3次元多様体の分岐被覆に関する数論的位相幾何学

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

3次元多様体の分岐被覆に関する数論的位相幾何学

植木, 潤

https://doi.org/10.15017/1543932

出版情報：Kyushu University, 2015, 博士（数理学）, 課程博士バージョン：

権利関係：Fulltext available.

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Arithmetic topology on branched covers of 3-manifolds

Jun Ueki

Graduate School of Mathematics Kyushu University

September, 2015

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number theory and 3-dimensional topology. In this thesis, based on the analogies between primes and knots, number rings and 3-manifolds, we study analogues of id`elic class field theory, genus theory, Iwasawa theory and Galois deformation theory in the context of 3-dimensional topology. We establish various foundational analogies in arithmetic topology.

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Acknowledgments

The author would like to express his sincere gratitude to his family for their successive warm and patient help, Masanori Morishita for the best introduction to his field as a pioneer of arithmetic topology and very strong encouragement, and Mikio Furuta for instructive advices form topological viewpoint and assistance in finding a subject the author really want to do. In addition, he would like to appre- ciate Jonathan Hillman for helpful advise on the proof of Chapter 4, Theorem 3.11, Yasushi Mizusawa for useful information on Kida’s formula, Teruhisa Kadokami for discussing significance of the notion of a branchedZp-cover, and Tsuyoshi Ito for useful suggestion on Iwasawaµ-invariants. Moreover, He is grateful to Tatsuro Shimizu for the first instruction on how to attack the problems in research, Yu-ichi Hirano for the first instruction to Iwasawa theory, Tomoki Mihara for suggestion on the existence theorem of class field theory and a lot of fruitful discussions onp- adic objects. He would also like to thank Gregory Brumfiel, Ted Chinberg, Takashi Hara, Shinya Harada, Yuki Imoto, Tetsuya Ito, Yuichi Kabaya, Takahiro Kitajima, Takahiro Kitayama, Makoto Matsumoto, Hitoshi Murakami, Kanako Nakajima, Hi- rofumi Niibo, Sachiko Ohtani, Takayuki Okuda, Adam Sikora, Kazuki Tokimoto, Kohei Yahiro, Seidai Yasuda, and Don Zagier for useful comments. The author is partially supported by Grant-in-Aid for JSPS Fellows (25-2241).

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Notation and convention

The symbol Zdenotes the ring of integers. The symbols Q, R, andC denote the fields of rational numbers, real numbers, and complex numbers respectively.

The symbolFq denotes the field with q-elements. For a prime number p, the symbolsZpandQpdenote the ring ofp-adic integers and the field ofp-adic numbers respectively.

A number fieldk means a finite extension ofQcontained in C. For a number fieldk, the symbolOk denotes the ring of integer.

A 3-manifold means oriented, connected, closed, and equipped with a base point, unless otherwise mentioned. For a manifold M, the symbolH_i(M) denotes i-th homologies with coeﬃcients in Z. In addition, for a fixed CW-structure of PL-structure onM, the symbolsC_i(M), Z_i(M), and B_i(M) denote the groups of i-chains, i-cycles, andi-boundaries with coeﬃcients inZ.

The symbol Sⁿ denotes the n-sphere. A 3-manifold M is called an integral homology 3-sphere (ZHS³) ifHi(M)∼=Hi(S³) for alli. It is equivalent to say that

#H1(M)<∞. A 3-manifold M is called a rational homology 3-sphere (QHS³) if Hi(M,Q)∼=Hi(S³,Q) for alli. It is equivalent to say that H2(M,Q) = 0.

A branched cover of a 3-manifold means one branched over a link. For a branched cover h : N → M and a link L in M, h⁻¹(L) denotes the link in N defined by the preimage. When h is Galois (regular), Gal(h) = Deck(h) denotes the Galois group, that is, the group of covering transformations ofN overM.

For a groupGand its subgroupH, we writeH < G.

For a group Gand aG-moduleA, we write Hbⁱ(G, A) =Hbⁱ(A) for simplicity, if there is no ambiguity. In addition,A^G andAG=A/IGAdenote theG-invariant subgroup and theG-coinvariant quotient respectively, whereIG= (g−1|g∈G)<

Z[G] is the augmentation ideal.

For a local ringR, we denote bym_R the maximal ideal ofR. For an integral domaink, we denote by char(k) the characteristic of k.

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Introduction

0.1. Historical background

The development of modern algebraic number theory may have its origin in the work of Gauss about two hundred years ago. In his Disquisitiones Arithmeticae (1801), Gauss proved the quadratic reciprocity law and initiated the theory of genera for binary quadratic forms, which form the main body of the arithmetic of quadratic number fields today. After Gauss’ work, the connection between reciprocity laws and number field extensions has been investigated extensively by Kummer for cyclotomic and Kummer extensions and by Hilbert for general number fields, and finally culminated in class field theory by Takagi, E. Artin and Chevalley.

