Adelic Solenoid I: Structure and Topology

Home , Solenoid (mathematics)

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A steivrelmto all of limit inverse the as tdigteadelic the studying y Q he edekTheorem: hendieck hc eietf with identify we which ∗ ebnlswhose bundles ne ePcr group Picard he K r f“iuecircle” “diffuse of ort rn a new has –ring e fterationals the of les oas( ionals esolenoid he Q lamination , )with +) S class Q 1 is n 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 projective 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 completion 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 adèle 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, homeomorphic 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 ,... 1) = φ(y)

such that y = (...,bn,...,bm,... 0). We have proved that ker(π1) ⊂ Im(φ). Be- cause π1 is an epimorphism and φ is a monomorphism we have the ﬁrst item. • For the second item, the second exact sequence follows exactly from the same ar- n guments as the ﬁrst. Because π1 = z ◦ πn, we have the right commutative square. The left square is trivial (diagram chasing).

1 We define the baseleaf as the image of the monomorphism ν : R → SQ defined as follows: 1 1 1 pm,n 1 1 SQ = lim S / ...S / S / ...S (2) O O O O ν t 7→eit/n t 7→eit/m t 7→eit ? = = = R / ... R / R / ... R In particular, the immersion ν is a group morphism and comparing the diagrams (1) and (2), we have ν(2πn)= φ(n) for every integer n. Define:

ˆ 1 exp : Z × R → SQ such that exp(a, θ)= φ(a).ν(θ).

4 Lemma 2.2. The following is a short exact sequence of topological abelian groups:

ι ˆ exp 1 0 / Z / Z × R / SQ / 1 such that ι(a)=(a, −2πa).

1 ia Proof. The exponential is an epimorphism: Consider x ∈ SQ and a ∈ R such that e = iθ ia −1 π1(x). Because π1 ◦ ν = e we have that π1(ν(a)) = e = π1(x); i.e. π1(ν(a) x) = 1. By Lemma 2.1 there is an adelic integer b ∈ Zˆ such that φ(b)= ν(a)−1x; i.e. x = φ(b)ν(a)= exp(b, a). The exponential kernel: Suppose that exp(a, θ) = φ(a)ν(θ) = 1. Then φ(a) = ν(−θ) −iθ and composing with π1 we have 1 = e and θ =2πk for some integer k. Then

1 = exp(a, 2πk)= φ(a)ν(2πk)= φ(a)φ(k)= φ(a + k)

Because φ is monomorphism we have that a + k = 0. We conclude that a is an integer and θ = −2πa

1 1 ˆ Corollary 2.3. 1. π1 : SQ → S is a ﬁber bundle with ﬁber isomorphic to Z and monodromy the shift T (x)= x +1.

2. exp is a local homeomorphism.

3. Restricted to a leaf, π1 is a local homeomorphism.

1 4. SQ is the dynamical suspension of the shift T (x)= x +1.

1 5. SQ is foliated by dense R-leaves. Proof. 1. If diam(U) < 2π then U is a trivializing neighborhood of S1.

2. Z acts as translations by ι(Z) and because ι(Z) is discrete in Zˆ × R then Z acts proper and discontinuously. We conclude that exp is a local homeomorphism.

3. By deﬁnition π1 is an open continuous epimorphism. Restricted to a leaf and a trivializing neighborhood π1 is one to one. 4. (x, 2π)+ ι(1) = (x +1, 0) so (x, 2π) ∼ (x +1, 0).

5. The foliation Zˆ × R is invariant under translations by ι(a) for every integer a hence it induces a foliation in the solenoid. Z is dense in its proﬁnite completion Zˆ and so is every coset of Zˆ/Z. By the preceding item, we have that every R-leaf is dense in the solenoid.

