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

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 neighborhoods. 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

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. We have not characterized, however, the notions of “distance” and “angle”. It turns out that these ideas are not always present in spaces that still exhibit observable symmetries. Even in spaces in which these notions are defined, they fail to arise in obvious or natural ways (e.g. Riemannian metrics on manifolds). There may also be conceptually different notions of distance and angle defined on the same topological space. Hence we need a way to discuss symmetries that remains independent of distance and angle.

2 Two types of topological spaces will be considered in this dissertation. In Chap- ter 2, we look at symmetries of very general spaces called Euclidean Neighborhood

Retracts (ENRs). In the remaining chapters, we look at symmetries in the subset of ENRs consisting of orientable manifolds. A manifold of dimension 2 is a space that locally resembles the plane. A manifold of dimension 3 is a space that locally resembles three dimensional space. As for symmetries, we will only consider those that are ﬁnite cyclic. A ﬁnite cyclic symmetry of a topological space X is a map f : X X such that f m is the identity map 1 : X X for some m. The smallest → → such m that works is called the period or order of f.

As the title suggests, the theme to be considered here is the number theory of

ﬁnite cyclic actions. Here we exit the realm of classical topology and focus on algebraic topology. In algebraic topology, algebraic structures (like groups or modules) are assigned to topological spaces and continuous maps of topological spaces. These algebraic structures are used to diﬀerentiate between the topological spaces themselves. For example, if the same procedure is used to assign an algebraic structure to topological spaces X and Y , but the resulting algebraic structures are not the same, then we can conclude that X and Y are not equivalent topological spaces. It is important to note that the converse is almost never true. Equivalent algebraic structures often arise from topological spaces which are not equivalent.

It is important to note that equivalence is not the same as identity. As the goal is to study the totality of topological spaces, we need to sanctify the important features and condemn the other ones to oblivion. For topologists, the hallowed structure is the neighborhood structure. Hence, we say that two topological spaces X and Y

3 are equivalent (more speciﬁcally, homeomorphic) if there is a one-to-one and onto continuous function X Y that gives a one-to-one correspondence between their → neighborhood structures. We will frequently use another form of equivalence here.

Two topological spaces X and Y are said to be homotopic if X can be continuously deformed onto Y . The notion of homotopy also extends to pairs of maps f,g : X Y . → Thus, homotopy equivalence is the general notion of equivalence for symmetries of a topological space. In Chapter 3, we will investigate the mapping class group of a surface. This group contains all of the symmetries up to isotopy, which is just a stronger version of homotopy. In Chapter 4, we will look at a notion of equivalence for symmetry groups acting on manifolds which is called equivariant bordism. One can think of equivariant bordism as a generalization of homotopy.

Now, algebraic structures assigned to topological spaces and their maps that preserve the topological notion of equivalence are called algebraic invariants. The three most commonly used algebraic invariants are the fundamental group functor, the homology functor, and the cohomology functor. All three of these invariants will be used here in some fashion; each plays an important part in the number theory of cyclic actions. The remainder of this section contains an overview of each of the ways these invariants are used in this dissertation to attach number theory to ﬁnite cyclic actions. A brief summary of the results of this dissertation is given as well as the historical context of these results.

Let N be the natural numbers and suppose q N 0 . The rational homology ∈ ∪{ } functor assigns to each topological space X a vector space Hq(X; Q). If X and Y are topological spaces and f : X Y a continuous map, it assigns to f a vector → 4 space homomorphism (i.e. linear transformation) H (f) : H (X; Q) H (Y ; Q). q q → q There are many numerical invariants of vector space homomorphisms and we can exploit them to find numerical invariants of continuous maps. For example, the trace function assigns to a linear transformation the sum along the diagonal of any matrix representation. Since the homology functor assigns to each map infinitely many linear transformations, this gives us too much information. The trace data is alternatively coded in the Lefshetz number which can be defined abstractly for a continuous function f : X X as: →

∞ Λ(f)= ( 1)qTr(H (f)) − q q=0 X This sum need not always be defined. It is certainly defined on any space in which the homology is finite for each q and vanishes for all q larger than some Q N. ∈ In fact, this is the case with the only spaces considered here: compact ENRs and compact manifolds. In Section 1.2, we will discuss the important properties of the

Lefshetz number and its relative, the ﬁxed point index. At the moment, it is only really necessary to know that Λ(f) is an integer.

The most important theorem about the Lefshetz number is called the Lefshetz

Fixed Point Theorem. It states that if Λ(f) = 0, then f must have at least one 6 ﬁxed point (i.e. a point x such that f(x) = x). Intuitively then, one thinks of the

Lefshetz number as a numerical invariant which “counts” the number of fixed points of f. This is not completely valid because it occurs, for example, that a map can have fixed points but its Lefshetz number is zero! Nevertheless, it provides a useful framework for answering questions about fixed points. The Lefshetz number can also

5 provide information about periodic points of a map f : X X: points for which → f k(x)= x for some k > 1 but f j(x) = x for all j

s = (Λ(f), Λ(f 2), Λ(f 3),..., )

This sequence of integers, called the Lefshetz sequence of f, can be interpreted as the sequence of ﬁxed point of iterates of f. However, this tells us nothing about the periodic points because not every point ﬁxed by f k is a periodic point of period k.

Something fixed by f k might also be fixed by f d where d k. To solve this, we can | examine the Möbius inverse sequence of s, denoted M : N Z: s → n M (n)= µ(d)s s d Xd|n where µ : N Z is the Möbius function. It will be discussed in Section 2.1 why →

Ms(k) can be considered intuitively to be the the number of periodic points of period k.

Albrecht Dold investigated these sequences in [11]. He showed that if X is a compact ENR and f : X X is a continuous map with Lefshetz sequence s : N Z, → → then n M (n) for all n N. In this thesis, we consider the number theory of integer | s ∈ sequences s : N Z such that n M (n) for all n N (called a Dold sequence). When → | s ∈ this is applied to the study of ﬁnite order symmetries, i.e. maps f : X X such → that f m = 1 for some m N, we see that the sequences of greatest interest will be ∈ those that are periodic. The main results of this part of the dissertation thus revolve around the number theory of periodic Dold sequences.

6 Some previous work has been done on the number theory of Dold sequences. In

[4], it was shown that a Dold sequence is either bounded or asymptotic to ex. Here, this result is extended to show that a Dold sequence is bounded if and only if it is periodic. In fact, one can even show that a Dold sequence is periodic if and only if Ms(k) = 0 for all but ﬁnitely many k. The period of a Dold sequence such that

Ms(k) = 0 for all but ﬁnitely many k is the least common multiple of the natural numbers k for which M (k) = 0. This work is proven in great detail in Section 2.2. s 6 Most of proofs use only elementary number theory. However, Dirichlet’s theorem on primes in an arithmetical progression is needed to prove the fundamental theorem of periodic Dold sequences.

There are only two sections of this thesis where we consider Dold sequences that are not periodic. The first is in Section 2.3 in which we look at limit points of Dold sequences. This is presented as a possible approach to finding a topological version of the Shub-Sullivan theorem. This theorem states that for a C∞ map f : X X → on a smooth manifold X, if the sequence (Λ(f), Λ(f 2),...) is unbounded then f has infinitely many periodic points. The work of Section 2.3 suggests a way to approach this problem when the hypotheses on f and X are relaxed to the continuous category.

Unbounded Dold sequences are also considered in Section 3.8. In this section, all Dold sequences of diﬀeomorphisms on the torus are computed. This turns out to be a fairly straightforward exercise.

The remainder of Chapter 2 focuses on a geometric approach to periodic Dold sequences. In [11] again, Dold proved that every Dold sequence is the Lefshetz sequence of some continuous function f : X X on a ﬁnite simplicial 2-complex X. Moreover, → 7 f can be chosen so that it has exactly M (k) periodic points of period k, each with | s | local Lefshetz number Sign(M (k)/ M (k) ). The maps constructed in [11] are never s | s | periodic. Hence it is necessary to ask which periodic Dold sequences are realized as the Lefshetz sequence of some periodic map on a simplicial complex. Unfortunately, this answer to this question remains unknown. Sections 2.4-2.6 establish the case for maps of “surface-like” simplicial two complexes(deﬁned in Section 2.4). To contrast our approach with the approach discovered by Dold, his original argument is outlined in Section 2.5.

Geometric realization is again the theme in Chapter 3: Applications to Maps of

Surfaces. Here we determine the number theory of those periodic Dold sequences which are the Lefshetz sequences of periodic orientation preserving diffeomorphisms on closed orientable surfaces. The goal of this investigation is to use periodic Dold sequences to reveal information about maps of finite order in the mapping class group of a surface. The mapping class group of a closed orientable surface is the group of orientation preserving diffeomorphisms modulo isotopies. Elements of the mapping class group are equivalence classes called mapping classes. Much is known about

ﬁnite order mapping classes. For example, Nielson proved the useful result that every mapping class of order m has a representative that is an orientation preserving diﬀeomorphism of order m.

The question considered in this dissertation is also of a practical nature. In general, mapping classes are diﬃcult to represent. It is known that every mapping class is a composition of a ﬁnite number of easily described mapping classes called Dehn twists.

The problem is that it is quite diﬃcult to ﬁnd a Dehn twist decomposition of even the

8 simplest ﬁnite order mapping classes. Hence, one wonders how to eﬃciently perform computations in the mapping class group. There is a well-known short exact sequence that to some extent reduces the problem to computing with matrices:

β (1) T (S) α Mod(S) Sp(2g, Z) (1) → −→ −→ → where Mod(S) is the mapping class group of the closed orientable surface S of genus g, Sp(2g, Z) GL(2g, Z) is the integral symplectic group, and T (S) is the Torelli ⊂ group. It is known that if the genus of S is at least 2, then T (S) contains no torsion.

However, this does not mean that the study of finite order mapping classes in Mod(S) is equivalent to studying finite order mapping classes in Sp(2g, Z). Indeed, we show by example (in Section 3.1.2) that there are finite order elements in Sp(2g, Z) which cannot be the image of some finite order mapping class in Mod(S).

It is desirable, then, to have a way of differentiating the “fake” finite order mapping classes (i.e. ones that are not finite order but induce maps of finite order) from the

“real McCoys”(finite order orientation preserving diffeomorphisms). This dissertation presents a method by which the distinction can be made up to similarity over the complex numbers. Moreover, we will show that this problem is equivalent to the problem of determining the periodic Dold sequences which are realized by periodic orientation preserving diffeomorphisms on a surface. This argument is presented in

Section 3.2.

To solve this realization problem on surfaces, we look at the history of ﬁnite group actions on surfaces. The ﬁrst result relevant to this thesis is the famous Hurwitz

84(g 1) theorem. This theorem states that the largest group that can act on a closed −

9 orientable surface of genus g has order 84(g 1). Numerous authors have expanded − upon and refined this result. In the 1960’s, the so-called Hurwitz problem began to be viewed in a different direction: Given a group G, what is the smallest genus surface on which G can act as a group of orientation preserving diffeomorphisms. Early progress on this problem was made by Harvey [19]. Using some results of Macbeath, Harvey answered the question for the class of finite cyclic groups. The solution for finite abelian groups was discovered nearly simultaneously. This effort, also building upon the results of Macbeath, is due to Maclachlan (see [29]). The most comprehensive theorem on the subject was not discovered until 1986. This surprising theorem states that for every finite group G, there is a g N and an N N such the group G acts s ∈ ∈ on the surface of genus g g if and only if g = Nk + 1 for some k N. The number ≥ s ∈ gs is called the stable genus and the number N is called the stable genus increment.

The game is to determine these two parameters for a given group. Ravi Kulkarni, who proved this theorem in [26] took care of a large number of these using the highly controversial classiﬁcation of ﬁnite simple groups. Although this is only a tangential topic for our purposes, a new proof for the case where G is a cyclic group of prime order is provided in Section 3.3.

Once again, the focus of this dissertation is not to add to the literature of stable genus increments. The history is presented because proofs of these results provide the number theory necessary to solve the realization problem of periodic Dold sequences of surface maps. There are two fundamental theorems from which this number theory ultimately arises. Let H denote the complex upper half plane. With the appropriate metric, this exhibits a hyperbolic geometry. A generalization of the Riemann Mapping

10 Theorem states that all surfaces of genus g 2 are of the form H/Γ where Γ is a ≥ discrete group of orientation preserving isometries of H (i.e. a Fuchsian group) that

H acts without ﬁxed points. By covering space theory, π1( /Γ) ∼= Γ. This is the ﬁrst of the theorems used by Harvey, Maclachlan, and Kulkarni.

The second important theorem relates the study of finite group actions on surfaces to the study of Fuchsian groups. It says that if G is a finite group of orientation preserving diffeomorphisms of a closed orientable surface S, then G is the quotient of two Fuchsian groups Γ and Γ′, Γ′ ⊳ Γ. The group Γ′ acts on H without fixed points and S is simply H/Γ′.

As yet, we have not seen where the number theory comes in. The standard trick is to compare the hyperbolic areas of fundamental regions for Γ and Γ′. This comparison leads to a Diophantine equation in variables that arise from a presentation of Γ. The idea championed be these authors is to add enough hypotheses to the group G so that the Diophantine equation can be minimized with respect the genus variable. Our approach is the same. What number theoretic hypotheses can be added to a periodic

Dold sequence of period m so that it appropriately solves the Diophantine equation?

The hypotheses are precisely determined in Section 3.7 and are subsequently shown to be necessary and suﬃcient.

The research presented here originated with the study of a numerical invariant called the equivariant signature. As will be shown, this invariant is a bordism invariant. A bordism class of closed, not necessarily connected, manifolds is mapped into the complex numbers C. The invariant is essentially a generalization of the

Hirzebruch signature, which in itself is an application of the signature invariant for

11 matrices. While the Hirzebruch signature is only deﬁned for manifolds of dimension

4n, the equivariant signature is deﬁned for all even dimensional manifolds. If the manifold is of dimension 2n, gcd(2, n) = 1, it can be shown that equivariant signature is a purely imaginary algebraic number. On the other hand, if the dimension is

4n, it can be shown that the equivariant signature is a real algebraic number.

The major theorem about the equivariant signature is the Atiyah-Singer g-Signature

Theorem. It states the the equivariant signature can be computed from the way in which the diﬀerential of the group acts on the normal bundle to the ﬁxed point set.

