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The Number Theory of Finite Cyclic Actions on Surfaces a Dissertation Submitted to the Graduate Division of the University of Ha THE NUMBER THEORY OF FINITE CYCLIC ACTIONS ON SURFACES A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS DECEMBER 2006 By Micah Whitney Chrisman Dissertation Committee: Robert D. Little, Chairperson Chris Allday Karl Heinz Dovermann Hugh M. Hilden George Wilkens Lynne Wilkens We certify that we have read this dissertation and that, in our opinion, it is satisfactory in scope and quality as a dissertation for the degree of Doctor of Philosophy in Mathematics. DISSERTATION COMMITTEE —————————————— Chairperson —————————————— —————————————— —————————————— —————————————— —————————————— ii Acknowledgements Any mathematical exposition is indebted to those previous authors who have taken the care to write their arguments clearly. In the present case, I would like to thank S. Katok, A. Dold, and J. Ewing for the works which are cited in the bibliography. Also, I am grateful for the typed lecture notes of B. Farb on mapping class groups. While not otherwise cited here, the collection provided early directions for this research. These notes were given to me by Eric Guetner. I am also very appreciative to Prof. Guetner for several informative and supportive conversations which helped during times when progress was stalled. I would also like to thank Mike Hilden for answering my innumerable questions about Fuchsian groups and branched coverings. As for questions involving anything from differential geometry to representation theory, I would like to thank George Wilkens for his thorough, prompt, and insightful replies. Thanks are also due to Ron Brown and Claude Levesque who took a look at some of the algebraic number theory presented here. Most of all, I would like to express my deep gratitude to my adviser, Robert Little. He has been a continual source of guidance and support for the past five and a half years. Thank you for sharing your mathematical, professional, and personal expertise. Also, I am very appreciative of his careful and comprehensive editing of this work. Finally, I would like to thank my family and friends for their encouragement during the completion of this dissertation. I am grateful to my parents, David and Jennifer, iii who have always encouraged my scientific interests. A special thanks is due to Jessi, for her patience. iv Abstract Let s : N Z be an integer sequence. To this sequence we associate the M¨obius → inverse sequence, denoted M : N Z, which is defined as follows: s → n M (n)= µ(d)s s d Xd|n Let X be a Euclidean Neighborhood Retract (ENR) and f : X X a continuous → map. Denote by Λ(f) the Lefshetz number of f. The Lefshetz sequence of f is defined to be: (Λ(f), Λ(f 2), Λ(f 3),...) A. Dold has proved that if the fixed point set of f n is compact for all n and s : N Z → is the Lefshetz sequence of f, then n M (n) for all n. In this thesis, we investigate | s the number theoretic consequences of sequences which satisfy this property(called Dold sequences). In particular, we will investigate periodic Dold sequences. The main theorem states that a Dold sequence is periodic with period m if and only if M (n) = 0 for almost all n N and m = lcm k N : M (k) = 0 . Moreover, it is s ∈ { ∈ s 6 } shown that a Dold sequence is bounded if and only if it is periodic. This extends a result of Babenko and Bogaty˘ı. The analysis of periodic Dold sequences is then applied to the study of mapping class groups of surfaces. Fuchsian groups are used to find all periodic Dold sequences of periodic orientation preserving maps on a surface. The solution of this realization problem provides some insight into the defect of the surjection Mod(S) Sp(2g, Z) → from preserving Nielsen-Thurston type. v Finally, algebraic number theory is used to determine a necessary and sufficient condition that a Zp-action on a surface extends to the handlebody which it bounds. This analysis results from investigating the Atiyah-Singer g-Signature Theorem. The main theorem states that the equivariant signature is 0 if and only if action extends. vi Contents Acknowledgements ................................... ............. iii Abstract ........................................... ............... v List of Figures ...................................... .............. x Chapter 1: Introduction .............................. ............. 1 1.1 Overview.................................. 1 1.2 LefshetzNumbers ............................. 14 1.2.1 EuclideanNeighborhoodRetracts . 14 1.2.2 TheLefshetzIndex. 16 1.3 Trace Formulas for Zm-Actions...................... 21 1.4 The G-Signature.............................. 25 1.4.1 The G-Signature for Inner Product Spaces . 25 1.4.2 The G-Signature for G-Manifolds................. 33 Chapter 2: Dold Sequences ............................. ........... 37 2.1 DoldSequences. .............................. 37 2.2 PeriodicDoldSequences. 38 2.3 LimitPointsofDoldSequences . 46 2.4 GeometricRealization. .. .. 50 2.5 Dold’sRealizationTheorem. 52 2.6 Periodic Realization of s : N Z ..................... 54 → vii 2.6.1 ProofofLemma32......................... 