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the Group of Homeomorphisms of a Solenoid Which Are Isotopic to The . The Group of Homeomorphisms of a Solenoid which are Isotopic to the Identity A DISSERTATION Submitted to the Centro de Investigaci´on en Matem´aticas in partial fulfillment of the requirements for the degree of DOCTOR IN SCIENCES Field of Mathematics by Ferm´ınOmar Reveles Gurrola Advisors Dr. Manuel Cruz L´opez Dr. Xavier G´omez–Mont Avalos´ GUANAJUATO, GUANAJUATO September 2015 2 3 A mi Maestro Cr´uz–L´opez que me ensen´oTodo˜ lo que S´e; a mi Maestra, que me Ensenar´atodo˜ lo que No s´e; a Ti Maestro Cantor, mi Vida, mi Muerte Chiquita. 4 Abstract In this work, we present a detailed study of the connected component of the identity of the group of homeomorphisms of a solenoid Homeo+(S). We treat the universal one–dimensional solenoid S as the universal algebraic covering of the circle S1; that is, as the inverse limit of all the n–fold coverings of S1. Moreover, this solenoid is a foliated space whose leaves are homeomorphic to R and a typical transversal is isomorphic to the completion of the integers Zb. We are mainly interested in the homotopy type of Homeo+(S). Using the theory of cohomology group we calculate its second cohomology groups with integer and real coefficients. In fact, we are able to calculate the associated bounded cohomology groups. That is, we found the Euler class for the universal central extension of Homeo+(S), which is constructed via liftings to the covering space R × Zb of S. We show that this is a bounded cohomology class. In particular, we find an analogue of the Poincar´erotation number. 5 Acknowledgments I appreciate the financial support of CONACyT during all these years as a graduate student at CIMAT. It has been a pleasure to work in such an ideal place for mathematics research. I hope this will be the “tip of the iceberg” of some future relevant developments. 6 Introduction At the end of the 19th century, Henri Poincar´e([Poi]) introduced an invariant of major importance for the study of the dynamics of homeomorphisms of the unit circle S1, this is the well known rotation number. There are some equivalent ways of defining this topological invariant. Our interest lies in the work of E. Ghys ([Ghy]). At the beginning of this century he found the rotation number using the language of cohomology of groups. 1 Consider the group Homeo+(S ) of of homeomorphisms of the circle which preserves the 1 1 orientation and Homeo^+(S ) the group of lifts with respect to the universal covering π : R −→ S . There is a surjective homomorphism 1 1 p : Homeo^+(S ) −→ Homeo+(S ) with kernel isomorphic to Z. Moreover, p is a universal covering map and this is a universal central extension 1 p 1 0 −→ Z −→ Homeo^+(S ) −→ Homeo+(S ) −→ 1. It is a consequence of a general result of Thurston ([Thu] and [Tsu]) that 2 1 H (Homeo+(S ), Z) ' Z and a generator eu of this group is known as the Euler class for the given extension. Moreover, the last is a universal central extension and from the theory of universal central extensions (see 1 for example [Mil]), the kernel is isomorphic to the Schur multiplier H2(Homeo+(S ), Z). Using the universal coefficient theorem we can derive the second cohomology as well. In particular, it can be shown that the Euler class is bounded, in the sense of the cocycles being 2 1 bounded maps and represents a generator for the bounded cohomology group Hb (Homeo+(S ), Z). 1 ∗ 2 If φ : Z −→ Homeo+(S ) is any homomorphism, the corresponding class φ (eu) ∈ Hb (Z, Z) ' R/Z is the rotation number of the homeomorphism φ(1). The main objective of this thesis is to extend these results to the case of the group of homeo- morphisms isotopic to the identity of the universal one dimensional solenoid S, which is a compact connected Abelian topological group and also a foliated space. The universal one–dimensional solenoid can be defined as S := lim /n . ← R Z S is a compact connected Abelian group, with a canonical inclusion P : R −→ S, such that ∼ L0 := P(R) is the arc–connected component of the identity element in S, and L0 = S. The projection onto the first coordinate S −→ S1 is a Zb–principal fiber bundle, with fiber b := lim /n ; Z ← Z Z where Zb is the profinite completion of the integers Z, which is a compact Abelian totally discon- nected topological group. Also, it is a perfect topological space. That is, Zb is isomorphic to the Cantor topological group. Moreover, S is a foliated space whose leaves are homeomorphic to R and every transversal is isomorphic