As our interest is the finite symmetries of surfaces, this action can be codified in a relatively easy manner. For a finite group acting on a closed orientable surface as a group of orientation preserving diffeomorphisms, the fixed point set is just a finite union of points. At each point, the normal bundle is identified with the tangent plane at that point. The tangent plane can be identified with the complex plane C. For a cyclic group of order m, the differential of a generator acts on C as multiplication by

λ, where λ = 1 is some m-th root of unity. Deﬁne λ = e2πi/m and: 6

λj +1 α = j/m λj 1 −

Now, let ψ : S S be an orientation preserving diﬀeomorphism of ﬁnite order m on → a closed orientable surface (i.e. an element of the group acting on S). Suppose that

ψ has aj fixed points such the action of the differential of ψ on the tangent plane is multiplication by λj. The total slice type of ψ is defined to be the m 1-tuple: −

(a1, a2,..., am−1)

12 Using this notation, the Atiyah-Singer g-Signature Theorem becomes:

m−1

σ(ψ)= ajαj j=1 X where σ(ψ) denotes the equivariant signature of ψ.

One major area of investigation in the equivariant topology of manifolds involves determining which fixed point sets and which slice types are possible on a given manifold. Even in the case of well-understood spaces like CP 2, this problem is quite difficult. In the case of finite cyclic actions on surfaces, this question reduces to asking which (m 1)-tuples are the total slice types of some periodic map on surface. −

Moreover, is it possible to determine some relations amongst the aj? We investigate the case in which m = p, p an odd prime. In Section 4.4, we prove the new result that for Zp-actions on surfaces, if the equivariant signature is 0, then aj = ap−j for all j. This theorem results from an analysis of the algebraic number theory of the set

α ,..., α which is presented in Section 4.2. { 1/m (p−1)/p}

The most surprising consequence of this relation amongst the ajs is that it reﬂects back on the symmetry of manifolds. In particular, it can be used to show that

σ(ψ) = 0 if and only if the Zp-action on the surface extends to the handlebody which it bounds. The problem of determining conditions on which an action extends to the handlebody has received some attention in recent years, but to the knowledge of the author, it has not been investigated from the standpoint of the equivariant signature.

This may be due to the fact that the method does not appear to generalize easily to the case of arbitrary Zm-action on surfaces. The algebraic number theory in this case is vastly more daunting.

13 As previously stated, it was the study of the equivariant signature which eventually led to the development of the other projects considered in this thesis. This point of view is explored in the remainder of Chapter 1. In Sections 1.2 and 1.4, we will review the definitions of the Lefshetz number, ENR, and the equivariant signature that will be used throughout the dissertation. The connection between the two invariants is indeed forged by algebraic number theory. The fundamental relationship for our work is considered in Section 1.3. This chapter should truly be reviewed as intro- ductory, although there has been no attempt to be comprehensive. The focus here is establishing the theorems and definitions relevant to the thesis using only elementary notions from linear and abstract algebra. Our approach does not add significantly to the length of the exposition.

1.2 Lefshetz Numbers

1.2.1 Euclidean Neighborhood Retracts

Let X be a topological space, A X and i : A X be the inclusion. Recall ⊂ → that A is said to be a retract of X if there exists a continuous map r : X A such → that r i = 1 . The space A is said to be a neighborhood retract in X if there ◦ A is a neighborhood W X, A W , such that A is a retract of W . We will focus ⊂ ⊂ on a special kind of neighborhood retract called a Euclidean Neighborhood Retract

(ENR).

Let A Rn. If A is a neighborhood retract in Rn, what properties must A ⊂ have as a subspace of Rn? Suppose W is an open set of which A is a retract. Let

14 r : W A W be the retract. Since A = x W : r(x) = x , A is closed in W . → ⊂ { ∈ } Hence, A = A¯ W , where the closure is the closure in Rn. This means that every ∩ neighborhood retract in Rn is of the form A = C O where C is closed in Rn and O ∩ is open in Rn. Subsets with this property in a topological space are said to be locally closed.

In Rn, it can be shown that all locally closed sets are locally compact in the relative topology. Hence, the special property of being a retract in Rn necessitates that the space itself possess a global topological property. This suggests that the concept can be unmoored from euclidean space and deﬁned for all topological spaces. Indeed, this is true. A topological space A is said to be a Euclidean Neighborhood Retract if A is homeomorphic to a set A′ Rn such that A′ is a neighborhood retract. Now, suppose ⊂ that A is homeomorphic to B Rn and C Rm where B is a neighborhood retract. ⊂ ⊂ For ENR’s to be well-deﬁned, one must have that C is also a neighborhood retract.

Using the locally closed property of B and C and the Tietze extension theorem, one can show that this is indeed the case ([11], IV.8.5). Thus, ENR is a well-deﬁned property for topological spaces to possess.

There are several nice suﬃcient criteria for a topological space to be an ENR. One of particular utility states that if a topological space X is covered by ﬁnitely many open sets X1,...,Xn such that each Xi is an ENR, then X is also an ENR (see

[11]). This implies that every compact manifold is an ENR. Another useful criterion is due to Borsuk. It states that if a topological space is locally compact and locally contractible (i.e. every point in X has a neighborhood U containing a neighborhood

V of x that is contractible to a point in U) then any embedding of X into a Euclidean

15 space is necessarily an ENR (see [11],IV.8.12 or [7], E.3). This surprising result implies that any ﬁnite simplicial complex is an ENR.

The focus on ENRs is due to the fact that they are a sufficiently large class of topological spaces on which the Lefshetz number is defined and satisfies the Lefshetz

Index Theorem. This also works for a still larger class of topological spaces called absolute neighborhood retracts (ANRs)(see [8]). Indeed, every ENR is an ANR but not every ANR is an ENR.

1.2.2 The Lefshetz Index

Lefshetz numbers have both a geometric and an algebraic side. The geometric part allows us to conclude things like (Λ(f) =0 = f has a ﬁxed point). The algebraic 6 ⇒ side uses linear algebra to make these numbers easily computable. The geometric side is encoded in the ﬁxed point index whereas the algebraic side is encoded in the

Lefshetz index.

Let V Rn be open and f : V Rn a continuous map. Let ι : V Rn be the ⊂ → → inclusion. Suppose that K V is compact. Denote by u the fundamental class of ⊂ K V around K. This element is characterized by the property that for all x K, The ∈ image of u under the map H (V,V K) H (V,V x ) is the local orientation k n \ → n \{ } at that point.

Now, let F denote the ﬁxed point set of f. Then (ι f) : V Rn maps V F − → \ into Rn 0 . Consider the following map. \{ }

(ι f) : H (V, V F ) H (Rn, Rn 0 ) = Z − ∗ n \ → n \{ } ∼ 16 Let u denote the local orientation of Rn about the origin. The map (ι f) sends 0 − ∗ uF to some multiple of u0. The fixed point index of f is defined to be the integer If that satisfies the following equation:

(ι f) (u )= I u − ∗ F f · 0

An immediate consequence of this deﬁnition is that if f has no ﬁxed points (i.e.

F = ), then I = 0. ∅ f The fixed point index enjoys many properties that aid in computation: additivity, homotopy invariance, and commutativity. Let f : V Rn be as above and define F → to be the fixed point set of f. Suppose that F is compact. For proofs of the following facts, the reader is refered to [11], or [8].

1. (Additivity) Suppose that F is covered by ﬁnitely many disjoint open sets

n V1, ...,Vn. Then If = i=1 If|Vi . P 2. (Homotopy Invariance) If g : V Rn is continuous and f g, then I = I . → ∼ f g

3. (Commutativity) Let U Rnf and U Rng . Suppose f : U Rng and f ⊂ g ⊂ f → g : U Rnf . Then f g : g−1(U ) Rng and g f : f −1(U ) Rng have g → ◦ f → ◦ g →

homeomorphic ﬁxed point sets. Moreover, If◦g = Ig◦f .

It is the third criterion which allows for an extension of the ﬁxed point index to ENRs. Let Y be any topological space, V Y an open set and f : V Y a ⊂ → continuous function. Suppose furthermore that f factors through a space X Rn, ⊂ X open in Rn. In other words, suppose that there exists maps α : V X and → 17 β : X Y such that f = βα: →

V A AA f α AA AA / X β Y Suppose that V Y is an ENR, V open in Y . Let A Rn be a neighborhood retract ⊂ ⊂ that is homeomorphic to V by a homeomorphism h : V A. Then there is an open → set X Rn and a map r : X A such that r i = 1 . This gives the following ⊂ → ◦ X sequence of maps:

−1 f V h A i X r A h V Y −→ −→ −→ −−→ −→

Define β : X Y by β = f h−1 r and α : V X by α = i h. Since β α = f, we → ◦ ◦ → ◦ ◦ have that a decomposition exists for every ENR. We define the fixed point index of f, denoted I , to be I . Note that αβ : β−1(V ) X is a map of Euclidean spaces f αβ → and hence Iαβ is already defined.

Theorem 1. Suppose Y is a topological space, V Y is an ENR, and f : V Y is ⊂ → continuous. Furthermore suppose that the ﬁxed point set of f is compact. Let:

β V α1 X 1 Y −→ 1 −→

β V α2 X 2 Y −→ 2 −→

be two decompositions of f(i.e. f = β1α1 = β2α2). Then Iα1β1 = Iα2β2 . Hence, the

ﬁxed point index is well-deﬁned on ENRs.

Proof. See [11],(VII.5.10).

It can be shown that this extended deﬁnition satisﬁes additivity, homotopy invariance, and commutativity. For the precise statement of these see [11]. For a

18 generalization to ANRs, see [8]. Additivity will be used extensively in subsequent sections as it allows for an easy computation of Lefshetz numbers of periodic maps on surfaces. Homotopy invariance will come into play as we wish to use the sequence of Lefshetz numbers as an invariant of the mapping class group.

The algebraic side of the Lefshetz Fixed Point theorem comes from the trace of the induced map in homology. Let Y be any topological space and f : Y Y a → continuous map. Suppose Hq(Y ; Q) is ﬁnitely generated for all q and nonzero for only

finitely many q. Then each level of homology can be considered as a finite dimensional rational vector space. The induced map H (f) : H (Y, Q) H (Y, Q) is thus a linear q q → q transformation. Recall that the trace function Tr : M(dim H (Y ; Q), Q) Q is q → independent of any matrix representation of Hq(f). Hence, we can define the trace of Hq(f), denoted Tr(Hq(f)) to be the trace of any matrix representation of Hq(f).

The Lefshetz number of f is thus deﬁned to be the ﬁnite sum:

Λ(f)= ( 1)qTr(H (f)) − q q X Lefshetz numbers are certainly defined for compact manifolds as their homologies vanish after their top homology and are finite dimensional over Q elsewhere. For compact ENRs, the homology is finitely generated at all levels and is nonzero for only finitely many levels. This is due to the fact that the euclidean neighborhood of which a compact ENR is a retract can be taken to be a compact CW -complex. The

CW -complex has the desired properties and the homology of the ENR appears as a direct summand of the homology of the CW -complex. Thus, Lefshetz numbers are deﬁned for all compact ENRs.

19 The connection between the geometric nature of the ﬁxed point index and the algebraic nature of the Lefshetz index is provided by the Lefshetz Index Theorem.

The statement of the following theorem is copied directly from [11]. The reader is referred to the excellent proof of this classic theorem that can be found there.

Theorem 2. Let Y be an ENR and K Y a compact subset. Let f : Y K be ⊂ → continuous. Then H(f K) : H(K; Q) H(K; Q) has ﬁnite rank and I = Λ(f K). | → f |

An immediate consequence of this theorem is that Λ(f) is an integer when the above hypotheses are satisﬁed. Another easy consequence is one which we will use extensively in Chapter 3. Let D2 denote the closed unit disk in the plane and let f : D2 D2 be any continuous map. Since D2 is clearly a compact ENR, it satisﬁes → the hypotheses of the Lefshetz Index Theorem. Thus, If = Λ(f). It is clear that

2 Q Q H0(f) is necessarily the identity on H0(D ; ) ∼= . Hence, If = Λ(f) = 1. This fact is especially useful when investigating finite order homeomorphisms on a closed orientable surface. In this case, the map has a finite number of fixed points. For each fixed point x, there is a coordinate neighborhood of x on which the map acts as a rotation of the disk. Using the same argument as above, we see that the local

Lefshetz number of this ﬁxed point is 1. Hence, we can interpret the Lefshetz number of such a homeomorphism as the number of ﬁxed points of that map.

The Lefshetz number can be used to deﬁne another invariant of maps of ENRs.

If Y is an ENR and f : Y Y is continuous, we deﬁne the Lefshetz sequence to be: →

(If , If 2 , If 3 ,...)

k The name for this sequence obviously comes from the case in which If k = Λ(f ) for

20 all k N. This occurs, for example, if Y is a compact ENR. In the applications given ∈ here, this will always be the case.

As the focus of this dissertation is on maps of finite order on surfaces, it is reason- able to investigate Lefshetz sequences which are periodic. This investigation occurs in Chapter 2. In the next section, we look at trace formulas for finite cyclic actions on finitely generated free Z-modules. The results presented there provide a way to recover information about the induced map in homology from the Lefshetz sequence.

The results are also useful in understanding the basic concepts behind the equivariant signature.

1.3 Trace Formulas for Zm-Actions

Suppose we are given a finitely generated free Z-module A. Furthermore, suppose that there is a cyclic group G of order m acting on A as a group of isomorphisms. Let g G be a chosen generator of G and let λ = e2πi/m. Define V = C A. The action ∈ ⊗ of G on A extends to V via the rule g(α x) = α gx. Since G acts as a group of ⊗ ⊗ isomorphisms, we can define a linear transformation θ : V V to be multiplication g → by g. We will investigate the action on V using a matrix representation of θg. The nature of the minimal and characteristic polynomials of these actions is summarized in the following lemma.

21 Lemma 3. With the conventions established above, we have the following:

1. The minimal polynomial of θ divides xm 1 but does not necessarily equal g − xm 1. −

2. θg is diagonalizable and for each eigenvalue of θg, the geometric multiplicity

equals the algebraic multiplicity.

3. Let m 0 be the multiplicity of λj as an eigenvalue of θ for 0 j m 1. If j ≥ g ≤ ≤ − λk is an algebraic conjugate of λj (i.e. any other root of the minimal polynomial

j of λ ), then mj = mk.

4. Let Φd(x) be the d-th cyclotomic polynomial and let φj(x) = minλj (x). Then

if m = 0, φ (x) = Φ (x) for some d m. Deﬁne m 0 to be the common j 6 j d | d ≥

multiplicity of all the roots of Φd(x) in the characteristic polynomial. Thus, the

characteristic polynomial of θg is:

md charθg (x)= (Φd(x)) Yd|m

5. The relationship between m, dimZ(A), and the multiplicities md is given by:

dimZ(A)= mdϕ(d) Xd|m where ϕ is the Euler ϕ function.