In the late of 1950’s, Iwasawa introduced his theory onZp-extensions (pbeing a prime number) and studied the precise structure of p-ideal class groups using infinite number field extensions and p-adic method (which originated in the work of Kummer) eﬀectively. In the middle of 1980’s Hida introduced Iwasawa theoretic method in the study of p-adic Hecke algebras, and it motivated Mazur to invent the deformation theory forp-adic Galois representations, which can provide a framework to generalize the Iwasawa theory in a non-abelian direction.

Knot theory also has its origin in the work of Gauss on linking numbers, and it is quite interesting and surprising that there have been developments in knot theory / 3-dimensional topology which are analogous to those mentioned above in number theory. It is classically known that there are analogous features between the theory of covers of topological spaces and that of extensions of fields, which was ultimately unified to the notion of Galois category by Grothendieck ([Gro63]).

However, beyond such a general similarity, there are found more precise and inti- mate analogies between number theory and 3-dimensional topology, based on the homotopical analogies between primes and knots, number rings and 3-manifolds.

In fact, the 3-dimensional view of a number ring was recognized for the first time when J. Tate, M. Artin and J.-L. Verdier gave a topological interpretation of class field theory as a sort of 3-dimensional Poincar´e duality in Galois/´etale cohomology ([Tat63], [AV64], [Maz73]). The analogy between a prime and a knot was firstly pointed out by B. Mazur, during that time, in the middle of 1960’s. The interesting analogies between Iwasawa theory and Alexander–Fox theory were also noticed by Mazur ([Maz64]). After a long silence, in the late of 1990’s, Kapranov ([Kap95]), Reznikov ([Rez97], [Rez00]), and Morishita ([Mor10], [Mor12]) started to study these analogies in a systematic manner. See the M²KR-dictionary in Section 0.2 below. A new research area bridging number theory and 3-dimensional topology is called Arithmetic Topology. A basic reference is Morishita’s book ([Mor12]).

However, there still remain lots of problems even in the foundational level.

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The purpose of this thesis is to enrich and deepen the foundation of arithmetic topology. We study various basic materials, with which were not treated in [Mor12], and give many new results. Although the exposition of [Mor12] is restricted to the case where the base space (resp. field) is the 3-sphere (resp. the rational number field), we try to make a base 3-manifold (resp. number field) as general as possible and deal with the relative situations.

0.2. M²KR-dictionary

We list basic analogies between number theory and 3-dimensional topology in the following dictionary. These analogies were pointed out by Mazur, Kapranov, Reznikov, and Morishita. We call itM²KR-dictionary for short in this thesis.

Number theory 3-dimensional topology

SpecFq =K(Zb,1) S¹=K(Z,1) circle π^´^et₁(SpecFq) =⟨σFrob⟩ ∼=Zb π1(S¹) =⟨l⟩ ∼=Z p-adic integer ring SpecOp≃SpecFq tubular neighborhoodVK ≃S¹

p-adic number field boundary torus Speckp≃SpecOp−SpecFq ∂VK ≃VK−S¹

number ring SpecOk 3-manifold M

prime ideal p: SpecFp ,→SpecOk knot K:S¹,→M prime idealsS={p1, ...,pr} link L={K1, ..., Kr}

extensionF/k branched coverf :N →M

´

etale fundamental group π₁^´^et(SpecOk) fundamental groupπ₁(M) π^´₁^et(SpecOk−S) link groupπ₁(M−L) geometric point SpecC,→SpecOk base point {pt},→M ideal groupI(k) 1-cycle groupZ₁(M) k^×→I(k); a7→(a) C₂(M)→Z₁(M); c7→∂c principal ideal group P(k) 1-boundary groupB₁(M) ideal class group Cl(k) =I(k)/P(k) 1st-homologyH1(M) =Z1(M)/B1(M)

Fact: # Cl(k)<∞ Assume: #H1(M)<∞(i.e. M: QHS³), or consider torsion subgroup H1(M)tor

Artin reciprocity Hurewicz isomorphism

π₁^´êt(SpecOk)âb∼= Gal(k_abûr/k)∼= Cl(k) π₁(M)âb∼= Gal(M_ab/M)∼=H₁(M) k_abûr/kHilbert class field Mab→M maximal abelian cover

unit groupO_k^× 2-cycle groupZ2(M),

or 2nd-homologyH2(M) (∼=H1(M)free) Here we assume that a number field is a finite extension of Q contained in C, and that a 3-manifold is oriented, connected, closed, and equipped with base points, and a branched cover of 3-manifolds is branched over a link. We note that any 3-manifold can be obtained as a branched cover overS³by Alexander’s theorem ([Ale20]).

0.3. The contents of this thesis

Here are the contents of this thesis. In Chapter 1, we recollect Hilbert theory and its analogue over a number field and a 3-manifold in a parallel manner. They describe in a group theoretic way how primes and knots decompose in an extension of number fields and in a branched cover of 3-manifolds.

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In Chapter 2, we study an id`elic class field theory for 3-manifolds. Class field theory given by Artin–Takagi controls all the abelian extensions of a given number filed k, in which id`ele theory arranged by Chevalley sums up all the local theory ofk_p corresponding to prime idealp and describes the global theory (Section 2.1).