5 3 Limit periodic functions

Definition 3.1. Consider a metric space (X,d) and a function f : R → X. f is limit periodic if for every ε> 0 there is a natural number N such that d (f(x +2πn), f(x)) <ε for every x ∈ R and every n ∈ NZ. Do not confuse this concept with the more general one of almost periodic functions based on relatively dense sets [Bo]. An interesting discussion relating limit periodic functions, solenoids and adding machines can be found in [Be]. Usually the concept is defined for real or complex valued functions. However, we will need the generality of considering a metric space. Lemma 3.1. Consider a metric space (X,d) and a function f : R → X. There is a unique function fˆ continuous on the fibers such that the family of its restrictions to the fibers is uniformly equicontinuous and f = fˆ◦ ν iff f is limit periodic.

Proof. Suppose that f is limit periodic. For every real number x, deﬁne the map gx : Z → X such that gx(n) = f(x +2πn). We will prove that gx is uniformly continuous respect to the proﬁnite topology on Z hence it admits a unique extension to Zˆ. Consider ε > 0. There is a natural number N such that d (f(y +2πn), f(y)) < ε for every y ∈ R and every n ∈ NZ. In particular, for every real number x we have:

d (gx(n),gx(m)) = d (f(x +2πn), f(x +2πn +2π(m − n))) <ε for every pair of integers such that m − n ∈ NZ; i.e. The family gx is uniformly equicontinuous respect to the profinite topology. Consider its unique continuous extensiong ˆx to Zˆ and define F : Zˆ × R → X such that F (z, x)=ˆgx(z). Uniqueness of the extension implies that the relation gx(n + m) = gx+2πm(n) extends tog ˜x(z + m)=˜gx+2πm(z) and we have F (z + m, x)= F (z, x +2πm). By Lemma 2.2, F defines a unique function f˜ on the solenoid and by construction it is continuous on the fibers. Conversely, given ε> 0 there is a natural number N such that:

d fˆ◦ exp(n, x), fˆ◦ exp(m, x) <ε for every real number x and every pair of integers such that m − n ∈ NZ. Then: d (f(x +2πn), f(x)) = d fˆ◦ ν(x +2πn), fˆ◦ ν(x) = d fˆ◦ exp(n, x), fˆ◦ exp(0, x) <ε for every real number x and integer n ∈ NZ; i.e. f is limit periodic. Corollary 3.2. A continuous limit periodic function is uniformly continuous. Proof. In the previous Lemma, if f is continuous then, because ν(R) is dense in the solenoid, fˆ is also continuous. The solenoid is compact hence fˆ is uniformly continuous and so is f. Lemma 3.3. Consider a metric space (X,d) and a function f from the solenoid to X such that its restrictions to the ﬁbers is a uniformly equicontinuous family. For every ε> 0, there is a function f ′ such that d(f, f ′) <ε and it factors through some canonical projection πn. Moreover, if f is continuous and (X,d) is a locally euclidean, we can choose f ′ to be continuous.

6 Proof. There is a natural number N such that:

f exp(z + NZˆ, x) ⊂ Bε (f(exp(z, x))) for every z ∈ Zˆ and x ∈ R. Consider the lifting s of πN such that s(1) = e, s is continuous 1 ′ on S −{1} and is continuous from one of the two directions at 1. Deﬁne f := f ◦ s ◦ πN . If f is continuous, taking N big enough we may suppose that exp(NZˆ, 0) ⊂ U where U is a local euclidean chart at e. Consider a continuous function h : X → X such that h f(exp(NZˆ, 0)) = f(e) and is the identity outside U. Now, take as f ′ the function h ◦ f ◦ s ◦ πN . The function h corrects the discontinuity at 1 preserving the condition d(f, f ′) <ε.

Corollary 3.4. 1. Every limit periodic function is the uniform limit of periodic functions.

2. If (X,d) is locally euclidean, then every continuous limit periodic function is the uniform limit of continuous periodic functions.

Proof. Follows from direct application of the Lemmas above.