Proof. The first fact is easily verified. To see that equality does not hold, consider the following matrix as defining a Z action on Z Z: 3 ⊕ 1 1 − 1 0 − 22 For the second fact, note that the m roots of xm 1 are distinct and thus the − roots of the minimal polynomial of θg have multiplicity 1. The Jordan canonical form must then have blocks of size 1 1. For each eigenvalue, the sum of the orders of the × corresponding Jordan blocks is the algebraic multiplicity. Since θg is diagonalizable, the dimension of the eigenspace must equal the algebraic multiplicity.

The third fact is a consequence of the fact that θg acts on A as well. Let

e ,...,e be a basis for A. Then 1 e ,..., 1 e is a basis for V . Note { 1 m} { ⊗ 1 ⊗ m} that:

g(1 e )=1 ge = 1 (α e + . . . + α e ) ⊗ i ⊗ i ⊗ 1 1 m m = α (1 e )+ . . . + α (1 e ) 1 ⊗ 1 m ⊗ m where α Z for all 1 i m. So there is a matrix representation in which all of i ∈ ≤ ≤ the entries are integers and hence the characteristic polynomial is in Z[x]. A little

Galois theory then shows that if λj is a root of the characteristic polynomial and λk is one of its conjugates, then λk must also be a root. The minimal polynomial of the

k algebraic number λ divides the characteristic polynomial of θg, so the multiplicity of each of the conjugate eigenvalues must be the same. Statement (4) is now clear from the above parts. Finally, the number of roots of Φd is ϕ(d) and the number of times each of them shows up is md. These must add to the degree of the characteristic polynomial.

The trace of θg is merely the sum of the eigenvalues. We know that the eigenvalues can be arranged as primitive roots of unity, so we only need to determine the sums of the primitive roots of unity.

23 Lemma 4. Let m 2 be a natural number and let ξ be a primitive m-th root of ≥ unity. Then the trace of the algebraic number ξ over C is given by:

2πij/m trC(ξ)= e = µ(m) (j,mX)=1 where µ is the M¨obius function (deﬁned explicitly in Section 2.1).

Proof. Let Φ (x) be the m th cyclotomic polynomial. The degree of Φ (x) is ϕ(m) m − m ϕ(m)−1 and the coeﬃcient of x in Φ (x) is trC(ξ). We will show that this coeﬃcient is m − given by µ(m). First suppose that m = p p ...p where p is a prime for 1 k s − 1 2 s k ≤ ≤ and p = p for i = j. If s = 1: i 6 j 6

m−2 m−1 Φm(x)=1+ x + . . . + x + x and thus, tr(ξ)= 1 = µ(m). We proceed by induction on s. Suppose the result is − true for s 1. By [21] we have: −

ps Φp1···ps−1 (x ) Φp1p2···ps (x)= Φp1···ps−1 (x)

Using long division, one can show:

Φ (x)= x(ps−1)ϕ(p1p2...ps−1) ( 1)s−2x(ps−1)ϕ(p1p2...ps−1)−1 + f(x) p1p2···ps − − where the degree of f(x) is smaller than ϕ(p p p p ) 1. Thus, tr(ξ)=( 1)s = 1 2 ··· s−1 s − − µ(m) and the theorem is proved by induction when m is square free. Now, suppose that m = pa1 pas where the p are distinct primes and a > 1 for some j. Also by 1 ··· s k j [21], we have that:

a1−1 as−1 p1 ···ps Φm(x) = Φp1···ps (x )

24 Every power of x on the right hand side of the equation is a multiple of pa1−1 pas−1. 1 ··· s Now, ϕ(m) 0 (mod pa1−1 pas−1), so xϕ(m)−1 occurs with coeﬃcient 0 on the left ≡ 1 ··· s hand side. Thus, tr(ξ)=0= µ(m) and the proposition is proved.

Combining all of the above, we have the following formula for the trace in terms of the variables md.

Theorem 5.

Tr(θg)= mdµ(d) Xd|m The following result lists some special cases which have been useful in working through examples.

Corollary 6. Let p and q be distinct primes and let g G and A be as above. ∈

1. If m = pk for some k 1, then Tr(θ )= m m . ≥ g 1 − p

2. If m = pq, then Tr(θ )= m m m + m . g 1 − p − q pq

1.4 The G-Signature

1.4.1 The G-Signature for Inner Product Spaces

In this section, we deﬁne the g-signature invariant for inner product spaces admitting invariant G-actions. This invariant is typically constructed using representation theory. Our construction uses only elements from standard linear algebra. Another approach can be found in [10].

25 Let m 2 and let G = Z be the cyclic group of order m. Let A be a ﬁnitely ≥ m generated free abelian group with Zm acting as a group of isomorphisms. We are also given a unimodular, bilinear, Zm-invariant, symmetric or skew-symmetric map:

Φ : A A Z × −−−−−→

In other words, Φ(gx,gy) = Φ(x, y) for all x, y in A and g G. Equivalently, ∈ we require Φ to be an inner product. This means that the maps x Φ(x, ) and → · y Φ( ,y) are both isomorphisms from A Hom(A, Z). → · → Now, the inner product space A can be made into a complex vector space by tensoring it with C over Z. Let V = A C. Note that G acts on V . Deﬁne ⊗

ΦC : V V C × −−−−−→

ΦC(x α,y β)= αΦ(x, y)β ⊗ ⊗

ΦC is Hermitian if Φ is symmetric:

ΦC(x α,y β) = αΦ(x, y)β ⊗ ⊗ = βΦ(y, x)α

= βΦ(y, x)α

= ΦC(y β, x α) ⊗ ⊗

Similarly, ΦC is skew-Hermitian if Φ is skew-symmetric.

Deﬁne a map θ : V V by x gx. In general, we can deﬁne a map ρ : Z g → → m → GL(V ) by g θ . The eigenvalues of θ can be easily determined. This is due to → g g the fact θm = 1. Then the minimal polynomial for θ divides xm 1 (Note that the g g − 26 minimal polynomial need not be xm 1. For example, G could act identically on A). − Hence, the eigenvalues of θ are the powers of λ = e2πi/m. The polynomial xm 1 g − has m roots of multiplicity 1, so the Jordan canonical form of the matrix associated with θg is diagonal and hence V is a direct sum of the eigenspaces of θg. Let Vj be

j the λ -eigenspace of θg. Then m

V = Vj j=1 M Now, ΦC is orthogonal under this decomposition, i.e. ΦC(v , v )=0if v V , j k j ∈ j v V and j = k. k ∈ k 6

ΦC(vj, vk) = ΦC(gvj, gvk)

j k = ΦC(λ vj,λ vk)

k j = λ λ ΦC(vj, vk)

k j But λ λ = 1 only when j = k and we are forced to conclude that ΦC(vj, vk)=0.

We now deﬁne a linear transformation L : V V that is invariant on the sub- → spaces Vj, self-adjoint, and has only real eigenvalues. This is done in two cases: Φ is symmetric and Φ is skew-symmetric. Let < , > be a Hermitian Z invariant inner · · m product on V . It can be shown that < , > is orthogonal on the subspaces V in a · · j ′ similar fashion to that used above on ΦC. Deﬁne L by:

′ < L v1, v2 >= ΦC(v1, v2) (1.1)

Recall that V = V V ⊥. If w V , we must have that w span V . Thus j ⊕ j 6∈ j ∈ k6=j{ k} ′ ′ ⊥ ⊥ ′ < L v,w >= ΦC(v,w) = 0 and L v (V ) = V . This gives us that L V V . ∈ j j j ⊂ j

27 Suppose that Φ is symmetric and let L = L′. Let µ be an eigenvalue of L and w an eigenvector associated with µ. Since ΦC is Hermitian,

µ = µ=< Lw,w > = ΦC(w,w)

= ΦC(w,w)

Hence, µ is a real number. To see that L is self-adjoint, just use the deﬁnition. Let u, v V . Since ΦC is Hermitian, ∈

< Lu,v >= ΦC(u, v)= ΦC(v,u)= =

Since L is self-adjoint, it is normal. Consider the restriction of L to the subspace

+ Vj. By the spectral theorem, Vj is a direct sum of the eigenspaces of L. Let Vj be the

− direct sum of the eigenspaces of L that have positive eigenvalues and Vj the direct sum of the eigenspaces of L that have negative eigenvalues. Since L has only nonzero eigenvalues, V = V + V −. j j ⊕ j If Φ is skew-symmetric, deﬁne V + and V − via the map L = iL′. The details of j j − the virtually identical construction are omitted.

Finally, we are ready to define the g-signature invariant. The definitions are given in the standard representation theory terminology. For our purposes, we may take the second equality in Definition 8, as the official definition. It is from this definition that we will prove the properties of the equivariant signature that are relevant to this dissertation.

28 Deﬁnition 7. The equivariant signature σ(Zm, A, Φ) is the element in the virtual representation ring deﬁned by

m−1 σ(Z , A, Φ) = V + V − m j − j j=0 X Deﬁnition 8. The g-signature is the character of the above representation evaluated at g.

m−1 σ(g, A, Φ) = tr(gV +) tr(gV −) j − j j=0 X m−1 + − j = (dimC V dimC V )λ j − j j=0 X It follows immediately from the deﬁnition that the image of σ is contained in Z[λ].

Using Lemma 3, we can say a little more about the image of the equivariant signature.

We present a new elementary proof of the following theorem of Berend and Katz [5].

Their work extends much further than the result given below. In fact, much is known about the image of the signature map. The earliest result, due to Ewing [13], deals with the G = Zp case, p and odd prime. Berend and Katz [5] solved the realization problem for G = Z , m 2. The ideas contained in the proofs of these earlier results m ≥ play a large role in the work presented here.

Theorem 9 (Berend-Katz). Let A be a ﬁnitely generated Z-module with a cyclic group G of order m acting on A as a group of isomorphisms. Let Φ be a unimodular symmetric bilinear form on A. Then:

1. σ(g, A, Φ) Z +2Z[λ] ∈

2. If Φ is either positive deﬁnite or negative deﬁnite, then σ(g, A, Φ) is an integer.

29 + − Proof. Since Φ has a nonzero determinant, dim(Vj) = dim(Vj ) + dim(Vj ). Thus, dim(V ) dim(V +) dim(V −) (mod 2). Let λ = e2πi/m and let Φ denote, as in j ≡ j − j d Lemma 3, the d-th cyclotomic polynomial, where d m. Deﬁne J = j Z : Φ (λj)= | d { ∈ d 0, 0 j m 1 for all d m. By Lemma 3, J = ϕ(d) and dim(V ) = m for all ≤ ≤ − } | | d| j d j J. This gives: ∈ m−1 k j dim(Vk)λ = md λ j∈J Xk=0 Xd|m Xd Let Kd be the splitting ﬁeld of Φd over Q. Label roots of Φd(x) as λ1,d,...,λϕ(d),d.

It is known that GalQKd is the cyclic group on ϕ(d) elements. So this group must be the group of automorphisms of the form α : λ λ , 1 j ϕ(d). Thus, j 1,d → j,d ≤ ≤ λj = tr(λ ). But since Q is separable, tr(λ ) equals the negative of the j∈Jd 1,d 1,d P coeﬃcient of x in Φd(x), which is an integer. Finally,

n−1 σ(g, A, Φ) = (dim(V +) dim(V −))λj (1.2) j − j j=0 X n−1 dim(V )λj (mod 2Z[λ]) (1.3) ≡ j j=0 X m tr(λ ) (mod2Z[λ]) (1.4) ≡ − d 1,d Xd|n This implies that the ﬁrst assertion is true. If Φ is positive or negative deﬁnite, then the eigenvalues of the matrix representation of Φ are either all positive or all

+ − negative. Then either dim(Vj) = dim(Vj ) for all j or dim(Vj) = dim(Vj ). Thus, the equivalence in equation (1.3) can be replaced by and equal sign (with appropriate signs placed in front of dim(Vj)). The result is clearly an integer.

In Section 4.4, a complete geometric interpretation is given of the case when the equivariant signature of a map of prime order on a surface is 0. A suﬃcient condition

30 for the equivariant signature to vanish is given in Theorem 10. It is useful for the statement of this theorem to establish a domain for the signature function. This is accomplished by deﬁning an equivalence relation, called Witt-equivalence, on the set of inner product spaces over Z that admit an invariant G-action. First, we deﬁne the notion of a split inner product space.

Let A be a ﬁnitely generated free Z-module of dimension n and Φ : A A Z × → an inner product on A. Such a module is said to be split if there exists submodules

N, K A such that A = K N and N = N ⊥. Now, let G be a group acting on ⊂ ⊕ A and let ρ : G GL (A) be the regular representation of G. Moreover, suppose → n that G is Φ-invariant. A collection of objects that satisﬁes these hypotheses will be denoted by the triple (G, A, Φ). We will say that A is equivariantly split if there are

ρ-invariant submodules K, N A such that A = K N (equivariantly, of course) ⊂ ⊕ and N = N ⊥. Milnor wrote an excellent text [31] on symmetric bilinear forms that discusses splitness at great length. The notion of equivariant splitness is well-known and the basic theoretical necessities extend easily from Milnor’s exposition. We will not use anything more than the deﬁnition here.

Now, (G, A, Φ) and (G, A′, Φ′) are said to be Witt-equivalent if there are equivariantly split inner product spaces S and S′ such that A S = A′ S′. The following ⊕ ∼ ⊕ theorem shows that the equivariant signature is well-deﬁned on Witt-equivalence classes. Note the use of Lemma 3 in the proof.

Theorem 10. If (G, A, Φ) is equivariantly split and g G, then σ(g, A, Φ)=0. ∈

31 Proof. Let k be the dimension of A and deﬁne V = A C. By Lemma 3, we have ⊗ the following formula for V :

k = m1 + mdϕ(d) Xd|n

Decompose V into eigenspaces Vλj of ρ(g). Since A, and therefore V , is split, there are ρ-invariant inner product spaces K and N such that V = K N and N = N ⊥. ⊕

The bilinear form βC restricts to a bilinear form on each Vλj . We will show that N

j intersects each Vλ and hence that each Vλj is split. Applying Deﬁnition 8 then shows that σ(g, A, Φ) = 0.

m−1 Since N is ρ-invariant, we can decompose N into eigenspaces N j , N = N j . λ ⊕j=0 λ

Then N j V j . By hypothesis, we have that dim(N)= k/2. Using Lemma 3 again, λ ⊂ λ we have the following formula for N:

k = m′ + m′ ϕ(d) 2 1 d Xd|n Subtracting the two equations gives:

0=(m 2m′ )+ (m 2m′ )ϕ(d) (1.5) 1 − 1 d − d Xd|n ′ We claim that m 2m for all d n. Suppose d n, 1 d n. Since N j N, d ≥ d | | ≤ ≤ λ ⊂ ⊥ ⊥ we must have that N j N j . Thus, the largest N j could be is N j . In this case, λ ⊂ λ λ λ m =2m′ . Hence, m 2m′ for all d n. This implies that each of the terms in the d d d ≥ d | sum in equation 1.5 are all 0. Therefore, m =2m′ for all d n and we conclude that d d | each Vλj is a split inner product space in the non-equivariant sense. In other words, each Vλj is a direct sum of hyperbolic planes. Then Deﬁnition 8 implies σ(g, A, Φ) = 0.