An id`ele theory for 3-manifolds was originally suggested by Sikora and another formulation was given by Niibo. We develop the latter and prove an analogue the global class field theory. A local theory in a 3-manifold is the Galois theory of branched covers over the tubular neighborhoodV_K of a knotK (Section 2.2). We first introduce the notion of a very admissible link K in M as an analogue of the set of all the primes, and construct it (Section 2.3). We also introduce a maximal K-branched cover as an analogue of an algebraic closure k of k and discuss roles of base points (Section 2.3). Then following [Nii14], we define id`ele class group CM,K, which sums up all the local theories of toriVK corresponding to knotsK in K (Section 2.4), and prove

Theorem A (2.5.4, The global reciprocity law). There is a canonical map ρM,K : CM,K → Gal(M,K)^ab which induces an isomorphism CM,K/C_N,h−1(K) ∼= Gal(h)for each abelian coverh:N →M branched over a finite link L⊂ Kand is compatible with local theories.

This mapρM,Kis called the global reciprocity map (Section 2.5). In addition,CM,K

admits two natural topologies calledthe standard topology andthe norm topology (Sections 2.6, 2.7) and satisfies

Theorem B (2.7.4, The existence theorem). There are bijections among the set of finite abelian covers of M branched over a finite link in K, the set of open sets of finite indices with respect to the standard topology, and the set of open sets with respect to the norm topology.

We further discuss the norm residue symbols (Section 2.8), calculate the Tate cohomologies of id`eles and compare with the axiom of class field theory (Section 2.9).

In Chapter 3, we study a theory of genera for 3-manifolds. We generalize Morishita’s work [Mor01] and establish an analogue of the relative genus formula given by Furuta in [Fur67] for a Galois branched coverh:N→M over a rational homology 3-sphere (QHS³). If his abelian, then the genus coverh^g :N^g →N is the maximal unbranched cover such thath◦h^g is abelian.

Theorem C (3.2.2 (1), The relative genus formula). If branch linkL =⊔K_i consists of null homologous components, thenthe genus numberg_h:= deg(h^g)and the branched indices e_i ofK_i satisfy

g_h= #H₁(M)Π_ie_i/deg(h).

This formula is essentially used in Chapter 6. We give a direct proof for Galois cases, a parental alternative proof wish use of id`ele theory, and a further study on the coinvariant groupH1(N)G forG= Gal(h).

Latter chapters are dedicated for Iwasawa theoretic studies. We try to refine Mazur’s analogy between Iwasawa theory and Alexander–Fox theory ([Maz64]), whereZp-extensions andZ-covers were dealt with.

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Iwasawa theory Alexander–Fox theory Zp-extension ofk Z-cover overM−L, or

branched Zp-cover over (M, L)

Iwasawa module link module

Iwasawa polynomial Alexander polynomial

In Chapter 4, we treat some finite cases and obtain suggestions. In Section 4.1, we recall several Iwasawa’s theorems on ideal class groups and unit groups in finite extensions ([Iwa56a], [Iwa56b]). The theorems on ideal class groups are toy-cases of Iwasawa’s class number formula. The assertion on unit groups relates Tate cohomology and used in Yokoi’s genus formula ([Yok67]). In Section 4.2, we translate Iwasawa’s theorems into the context of 3-dimensional topology, and obtain a new insight into the analogy between units and 2-cycles (Remark 4.2.7).

It might be remarkable that we have

TheoremD (4.2.6). If we fix CW-structures on a branched Galois cover, then the Tate cohomology of 2-cycles is a topological invariant of branched covers (inde- pendent of the choice of CW-structures).

In Section 4.3, we observe how our new dictionary works in translation of Yokoi’s formula.

In Chapter 5, we refine an analogue of Iwasawa’s class number formula originally studied by Morishita and others forZ-covers ([HMM06], [KM08]). We first recall Iwasawa’s formula forZp-extensions ([Iwa59]) together with the structure theorem of compact Λ-modules (Section 5.1). Next, we introduce the notion of abranched Zp-cover as an inverse system of cyclic branchedp-covers, which is not necessarily obtained from aZ-cover (Section 5.2), and prove

TheoremE (5.3.1, An Iwasawa type formula). Every branchedZp-coverMf= {Mn}nofQHS³admitsλ, µ∈Nandν∈Zsuch that the cardinality of thep-torsion subgroup satisfies#H₁(M_n)_[p]=p^λn+µpⁿ^+ν for alln≫0.

These λ, µ, ν are calledthe Iwasawa invariants. This formula can be deduced to the case ofZ-covers. However, we prove Sakuma’s exact sequence (Proposition 5.3.3) for branchedZp-covers in Section 5.4, and deduce an alternative proof of the formula in a parallel manner to Section 5.1, together with a proposition on a certain direct limit module (Proposition 5.3.5). Further generalization to non-QHS³cases is discussed briefly in Section 5.5.

