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Adelic Solenoid I: Structure and Topology

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Adelic Solenoid I: Structure and Topology Adelic solenoid I: Structure and topology Juan M. Burgos, Alberto Verjovsky ∗ August 6, 2019 2010 Mathematics Subject Classification. Primary: 11R56, 32L05, 55M25, 57R30. Secondary: 22Bxx. Key Words: Adelic, Solenoid, Birkhoff-Grothendieck Abstract Topologically the adelic Riemann sphere is the suspension of the adelic solenoid and because of this relation, here we study the adelic solenoid by studying the adelic Riemann sphere topology. The main result is the Birkhoff-Grothendieck Theorem: A holomorphic vector bundle splits as a sum of holomorphic line bundles whose Chern character is now a rational number. As a consequence, the Picard group is isomorphic to the additive group of rational numbers and the K–ring has new elements that factor the tautological class. 1 Introduction Since its creation by Claude Chevalley and Andr´eWeil the ring of ad`eles of the rationals ′ AQ = R × Qp has played a fundamental role in number theory, for instance in class Qp field theory [RV], [Ta] and the Langlands program. The canonical diagonal inclusion i : Q → AQ embeds Q as a discrete cocompact subgroup of AQ which we identify with Q. The quotient AQ/Q is the ad`ele class group with its additive structure is a compact abelian group and its Pontryagin dual is the additive group of the rationals (Q, +) with the discrete topology. There is another description of AQ/Q as the inverse limit of all n 1 1 arXiv:1603.05676v4 [math.CV] 2 Aug 2019 finite coverings pn(z) = z (z ∈ S , n ∈ Z) of the circle S . This is a one dimensional solenoidal compact abelian group in the sense of Pontryagin. It is a sort of “diffuse circle” (a lamination, a current or a foliated cycle in the sense of Sullivan [Su2]) and in fact we denote this group as 1 ∨ SQ = Q = Pontryagin dual of Q to convey the idea that it is a generalization of a circle. This solenoid is a lamination 1 with dense leaves which are densely immersed copies of the real line. The solenoid SQ is also called the algebraic universal covering space of the circle S1. The Grothendieck–Galois ˆ 1 group of the covering is Z, the ´etale fundamental group of SQ [SGA1]. ∗Grant support listed here. 1 Concerning dynamical systems theory, in [Su] (see also [Su2]) D. Sullivan studies the linking between universalities of Milnor-Thurston, Feigenbaum’s (quantitative) and Ahlfors-Bers. As he points out in his second example, the dynamical suspension of the 1 square map from the diadic solenoid S2 to itself is the basic solenoidal surface required in the dynamical theory of Feigenbaum’s Universality [Fe]. This object can be seen as a ∗ quotient of ∆Q, the inverse limit of the punctured disk by the inverse system of coverings considered before. This is a two dimensional solenoid with hyperbolic leaves and so is L. This work was continued, for instance, in the use of 3-dimensional hyperbolic laminations by Misha Lyubich and Yair Minsky in [LM]. If instead of the circle we consider the same inverse system of coverings on the Riemann 1 sphere, its inverse limit is the adelic Riemann sphere and we denote it by CPQ. Now the finite coverings ramify on 0 and ∞ giving two cusps in the inverse limit. Topologically, it is the suspension of the adelic solenoid and because of this relation, here we study the adelic solenoid by studying the adelic Riemann sphere topology. We prove the analog of the Birkhoff-Grothendieck Theorem [Bi], [Gr]: A holomorphic vector bundle splits as a sum of holomorphic line bundles whose Chern character is now a rational number. In particular, the Picard group is isomorphic to the additive group of rational numbers instead of the integers as in the usual case. The K–ring has also new elements that factor the tautological class x: Z[xq / q ∈ Q] K CP 1 ∼= Q h (xq − 1)(xp − 1) / q,p ∈ Qi Note that, as it was expected, the K–ring of the Riemann sphere embeds in this ring: Z[x] K(CP 1) ֒→ K CP 1 =∽ h (x − 1)2i Q where x is the image of the class represented by the tautological complex line bundle. For the proof of these results, first we must state some preliminaries about the adelic solenoid algebraic structure and limit periodic maps. We also develop the notion of degree for a continuous map from the adelic solenoid to itself. It is a homotopy invariant taking values in the rational numbers and extends the usual degree in circles. This is the first part of a series of two papers. The next part generalizes Ahlfors-Bers theory to the solenoidal context. 