to Zb. 7 Let Homeo+(S) be the component of the identity of Homeo(S) and let Homeo^+(S) be the group of lifts of homeomorphisms in Homeo+(S) via the covering Π : R × Zb −→ S, defined as the quotient map of the diagonal Z–action γ · (x, k) −→ (x − γ, k + γ), (γ ∈ Z). There is a surjective homomorphism p : Homeo^+(S) −→ Homeo+(S), with kernel isomorphic to Z. In fact, Z will be identified with the subgroup of deck transforma- tions ∆(Z) ⊂ Homeo^+(S). As we said before, the main concern of this work is to emulate the theory as presented by Ghys to the case of homeomorphisms of the solenoid S. In first place, we prove that Homeo+(S) has the homotopy type of the group of translations on the base leaf L0 ,→ Homeo+(S). Then, we describe a central extension 0 −→ Z −→ Homeo^+(S) −→ Homeo+(S) −→ 1 and prove that it is a universal central extension. This can be done by extending the fact that Homeo+(S) is uniformly perfect, as shown in [AP], to the group of lifts Homeo^+(S). Therefore we have the Schur multiplier H2(Homeo+(S), Z) ' Z. Thus, using the universal coefficient theorem, there is an Euler class 2 eu ∈ H (Homeo+(S), Z) ' Z. 2 This class being bounded represents a generator of Hb (Homeo+(S), Z). Consider the rotation element ρ : Homeo+(S) −→ S as introduced in the work of A. Verjovsky and M. Cruz–L´opez (see [CV]). For f ∈ Homeo+(S) take a lift of ρ(f) to the covering R × Zb. This give us an element τ ∈ R × Zb and fixing a height on the covering we can describe a homogeneous quasimorphism T : Homeo^+(S) −→ R. Thus, we have a quasicorner C(T, p) defined by T Homeo^+(S) / R p Homeo+(S). Since the set of equivalence classes of quasicorners QC(Homeo+(S)) is in a bijective correspon- 2 dence to Hb (Homeo+(S), R), the result is that the class [C(−T, p)] ∈ QC(Homeo+(S)) is related R 2 2 to the class eub ∈ Hb (Homeo+(S), R). Therefore Hb (Homeo+(S), R) ' R. 8 Finally, if we consider the projection of T : Homeo^+(S) −→ R to the circle we obtain an element 1 % : Homeo+(S) −→ S , ∗ 2 and we prove that for every homomorphism ϑ : Z −→ Homeo+(S), the class ϑ (eub) ∈ Hb (Z, Z) 2 is the number %(ϑ(1)). Moreover, in view of Char(S) ' Q we have that Hb (Q, Z) ' S. Thus, for ∗ every homomorphism ϕ : Q −→ Homeo+(S), the class ϕ (eub) is the rotation element of ϕ(1). In the first chapter we review the theoretical ingredients as cohomology, homology and bounded cohomology of groups. We also study the connection between the second bounded cohomology of groups and the theory of quasicorners. We put some emphasis in the case of the 1 group Homeo+(S ) and how to derive the bounded cohomology class 1 2 1 eub(S ) ∈ Hb (Homeo+(S ), Z). In the last section we review the work of E. Ghys ([Ghy]). In the second chapter we present the general results of Homeo(S). As a consequence of the work of C. Odden in the two–dimensional case (see [Odd]), there is a decomposition ∼ Homeo(S) = HomeoL0 (S) ×Z Zb; where HomeoL0 (S) is the subgroup of homeomorphisms of S which preserves the base leaf, the subgroup Zb ,→ Homeo+(S) representing the translations on the fiber and the quotient HomeoL0 (S) ×Z Zb = (HomeoL0 (S) × Zb)/Z due to a generalized diagonal action (see section 2.3). Also, we are able to recognize the isotopy classes of elements in HomeoL0 (S) as ∼ HomeoL0 (S)/Homeo+(S) = Aut(S). At the end of this chapter we mention the result of Homeo+(S) being uniformly perfect as in [AP]. The third chapter presents a complete study of the covering p : Homeo^+(S) −→ Homeo+(S), we specify the construction of an universal extension for Homeo+(S) and prove that this group has the homotopy type of the subgroup of translations over the base leaf. Also, we calculate its second cohomology groups and find an invariant of the dynamics as we mentioned before. CONTENTS Abstract . 4 Acknowledgments . 5 Introduction . 6 1. Homeomorphisms of the circle and the Euler class ................. 10 1.1 Cohomology of groups . 10 1.2 Homology and universal central extensions . 13 1.3 Bounded cohomology of groups . 15 1.4 Homeomorphisms of S1 which preserve orientation . 20 2. Homeomorphisms of the solenoid ........................... 25 2.1 The solenoid . 25 2.2 Continuous functions preserving the zero element . 28 2.3 The group of homeomorphisms of the solenoid . 29 2.4 Isotopy classes of Homeo(S).............................. 31 2.5 Uniform perfectness of Homeo+(S).......................... 36 3. The connected component of the identity ...................... 39 3.1 Lifts of elements in Homeo+(S)...........................
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