32 1.4.2 The G-Signature for G-Manifolds

To define the equivariant signature for G-manifolds, we need to associate an inner product which is invariant with respect to the group action. This is accomplished via the cup product and the Poincaréduality isomorphism. More specifically, let M be a smooth closed orientable manifold of dimension 2k. Let G be a group of orientation

k preserving diﬀeomorphisms of M. Let A = H (M; Z)/Ext(Hk−1(M), Z). Since the

k homology of M is ﬁnitely generated, Ext(Hk−1(M), Z) is just the torsion in H (M; Z)

We deﬁne a bilinear form Φ : A A Z by: × →

Φ(x, y)=< x,Dy > where D : Hk(M; Z) H (M; Z) is the Poincar´eduality isomorphism and: → k

< , >: Hk(M; Z) H (M; Z) Z · · × k → is the scalar or Kronecker product. It is easy to see that Φ is a bilinear form. The form Φ is skew-symmetric if k is odd and symmetric if k is even. To show that Φ is an inner product, we use the universal coefficient theorem (see [36]). The content of this theorem is that the map x < x, > from A to H∗(M) is an isomorphism. Since D is → · k an isomorphism, we must have that the map x Φ(x, ) is an isomorphism. Thus, Φ → · is an inner product. Note that Φ is often written in the form Φ(x, y)=< x y, [M] >, ∪ where [M] denotes the fundamental homology class of M and is the cup product. ∪ The fact these these definitions are equivalent follows from the standard formulas relating all the different cohomology products.

33 Now, suppose that G is a cyclic group of order m. If g G, then g∗ = Hk(g) : ∈ Hk(M) Hk(M), is an isomorphism of A. The group of such elements deﬁnes a →

Zm-action on A. The following computation shows that this action is invariant with respect to Φ:

Φ(g∗x, g∗y) = ∪ = ∪ = < x y, g [M] > ∪ ∗ = < x y, [M] > ∪ = Φ(x, y)

With all of the ingredients laid out, we are prepared for the deﬁnition. For g G, we ∈ deﬁne the equivariant signature of g with respect to Φ to be:

∗ k σ(g)= σ(g ,H (M; Z)/Ext(Hk−1(M), Z), Φ)

The equivariant signature of a G-invariant inner product space was shown to be well-deﬁned on the equivariant Witt ring (graded). We can also identify a corresponding domain for the equivariant signature of G-manifolds. Let G be a group acting on two smooth closed oriented G-manifolds M1 and M2 of dimension n. M1 and M2 are said to be bordant if there is a smooth compact oriented G-manifold B of dimension (n + 1) such that ∂B = M ( M ) and the restriction of the action to 1 ∪ − 2 M is the original G-action on M . Here, M just means M with the opposite ori- i i − 2 2 entation. This relation is called equivariant bordism, or G-bordism, and is indeed an equivalence relation (see [9]). The disjoint union of two G-bordism classes deﬁnes an

34 associative and commutative operation. In fact, it is a group operation. The inverse of M is M. The additive identity is the bordism class of G-manifolds of dimension − n which bound G-manifolds of dimension (n+1)(called equivariant boundaries). The following theorem shows that the equivariant signature is well-deﬁned on bordism classes. More exactly, it shows that the cohomology functor maps two equivariantly bordant manifolds to two Witt-equivalent triples. This theorem also fully establishes the suﬃciency of Theorem 85.

Theorem 11. Let M be a manifold of dimension 2k and G a cyclic group of order m acting on M as a group of diﬀeomorphisms. Suppose that M is an equivariant

k boundary. Then (G,H (M; Z)/Ext(Hk−1(M), Z), Φ) is equivariantly split. Moreover, for every g G, σ(g)=0. ∈

Proof. It is suﬃcient to prove the result when M is connected. Let M be the G- manifold guaranteed by the hypotheses, ∂M = M. The result will be establishedf in the case that H∗(M; Z) and H∗(M; Z) aref torsion free. The general result can be established by performing a similarf procedure on the appropriate quotients. We have the following long exact sequence of cohomology:

j∗ ∗ . . . Hk(M, M; Z) Hk(M; Z) Hk(M; Z) δ Hk+1(M, M; Z) . . . → → −→ −→ → f f f Deﬁne W = image(j∗ : Hk(M; Z) Hk(M; Z)). We will show that W = W ⊥ and → hence that A is split. f

Let g G and supposeg ˜ M = g. Clearly, j g =g ˜ j. Hence if w = j∗(u) W , ∈ | ◦ ◦ ∈ we have g∗(w)= g∗j∗(w)= j∗(˜g∗(w)) W . Thus, W is G-invariant. ∈

35 We now show that W W ⊥. Let w, w′ W . Then w = j∗(u) and w′ = j∗(u′). ⊂ ∈ Then:

Φ(j∗(u), j∗(u′)) = < j∗(u) j∗(u′), [M] > ∪ = ∪ ∗

The orientation on M is induced by the orientation on M. In other words, if [M, M] denotes the fundamental homology class in H2k+1(M, Mf; Z), then ∂∗[M, M] =f [M]

(by [36], 6.3.10, pg. 304). Here, ∂ : H (M, M; Z)f H (M; Z) is thef connecting ∗ 2k+1 → 2k homomorphism. Since the homology long exactf sequence is exact, we have j∗∂∗ = 0.

Thus, Φ(w,w′) = 0 and we conclude that W W ⊥. ⊂ The hard part of the proof is the last part: W ⊥ W . Suppose that v W ⊥. We ⊂ ∈ will show that δ∗(v) = 0 in Hk(M; Z) and hence v ker(δ∗) = im(j∗) = W . For all ∈ u Hk(M; Z), we have: ∈

f 0=Φ(v, j∗u) = < v j∗u, [M] > ∪

2 = ( 1)k < j∗u, v [M] > − ∩

2 = ( 1)k − ∗ ∩

Since there is no torsion, the universal coeﬃcient theorem implies that j (v [M]) = 0. ∗ ∩ Now consider the following commutative diagram:

δ∗ Hk(M; Z) / Hk+1(M, M; Z)

∩[M] =∼ f Hk(M) / H (M; Z) j∗ k This implies that δ∗(v) = 0 and hence that v im(j∗)= W . The fact that σ(g)=0 ∈ f for all g G follows from Theorem 10. ∈

36 Chapter 2 Dold Sequences

2.1 Dold Sequences.

Let s : N Z be any sequence. We associate to s : N Z a function M : N Z → → s → which is deﬁned by the following equation:

n M (n)= µ(d)s( ) s d Xd|n where µ is the M¨obius function:

1 , m =1 µ(m)= ( 1)s , m = p p ...p , p prime for 1 k s, p = p  − 1 2 s k ≤ ≤ j 6 k  0 , else

It is often useful to write Ms(n) in the following form [12]:

M (n)= ( 1)|τ|s(n : τ) s − τ⊂XP (n) where P (n) is the set of primes dividing n and if τ P (n), n : τ = n p−1. In ⊂ p∈τ 1983, Dold [12] proved the following theorem: Q

Theorem 12 (Dold). Let Y by an ENR and f : Y Y a continuous map. If Fix(f n) → is compact for some n> 1 and s : N Z is the Lefshetz sequence of f, then n M (n). → | s

In the case where Y is a ﬁnite discrete space and f is a permutation, Dold’s theorem has a very elementary proof. The idea of the proof gives us some intuition about the abstract function Ms(n). First note that if Y is discrete, H0(Y ; Z) is the

37 free product of Y copies of Z. A permutation f will induce a linear map f on | | ∗ homology which permutes the canonical basis as f permutes Y . Then Tr(f∗) is just the number of ﬁxed points of f. Hence, we have that I(f k) = Fix(f k) for all k. | | Now, deﬁne

Fix (f)= y Y : f n(y)= y but f k(y) = y for all k < n n { ∈ 6 }

An elementary argument shows that if s : N Z is the Lefshetz sequence of f, →

2.2 Periodic Dold Sequences

A periodic Dold sequence of period m is a Dold sequence with the property that there exists an m N such that s(k) = s(k + m) for all k. The period of a periodic ∈ Dold sequence s : N Z is the smallest m that satisfies its defining condition. The → following lemmas begin the task of analyzing periodic Dold sequences. It is shown that for all periodic Dold sequences of period m, Ms(k) = 0 if gcd(k, m) < k. In particular, Ms(k) = 0 for all k > m. This means that all the periodic Dold sequences of period m can be determined abstractly without the topological considerations. It is also shown that if s : N Z is a Dold sequence such that M (k) = 0 for all but → s 38 finitely many k, then s : N Z is necessarily periodic. This fact has some surprising → consequences.

Lemma 13. Let s : N Z be a periodic Dold sequence with period m. Suppose that → p is a prime with exponent r in some natural number k and gcd(p, m)=1. If p′ is a prime such that p′ p (mod m), then: ≡ k M (p′)r = M (k) s pr s Proof. Let k′ = k (p′)r. Then P (k) = P (k′) . Suppose that τ P (k) and p τ. pr | | | | ⊂ ∈ Deﬁne τ ′ P (k′) by τ ′ =(τ p ) p′ . Then: ⊂ \{ } ∪{ }

′ ′ k ′ r −1 k : τ = r (p ) q p ′ qY∈τ k = (p′)r−1 q−1 pr q∈τ′ qY6=p′ k q−1 (mod m) ≡ p q∈τ′ qY6=p′ k q−1 (mod m) ≡ q∈τ Y k : τ (mod m) ≡

In the case that p τ, deﬁne τ ′ = τ. Then k : τ k′ : τ ′ (mod m). Since in each 6∈ ≡ case τ = τ ′ , the result follows. | | | |

Lemma 14. Let s : N Z be a periodic Dold sequence with period m. Suppose that → k N is divisible by a prime p that does not divide m. Then M (k)=0. ∈ s

Proof. Let r be the exponent of p in k. Since gcd(p, m) = 1, we know by Dirichlet’s theorem [40] that there are inﬁnitely many primes p p (mod m). By Lemma 1, i ≡ 39 we have: k M pr = M (k) s pr i s But this implies that p M (k) for all i. Hence, M (k)=0. i| s s

While M : N Z is deﬁned in terms of the function s : N Z, we can also s → → write s in terms of Ms. This process is known as M¨obius inversion[39].

Theorem 15 (M¨obius Inversion). Suppose that f,g : N Z are functions satisfying → f(n)= µ(d)g(n/d). Then g(n)= f(d) for all n N. d|n d|n ∈ P P Lemma 16. Let s : N Z be a periodic Dold sequence with period m. If k is a → { i} ﬁnite set of n natural numbers and gcd(k , k )=1 for i = j, then, i j 6

n s k = . . . M (d d d ) i s 1 2 ··· n i=1 ! Y dX1|k1 dX2|k2 dXn|kn n Proof. Let k = i=1 ki. Using M¨obius inversion, we have s(k) = d|k Ms(d). If d k, then d canQ be written uniquely as a product d d d wherePd k for 1 | 1 2 ··· n i| i ≤ i n. Thus, the above formula is merely a rearrangement of the M¨obius inversion ≤ formula.

Theorem 17. Let s : N Z be a periodic Dold sequence with period m. Suppose → that k N and gcd(k, m)

Proof. Let m = pa1 pan for some primes p ,...,p having exponents a 1 for 1 ··· n 1 n i ≥ 1 i n. Let k = qb1 qbr for some primes q ,...,q having exponents b 1 ≤ ≤ 1 ··· r 1 n j ≥ for 1 j r. If q p for some j, then we know by the previous lemma that ≤ ≤ j 6∈ { i} M (k) = 0. Now, suppose that q p . For all primes in p q , insert them s { j}⊂{ j} { i}\{ j}

40 in the prime factorization of k with exponent 0. We relabel the primes qj so that p = q for 1 i n. The only remaining case is the case when the exponent b > a i i ≤ ≤ i i for some prime p (for otherwise, k m). Without loss of generality, we assume that i | this prime is pn.

′ an ′ bn ′ Let m = m/pn and k = k/pn . Then gcd(m ,pn) = 1. By Dirichlet’s theorem again, there exist inﬁnitely many primes α p (mod m′). Multiplying through the z ≡ n

′ bn−1 modular equation by k pn gives the following congruence modulo m:

k′α pbn−1 k′pbn k (mod m) z n ≡ n ≡

′ bn−1 Thus we have the equation s(k pn αz) = s(k). Note the the hypothesis bi > ai was used in this step. We will apply the last lemma to both sides of this equation.

Choosing αz so large that gcd(αz, m) = 1 we have:

′ bn−1 s(k pn αz) = Ms(d1d2d3) d |k′ bn−1 d |α X1 d2X|pn X3 z

= (Ms(d1d2)+ Ms(αzd1d2)) d |k′ bn−1 X1 d2X|pn = Ms(d1d2) d |k′ bn−1 X1 d2X|pn

Also, we have that:

′ bn s(k)= s(k pn )= Ms(d1d2) d |k′ bn X1 dX2|pn These equations can be simpliﬁed by noting that:

bn Ms(d1d2)= Ms(pn d1)+ Ms(d1d2) bn bn−1 dX2|pn d2X|pn

41 Then from the above we have:

′ bn−1 bn 0= s(k) s(k pn αz)= Ms(pn d1) − ′ dX1|k Now, given this setup, we induct on the number ν of distinct divisors of k′. More precisely, Let N = d N : d k′ . We induct on ν = N . Suppose that ν = 1. Then { ∈ | } | |

′ bn k = 1 and the above equation reduces to Ms(k) = Ms(pn ) = 0. Now, suppose the theorem is true up to ν 1, i.e. all k′ having ν 1 distinct divisors. Suppose that ≥ ≥ k′ has ν + 1 divisors. We have the relation:

bn Ms(pn d1)=0 ′ dX1|k For all d k′ with d = k, the number of divisors of d is less than or equal to 1| 1 6 1 ν. Hence M (pbn d ) = 0 for all d k,d = k. But by the relation above, we have s n 1 1| 1 6

bn ′ Ms(k)= Ms(pn k ) = 0. By mathematical induction, we are done.