We call a field simply a Zp-field if it can be obtained as a Zp-extension of a number field. Behaviors of Iwasawa invariants in extensions ofZp-fields are studied by Iwasawa ([Iwa73b], [Iwa81]). In Chapters 6–8, we formulate their analogues.

In Section 6.1, we first introduce the notion of abranched Galois cover of branched Zp-covers as a certain compatible system of branched covers on each layer of two inverse systems. Then in Sections 6.2 and 6.3, we recall Iwasawa’s theorem on µ-invariants and prove their analogues, using a formula obtained in Chapter 3.

In Chapter 7, we calculate the Tate cohomologies of various modules. In Section 7.1, we calculate that of 2-cycles in a branched Galois cover in an explicit manner, and compare with the results of units. In Section 7.2, we define analogues of S- ideals and others, and calculate their Tate cohomologies. In Section 7.3, we define chains of branched Zp-covers by the injective limits with respect to the transfers, and prove analogues of results in ap-extension of Zp-fields.

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In Chapter 8, as a goal of our Iwasawa theoretic study, we prove an analogue of Kida’s formula. It was originally an analogue of the Riemann–Hurwitz formula on Riemann surfaces forp-extensions ofZp-fields, and tells that ifµ= 0, thenλ’s resemble genera of Riemann surfaces (Section 8.1). Our formula is stated as follows:

TheoremF (8.2.1, Kida’s formula for branchedZp-covers). Letf :fN→Mfbe a branched Galoisp-cover of branchedZp-covers. Then, under proper assumptions, if µ_M_f= 0, then µ_N_e = 0, and the branch indicese_w of branch componentsw⊂N of f satisfy

λ_f_N−1 = deg(f)(λ_M_f−1) +∑

w

(ew−1).

In Iwasawa’s second proof of Kida’s formula in [Iwa81], following Chevalley’s method ofp-adic representations of a finite group, he formulated the case of degree pwith use of the Tate cohomologies of units, and deduced the case for degree p- power by Kida’s method [Kid80]. In our proof, we follow his argument (Section 8.3).

As Mazur pointed out ([Maz00], [Mor12, Chapters 13, 14]), from a view point of representation theory, Iwasawa theory and Alexander–Fox theory are concerned about 1-dimensional representations of knot groups and Galois (prime) groups respectively, and it would be interesting to pursue the analogies further for higher dimensional cases. In Chapters 10 and 11, we study deformations of knot groups representations following after the theory of Galois deformations ([Maz89]).

In Chapter 9, we study GL1-deformations. We recall a relation between deformations ofp-adic GL1-representations of prime groups Πp =π^´₁^et(SpecZ− {p}) and Iwasawa theory (Section 9.1), and a relation between deformations of GL1- representations of knot groups ΠK =π1(S³−K) and Alexander–Fox theory (Sec- tion 9.2).

In Chapter 10, we study SL2-deformations of knot groups. We first review a deformation theory forp-ordinary GL2-representations of prime groups Πp following Mazur. Next, by employing the pseudo-SL2-representations (Section 10.2), we prove the existence of the universal deformation of a given SL2-representation of a finitely generated group Π over a fieldkwhose characteristic is not 2 (Section 10.3).

We then show its connection with the character scheme for SL₂-representations of Π when k is an algebraically closed field (Section 10.4). We investigate examples concerning Riley representations of 2-bridge knot groups and give explicit forms of the universal deformations (Section 10.5). Finally we discuss the universal deformation of the holonomy representation of a hyperbolic knot group in connection with Thurston’s theory on deformations of hyperbolic structures (Section10.6).

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Hilbert theory for 3-manifolds

Hilbert theory deals with, in a group-theoretic manner, the decomposition law of a prime in a finite Galois extension of number fields. In this chapter, we present a topological analogue of Hilbert theory for 3-manifolds, which describes the decomposition law of a knot in a finite Galois branched cover of 3-manifolds. We follow the exposition of [Mor12, Chapter 5], where the base space isS³.

In Section 1.1, we review the Hilbert theory for number fields. In Section 1.2, we show its topological analogue for 3-manifolds, working with any base 3-manifold.

1.1. Hilbert theory for number fields

First, we recall the Hilbert theory for number fields. LetF/kbe a finite Galois extension ramified over a finite setSof prime ideals. Leth: SpecOF →SpecOk be the associated ramified cover. We putX := SpecOk−S,Y := SpecOF−h⁻¹(S), G:= Gal(Y /X) = Gal(F/k), and n := #G. Let p be a prime ideal ofk and let Sp := h⁻¹(p) = {P1, . . . ,Pr}, r = rp. We have OF ⊗_Ok OkP_i, where OkP_i denotes the Pi-adic completion of Ok. Fix an algebraic closure k_p of k_p and let x : Speck_p → X be the base point induced by the inclusion Ok[1/S] ,→ k_p. Let F_x(Y) = Hom_X(Speck_p, Y) = {y₁, . . . , y_n} and let ρ : G_S := π₁(X, x) → Aut(F_x(Y)) be the monodromy permutation representation which induces an isomorphism π1(X, x)/h_∗(π1(Y, yi)) ∼= Imρ ∼= G. Note that π1(X, x) and hence G acts on Sp transitively. We call the stabilizer DP_i = {g∈G|g(Pi) =Pi} of Pi the decomposition group of Pi. Since we have the bijection G/DP_i ∼= Sp,