4 Degree

1 The baseleaf map ν : R → SQ induce a coarser topology on the real line such that ν becomes an embedding. This will be called the leaf topology and it is generated by the following basic open sets:

U(x, ε, N) := I(x +2πNn,ε) n[∈Z where I(x, ε) is the interval with center x and radius ε. The real line with the leaf topology will be denoted by RL. Every continuous map from RL to itself is also continuous from R to itself. However, the converse is not true. A continuous map f from the solenoid to itself can be assumed to be baseleaf preserving just by multiplying it by f(1)−1.

Lemma 4.1. Consider a continuous baseleaf preserving function f from the adelic solenoid to itself. There is a unique rational number q and a unique continuous limit periodic function h such that f0(x) = qx + h(x) where f0 is deﬁned by the following commutative diagram: 1 f 1 SQ / SQ O O ν ν

? f0 ? R / R Proof. Respect to the leaf topology, the baseleaf is an embedding and because the solenoid is compact we have that f0 is uniformly continuous respect to the leaf topology; i.e. For

7 every ε> 0 and natural number λ there is a real number δ > 0 and a natural number N such that: f0 (U(x, δ, N)) ⊂ U (f0(x),ε,λ) (3) for every x ∈ R. Consider ε<π/2 and deﬁne gm : R → R such that gm(x)= f0(x +2πNm) − f0(x) for every integer m. We will prove that there is an integer k such that gm(R) ⊂ I(2πkm,ε) for every integer m. We will prove it in the following steps:

• Base case: Because of (3) we have that g1(R) ⊂ U(0,ε,λ). Because U(0,ε,λ) is a disjoint union of open intervals, g1 is continuous and R is connected, the image of g1 must be contained in one of these intervals; i.e. There is an integer k such that g1(R) ⊂ I(2πk,ε) and λ|k.

• Induction step: Suppose that gm(R) ⊂ I(2πkm,ε) for every natural m ≤ M. Be- cause gM+1(x)= gM (x +2πN)+ g1(x) and the inductive hypothesis, we have that gM+1(R) ⊂ I(2πk(M +1), π). By equation (3) we have gm(R) ⊂ U(0,ε,λ) for every integer m. Then,

gM+1(R) ⊂ I(2πk(M + 1), π) ∩ U(0,ε,λ)= I(2πk(M + 1),ε)

• Trivial case: g0(R)= {0} ⊂ I(0,ε).

• Negative integers: g−m(R) = −gm(R) ⊂ −I(2πkm,ε) = I(2πk(−m),ε) for every natural m.

We have proved a stronger version of equation (3): For every ε> 0 and natural number λ such that ε<π/2, there is a real number δ > 0, a natural number N and an integer k such that: f0 (I(x +2πNm, δ)) ⊂ I (f0(x)+2πkm,ε) (4) for every x ∈ R and every integer m. Let’s see that the quotient k/N is independent of the ε and λ chosen. Consider another 0 < ε′ < π/2 and λ′. There is a real number δ′ > 0 such that δ′ < δ, a natural number N ′ and an integer k′ such that:

′ ′ ′ ′ ′ ′ f0 (I(x +2πN m , δ )) ⊂ I (f0(x)+2πk m ,ε ) (5) for every x ∈ R and every integer m′. Choose m and m′ such that N ′m′ = Nm. Then,

′ ′ ′ ′ ′ ′ ∅= 6 f0 (I(2πN m , δ )) ⊂ I (f0(0)+2πkm,ε) ∩ I (f0(0)+2πk m ,ε ) and because ε,ε′ < π/2 we have that k.m = k′.m′ hence k/N = k′/N ′. Denote this ε,λ–independent rational by q. We claim that:

f0(x)= qx + h(x) where h is a continuous limit periodic function. In eﬀect, because f0 is continuous we have that h is continuous. It rest to show that it is limit periodic. Because we proved

8 that the rational q was ε,λ-independent, equation (4) implies the following: For every ε> 0 there is a natural number N such that:

h(x +2πNm) − h(x)= f0(x +2πNm) − f0(x) − 2πq Nm ∈ I(0,ε) for every x ∈ R and every integer m. This proves the claim. Moreover, this decomposition is unique for a linear limit periodic function must be zero. Deﬁnition 4.1. The rational number in Lemma 4.1 will be called the degree of f and will be denoted deg(f). Proposition 4.2. The degree is a homotopic invariant.