55 2.6.2 ProofofTheorem30. .. .. 58 Chapter 3: Applications to Maps of Surfaces . ........ 61 3.1 MappingClassGroups .......................... 61 3.1.1 The Thurston Classification Theorem . 62 3.1.2 Finite Order Mapping Classes and Their Dold Sequences .... 64 3.2 TheSimilarityAlgorithm . 68 3.3 Realization in the Special Case of Zp ................... 72 3.4 ReviewofFuchsianGroups . 85 3.4.1 Definition and Properties of Fuchsian Groups . .. 85 3.4.2 Parabolic, Hyperbolic, and Elliptic . .. 88 3.4.3 FundamentalRegions . 88 3.4.4 Periods, Presentations, and Signatures . ... 91 3.4.5 Uniformization ........................... 94 3.5 Surface Groups and Surface Kernel Homomorphisms . .... 96 3.6 TheTheoremsofHarvey ......................... 98 3.7 Realization of Periodic Dold Sequences on Surfaces . ....... 106 3.7.1 ProofofSufficiency . 111 3.7.2 ProofofNecessity . 117 3.8 TheSphereandtheTorus. 119 Chapter 4: On Zp-actions that extend to the Handlebody . 128 4.1 Slice Types and the G-SignatureTheorem. 129 viii 4.2 Rational Linear Independence and the αj/p’s .............. 135 4.3 SomeGeometry,Topology,andBordism . 144 4.3.1 Equivariantly Straightening the Angle . 144 4.3.2 Tubular Neighborhoods of G-Manifolds . 147 4.3.3 An Equivariant Bordism Theorem . 155 4.4 Applications to Zp-ActionsonSurfaces . 162 Chapter 5: Summary ................................... ........... 171 5.1 Chapter2SummaryandFuture . 171 5.2 Chapter3SummaryandFuture . 172 5.3 Chapter4SummaryandFuture . 174 References ......................................... ............... 177 ix List of Figures 1 Illustration of Dold’s simplicial 2-complex. ..... 53 2 Schematic of the basic s(1)=0case. .................. 58 3 Overhead view of the case when s(m) > N................ 60 4 A four-holed torus with threefold symmetry. .. 75 5 The Kosniowski generator with p = 3, j = 1, and q =2......... 77 6 Diagramfortheproofofsufficiency.. 110 7 Rotation of the torus by π has4fixedpoints. 126 8 Threefold symmetry of the two-holed torus . 134 9 A bent angle to be straightened equivariantly. ... 146 10 Unbending the angle in R R ..................... 146 + × + 11 A tubular neighborhood with two-sided collar. .... 155 12 B I withlabeledtwo-sidedcollar. 158 i × 13 Pasting V 1 with V 2 along the map ϕ............... 159 1 × 2 × 14 Assigning C∞ structure to C. ...................... 161 15 The n-manifold Z iswhatremainsaftergluing. 162 16 Construction of Z near two fixed points of ψ. ............. 169 x Chapter 1 Introduction 1.1 Overview The symmetries of planar objects have long fascinated artists and mathematicians. In the form of tesselations, their history can be traced back six thousand years to the ancient Sumerians. Their use continued with the ancient Greeks and Spanish Moors. Of course, few modern students are unfamiliar with the works of the Dutch painter M.C. Escher. Mathematicians became interested in the symmetries of planar objects as they provided some of the first examples of finite groups (sets with an associative operation containing inverses and an identity element). Today, tesselations and symmetries remain a hotbed of mathematical activity. The questions asked by mathematicians in the last two hundred years typically focus on much broader notions of symmetry and geometry. One can naively define a symmetry to be a one-to-one and onto continuous function from some given object to itself that preserves “distance” and “angle”. One also needs that the symmetry can be undone. In other words, a symmetry must posses an inverse symmetry: a one-to-one and onto continuous function from the object to itself that reverses the action of the original symmetry. Now, every object has an identity symmetry. It is simply the function that maps every point of the object to itself. Hence we see that a symmetry can be considered as an element of a group. 1 So instead of speaking intuitively about symmetry, mathematicians ask about how certain groups can act on certain objects. What is an object? In this dissertation, we will look at objects that are examples of topological spaces. One can think of a topological space as a set with an added neighborhood structure. Any point in the set is contained in a neighborhood. Every topological space must have at least two neighborhoods: the set itself and the empty set. Also, the union of any (possibly infinite) number of neighborhoods must be a neighborhood. Finally, finite intersections of neighborhoods must also remain neigh- borhoods. Topological spaces are very general and abstract. The easiest concrete example is that of the real numbers, R. The neighborhood structure is given by the open intervals, (a, b)= x R : a<x<b . The plane and three-dimensional space { ∈ } are other examples of topological spaces. In the most general context, then, the study of symmetries in mathematics is the study of groups acting on topological spaces. The groups are almost always assumed to be topological groups: topological spaces with a group structure so that the operations of multiplication and inversion are continuous. It is probably not clear why this level of abstraction is necessary.
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