Proposition 18. A periodic Dold sequence s : N Z with period m is completely → determined by the elements s(d) where d m. |

Proof. It is suﬃcient to show that the these numbers determine the s(k) for k < m.

Suppose that s : N Z is a periodic Dold sequence of period m and that the values → of s(d) for d m are known. For k N, k m, we use M¨obius inversion to obtain: | ∈ 6|

42 The following lemma shows how easy it is to write down the form of a periodic

Dold sequence.

Lemma 19. Let s : N Z be a periodic Dold sequence of period m. →

1. If gcd(k, m)=1, then s(k)= s(1).

2. If gcd(k, m)= d = k, then s(k)= s(d) 6

Proof. Suppose gcd(k, m) = 1. Then s(k) = j|k Ms(j) = Ms(1) = s(1). If gcd(k, m)= d

s(k)= Ms(j) = Ms(j)+ Ms(j) j|k j|k, j|k, X Xj6|m Xj|m

= Ms(j) Xj|d = s(d)

For example, all periodic Dold sequences of prime period p must look like this:

(a, a, . . . , a, a + pα,a,a,...,a, a + pα,...)

(p−1) times (p−1) times All periodic Dold sequences| {z of period} 4 must| {z look} like this:

(a, a +2α, a, a +2α +4β, ...)

The next theorem will be used in Chapter 3.

Theorem 20. Let G be a ﬁnite cyclic group acting on a compact ENR as a group of homeomorphisms. If f and g are generators of G and s : N Z and t : N Z are → → the Lefshetz sequences of f and g, respectively, then s = t.

43 Proof. Since G is cyclic, f k = g for some k,1 1. Since gj = f kj we have s(j) = t(kj).

Then:

s(j)= t(kj)= Mt(l) = Mt(l)+ Mt(l) l|kj l|kj l|kj X Xl|mt lX6|mt

= Mt(l) Xl|j = t(j)

Thus, s = t.

Theorem 21. Let s : N Z be a Dold sequence such that M (k)=0 for all but → s ﬁnitely many k. Let m = lcm k : M (k) = 0 . Then s : N Z is a periodic Dold { s 6 } → sequence with period m.

Proof. We must show that s(k+m)= s(k) for all k. If j m, then j k : M (k) =0 6| 6∈ { s 6 } and hence M (j) = 0. First suppose that k m. Then: s |

s(k + m)= Ms(j)= Ms(j)+ Ms(j)= Ms(j)= s(k) j|(k+m) j|k j|(k+m) j|k X X j6|Xk,j6|m X

Suppose that k m. Then: 6|

s(k + m) = Ms(j)+ Ms(j) j|m, j|k j|(k+m) X j6|Xk,j6|m

= Ms(j)+ Ms(j) j|Xm, j|k j6|Xm, j|k = Ms(j) Xj|k = s(k)

44 This shows that s : N Z is periodic. Suppose the period of s : N Z is m′. Then → → m′ m. By Theorem 17, we have that k m′ for all k such that M (k) = 0. Then by | | s 6 deﬁnition of m, we must have that m m′. Thus, the period of s : N Z is m. | →

Corollary 22. A Dold sequence s : N Z is periodic with period m if and only if → M (k)=0 for all but ﬁnitely many k N and m = lcm k N : M (k) =0 . s ∈ { ∈ s 6 }

In [4], it was proved that the Lefshetz sequence of a periodic map on a surface is either bounded or asymptotic to ex. The following theorem shows that for an ENR and a map f with Fix(f k) compact for all k, boundedness is equivalent to periodicity.

Recall that σ0(n) denotes the number of divisors of n. For example, σ0(4) = 3.

Theorem 23. Let s : N Z be a Dold sequence and M 0 an integer such that → ≥ s(k) M for all k. Then M (k)=0 for all but ﬁnitely many k and s : N Z is | | ≤ s → periodic.

Proof. Applying M¨obius inversion to s : N Z, we get M (n) = µ(k)s(n/k). → s k|n Then: P

n M (n) µ(k) s | s | ≤ | | k Xk|n n s ≤ k k|n X M 1 ≤ Xk|n = Mσ0(n)

Dividing by n on both sides gives:

M (n) σ (n) | s | M 0 n ≤ n 45 The expression on the right is either 0 or a natural number for all n. Hence, if we show that limn→∞ σ0(n)/n = 0, then the theorem follows. To do this, note that the number of divisors of n fall into two categories: those > √n and those √n. Those ≤ that are greater than √n look like n/k where k √n and hence there are less than ≤ ⌊ ⌋ √n of them. There are tautologically less than or equal to √n divisors of n less ⌊ ⌋ than √n. Hence, σ (m) 2√n and: 0 ≤ σ (n) 2√n 2 0 0 = ≤ n ≤ n √n

Then by the squeeze theorem, limn→∞ σ0(n)/n = 0. This implies that there exists an N such that M (n) = 0 for all n > N. By the previous theorem, s : N Z is s → periodic.

Corollary 24. Let X be an ENR and f : X X a continuous map. Suppose that → either X is compact or F ix(f k) is compact for all k. If the Lefshetz sequence of f is bounded, then it is periodic.

Proof. This follows immediately from Dold’s theorem.

2.3 Limit Points of Dold Sequences

Lemma 25. Let s : N Z be a Dold sequence. Then the set of limit points of the → M¨obius inverse sequence, M : N Z is contained in the set , 0, . s → {−∞ ∞}

Proof. Suppose that Ms has a ﬁnite limit point a. It is clear that a is an integer and hence that M (k) = a for inﬁnitely many k. Since k M (k) = a for each of these k, s | s we must have a = 0. The lemma follows.

46 If s : N Z is any sequence, then we will denote the set of limit points of → s : N Z by L(s). In the following, we analyze the relationship between the set of → limit points of s : N Z and the set of limit points of the M¨obius inversion sequence. → Some potential applications of this theory are given afterwards.

Proposition 26. Let s : N Z be a Dold sequence. →

1. If exactly one of is in L(M ), then that one is in L(s). ±∞ s

2. If L(M )= , then L(s)= . s {∞} {∞}

3. If L(M )= , then L(s)= . s {−∞} {−∞}

Proof. It suffices to establish the verity of the statements only when L(M ) and ∞ ∈ s when = L(M ). For the first, we divide the problem into two cases: all the {∞} s values taken by Ms are positive, and a finite number of the Ms values are negative.

Suppose then that all the values taken by M are positive and L(s). Deﬁne: s ∞ ∈

A = k N : M (k) > 0 { ∈ s }

Then A = . Let M > 0 be given. Let k A such that k > M. Then k M (k) | | ∞ ∈ | s and:

s(k)= M (j) M (k) k > M s ≥ s ≥ Xj|k Since for every M there is a k A such that k > M, the above inequality implies ∈ that is a limit point of s : N Z. ∞ → Now suppose that L(M ) and a ﬁnite number of the M are negative. Deﬁne: ∞ ∈ s s

J = Ms(k) k s.t.XMs(k)<0 47 and let A be as in the previous case. Let M > 0 be arbitrary and choose k A such ∈ that k > M J. Then we have: −

s(k) = Ms(j) Xj|k M (k)+ J ≥ s k + J ≥ > M

The statement follows as in the ﬁrst case.

Now suppose that L(M ) = and let M > 0 be arbitrary. The proof of this s {∞} statement is very similar to the last argument. Suppose ﬁrst that M (k) 0 for all k. s ≥ Then s(k) M (k). There are only ﬁnitely many k such that M (k) = 0, so choose ≥ s s K such that M (k) = 0 for all k>K. Choose N = max K, M . Then if k > N: s 6 { }

s(k) M (k) k > M ≥ s ≥

The result now follows. If M (k) 0 for ﬁnitely many k, deﬁne J as above. Let K s ≤ be such that M (k) > 0 for all k>K. Set N = max K, M J . Then if k > N: s { − }

s(k) M (k)+ J k + J > M J + J = M ≥ s ≥ −

This completes the proof.

Proposition 27. Let s : N Z be an unbounded Dold sequence. Then at least one → of is a limit point of M : N Z, i.e. M is unbounded. ±∞ s → s

Proof. If not, then 0 would be the only limit point of M : N Z and hence s : N Z s → → would be periodic, and thus bounded.

48 These theorems considered here become of greater interest when considered in light of the Shub-Sullivan Theorem. It reads:

Theorem 28 (Shub-Sullivan). Let M be a compact smooth manifold and f : M M → a C1 map such that the Lefshetz sequence of f is unbounded. Then the set of periodic points is inﬁnite.

R.F. Brown [14] has asked to what extent can the hypotheses on M and f be relaxed. With our work, the question can be considered in the following way. Let

X be an ENR. If f : X X is any continuous map, we will denote it’s Lefshetz → sequence by s(f) : N Z. Suppose that X has the property that for all continuous → functions f : X X, M (k) = 0 implies f has a periodic point of period k. Then → s(f) 6 the hypothesis of Schub and Sullivan automatically implies that Ms(f) is unbounded.

The property on X would then imply their theorem. However, a full description of spaces having this property is not known. A paper by Fagella and Llibre [15] shows that for a holomorphic map f : M M of a compact complex manifold M there → exists an N > 0 such that for all p > N, M (p) = 0 implies f has a periodic point s(f) 6 of period p. Of course, with our work, this is strong enough to prove a much weaker version of the Schub-Sullivan theorem. The minimal hypotheses on the space X and f used here seem promising at developing a stronger form for their theorem. This interesting possibility will not be considered further in this dissertation.

49 2.4 Geometric Realization

For the remainder of the chapter, we follow the example of Dold and consider realization of some Dold sequences on Euclidean neighborhood retracts (ENR). It was shown in [12] that given any Dold sequence s : N Z, one can construct certain → topological spaces with self-maps whose Lefshetz sequence is s : N Z. →

Theorem 29. If s : N Z is a Dold sequence, then there is a connected 2 dimensional → − simplicial complex Y and a map f : Y Y such that I(f n)= s(n) for all n. More- → over, f can be chosen so that M (k) is the number of periodic points of f k, each | s | with index 1. ±

Our concern is with periodic Dold sequences. The realization problem then, is to determine exactly which periodic Dold sequences are the Lefshetz sequences of periodic homeomorphisms on ENRs. We will not consider the problem in general, but only for the case that most resembles the case of periodic orientation preserving diﬀeomorphisms on surfaces.

Recall that an orientation preserving diffeomorphism ψ : S S of finite order m → on a surface S has a finite number of fixed points. Clearly, then the powers ψk : S S → for k 0 (mod m) will also have a finite number of fixed points. Hence, the Lefshetz 6≡ sequence always contains positive entries for k 0 (mod m). In fact, more can be 6≡ said. If s : N Z is the Lefshetz sequence of ψ, it must be that M (k) 0 for all → s ≥ k 0 (mod m). This follows from the fact that if x is a periodic point of period k, 6≡

50 then the set x, ψ(x), ψ2(x),...,ψk−1(x) contains exactly k points and is invariant { } under the action of ψ. This set is contained in the ﬁxed point set of ψk and each element has local Lefshetz index 1. This situation is therefore equivalent to the case where we have a permutation acting on a discrete set. In that case, Ms(k) is exactly the number of periodic points of period k and M (k) 0. The other “obvious” s ≥ property of Lefshetz sequences of periodic orientation preserving diﬀeomorphisms on surfaces is that

2 2 m k k Λ(ψ )= ( 1) Tr(Id )= ( 1) dimZH (S; Z)= χ(S) − k − k Xk=0 Xk=0 For ENRs, it is natural to attempt realization of periodic Dold sequences s : N Z → of period m by periodic homeomorphisms f : X X of period m such that the → following three conditions are satisﬁed:

1. The Lefshetz sequence of f is s : N Z. →

2. M (k) 0 for k 0 (mod k). s ≥ 6≡

3. Λ(f m)= χ(X).

It is clear that the last condition is redundant. However, our approach will be to realize ﬁrst all of the terms in the sequence except the m-th term, and then to adjoin the appropriate spaces so that the correct Euler characteristic is given. We aim to proving the following theorem.

Theorem 30. Let s : N Z be a periodic Dold sequence of period m. Furthermore, → suppose that M (k) 0 for all k 0 (mod m). Then there exists a ﬁnite connected s ≥ 6≡ compact 2-dimensional simplicial complex (and thus ENR), X, and a homeomorphism

51 f : X X of period m such that the above three conditions are satisﬁed. Moreover, → f : X X has exactly M (k) periodic points of period k for all k 0 (mod m). → s 6≡

2.5 Dold’s Realization Theorem

Our proof modiﬁes the ideas of Dold’s paper. In this section we outline his argument for the solution to the general realization problem for connected ENRs. The purpose of this is to highlight the diﬃculties which appear when the realizing maps are required to be periodic. We will also see that the constructions given by Dold never produce periodic maps. Hence, the work presented in subsequent sections is a necessary addition to the geometric realization program.

Let k 1 be given. Suppose that M (k) > 0. Let Σ(1, 2,...,n) denote the ≥ s group of permutations of an n-element set. Choose a σ Σ(1, 2,...,M (k)) that k ∈ s is a product of Ms(k)/k disjoint k-cycles. The permutation σk has exactly Ms(k) periodic points of period k. For this k, we form the quotient space:

S2 Z X = × Ms(k) k Z {∞} × Ms(k) We will identify S2 with the one point compactification of R2. A map on S2 can be defined by: 2x if x < 1 e(x)= 1−kxk2 k k if x 1 ∞ k k ≥ 1 Now, if Ms(k) < 0, we do the same thing on S except everywhere we have Ms(k), we replace it with M (k) . The map e is also defined on S1 as it is the one point | s | compactification of R. Clearly e : Sn Sn, n = 1, 2 has two fixed points: 0, → . The fixed point at will be “moved” later. Dold shows that the point 0 has ∞ ∞ 52 Figure 1: Illustration of Dold’s simplicial 2-complex. Two 2−cycles One 4−cycle permute circles. permutes circles. 1 2 3 4

s(1)= 1 s(2)= −3 s(3)= 1 s(4)= −7 Ms (1)= 1 M s(2)= −4 M s(3)= 0 M s(4)= −4

Lefshetz index ( 1)n, n = 1, 2 for all iterates of e. The map e σ induces a map − × k f : X X . The restriction of f to X has M (k) periodic points of period k k → k k k\{∞} | s | k, each of index Sign(Ms(k)).

Finally, if Ms(k) = 0, do nothing. The space Xk will be the empty topological space.