1 1 Proof. Consider a baseleaf preserving homotopy H : SQ × [0, 1] → SQ. By Lemma 4.1, on the baseleaf the homotopy has the following expression:

H0(x, t)= q(t)x + h(x, t) such that t 7→ q(t) ∈ Q is a continuous function hence constant for Q is totally disconnected and we have the result.

The following proposition gives the converse of Lemma 4.1. Proposition 4.3. For every rational q and continuous limit periodic real valued function h, there is a unique continuous baseleaf preserving map f from the solenoid to itself such that the following diagram commutes:

1 f 1 SQ / SQ O O ν ν

? f0 ? R / R where f0(x)= qx + h(x). Proof. Consider a rational p/q such that p and q are coprime natural numbers and deﬁne F : qZ × R → pZ × R such that F (qn, x)=(pn, fn(x)) where: p f (x)= f (x +2πqn) − 2πpn = x + h(x +2πqn) (6) n 0 q for every integer n. Because h is continuous and limit periodic, the function g : qZ×R → R such that g(n, z) = h(x +2πn) admits a unique continuous extensiong ˆ : qZˆ × R → R such thatg ˆ(a, x + 2πq)=g ˆ(a + q, x). Then, there is a unique continuous extension Fˆ : qZˆ × R → pZˆ × R of F such that: p Fˆ(qa, x)= pa, x +ˆg(qa, x) q and satisﬁes the same structural condition as F :

Fˆ(qa, x +2πq)= Fˆ(q(a + 1), x)+(−p, 2πp)

9 By Lemma 2.2, there is a continuous map f such that the following diagram commutes:

1 f 1 SQ / SQ O O exp exp

Fˆ qZˆ × R / pZˆ × R O O

? (p/q)x+h(x) ? R / R and we have the result. Proposition 4.4. deg(f) = deg(f ′) iﬀ f is homotopic to f ′. Proof. If f is homotopic to f ′, then they have the same degree by Proposition 4.2. ′ Conversely, consider their expressions on the baseleaf: f0(x) = qx + h(x) and f0(x) = ′ ′ qx + h (x). Deﬁne the homotopy H0(x, t) = qx + th(x)+(1 − t)h (x). The construction given in Lemma 4.3 extends to a continuous family of maps hence there is a homotopy ′ between f and f whose expression in the baseleaf is H0.

Deﬁne the baseleaf preserving continuous map zq from the solenoid to itself as the lifting of the map x 7→ qx through the baseleaf. Remark 4.1. Although they have the same notation, do not confuse this map with the one deﬁned in the section 5: One is from the solenoid to itself and the other is from the (algebraic) solenoid to C∗. The context will make the distinction clear. Among these maps, only the identity is leaf preserving and all the others permute leaves. Corollary 4.5. The homotopic classes of baseleaf preserving continuous maps from the solenoid to itself are in one to one correspondence with the rational numbers: 1 1 ∼ [SQ,SQ]L = Q and the correspondence is given by the degree.

