The p-adic solenoid Jordan Bell [email protected] Department of Mathematics, University of Toronto November 19, 2014 1 Definition We shall be speaking about locally compact abelian groups, and unless we say otherwise, by morphism we mean a continuous group homomorphism. n For p prime and n 2 Z≥0, p Z is a closed subgroup of the locally compact abelian group R, and the quotient R=pnZ is a compact abelian group. For n m n ≥ m, let φn;m : R=p Z ! R=p Z be the projection map, which is a morphism. n The compact abelian groups R=p Z and the morphisms φn;m are an inverse system, and the inverse limit is a compact abelian group denoted Tp, called the n p-adic solenoid, with morphisms φn : Tp ! R=p Z. Because the maps φn;m : n m n 1 R=p Z ! R=p Z are surjective, the maps φn : Tp ! R=p Z are surjective. n Let πn : R ! R=p Z be the projection map, which is a morphism. The projection maps πn are compatible with the inverse system φn;m, so there is a unique morphism π : R ! Tp such that φn ◦ π = πn for all n 2 Z≥0. If x; y 2 R are distinct, then for sufficiently large n we have πn(x) 6= πn(y). If π(x) = π(y) then πn(x) = φn(π(x)) = φn(π(y)) = πn(y), a contradiction. n Therefore π : R ! Tp is injective. Furthermore, the maps πn : R ! R=p Z 2 being surjective implies that the image π(R) is dense in Tp. 2 Pontryagin dual If G is a locally compact abelian group, we denote by G∗ the collection of morphisms G ! S1. We assign G∗ the coarsest topology such that for all g 2 G, the map γ 7! γ(x) is continuous G∗ ! S1, and with this topology, G∗ is a locally compact abelian group, called the Pontryagin dual of G. If φ : G ! H is a morpism of locally compact abelian groups, then φ∗ : H∗ ! G∗ defined by φ∗(θ)(g) = θ(φ(g)); θ 2 H∗; g 2 G; 1Alain M. Robert, A Course in p-adic Analysis, Chapter 1, x4, p. 29. 2Luis Ribes and Pavel Zalesskii, Profinite Groups, p. 7, Lemma 1.1.7. 1 ∗ ∗ is a morphism. Say φ is surjective, and φ (θ1) = φ (θ2) but that θ1 6= θ2. Then there is some h 2 H such that θ1(h) 6= θ2(h). Since φ : G ! H is surjective, there is some g 2 G such that φ(g) = h. But then ∗ ∗ θ1(h) = θ2(φ(g)) = φ (θ1)(g) = φ (θ2)(g) = θ2(φ(g)) = θ2(h); ∗ contradicting θ1(h) 6= θ2(h). Therefore, if φ : G ! H is surjective then φ : H∗ ! G∗ is injective. Let 1 j = : j 2 ⊂ ; pn Z pn Z Q which with the discrete topology is a discrete abelian group. 1 n ∗ Theorem 1. For prime p and n 2 Z≥0, the map Φn : pn Z ! (R=p Z) defined by 1 Φ (a)(x + pn ) = e2πiax; a 2 ; x + pn 2 =pn ; n Z pn Z Z R Z is an isomorphism of topological groups. j n n n Proof. Write a = pk , j 2 Z. If x + p Z = y + p Z, then x − y 2 p Z, so x − y = pnk for some k 2 Z. Then j n j n 2πiax 2πi n (p k+y) 2πik+2πi n y 2πiay n Φn(a)(x+p Z) = e = e p = e p = e = Φn(a)(y+p Z); showing that Φn is well-defined. Furthermore, one checks that indeed Φn(a) 2 n ∗ 1 (R=p Z) for each a 2 pn Z. It is apparent that Φn(a + b) = Φn(a) · Φn(b). Φ is continuous because 1 n n pn Z is discrete. If Φn(a) = Φn(b), this means that for all x + p Z 2 R=p Z, e2πiax = e2πibx, equivalently, that (a − b)x 2 Z for all x 2 R, whence a = b. Thus Φn is injective. n ∗ 1 ∗ Let γ 2 (R=p Z) . Define Γ : R ! S by Γ = γ ◦ πn, so that Γ 2 R . We take as given that because Γ 2 R∗, there is some y 2 R such that Γ(x) = e2πiyx for all x 2 R. In particular, for x = pn, on the one hand n n n Γ(p ) = γ(πn(p )) = γ(0 + p Z) = 1; and on the other hand n Γ(pn) = e2πiyp ; n 1 so yp 2 Z, i.e. y 2 pn Z, and it follows that γ = Φn(y). Therefore Φn is surjective. The open mapping theorem for topological groups states that if G; H are locally compact groups, f : G ! H is a surjective morphism, and G is σ-compact, then f is open. Z is discrete and countable, hence is σ-compact, so Φn is open. Therefore Φn is an isomorphism of topological groups. 