To prove the theorem, we adjoin X , for all k 1, to the real line R by identifying k ≥ with k R. The resulting space is called X . A map f : X X is deﬁned by: ∞ ∈ s s → s (x + 1) R if x R ∈ 1 ∈ f(x)= (x + 1) R if x Xk, x 1  − kxk ∈ ∈ k k ≥ f (x) X if x X , x 1  k ∈ k ∈ k k k ≤ Note that the middle row is responsible for “moving” the point at inﬁnity. The map f : X X turns out to have all of the requisite properties. A picture that is worth s → s a thousand words is given in Figure 1.

Dold’s construction does not produce a map on a compact ENR. However, in the

n case that “we only prescribe ﬁnitely many values of I(f ) or if almost all In(f) = 0”, he proves the following result:

Theorem 31. If s = s(n) N is an N-tuple of integers such that n M (n) for all n, { }n=1 | s 1 n N, then there is a compact connected simplicial complex K and a continuous ≤ ≤ map g : K K such that I(gn)= s(n) for all n, 1 n N. → ≤ ≤ 53 The idea of the proof is very simple. Construct the space Xs as above setting

Ms(k) = 0 for k > N. Now let α be an irrational number. Form K be identifying all the points that diﬀer by an integer multiple of α. The resulting space is compact.

Since α is irrational, the map induced by fs on K has no new periodic points. Hence, g : K K is the correct map. s s → s 2.6 Periodic Realization of s : N Z →

To prove Theorem 30, we ﬁrst construct a space and a map satisfying every hypothesis other than the restriction on the Euler characteristic. Let s : N Z → m−1 be a periodic Dold sequence of period m. Deﬁne N = k=1 Ms(k). Note that P Ms(k)=0 for all k such that gcd(k, m)

Lemma 32. If s : N Z is a periodic Dold sequence of period m such that M (k) → s ≥ 0, 1 k m 1, then there exists a connected compact two-dimensional simplicial ≤ ≤ − complex X and a periodic homeomorphism f : X X of period m such that N N N → N Λ(f k)= s(k), 1 k m 1, and χ(X )= N. ≤ ≤ − N

Now the Euler characteristic needs to be adjusted so that it is equal to s(m).

There is a simple divisibility condition that makes this possible.

Lemma 33. Suppose we have two periodic Dold sequences of period m whose repeating parts are given by:

s¯ =(a1, a2, a3,...,am−1, x)

t¯ =(a1, a2, a3,...,am−1, y)

Then x y (mod m). ≡

54 Proof. Simply compute:

x y = M (k) M (k)= M (m) M (m) − s − t s − t Xk|m Xk|m The last term is divisible by m.

The adjustment of the Euler characteristic is made by adjoining a multiple of m spheres (of varying dimension) to the space constructed in Lemma 32. In the next subsection, we prove Lemma 32. The proof of Theorem 30 is given immediately there- after.

2.6.1 Proof of Lemma 32

Just as in the proof of Dold’s theorem, we ﬁrst look for a permutation in some

Σ(1, 2,..., N) that permutes the numbers in the same way as prescribed by s : N →

Z. The intuition for this argument comes from thinking of Ms(k) as the number of periodic points of period k. By assumption, k M (k). The number of orbits (in this | s situation) of the set

Fix (f)= x X : f k(x)= x, f j(x) = x for j

of the action generated by f on X is Ms(k)/k. This is precisely how many k-cycles we will need in our permutation. Thus, the size of our permutation group must be

m−1 m−1 k kMs(k)/k. Thus, N = k Ms(k), as has been already deﬁned.

P For starters, denote by kPthe smallest k > 1 such that M (k) = 0. Find an 1 s 6 element σ Σ(1,...,M (k )) that is a product of M (k )/k disjoint k -cycles. k1 ∈ s 1 s 1 1 1 Note that this is possible since k M (k) for all k. Now let k be the smallest k>k | s 2 1 55 such that M (k) = 0. Choose an element s 6

σ Σ(M (k )+1,..., M (k )+ M (k )) k2 ∈ s 1 s 1 s 2 that is a product of Ms(k2)/k2 disjoint k2-cycles. We repeat this process, obtaining a sequence k1

M (k) = 0, and k is the smallest number greater than k such that M (k) = 0. s 6 i+1 i s 6

Furthermore, for each ki we obtain a

i−1 i−1 i σ Σ := 1+ M (k ), 2+ M (k ),..., M (k ) ki ∈ ki s j s j s j j=1 j=1 j=1 ! X X X X such that σ is a product of M (k )/k disjoint k -cycles. Now, note that N ki s i i i −

k,k>1 Ms(k) = Ms(1) = s(1), which is the number of ﬁxed points speciﬁed by

Ps : N Z. → We deﬁne σ Σ(1,..., N) to be the concatenation of the σ Σ in Σ(1,..., N). s ∈ ki ∈ ki Then σ contains a product of M (k)/k k-cycles for each k,1

56 Lemma 34. Let Y be a topological space, and let g : Sn−1 Y be a map. Deﬁne → Z = Dn Y to be the quotient space where a point x Sn−1 is identiﬁed with its ∨g ∈ image g(x) Y . Then there is a map ψ and a short exact sequence: ∈

ψ 0 H˜ (Y ) H˜ (Z) Ker(H (g)) 0 → n → n → n−1 →

Let Y = D2, g : S1 D2 be inclusion, and Z the space obtained by adding 1 → 1 a 2-cell to Y . Homology is of course done over the integers. Since Ker(H1(g)) is a subgroup of Z, the above ses is split exact and we have that H˜ (Z ) = H˜ (Y ) 2 1 ∼ 2 ⊕ Ker(H (g)). Since D2 is contractible, H (D2) = H (D2) = 0 . Hence, H (Z ) = Z. 1 1 ∼ 2 ∼ { } 2 1 ∼ Let g : S1 Z be the obvious inclusion and Z be the space obtained by adding a 2 → 1 2 2 2-cell to Z1. The above ses then shows that H2(Z2) ∼= Z . It is easy to see that XN

2 is the space ZN−1, where by Z0 we simply mean the space Y = D . An induction argument then shows that XN has homology: Z if q =0 Hq(XN ) = 0 if q =1, 2 ∼  N−1 6  Z if q =2

Now, (ρm 1)∗ = Id. Hence, the only effect on the Leftshetz number comes from × (1 σ ) . Suppose that k m and M (k) > 0. Then 1 σ has M (k)/k cycles of × s ∗ | s × s s length k. Since (1 σ )k =1 σk, there are M (k) points fixed by 1 σk that are not × s × s s × s fixed points of 1 σj for j

Λ([(ρ 1) (1 σ )]k) = Λ(1 σk)= M (j)= s(k) m × ◦ × s × s s Xj|k 57 Figure 2: Schematic of the basic s(1) = 0 case. 1 2 3 4 5

2π m

1 2 4 5 2 3 5 4 3 1

Moreover, we have that χ(X )=1+(N 1) = N. A schematic diagram of one N − of these X is drawn in Figure 2 with the repeating part of s : N Z given by N → (0, 2, 3, 0, 5).

2.6.2 Proof of Theorem 30

It is only necessary to correct the Euler characteristic of the the space XN con-

1 structed above. This is accomplished by adjoining copies of S if χ(XN ) is too large

2 and adjoining copies of S if χ(XN ) is too small. The following lemma, copied ver- batim from Dold’s book ([11],III.7.8), allows us to compute the homology of spaces obtained from these adjoins.

Proposition 35. Let X and Y be topological spaces, P X and Q Y . Denote ∈ ∈ by X Y denote the adjoining of X to Y at the points P and Q. If the closure of a ∨ point P in X has a neighborhood U in X whose inclusion map U X is homotopic → relative to P to the constant map U P , then: →

∼ (i , j ) : H(X,P ) H(Y, Q) = H(X Y,P = Q) ∗ ∗ ⊕ → ∨ and it follows that H˜ (X) H˜ (Y ) = H˜ (X Y ). ⊕ ∼ ∨ 58 We have the two cases s(m) > N and s(m) < N. Suppose ﬁrst that s(m) > N(i.e.

M (m) > 0). Let δ = s(m) N. On S1 C, mark the δ points exp(2πij/δ), s − ⊂ j = 0,..., δ 1. Take a disjoint union, U , of δ copies of S2 and label the copies − δ 0, 1, 2,..., δ 1. A map ϕ : U U is deﬁned by: − δ → δ δ k k + (mod δ) → m where the k-th copy of S2 is mapped to k + δ/m (mod δ), the remainder of k + δ/m

2 from division by δ. On XN , we identify the North pole of the k-th copy of S with the point exp(2πik/δ) and denote the resulting quotient space by Xs. Since Xs is

2 obtained from XN by subsequently adjoining copies of S , the homology of Xs is given by: Z if q =0 Hq(Xs) = 0 if q =1, 2 ∼  N−1+δ 6  Z if q =2 A map on X is defined first on the disjoint union of X and U , U X . Define s  N δ δ ∨ N ψ : U X U X by ψ U = ϕ and ψ X = f . Since m δ, ψ induces a δ ∨ N → δ ∨ N | δ | N N | homeomorphism fs = ψ on Xs that has period m.

Note that for k between 1 and m 1, Λ(f k)=Λ(f k ). This follows from the fact − s N k that the ﬁxed point set of each f is covered by disjoint open sets in Xs on which the

k Lesfshetz number can be computed in the same way as fN . To compute the Euler characteristic, we use the formula χ(X )=Λ(f m)=1+ N 1+ s(m) N = s(m). s s − − An “overhead” view of the quotient space in the case where m = 3 and δ = 6 is given in Figure 3.

If the situation is that s(m) < N (i.e. Ms(m) < 0, we do the same thing but with copies of S1. Let δ = N s(m). Repeating the above argument gives a space X and − s 59 Figure 3: Overhead view of the case when s(m) > N. 2 1

0 2 2 4 4 0 3 0

1 3 3 5 5 1

4 5 a homeomorphism f : X X with Λ(f k) = s(k) for all k, 1 k m 1. The s s → s s ≤ ≤ − homology of Xs is given by:

Z if q =0 Zδ if q =1 H (X ) =  q s ∼ ZN−1 if q =2   0 q =0, 1, 2 6  Hence, Λ(f m)= χ(X )=1 (N s(m))+ N 1= s(m) and the proof is complete. s s − − − 2

60 Chapter 3 Applications to Maps of Surfaces

3.1 Mapping Class Groups

In the previous chapter, we discussed the general number theoretic properties of periodic Dold sequences. Also, we investigated when certain periodic Dold sequences are the Lefshetz sequences of some periodic homeomorphism on a compact simplicial

2-complex. We considered only periodic Dold sequences which were “surface-like”.

In this chapter, we look at surfaces themselves. What number theoretic conditions must be placed on a periodic Dold sequence so that it is the Lefshetz sequence of some periodic homeomorphism (equivalently, diﬀeomorphism) of a connected closed orientable surface? This problem is solved in the last six sections of this chapter.

Also, particular attention is given to the case in which the period is of prime order.

The proof of the theorem in this special case uses the equivariant connected sum whereas the proof of the general case uses Fuchsian groups.

In the first two sections, we tie our realization problem to the study of mapping class groups, i.e. the group of orientation preserving diffeomorphisms on a surface modulo isotopies. In this first section, we take a look at the general theory of finite order mapping classes and the more general theory of mapping classes due to

Thurston. Throughout this chapter, a surface will be a closed connected and orientable 2-manifold.

61 3.1.1 The Thurston Classiﬁcation Theorem

Recall that the mapping class group of a surface S is the group of orientation preserving diffeomorphisms modulo the normal subgroup of diffeomorphisms isotopic to the identity. We will denote this group by Mod(S). This group was studied for much of the twentieth century but has experienced somewhat of a renaissance in the last thirty years. Lickorish showed that if S has genus g, then Mod(S) is finitely generated by 4g 1 easily described homeomorphisms called Dehn twists. It is also true that − the mapping class group admits a finite presentation. While this is all good news, it is

Z quite diﬃcult to ﬁnd presentations of these groups. For the torus, Mod(S) ∼= SL(2, ) (see [34]), so this case is not so bad. The case of the two-holed torus is much worse

(see [6]). The generators are very useful for doing a few practical things like showing that a homomorphism from Mod(S) is onto. In general, however, they are very hard to use. For example, writing even the simplest of periodic surface diﬀeomorphisms as a composition of Dehn twists is extraordinarily diﬃcult.

For our purposes, it is best not to work with this particular information about the mapping class group. We would much prefer to deal with information that typiﬁes the elements of the mapping class group. In fact, the following theorem says that no matter how complicated a composition of Dehn twists gets, it still must fall in one of three classes.

62 Theorem 36 (The Thurston Classiﬁcation Theorem [38]). Let S be a surface of genus g 2. If ψ Mod(S), then either: ≥ ∈

1. ψ is periodic, i.e. ψm =1 for some m> 1 and ψ ﬁxes an element of Teichm¨uller

space T(S),

2. ψ is pseudo-Anosov, or

3. ψ is reducible, i.e. ψ ﬁxes a family Γ of disjoint, separating, nonisotopically

trivial curves on S that is invariant under the action of ψ. Moreover, each of

the parts (which are not necessarily connected) of S Γ satisﬁes (1) or (2). \

A mapping class ψ is said to be pseudo-Anosov if there exists a representative homeomorphism f : S S and transverse measured foliations (Fs, µ ) (called the → s u stable foliation) and (F , µu)(called the unstable foliation) such that:

1 f(Fs)= Fs; f(Fu)= λFu λ for some λ > 1. Hence, a pseudo-Anosov map “stretches” in the unstable direction and “shrinks” in the stable direction.

Thurston describes a representation theory of a large class of mapping classes into

PSL(2, R) in his paper. Periodic mapping classes correspond to elliptic elements. Hy- perbolic elements correspond to pseudo-Anosov mapping classes. Reducible mapping classes correspond to parabolic M¨obius transformations. However, the class that he describes is once again dependent upon a Dehn twist decomposition of the map. It would be preferable to ﬁnd a representations of mapping classes in matrix groups

63 which does not ultimately rely on this underlying difficulty. The idea under investigation in this dissertation is figure out how well Lefshetz sequences differentiate between the types given by Thurston. This will be shown to have applications to the representation problem in matrix groups.

3.1.2 Finite Order Mapping Classes and Their Dold Sequences

Let S be a surface of genus g 2 and let ψ : S S be an orientation preserving ≥ → diffeomorphism of finite period m. Since S is compact and ψ is onto, I(ψ)=Λ(ψ). It is well-known that the fixed points of ψ are isolated [18]. The Lefshetz index theorem along with the fact that ψ is orientation preserving implies that the Lefshetz index of each fixed point is 1. In this case we then have that the Lefshetz number is equal to the number of fixed points. The sequences under investigation now take the form:

(Λ(ψ), Λ(ψ2), Λ(ψ3),..., Λ(ψm), Λ(ψ),...)