#DP_i = n/r is independent of Pi. Indeed, if Pj = g(Pj) for g ∈ G, then we have D_P_j =gD_P_ig⁻¹. Since eachg induces an isomorphism ˆg: F_P_i →^∼⁼ F_g(P_i₎, ˆg gives an automorphism of FPi overOp for each g∈D_P_i, and the correspondence g7→ˆggives an isomorphism DP_i ∼= Gal(FP_i/kp). The subfield ofF corresponding toDP_i is called the decomposition field ofPi and is denoted byZP_i. Furthermore, g∈DP_i induces the isomorphism g ofFP_i =OF/Pi overFp=Ok/pi defined by g(αmodPi) := g(α) modPi (α ∈ OF). The map g 7→ g gives a homomorphism DP_i →Gal(FP_i/Fp) whose kernelIP_i={g∈DP_i|g= id_F_P

i}is called the inertia group ofPi. Ifg(Pi) =Pj forg∈G, we haveIP_j =gIP_ig⁻¹ and hence #IP_i is independent ofPi. Pute=ep := #IP_i. The subfield ofF corresponding toIP_i is called the inertia field ofPi and is denoted byTPi.

F ⊃ TP_i ⊃ ZP_i ⊃ k 1−→^e I_P_i −→^f D_P_i −→^r G.

Here we have equalities #D_P_i =ef and #I_P_i =e for f := # Gal(FPi/Fp). We can show that the homomorphism D_P_i → Gal(FPi/Fp);g 7→ g is surjective, and

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there is an exact sequence

1→I_P_i→D_P_i→Gal(FPi/Fp)→1.

LetP_i,T :=P_i∩OTPi andP_i,Z :=P_i∩OZPi. Then we have the following theorem.

Theorem1.1.1. The extensionF/T_P_i is a ramified extension of degreeesuch that the ramification index of P_i over P_i,T is e. The extension T_P_i/Z_P_i is a cyclic extension of degreef such that the covering (inert) degree ofP_i,T overP_i,Z is f. The extension Z_P_i/k is an extension of degree r such that p is completely decomposed intor prime ideals containingPi,Z as one prime factor.

1.2. Hilbert theory for 3-manifolds

Now we study Hilbert theory for 3-manifolds. Let h : N → M be a finite Galois (regular) cover of connected oriented closed 3-manifolds branched over a link L ⊂M, and putX =M −L, Y =N−h⁻¹(L), G= Gal(h), andn = #G.

(The Galois group Gal(h) = Gal(Y /X) = Gal(N/M) = Deck(h) of his also called the deck transformation group ofh.) Let Kbe a knot in M which is a component of Lor disjoint from L, and suppose thath⁻¹(K) = K1∪. . .∪Kr is an r=rK- component link. For a tubular neighborhood VK of K, letVK_i be the connected component of h⁻¹(VK) containing Ki. (They are canonical up to isotopy, in any category.) Fix a base pointx∈∂VK. Supposeh⁻¹(x) ={y1, . . . , yn}. Letρ:GL= π₁(X, x) → Aut(h⁻¹(x)) be the monodromy permutation representation, which induces an isomorphismπ₁(X, x)/h_∗(π₁(Y, y_i))∼= Imρ∼=G. Note thatπ₁(X, x)∼= Gacts transitively on the set of knots S_K := {K₁, ..., K_r} lying overK. We call the stabilizer D_K_i ={g ∈G| g(K_i) =K_i} ofK_i thedecomposition group ofK_i. Since we have the bijection G/D_K_i ∼=S_K for each i, #D_K_i =n/r is independent of Ki. In fact, if g(Ki) = Kj, then DK_j =gDK_ig⁻¹. Since each g ∈ G induces a homeomorphism g|∂V_Ki : ∂V_K_i →^∼⁼ ∂V_g(K_i₎, g|∂V_Ki is a covering transformation of ∂V_K_i over ∂V_K for each g ∈D_K_i, and the corresponding g → g|∂V_Ki gives an isomorphismDK_i∼= Gal(∂VK_i/∂VK).The Fox completion of the subcover ofY over X corresponding toDK_i is called thedecomposition cover ofKiand is denoted by ZK_i. The map g 7→ g¯ = g|∂V_Ki induces an homomorphismDK_i → Gal(Ki/K), whose kernel IK_i = {g ∈ DK_i | g¯ = idK_i} is called the inertia group of Ki. If Kj = g(Ki) (g ∈ G), one has IK_j = gIK_ig⁻¹, and hence #IK_i is independent of Ki. Set e = eK := #IK_i. The Fox completion of the subcover of Y over X corresponding toIKi is called the inertia cover ofKi and is denoted by TKi:

N−→T_K_i −→Z_K_i −→M 1−→^e I_K_i −→^f D_K_i−→^r G.