2 n m Because the morphisms φn;m : R=p Z ! R=p Z are surjective, the mor- ∗ m ∗ n ∗ phisms φn;m :(R=p Z) ! (R=p Z) are injective. For m ≤ n, define ιm;n : 1 1 j j pn−mj 1 pm Z ! pn Z by ι pm = pm = pn 2 pn Z; this is an injective morphism. One checks that the following diagram commutes. ∗ φn;m (R=pmZ)∗ (R=pnZ)∗ Φm Φn 1 ιm;n 1 pm Z pn Z 1 The discrete groups pm Z and the morphisms ιm;n are a direct system. The localization of Z away from p is the abelian group j [1=p] = : j 2 ; m 2 ⊂ : Z pm Z Z≥0 Q We assign Z[1=p] the discrete topology. One proves that Z[1=p] with the maps 1 ιm : pm Z ! Z[1=p] defined by j j ι = m pm pm 3 1 1 is the direct limit of this direct system. The direct system ιm;n : pm Z ! pn Z n m is dual to the inverse system φn;m : R=p Z ! R=p Z. It follows that the Pontryagin dual of the limit of either system is isomorphic as a topological group to the limit of the other system. That is, ∗ ∼ ∗ ∼ Tp = Z[1=p]; (Z[1=p]) = Tp; as topological groups. 3 p-adic integers n m For n ≥ m, let n;m : Z=p Z ! Z=p Z be the projection map. With the discrete topology, Z=pnZ is a compact abelian group, as it is finite. Then n m n;m : Z=p Z ! Z=p Z is an inverse system, and its inverse limit is a compact abelian group denoted Zp, called the p-adic integers, with morphisms n : n Zp ! Z=p Z. Because the morphisms n;m are surjective, the morphisms n are surjective. n n Let λn : Z=p Z ! R=p Z be the inclusion map. Then the morphisms n Λn = λn ◦ n : Zp ! R=p Z are compatible with the inverse system φn;m : 3A direct limit of discrete abelian groups is the direct limit of abelian groups. On di- rect limits of abelian groups, cf. Luis Ribes and Pavel Zalesskii, Profinite Groups, p. 15, Proposition 1.2.1. 3 n m R=p Z ! R=p Z, so there is a unique morphism Λ : Zp ! Tp such that φn ◦ Λ = Λn for all n 2 Z≥0. Suppose that x; y 2 Zp are distinct and that Λ(x) = Λ(y). It is a fact that there is some n such that n(x) 6= n(y). Because λn is injective, this implies that Λn(x) 6= Λn(y), and this contradicts that Λ(x) = Λ(y). Therefore Λ : Zp ! Tp is injective. One proves that ker φ0 = Λ(Zp), so that 0 ! Zp ! Tp ! R=Z ! 0 is a short exact sequence of topological groups.4 It can be proved that for each m 2 Z>0 such that gcd(m; p) = 1, the p-adic solenoid Tp has a unique cyclic subgroup of order m, and on the other hand that there is no element in Tp whose order is a power of p, namely, Tp has no p-torsion.5 4 Further reading Garrett has written several notes on the p-adic solenoid.6 The p-adic solenoid occurs in several places in the books of Hofmann and Morris.7 For properties of the p-adic solenoid involving homological algebra, see the below references.8 4Alain M. Robert, A Course in p-adic Analysis, Chapter 1, Appendix, p. 55. 5Alain M. Robert, A Course in p-adic Analysis, Chapter 1, Appendix, pp. 55{56. 6Paul Garrett, Solenoids, http://www.math.umn.edu/~garrett/m/mfms/notes/02_ solenoids.pdf; Paul Garrett, Bigger diagrams for solenoids, more automorphisms, colimits, http://www.math.umn.edu/~garrett/m/mfms/notes/03_more_autos.pdf; Paul Garrett, The ur-solenoid and the adeles, http://www.math.umn.edu/~garrett/m/mfms/notes/04_ur_ solenoid.pdf 7Karl H. Hofmann and Sidney A. Morris, The Structure of Compact Groups, 2nd revised and augmented edition; Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups. 8 For Ext(Z; Tp) see Jean Dieudonn´e, A History of Algebraic and Differential Topol- ogy, 1900 { 1960, p. 94; see also J. M. Cordier and T. Porter, Shape Theory: Cate- gorical Methods of Approximation, p. 83; and http://mathoverflow.net/questions/4478/ torsion-in-homology-or-fundamental-group-of-subsets-of-euclidean-3-space 4.