5 Adelic Riemann sphere

Lemma 5.1. Consider a finite dimensional Lie group G and a continuous function f from the solenoid to G. There is ε′ > 0 such that for every ε′ >ε> 0, there is a continuous function f ′ homotopic to f such that d(f, f ′) < ε and it factors through some canonical projection πn. Proof. Consider a bi–invariant metric on the Lie group and a convex neighborhood V of the origin in the Lie algebra g such that the exponential restricted to V is a local ′ homeomorphism. Consider ε > 0 such that Bε′ (e) ⊂ exp(V ) and ε > 0 such that ε′ >ε> 0. By Lemma 3.3, there is a continuous function f ′ such that d(f, f ′) <ε and it factors through some canonical projection πn. Define the continuous function a from the solenoid to V ⊂ g such that f ′(p)f(p)−1 = exp (a(p)). Define the homotopy: H(p, t) := exp (t a(p)) f(p) Then f ′ is homotopic to f and we have the result.

10 In the previous Lemma, the finite dimension hypothesis on the Lie group is too restric- tive for we only need the existence of an exponential map. For example, in the infinite dimensional case, a loop group has exponential map but the Virasoro group does not. If instead of the circle we take the inverse limit of the coverings of C∗ just as before, ∗ we get the algebraic solenoid CQ. Again, taking the inverse limit of the ramified coverings 1 1 of the Riemann sphere CP , we get the adelic Riemann sphere CPQ. All of the ramifica- tions occur at 0 and ∞ hence topologically the adelic Riemann sphere is the topological suspension of the adelic solenoid. The same theory in section 2 can be constructed for the algebraic solenoid almost verbatim. The complex structure of the algebraic solenoid as a laminated object is induced ˆ ∗ ˆ by the exponential map exp : Z × C → CQ with the obvious complex structure on Z × C. A rank n holomorphic vector bundle over the adelic Riemann sphere is a vector bundle whose clutching map is a holomorphic function from the algebraic solenoid to GL(n, C). m/n m We will denote simply as z the map (z 7→ πn(z) ) from the algebraic solenoid to C∗, where m and n are natural numbers (See remark 4.1). We will denote as O(q) the holomorphic line bundle whose clutching map is zq and q is a rational number. The following is the adelic version of the celebrated Birkhoff-Grothendieck Theorem: Theorem 5.2. Consider a rank n holomorphic vector bundle V over the adelic Riemann 1 sphere CPQ. There is a set of rational numbers q1,...qn such that: ∼ V = O(q1) ⊕ ... O(qn) isomorphic as holomorphic vector bundles. ∗ Proof. The clutching map of V is a holomorphic map f from the algebraic solenoid CQ to ′ 1 G = GL(n, C) and consider its restriction f to the solenoid SQ. By Lemma 5.1, there is ′′ ′ a continuous homotopic map f ≃ f that factors trough some canonical projection πm; 1 ′′ i.e. There is a continuous map g : S → G such that f = g ◦ πm. Apply the π1 functor to this map: π ∼ 1 1(g) ∼ Z = π1(S ) −−−→ π1(G) = Z ∗ Then, π1(g) acts by multiplying by some integer n. Define the holomorphic mapg ˆ : C → n G such thatg ˆ(z) = diag(z , 1,... 1) and the pullback by the projection fm :=g ˆ ◦ πm. −1 Define the holomorphic map h := fm · f . By construction, h factors through the µ universal cover group G˜ −→ G and the pullback of the lifting by the exponential map is a holomorphic map a from the algebraic solenoid to the Lie algebra; i.e. h(p)= µ (exp(a(p))) and a(p) ∈ g. We summarize the construction in the following commutative diagram:

exp ˜ g rr/9 G O rrr h˜rr a rrr µ rrr ∗ h CQ / G Deﬁne the holomorphic isotopy (continuous map such that it is holomorphic on each t): H(p, t)= µ (exp(t a(p))) f(p)

Then, fm is holomorphic isotopic to f and we have: V ∼= O(n/m) ⊕ O(1/m) ... O(1/m)

11 In particular, for every holomorphic line bundle L there is a rational number q such that L ∼= O(q) and, as the next Lemma shows, this number is unique.

