can be written as: where x = x 0 + x 1 ζ p + · · · + x p−1 ζ p−1 p for an x ∈ Z[ζ p ]. Using (4.4), one gets a symmetric bilinear form on H 2 (M 4 ; Z): It remains to notice, that M 4 ∈ M 4 is a smooth manifold and (4.9) is a positive definite diagonalizable intersection form. Item (i) of corollary 4.5 follows.