Here we have equalities #DK_i =ef andIK_i =eforf := Gal(Ki/K).By comparing their orders, we see that the homomorphismDK_i →Gal(Ki/K);g7→g¯is surjective, and there is an exact sequence

1→IK_i →DK_i →Gal(Ki/K)→1.

Let K_i,T be the image of K_i under N → T_K_i, and let K_i,Z be the image of K_i,T underT_K_i →Z_K_i. Then we have the following.

Theorem 1.2.1. The map N→T_K_i is a branched cover of degree e such that the branching index ofK_i overK_i,T ise. The mapT_K_i→Z_K_i is a cyclic cover of

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degree f such that the covering degree ofKi,T overKi,Z isf. The mapZK_i →M is a cover of degree r such that K is completely decomposed into anr-component link containingKi,Z as a component.

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Id` elic class field theory for 3-manifolds

Id`elic class field theory is not only a sophisticated reformulation of the classical ideal-theoretic class field theory but also has various theoretical advantages, and its construction is based on combining local class field theory at all primes of a number field. As M²KR dictionary tells us, a topological analogue of the ideal- theoretic class field theory may be given by the Hurewicz isomorphism. In this chapter, we study a topological analogue of id`elic class field theory for 3-manifolds.

In Section 2.1, we review the local class field theory for local fields and idèlic class field theory for number fields. In Section 2.2, we give a topological analogue of local class field theory for a 2-dimensional torus. In Section 2.3, we introduce the notion of very admissible link K in a 3-manifold M, which may be regarded as an analogue of the set of primes of a number field, and prove its existence. In Section 2.4, we introduce the notion of a universalK-branched cover, which may be regarded as an analogue of an algebraic closure of a number field, and we also discuss the role of base points. In Section 2.5, we introduce the notion of an idèle for a pair (M,K) and establish a topological analogue of idèlic class field theory for (M,K), by combining local class field theory for tori. In Sections 2.6 and 2.7, we introduce certain topologies on our idèle class group, called the standard topology and the norm topology, and then establish a topological analogue of the existence theorem for (M,K). In Section 2.8, we discuss an analogue of the norm residue symbol for a 3-manifold. In Section 2.9, we calculate the Tate cohomology of our idèle class group in light of the comparison with the axiom of class field theory.

This chapter is based on [NU].

2.1. Id`elic class field theory for number fields

In this section, we briefly review the id`elic class field theory for number fields, whose topological analogues will be studied in later sections. We consult [KKS11]

and [Neu99] as basic references for this section.

2.1.1. Local theory. We firstly review the local theory. Letk be a number field, that is, a finite extension of the rationalsQ, and letp⊂ Ok be a prime ideal of its integer ring. Then, for a local field kp, we have the following commutative diagram of splitting exact sequences.

0 //O^×p //

k^×_p ^v^p //

ρ_p

Z //

0

0 //Gal(kâb_p /kûr_p ) //Gal(kâb_p /k_p) //Gal(kûr_p /k_p) //0

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Here,O^×p is the local unit group,vpis thevaluation,k^ab_p /kpis the maximal abelian extension, and k_p^ur/kp is the maximal unramified abelian extension. The mapρp

is called thelocal reciprocity homomorphism, which is a canonical injective homomorphism with dense image, and controls all the abelian extensions of the local fieldkp. In the lower line,I_pâb= Gal(kâb_p /kûr_p ) is the abelian quotient of the inertia group, and we have Gal(kûr_p /kp)∼= Gal(Fp/Fp)∼=Zb.

The theory of a local field is rather complicated. There are non-abelian extensions, and there are notions of wild and tame for ramifications. For the tame quotients, we have an exact sequence

1→I_p^t=⟨τ⟩ →π₁^t(Spec(kp)) =⟨τ, σ|τ^q⁻¹[τ, σ]⟩ →Gal(Fp/Fp) =⟨σ⟩ →1, whereτ andσ=σFrob are called the monodromy and the Frobenius, respectively.

2.1.2. Definitions. Next, we review the global theory. Let k be a number field. We define theid`ele group Ik of k by the following restricted product of k_p^× with respect to the local unit groupsUp over all primespofk:

I_k:=∏⨿

p

k^×_p =



(a_p)_p ∈ ∏

p: prime

k_p^×

v_p(a_p) = 0 for almost all finite primesp



. Since we have v_p(a) = 0 for a ∈ k^× and for almost all finite primes p, k^× is embedded into I_k diagonally. We define the principal id`ele groupP_k of k by the image ofk^× inI_k, and theid`ele class groupofkby the quotientC_k:=I_k/P_k.

Then, the homomorphism to the ideal groupφ:Ik →I(k); (ap)p7→∏

pp^v^p^(a^p⁾ induces an isomorphism

Ik/(Uk·Pk)∼= Cl(k), where Uk = Kerφ = ∏

pU_p denotes the unit id`ele group and Cl(k) denotes the ideal class group ofk.