2 Adelic solenoid In what follows we will identify the group U(1) with the unit circle S1 = {z ∈ C : |z| =1} and the finite cyclic group Z/nZ with the group of nth roots of unity in S1. By covering space theory, for any integer n =6 1, it is defined the unbranched covering 1 1 n + space of degree n, pn : S → S given by z 7−→ z . If n, m ∈ Z and n divides m, then 1 1 m/n there exists a covering map pn,m : S → S such that pn ◦pn,m = pm where pn,m(z)= z . th We also denote with the same letters the restriction of pn and pn,m to the n roots of unity. In particular we have the relation: pn,m ◦ pm,l = pn,l 2 1 This determines a projective system of covering spaces {S ,pn,m}n,m≥1,n|m whose projec- tive limit is the universal one–dimensional solenoid or adelic solenoid S1 := lim S1. Q p ←−n 1 Thus, as a set, SQ consists of sequences (zn)n∈N,z∈S1 which are compatible with pn i.e. pn,m(zm)= zn if n divides m. 1 πn 1 The canonical projections of the inverse limit are the functions SQ → S defined by πn (zj)j∈N = zn and they define the solenoid topology as the initial topology of the family. The solenoid is an abelian topological group and each πn is an epimorphism. In particular each πn is a character which determines a locally trivial Zˆ–bundle structure where the group Zˆ := lim Z/mZ p ←−n is the profinite completion of Z, which is a compact, perfect and totally disconnected abelian topological group homeomorphic to the Cantor set. Being Zˆ the profinite com- pletion of Z, it admits a canonical inclusion of Z ⊂ Zˆ whose image is dense. We have an ˆ φ 1 ˆ φ 1 π1 1 inclusion Z → SQ and a short exact sequence 0 → Z → SQ → S → 1. 1 The solenoid SQ can also be realized as the orbit space of the Q–bundle structure Q ֒→ A → A/Q, where A is the ad`ele group of the rational numbers which is a locally ∼ 1 compact Abelian group, Q is a discrete subgroup of A and A/Q = SQ is a compact Abelian group (see [RV]). From this perspective, A/Q can be seen as a projective limit 1 whose n–th component corresponds to the unique covering of degree n ≥ 1 of SQ. By considering the properly discontinuously diagonal free action of Z on Zˆ × R given by n · (x, t)=(x + n, t − n), (n ∈ Z, x ∈ Zˆ, t ∈ R) 1 ˆ the solenoid SQ is identified with the orbit space Z ×Z R. Here, Z is acting on R by covering transformations and on Zˆ by translations. The path–connected component of 1 the identity element 1 ∈ SQ will be called the baseleaf [Od] and it is a densely immersed copy of R. 1 Hence SQ is a compact, connected, abelian topological group and also a one-dimensional lamination where each “leaf” is a simply connected one-dimensional manifold, homeomor- phic to the universal covering space R of S1, and a typical “transversal” is isomorphic ˆ 1 ∞ to the Cantor group Z. The solenoid SQ also has a leafwise C Riemannian metric (i.e., C∞ along the leaves) which renders each leaf isometric to the real line with its standard metric dx. So, it makes sense to speak of a rigid translation along the leaves. The leaves also have a natural order equivalent to the order of the real line hence also an orientation. Summarizing the above discussion we have the commutative diagram: 1 1 1 pm,n 1 1 SQ = lim S / ...S / S / ...S (1) O O O O φ l 7→e2πil/n l 7→e2πil/m 0 7→1 ? ? pm,n ? ? Zˆ = lim Z/nZ / ... Z/nZ / Z/mZ / ... {0} 3 where Zˆ is the adelic profinite completion of the integers and the image of the group ˆ 1 monomorphism φ : (Z, +) → (SQ, ·) is the principal fiber. We notice that πn(x) = πn(y) −1 −1 implies πn(y x) = 1 and therefore y x = φ(a) where a ∈ nZˆ for some n ∈ Z ⊂ Zˆ. Lemma 2.1. The following is a short exact sequence of topological abelian groups: ˆ φ 1 π1 1 0 / Z / SQ / S / 1 and we have the commutative diagram: ˆ φ 1 π1 1 0 / Z / SQ / S / 1 O O O = pn ? ˆ φ 1 πn 1 0 / nZ / SQ / S / 1 Proof. • By definition the following diagram commutes: 1 π1 1 SQ / S O O φ 0 7→1 ? ? Zˆ / {0} In particular π1 ◦ φ = 1 and Im(φ) ⊂ ker(π1). Suppose that π1(x) = 1. Then 2πibn/n 2πibm/m x =(...,an,...,am,... 1)=(...,e ,...,e ,..
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