Lemma 5.3. Consider rational numbers q and q′. Then, O(q) ∼= O(q′) iﬀ q = q′. Proof. There is a holomorphic isotopy h between the clutching maps of O(q) and O(q′); ′ i.e. h(p, 0) = pq and h(p, 0) = pq Following the proof of Theorem 5.2, for every t there is a holomorphic isotopy from (p 7→ h(p, t)) to zq(t) and these isotopies vary continuously with respect to t: H(p,t,s) := µ (exp(s a(p, t))) h(p, t) Then, H(p, t, 1) = pq(t) is a holomorphic isotopy hence (t 7→ q(t)) is a continuous map such that q = q(0) and q′ = q(1). Because Q is totally disconnected, the map is constant and q = q′.

Deﬁnition 5.1. For every holomorphic line bundle L over the adelic Riemann sphere, deﬁne its Chern character ch(L) as the unique rational number q such that L =∼ O(q). Corollary 5.4. The Picard group of the adelic Riemann sphere is isomorphic to the rational additive abelian group: 1 ∼ P ic CPQ = Q and the isomorphism is given by the Chern character.

Proof. If L =∼ O(q) and L′ ∼= O(q′), then L ⊗ L′ =∼ O(q) ⊗ O(q′) ∼= O(q + q′).

The analog of the Picard group for higher rank vector bundles is the K–ring:

Corollary 5.5. The K–ring of the adelic Riemann sphere is the following:

Z[xq / q ∈ Q] K CP 1 =∼ Q h (xq − 1)(xp − 1) / q,p ∈ Qi Proof. The proof is almost verbatim to the usual one for CP 1. By Theorem 5.2, we have an ring epimorphism q 1 ξ : Z[z / q ∈ Q] → K CPQ such that zq 7→ O(q). Consider the homotopy:

za 0 cos(π t/2) −sen(π t/2) zb 0 cos(π t/2) sen(π t/2) F (z, t)= 0 1 sen(π t/2) cos(π t/2) 0 1 −sen(π t/2) cos(π t/2)

Evaluating t = 0 and t = 1 we have that the following clutching functions are holomorphic isotopic: za+b 0 za 0 ∼ 0 1 0 zb In particular we have the following isomorphism as holomorphic vector bundles:

O(a + b) ⊕ O(0) ≃ O(a) ⊕ O(b)

12 This implies that (za − 1)(zb − 1) = za+b − za − zb + 1 is annihilated by the map ξ for every pair a, b of rational numbers and the epimorphism ξ factors through the quotient: Z[zq / q ∈ Q] ξˆ : → K CP 1 h (zq − 1)(zp − 1) / q,p ∈ Qi Q To see that ξˆ is in fact an isomporhism, we just need to show an inverse. Deﬁne the map: Z[zq / q ∈ Q] ν : K CP 1 → Q h (zq − 1)(zp − 1) / q,p ∈ Qi as the ring morphism such that ν(O(q)) = zq. It is clear that ν is well deﬁned and it is the inverse of ξˆ.

1 1 As it was expected from the continuous surjective map CPQ → CP and the fact that K is a contravariant functor, we have the inclusion: 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. It is interesting to note the rational factors of the tautological class in the adelic case. We say that a complex function of the adelic Riemann sphere is holomorphic if it is holomorphic on the algebraic solenoid and admits a continuous extension to the cusps.

Proposition 5.6. A holomorphic function of the adelic Riemann sphere is constant:

1 ∼ Hol(CPQ) = C

Proof. Consider a holomorphic function f of the adelic Riemann sphere and its pullback g by the baseleaf ν. Then, g is a holomorphic limit periodic respect to x (z = x + iy) function such that: lim ||f(0) − g|Im(z)≥y ||∞ =0 y→+∞

lim ||f(∞) − g|Im(z)≤y ||∞ =0 y→−∞ Then, g is a bounded entire function hence it is constant. Because the baseleaf is dense in the adelic Riemann sphere, by continuity f is constant.

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