2.1.3. Standard topology. The id`ele class group C_k equips the standard topology, which is the quotient topology of the restricted product topology on the id`ele groupI_k of the local topologies, defined as follows.

We firstly consider on O^×p the relative topology oflocal norm topology of k_p^×, and re-define thelocal topologyonk^×_p as the unique topology such that the inclusion O^×p ,→k^×_p is open and continuous. (For this local topologies, I_k is the restricted products with respect to the family of open sets{

O^×p ⊂k_p^×}

p.)

Next, for each finite set of primesT which includes all the infinite primes, we consider the product topogy onG(T) =∏

p∈Tk^×_p ×∏

p̸∈TO^×p.

Then, we define the standard topology onIk so that each subgroup H < Ik is open if and only ifH∩G(T) is open for everyT.

This standard topology on C_k diﬀers from the one defined as the quotient topology of relative topology of product topology of the local topologies on I_k <

∏k_p^×, and it is finer than the latter.

2.1.4. Norm topology. For a finite abelian extension F/k, the norm map N_{F /k}:C_F →C_k is defined as follows.

Let p be a prime of k and F_p^× := ∏

P|pF_P^×. Each αp ∈ F_p^× defines a kp- linear automorphism α_p : F_p^× → F_p^×; x 7→ α_px, and the norm of α_p is defined by N_F_p_/k_p(α_p) = det(α_p). It induces a homomorphism N_F_p_/k_p : F_p^× → k^×_p, and

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the norm homomorphism NF /k : IF → Ik on the idèle groups. SinceNF /k sends the principal idèles to principal idèles, it also induces the norm homomorphism NF /k:CF →Ck on the idèle class groups.

For a number fieldk, the id`ele class groupC_kequips thenorm topology, so that it is a topological group, and the family ofN_{F /k}(C_F) is a fundamental system of neighborhoods of 0, whereF/kruns through all the finite abelian extensions of k.

Proposition2.1.1. A subgroupHofCk is open and of finite index with respect to the standard topology if and only if it is open with respect to the norm topology.

2.1.5. Global theory. Here is a main theorem of id`elic class field theory for number fields (cf. [Neu99,§6], Theorem 6.1):

Theorem2.1.2 (Id`elic class field theory for number fields). Letk be a number field and letk^ab denote the maximal abelian extension ofk which are fixed inC. (1) (Artin’s global reciprocity law.) There is a canonical surjective homomorphism, called the global reciprocity map,

ρk :Ck→Gal(k^ab/k) which has the following properties:

(i)For any finite abelian extension F/k inC,ρ_k induces an isomorphism Ck/N_{F /k}(CF)∼= Gal(F/k).

(ii)For each prime p ofk, we have the following commutative diagram k^×_p

ι_p

ρ_kp

//

⟲

Gal(k_p^ab/kp)

C_k _ρ

k //Gal(k^ab/k),

whereι_p is the map induced by the natural inclusionk^×_p →I_k. (2) (The existence theorem.) The correspondence

F 7→ N =N_{F /k}(CF)

gives a bijection between the set of finite abelian extensions F/k in C and the set of open subgroups N of finite indices of Ck with respect to the standard topology.

Moreover, the latter set coincides with the set of open subgroups ofCk with respect to the norm topology.

In the proof, we use thenorm residue symbol ( , F/k) :C_k ↠Gal(F/k). For this map, we have Ker(, F/k) =N_{F /k}(CF).

2.2. Local class field theory for a 2-dimensional torus

In this section, we study the local theories in 3-manifolds. LetK be a knot in its tubular neighborhoodVK in a 3-manifold M. In our context, the local theory for 3-manifolds is nothing but the Galois theory for the covers of ∂VK, which is dominated by an abelian group π₁(∂V_K) = ⟨µ_K, λ_K|[µ_K, λ_K]⟩ ∼= H₁(∂V_K) ∼=Z². (In a sense, the tame case of a local field is a “quantized” version of this case.) For

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each manifoldX, let Gal(X/X) denote the Galois group of the universal cover. Wee have the following commutative diagram of exact sequences.

0 //⟨µK⟩ //

H1(∂VK) ^v^K //

Hur.

H1(VK) =⟨λK⟩ //

0

0 //Gal(∂VgK/∂VfK) //Gal(∂VgK/∂VK) //Gal(VfK/VK) //0 By an isomorphism ∂_∗ :H2(VK, ∂VK) =⟨[DK]⟩→ ⟨^∼⁼ µK⟩, classes of meridian disks D_K can be seen as analogues of local units. The map v_K is the one induced by the natural injection ∂V_K ,→ V_K, which is an analogue of the valuation map v_p in number theory. The vertical maps are the Hurewicz isomorphisms. In the lower line,I_K := Gal(∂Vg_K/∂Vf_K) is the inertia group, and we have Gal(Vf_K/V_K)∼= Gal(K/K)e ∼=Z.

2.3. Very admissible links

In this section, in order to build an id`ele theory in 3-dimensional topology, we introduce the notion of avery admissible link Kin a 3-manifoldM, as an analogue object of the set of all the primes in a number fieldk. It was originally introduced by Niibo in [Nii14] as a very admissible knot set. We prove the existence, and give some remarks.