Since S is a Euclidean neighborhood retract, Dold’s theorem tells us that this sequence is in fact a periodic Dold sequence of period m′, where m′ m. We will show that when | g 2, ψ, H (ψ), and the Lefshetz sequence of ψ all have the same period. This fact ≥ 1 will be proved after we introduce an important short exact sequence whose properties we wish to study.

The integral homology functor induces a map from Mod(S) into GL(2g, R). The intersection form induces a skew-symmetric real bilinear form on H1(S; Z) that is preserved by orientation preserving maps. With respect to the canonical basis of

64 H1(S; Z), we write the matrix of the intersection form as:

0 I J = I 0 − where I is the g g identity matrix. The observations of this paragraph imply that if × t A is the matrix of ψ∗ written with respect to this basis, then AJ A = J. We denote by Sp(2g, Z) the set of all such invertible A that have entries in the integers. This gives a map of β : Mod(S) Sp(2g, Z). It is possible to ﬁnd generators for Sp(2g, Z) → and show that these generators are achieved by certain compositions of Dehn twists.

Hence, β is onto. The kernel of β i.e. the set of those mapping classes which induce the identity on homology, is called the Torelli group. We denote the Torelli group by

T (S). This gives us the following short exact sequence:

β (1) T (S) α Mod(S) Sp(2g, Z) (1) → −→ −→ →

We now prove that the order of the map, the order of its induced map and the order of its mapping class are all the same when g 2. First, we have the following ≥ easily proved lemma:

Lemma 37. Let A, B, C be groups and suppose we have the following short exact sequence:

β (1) A α B C (1) → −→ −→ →

Suppose furthermore that A contains no nonidentity elements of ﬁnite order. Then if b B has order m, then β(b) also has order m. ∈

Proof. Suppose that b B has order m. Then β(b)m = 1 and β(b) has order m′ m. ∈ | Since β(bm′ ) = 1, we have bm′ Im(α). Then for some a A, α(a) = bm′ . The ∈ ∈ 65 injectivity of α and the absence of torsion in A implies that a = 1. Hence, bm′ = 1 and m m′. Thus, m′ = m. |

It is clear, then, that it is of utmost importance to know that the Torelli group contains no torsion. This, in fact, is a theorem of Hurwitz:

Theorem 38 (Hurwitz). Let S be a surface of genus g 2 and ψ Mod(S) be a ≥ ∈ mapping class of ﬁnite order m, m> 1. Then ψ∗ does not act trivially on H1(S; Z).

In other words, the Torelli group contains no elements of ﬁnite order.

Proof. By the Nielsen Realization theorem, there is an orientation preserving diﬀeo- morphism g : S S of order m that is isotopic to ψ. The ﬁxed points of g are → isolated and each have local Lefshetz index 1. As the Lefshetz number is the sum of the local Lefshetz numbers, Λ(g) 0. If ψ acts trivially on homology, then ψ ≥ ∗ is the 2g 2g identity matrix. Computing the Lefshetz number in this way gives × Λ(ψ)=2 2g < 0. Hence, the Torelli group contains no torsion. −

These two results together imply that every element of ﬁnite order in Mod(S) has the same order in Sp(2g, Z). Now, every orientation preserving diﬀeomorphism of

ﬁnite order (as will be discussed in Section 3.5) can be considered as an automorphism of a Riemann surface. In this case, the restriction of β is an injection Aut(S) → Sp(2g, Z). Thus, ψ, the mapping class of ψ, and β(class of ψ) all have the same period in their respective groups.

Finally, it is easy to see why the period of the Lefshetz sequence and the period of

ψ are the same if g 2. This is because ψk is also a periodic map and cannot be in ≥ the Torelli group unless k = m. For each of these periodic maps, Λ(ψk) 0 if k 0 ≥ 6≡ 66 (mod m) and Λ(ψk) < 0 if m k. Hence, the period of the Lefshetz sequence must be | exactly m.

The next theorem shows that the problem of determining the similarity classes of elements of finite order (for g 2) in GL(2g,H (S, Z) C) which are achieved by ≥ 1 ⊗ orientation preserving diffeomorphisms of finite order is equivalent to the problem of determining all periodic Dold sequences which are achieved by orientation preserving diffeomorphisms of finite order.

Theorem 39. Let S be a surface of genus g 2 and suppose ψ : S S is an ≥ → orientation preserving diﬀeomorphism of ﬁnite order m. Then the conjugacy class of

H (ψ) : H (S; Z) H (S; Z) in GL(2g,H (S; Z) C) is uniquely determined by the 1 1 → 1 1 ⊗ integers Λ(ψk), where k m. |

Proof. To prove this, we need only show that the eigenvalues of H1(ψ) are completely determined by the Λ(ψk) where k m. We know by Dold’s theorem that the Lefshetz | sequence s : N Z of ψ is a periodic Dold sequence. By the above, the period of this → periodic Dold sequence is m. Hence, all of the values of s(k) are known for all k m. | Theorem 18 then shows that s is completely determined by the integers Λ(ψk) for k m. Thus, by the Lefshetz Fixed Point Theorem, Tr(ψk)=2 Λ(ψk) is known for | ∗ − k k all k. By Theorem 3, ψ∗ is diagonalizable for all k. Thus, Tr(H1(ψ )) is just a sum of the powers of the eigenvalues of ψ. These are otherwise known as the power sums of the characteristic polynomial f(x) of H1(ψ). The technique of Newton sums [8] now tells us that the coeﬃcients of the characteristic polynomial are uniquely determined.

As this uniquely determines the eigenvalues, we are done.

67 According to this proof, we at least need to know how to factor the characteristic polynomial of H1(ψ) in order to determine the conjugacy classes. In our case, this is not particularly prohibitive because the minimal polynomial must divide xm 1. In − the next section, we provide a more direct algorithm for ﬁnding these eigenvalues.

As a motivating example, we will show that there is an orientation preserving diffeomorphism of the two-holed torus that has infinite order but whose induced map in homology has order 3. Thus, it is valuable to have a technique of finding only those similarity classes in GL(2g, (H (S, Z) C)) that are achieved by orientation 1 ⊗ preserving diffeomorphisms of finite order. Consider the following 4 4 matrix: × 1010 −0100 A =  1000  −  0001      It is easy to check that A has order 3 and that AJ tA = J. Thus, A Sp(4, Z) ∈ and hence β(ψ¯) = A for some orientation preserving diffeomorphism ψ : T T . 2 → 2 Suppose that ψ has order 3. Then ψ has a finite number of fixed points, each of index 1, and the Lefshetz number is identically the number of fixed points of ψ.

Now, Tr(A) = 1, and hence, Λ(ψ)=2 1 = 1. It is well-known, however, that no − diﬀeomorphism of prime order on a compact manifold can have exactly 1 ﬁxed point

(see [2] or [18]). Thus, every diﬀeomorphism whose image is A must have inﬁnite order.

Our goal, then, is to determine which periodic Dold sequences are “real” and which ones are “fake”. The real periodic Dold sequences are those sequences which are the Lefshetz sequences of some ﬁnite order orientation preserving diﬀeomorphism on a surface. We will henceforth refer to this problem as our realization problem.

68 3.2 The Similarity Algorithm

In the last section, we demonstrated the connection between periodic Dold sequences and complex similarity classes in the integral symplectic group. In this section, we use our previous results on the trace formulas of Zm-actions (see Section 1.3) to provide a precise algorithm for making this connection.

Using the notation from Section 1.3, recall that we can write Tr(θg) in terms of the σ (m) variables m . Now, suppose that k m, k < m. Since θ is diagonalizable, it 0 d | g k k is easy to see that θg has eigenvalues which are also m-th roots of unity. Also, Tr(θg ) can be expressed as a linear combination over C of the variables md. For k = m, we

m look at Tr(θg )= Tr(1) = dimZ(A). As before, this gives us the equation:

dimZ(A)= ϕ(d)md Xd|m

All in all, we have σ0(m) equations in σ0(m) unknowns. We order the divisors of m by size:

1= d1, d2,...,dσ0(m) = m

This gives us a matrix equation:

Tr(θg) m1 d2 1 µ(k2) ... µ(m) Tr(θg ) md2  .  = ......  .  .   .   1 ϕ(d2) ... ϕ(m)    Tr(1)   mdσ (m)       0     

We denote by B the σ (m) σ (m) matrix in the above equation. Note that B G 0 × 0 G does not depend on the dimension of A. Also note the ﬁrst and last rows consist only

69 of integers. The following theorem shows that this is in fact true for all rows. In fact, it provides formulae for computing each of the entries of BG.

Proposition 40. The entries of BG are all integers.

Proof. Let D be the diagonal representation of θg. We consider a block of D consisting of a primitive d-th root of unity, d m, λ = e2πi/d and all of its algebraic conjugates. | d j Deﬁne Dλd = diag(λd : gcd(j, d) = 1). We wish to show that any power of Dλd has an integer trace. Since md merely counts the number of such blocks in D, the trace of D will then be a sum of integer multiples of the variables md.

If Dλd is raised to some power relatively prime to d, the sum will still be µ(d).

The other case is that Dλd is raised to some power b such that gcd(d, b) > 1. We ﬁrst consider the case where b d. Deﬁne d′ = d/b. Then the entries of Db are of the form | λd e2πik/d′ where 1 k

× × Now, we deﬁne φ : Z Z ′ by φ(g)=g ˆ. This map is well-deﬁned and a d → d × ′ homomorphism. We can also show that φ is onto. Chooseg ˆ Z ′ where 1 gd such that p g (mod d′). Then gcd(p,d) = 1. ≡ Choose x

× × × × ′ This implies that Z /ker(φ) = Zd ′ and hence Z /ker(φ) = Zd ′ = ϕ(d ). The d ∼ | d | | | 2πi/d b set of powers of e in the diagonal of Dλd is the set:

W = w :1 s

b Dλd . Thus, ϕ(d) Tr(D b )= µ(d ′) λd ϕ(d ′)

b It follows that Tr(Dλd ) is an integral linear combination of the variables md and hence that all of the entries of BG are integers.

The ﬁnal case is the general situation in which gcd(b, d) > 1. Using methods similar to those above, it can be shown that in this case:

b ϕ(d) d Tr(Dλ )= µ d d gcd(d, b) ϕ gcd(d,b) Thus the proposition is proved.

Now, suppose that S is a surface of genus g 2 and that ψ : S S is an ≥ → orientation preserving diﬀeomorphism of S of ﬁnite order m > 1. We can apply the above analysis to A = H1(S, Z) and G =< H1(ψ) >. We have already shown that G = m. The fact that ψ is orientation preserving implies that Λ(ψk)=2 | | − Tr(H (ψk)) for all k, 1 k m. Putting this into the above matrix equation gives 1 ≤ ≤ a form in which the multiplicities of the eigenvalues, and thus the conjugacy class of

H1(ψ), can be determined from the Lefshetz sequence:

2 Λ(ψ) m1 − d2 1 µ(d2) ... µ(m) 2 Λ(ψ ) md2  − .  = ......  .  .   . 1 ϕ(d2) ... ϕ(m)  2 χ(S)   m     dσ0(m)   −   B      G   | 71 {z } Let p and q be primes, p = q. Some examples of the matrices B are given below: 6 G

1 1 BZp = − 1 p 1 − 1 1 1 1 1 (p −1) −1 1 p BZpq =  − − −  1 1 (q 1) 1 q − − −  1 (p 1) (q 1) (p 1)(q 1)   − − − −  All of the linear algebra and algebraic number theory has nowbeen removed from the problem and we can proceed to determining which periodic Dold sequences are realized. Once this is known, all of the above work has completely reduced the topological problem to a combinatorial one.

3.3 Realization in the Special Case of Zp

The construction of the diﬀeomorphisms in this special case require the use of the equivariant connected sum. We begin with an overview of this topic and then proceed to some lemmas that relate the number of ﬁxed points of a map to the order of the map and the genus of the surface.

Let G = Z be the finite cyclic group of order m and let ψ : S S and ψ : m 1 1 → 1 2 S S be o.p. diffeomorphisms of order m on smooth closed and oriented (but not 2 → 2 necessarily connected) surfaces S1 and S2, respectively. There are monomorphisms of groups Ψ : Z Diff+(S ) such that Ψ (1) = ψ . We define the equivariant i m → i i i connected sum of Ψ1 and Ψ2, denoted Ψ1#Ψ2, following [30]. Considering G as a discrete space, we choose imbeddings of η : D2 G S such that: i × → i

ψ η (x, g)= η (x, g + 1) i ◦ i i 72 for all x D2 and all g G. In other words, the powers of ψ permute the copies of D2 ∈ ∈ i on the surface. It is important to note that such imbeddings exist because the orbit of all but a finitely many of points of S has size m. We define S′ = S int(η (D2 G)). i i i \ i × Let H : ∂S′ ∂S′ be an orientation reversing diffeomorphism such that: 1 → 2

H ψ ∂S′ = ψ ∂S′ H ◦ 1| 1 2| 2 ◦

Define the surface S to be the quotient of the disjoint union of S1 and S2 modulo the equivalence relation defined by H. The resulting surface can be given a smooth structure using standard collaring techniques. Now, the action of G on η (∂D2 G) 1 × agrees with the action of G on η (∂D2 G). Thus, we can define a G-action on the 2 × smooth closed orientable surface S that is generated by:

ψ (x) if x S (ψ #ψ )(x mod H) , 1 1 1 2 ψ (x) if x ∈ S 2 ∈ 2 This action is an equivariant connected sum. The construction can be made with any finite group G and can also be made with non-orientable manifolds. There are some issues about whether this construction is well-defined. It turns out that it is well defined in the category of equivalence classes of actions that preserve orientations on smooth closed oriented manifolds. For more details on these issues, see [30]. It is also clear that the equivariant connected sum will be associative in this category.

Hence, if Ψ : G Diﬀ+(S) is a monomorphism, we deﬁne for all k N: → ∈

k#Ψ = Ψ#Ψ# . . . #Ψ k-times

It is not too difficult to see that if| the genus{z of a} connected surface S1 is g1 and the genus of a connected surface S is g , then the genus of S is g + g + m 1. Also, 2 2 1 2 − 73 if ψ1 has p1 fixed points and ψ2 has p2 fixed points, then ψ1#ψ2 has p1 + p2 fixed points.

Now that we have a way to combine known actions, we need some known actions with which to build. The ﬁve examples below are all we need to accomplish our goal.