Id`ele theory sums up all the local theories and describes the global theory. In number theory, each prime equips local theory, and in an extension of number field F/k, every prime ofF is above some prime ofk, In 3-dimensional topology, local theories are the theories of covers over the tubular neighborhoods of knots. We define a very admissible link as follows.

Definition2.3.1. LetM be a closed, oriented, connected 3-manifold, and let K be a finite or infinite link inM equipped with a tubular neighborhood. We say K is anadmissible link of M if the components ofK generatesH1(M). We sayK is avery admissible link ofM if for any finite abelian coverh:N →M branched over a finite link inK, the components of the link h⁻¹(K) generatesH₁(N).

In the following, a link means a finite or infinite link equipped with a tubular neighborhood. For a linkL in a 3-manifold M, we denote by VL =⊔K⊂LVK the tubular neighborhood, whereK runs through the components of K, and VK is a tubular neighborhood of K. We denote the meridian by µK ∈ H1(∂VK), and fix an element λK ∈ H1(∂VK) such that µK and λK form a basis of H1(∂VK). In this paper, we call such λK a longitude of K. For a branched cover h: N →M, for each component ofh⁻¹(K) inN, we fix a meridian and a longitude which are components of the preimages of those ofK.

Lemma 2.3.2. Let M be a closed, connected, oriented 3-manifold andLa link in M. Then, there is a link L in M such thatL⊂ L and for any finite coverh: N →M branched over a finite sublink ofL,H1(N)is generated by the components of the preimageh⁻¹(L).

Proof. The set of all the finite branched covers of M branched over finite sublinks of L is countable, and can be written as{h_i:N_i→M}i∈N, where h₀ = id_M. Indeed, for a finite sublink L^′ ⊂ L, each branched cover branched over L^′

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can be obtained as the Fox completion of a cover of the exterior XL^′ :=M \L^′. Each cover ofXL^′ corresponds to each subgroup ofπ1(XL^′) by Galois theory. Each group π1(XL^′) is finitely generated, and the set of its subgroups of finite indices is countable. Since the set of finite links ofL is countable, so the set of branched covers.

We construct an inclusion sequence of links {L_i}_i as follows: First, we put L₋₁ = L. Next, for i ∈ N, let L_i₋₁ be given. Since N_i is compact, H₁(N_i) is finitely generated, and we can take a linkfL_i inN_isuch that it includesh⁻_i¹(L_i₋₁), its components generates H1(Ni), and its image hi(fLi) is again a link inM. We putL_i=h_i(fL_i). Then, the unionL:=∪iL_isatisfies the expected condition. □ Theorem 2.3.3. Let M be a closed, connected, oriented 3-manifold, and L a link inM. Then, there is a very admissible linkK which includesL. Moreover, we can remove the condition “abelian” in the definition.

Proof. We construct an inclusion sequence of links{Ki}ias follows: First, we putK−1=L. Next, fori∈N, letKi−1be given, and letKibe a link obtained from Ki−1 by the above Lemma 2.3.2. Then the union K :=∪Ki is a very admissible

link, and the condition of abelian is removed. □

LinksLandKin the lemma and theorem above may be taken smaller than in the constructions. It may be interesting to ask whether they can be finite.

LetM =S³. The unknot is very admissible link. IfLis the trefoil, by taking branched 2-cover, we see that K1 is greater thanL. We expect thatK has to be infinite.

Next, let M be a 3-manifold, and L a minimum admissible link (L can be empty). For an integral homology 3-sphere M, we have K = L = ϕ. For a lens spaceM =L(p,1) orM =S²×S¹, we can take a knot (simple loop)K=L=K.

In the latter sections of this paper, we assume that a very admissible linkK is an infinite link. However, our argument are applicable for finiteK also.

Remark2.3.4 (variants). (1) We will discuss a weaker condition on an admissible link in the end of Section 5,Remark2.5.9.

(2) LetL be an infinite link such that any (ambient isotopy class of) finite link in M is contained in L. There exists such a link. Indeed, since the classes of finite links are countable, by putting links side by side inS³=R³∪ {∞}, we obtain such a link L, with one limit point at ∞. By using the metric of R³, we can take a tubular neighborhood ofL. If we start the construction form such an infinite link, then we obtain a special very admissible link K, which controls all the branched covers ofM branched over any finite links inM.

(3) We could remove the condition of “abelian” on the covers in the definition and the constructions. In addition, we can replace every “the components generatesH₁” by “the components with paths to the base point generatesπ₁”. Then we obtain a very admissible link K which may be used in the future work of non-abelian class field theory for 3-manifolds.

Strictly speaking, a very admissible link is an analogue of the set of finite primes.

According to [Mor12], counterparts of infinite primes are ends of 3-manifolds. F.

Hajir also studies cusps of hyperbolic 3-manifolds as analogues of infinite primes of number fields in [Haj12]. In this paper, however, since we deal with closed manifolds, the counterpart of the set of infinite primes is empty.

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