Example 1. Deﬁne the 2-sphere, S2 C R by: ⊂ ×

S2 = (x + iy, z) : x2 + y2 + z2 =1 { }

We define θ : S2 S2 to be counter-clockwise rotation about the z-axis by 2π/p. A p → monomorphism S2 : Z < θ > Diff+(S2) can be defined by 1 θ . p p → p ⊂ → p

Example 2. We define the torus, X C R to be the surface of rotation of the ⊂ × circle 1 C = (iy, z) C R : y, z R, (y 2)2 + z2 = ⊂ × ∈ − 4 about the z-axis. We define τ : X X to be rotation about the z-axis by an angle p → of 2π/p. A monomorphism T : Z < τ > Diff+(X) can be defined by 1 τ p p → p ⊂ → p

Example 3. Taking the equivariant connected sum of Tp with itself j times where j 1, gives the monomorphism Q = (j) T . The resulting surface has genus ≥ j # p (j 1)p + 1 and the specified generator has no fixed points. As will be shown later, − a surface of genus g has a diffeomorphism of odd prime order p without fixed points if and only if g 1 (mod p). A picture is given in Figure 4 in the case where p =3 ≡ and j = 2.

74 Figure 4: A four-holed torus with threefold symmetry.

Example 4. In C R, we take the disjoint union of X and S2 as deﬁned above. × The equivariant connected sum

W = T #Θ : Z <ω > Diﬀ+(S) p p p p → p ⊂

defines a Zp action on a surface of genus p such that the specified generator has exactly 2 fixed points.

Example 5. We now begin the construction of the most important examples. These examples are originally due to Kosniowski. Among other things, they form part of his generators for the unoriented and oriented Zp-bordism ring [25]. He also used them to give a geometric argument that all but a ﬁnite number of genera of closed connected orientable 2-manifolds have a smooth Zp-actions (see [24]). Later, Ewing

[13] used them to show that for odd primes p, the image of the Atiyah-Bott map, ab : O (Z ) W (Z, Z ), from the oriented Z bordism ring to the graded Witt ∗ ∗ p → ∗ p p− ring of Zp-invariant symmetric and skew-symmetric bilinear forms, is onto if and only if the ﬁrst factor of the ideal class group of Q(e2πi/p) is odd. Berend and Katz [5] used them in solving realization problems of the equivariant signature of Zm-actions.

75 The construction follows [25]. Suppose that 1 j (p 1)/2. Deﬁne q by ≤ ≤ − jq 1 (mod p). Let D2 be the closed disk in C. In C R we take the q disks ≡ − × D = D2 0 ,...,D = D2 q 1 . To these disks we will attach the tops and 0 ×{ } q−1 ×{ − } bottoms of twisted rectangles in the following manner. First we mark 4p points on the boundary of each disk Da:

X(a, k)=(e2πik/4p, a) for k even, 0 k 4p 2 ≤ ≤ −

Y (a, k)=(e2πik/4p, a) for k odd, 1 k 4p 1 ≤ ≤ −

Now, we take (q 1)p copies of I I with the following indexing. For a even, − × 0 a q 1, let B (4t, 4t +1) = I I for 0 t p 1. For a odd, 0 a q 1, ≤ ≤ − a × ≤ ≤ − ≤ ≤ − let B (4t 2, 4t 1) = I I for 1 t p. We identify each vertex of the the square a − − × ≤ ≤ with one of the points X or Y :

a even, 0 a q 1; B (4t, 4t + 1) ≤ ≤ − a 0 1 Y (a +1, 4t + 1) 1 1 X(a +1, 4t) × ↔ × ↔ 0 0 X(a, 4t) 0 1 Y (a, 4t + 1) × ↔ × ↔

a odd, 0 a q 1; B (4t 2, 4t 1) ≤ ≤ − a − − 0 1 Y (a +1, 4t 1) 1 1 X(a +1, 4t 2) × ↔ − × ↔ − 0 0 X(a, 4t 2) 0 1 Y (a, 4t 1) × ↔ − × ↔ −

The switching of X and Y at the top of each I I corresponds to twisting the × rectangles. Now we identify each I 0 and I 1 of B with the two corresponding × × a 76 Figure 5: The Kosniowski generator with p = 3, j = 1, and q = 2.

(short) arcs on Da and Da+1. During this identiﬁcation, we perform each twist in the clockwise direction. The result in the smallest case, where p = 3, j = 1, and q = 2 is pictured in Figure 5.

This surface is always a connected compact orientable 2-manifold with boundary.

A straightforward combinatorial argument suﬃces to show that the boundary is S1.

This surface has a Zp-action which is deﬁned by a rotation of 2π/p about the R axis.

In this action, the boundary circle is mapped to itself. This map extends naturally to a disk. We form the Kosniowski surface Kj,p by attaching a disk along the boundary

1 S . Kj,p can be made smooth in such a way that the action is also smooth. Using a triangulation, one can show that the genus of this closed surface is (p 1)(q 1)/2. − − In the future, we will refer to the 2π/p rotation as the map κ : K K . It is j,p j,p → j,p clear that κj,p is orientation preserving. Also, note that κj,p has exactly q + 1 fixed points. It is easy to see that dκj,p acts on the normal bundle to q fixed point as multiplication by λ. At the remaining fixed point, a combinatorial argument shows

j that dκj,p acts as multiplication by λ . The monomorphism described here will be

77 denoted as follows:

C : Z < κ > Diﬀ+(K ) j,p p → j,p ⊂ j,p where Cj,p(1) = κj,p.

Now we are ready to use the equivariant connected sum to find diffeomorphisms of odd prime order with a specified number of fixed points. In fact, the first proposition shows that this problem can be reduced to solving a simple Diophantine equation.

Proposition 41. Let S be a closed connected orientable surface of genus g and let p be an odd prime. S has a diﬀeomorphism of order p with exactly Λ 2 ﬁxed points ≥ if and only if there is a y 2N such that: ∈

2g = py + (Λ 2)(p 1) − −

Proof. Suppose that S has a diffeomorphism ψ of order p that has at least one fixed point. Then ψ has at least two fixed points. Let S′ be the orientable surface that is the orbit space of the <ψ> action on S. Using the Riemann-Hurwitz equation, we have:

χ(S)= pχ(S′) (p 1)Λ(ψ) − − If the genus of S′ is g′ we have (2 2g)= p(2 2g′) (p 1)Λ(ψ) and hence: − − − − 2g = p(2g′ 2)+(p 1)Λ(ψ)+2 − − = (2g′)p 2p +(p 1)Λ(ψ)+2 − − = (2g′)p +(p 1)Λ(ψ) 2(p 1) − − − = (2g′)p +(p 1)(Λ(ψ) 2) − −

Since g′ 0, this constitutes a solution to the equation. ≥ For the converse, suppose that Λ is even and that there is a y 2N such that: ∈ y Λ g = p + 1 (p 1) 2 2 − − 78 Consider the equivariant connected sum given by:

y Λ 2 Tp # Sp 2 # " 2 # # This action is on a surface of genus

y Λ 1 p +1+ 1 (p 1)+(p 1) = g 2 − 2 − − − The generator speciﬁed above has exactly Λ ﬁxed points. Suppose now that Λ is odd.

In this case, we have that:

y Λ 2 g = p + − (p 1) 2 2 − Consider the following equivariant connected sum:

y Λ 3 2 Tp # − Sp #C p−1 2 ,p 2 # " 2 # # The action is on a surface of genus:

y Λ 3 p 1 1 p +1+ − 1 (p 1)+(p 1) + − +(p 1) = g 2 − 2 − − − 2 − The generator that we speciﬁed above has Λ = Λ 3 + 3 ﬁxed points. −

From this we can also get the solution to the so-called stable genus increment for Zp. Much work has been done on this topic in recent years (see [16], [19], and

[26]). The case for Zp has been known for some time. Since our work gives a slightly diﬀerent proof of this theorem, we will include it here.

Theorem 42. Let p be a prime. There are only finitely many closed connected orientable surfaces that do not have a diffeomorphism of order p. More specifically, if g max p, (p 1)2/2 , then the class of smooth closed connected orientable surfaces ≥ { − } of genus g has a diffeomorphism of order p.

79 Proof. Every surface admits an involution, so we may assume that p is odd. We will show that for every prime p, there is a constant R such that for g R, the equation ≥ p 1 g = − x + pz 2 has a solution (x, z) in the natural numbers. Let a = (p 1)/2. First we choose − R p. For all g p, there are natural numbers q and r, 0 r p 1, such that ≥ ≥ ≤ ≤ − g = pq + r. We will show that for each r, there is a R such that if g R is in r ≥ r the residue class of r then there is a solution to the above equation. If we choose

R = max ( R p ), then g R implies that there is a solution to the above 0≤r≤p−1 { r}∪{ } ≥ equation.

Now gcd(a, p) = gcd(r, p) = 1 and hence a has an inverse modulo p and r 0 6≡ (mod p). Deﬁne n N, 1 n p 1, by ∈ ≤ ≤ −

n a−1r (mod p) ≡

Then there is an integer b such that an = r + pb. Note that b and n depend only on r and p, and thus b and n depend only on the residue class of g modulo p. Since an 0 and 0 r

g = pq + r = p(q b + b)+ r − = p(q b)+(pb + r) − = p(q b)+ an −

This will give us a solution in N N if q b 0, or equivalently, if g an. Thus, × − ≥ ≥ if we let Rr = an, we are done with the ﬁrst part.

80 Now we apply this to ﬁnding a lower bound on the genus to guarantee that p is admissible. We will show that:

0 , r =0 n(r)= p 2r , 1 r (p 1)/2  − ≤ ≤ − 2(p r) , (p + 1)/2 r p 1  − ≤ ≤ −  0 , r =0 R = (p 1)(p 2r)/2 , 1 r (p 1)/2 r  (−p 1)(−p r) ,(p≤+ 1)≤/2 −r p 1  − − ≤ ≤ − Note that (p 2)a 1 (mod p) and hence n(r) 2r (mod p). The above −  ≡ ≡ − formula for n(r) follows from this fact and the bound on n(r), 0 n(r) p 1. The ≤ ≤ − formula for Rr now follows from its deﬁnition. The maximum value of Rr is achieved at r =(p + 1)/2. At this value of r, we have R =(p 1)2/2. (p+1)/2 −

We can now determine all periodic Dold sequences that are realized by orientation preserving diﬀeomorphisms of odd prime order on surfaces. Let p be an odd prime.

We divide the problem into three cases:

1. Actions on S2.

2. Free actions on surfaces of any genus g 1. ≥

3. Actions with 2 or more ﬁxed points on surfaces of genus g 1. ≥

The solutions of the ﬁrst two are very simple and they are presented here only for completeness. The third is also an application of the Riemann-Hurwitz (see

[18])equation, but it appears to need our work to ﬁnish it oﬀ. First we need a lemma.

81 Lemma 43. The only periodic Dold sequences of period 1 that are realized by o.p. maps of ﬁnite period on an closed connected orientable surface are the sequence of all

2’s and the sequence of all 0’s.

Proof. Suppose that an o.p. map of ﬁnite period m has a periodic Dold sequence of period 1. Then the m-th term in the Lefshetz sequence is the Euler characteristic.

Since all the Euler characteristics are negative if g 2, we must have g 1. ≥ ≤

Proposition 44. A periodic Dold sequence s : N Z of odd prime period p is → realized as the Lefshetz sequence of an o.p. diﬀeomorphism on S2 if and only if it is the constant sequence of 2’s. In fact, the only Lefshetz sequence of any mapping class in the mapping class group of S2 is the constant sequence of all 2’s.

Proof. The sequence of 2’s is certainly realized by Θ : Z < θ >. Conversely, the p p → p fact that ψ is orientation preserving implies that Λ(ψ)=1 0 + 1 = 2 and the result − follows by our previous determination of all periodic Dold sequences.

Proposition 45. If a periodic Dold sequence s : N Z of odd prime period p → is realized as the Lefshetz sequence of a ﬁxed-point free o.p. diﬀeomorphism on a surface of genus g 1, then it is the sequence: ≥

s = (0, 0,..., 0, 2pk, 0, 0,..., 2pk,...)

(p−1)−times where k 0 is such that pk =1| {zg. Conversely,} given a sequence as above with k 0, ≤ − ≤ it is a periodic Dold sequence and there is a fixed-point free orientation preserving diffeomorphism ψ : S S of odd prime order p on any surface S of genus g = → (2 s(p))/2 − 82 Proof. Suppose that you are given the Lefshetz sequence s : N Z of ψ. Since → ψ is fixed-point free, Λ(ψ) = 0. Also, Λ(ψp) = 2(1 g). Dold’s theorem implies − that p M (p) = s(p) s(1) = s(p) and hence, p (1 g). Note that it is completely | s − | − unnecessary to use Dold’s theorem to prove this last fact. An easier approach is to note that with the notation of Theorem 3, 0 = Λ(ψ)=2 (m m ) and hence: − 1 − p

2g = m +(p 1)m =(m m )+ pm =2+ pm 1 − p 1 − p p p

Now suppose we are given a sequence as above. Deﬁne g = (2 s(p))/2=1 kp. − − Then g 1 (mod p) and by Example 3, we know that the given generator, ψ, of ≡

Q1−k is a diﬀeomorphism on the surface of genus g. The Lefshetz sequence of ψ is a periodic Dold sequence having 0’s everywhere but at Λ(ψi) where i 0 (mod p). ≡ Since Λ(ψp)=2 2g = s(p), we are done. −

Proposition 46. A periodic Dold sequence s : N Z of odd prime period p is realized → as the Lefshetz sequence of an o.p. diffeomorphism on a surface of genus g 1 with ≥ at least 2 fixed points if and only if the following conditions are satisfied.

1. s(i) 2 for all i 0 (mod p) ≥ 6≡

2. s(i) 2N 0 for i 0 (mod p) ∈ − ∪{ } ≡

3. M (p)/p (2 M (1)) s ≤ − s

Equivalently, the realization occurs if and only if s : N Z is the sequence →

(a, a, . . . , a, a + pk,a,a,...,a + pk,...)

(p−1)−times where a 2, a + pk |2N {z 0 }, and a + k 2. ≥ ∈ − ∪{ } ≤ 83 Proof. Suppose s : N Z is a periodic Dold sequence satisfying the above conditions. → Then M (1) 2 and M (p) = s(p) s(1) 2. Since p M (p), there is an x 0 s ≥ s − ≤ − | s ≥ such that:

px = p(2 M (1)) M (p) − s − s

By the hypotheses, we have that x N 0 . Deﬁne g = (2 s(p))/2. Then ∈ ∪{ } − s(p)=2 2g. Rearranging the above equation gives: −

px = 2p ps(1) s(p)+ s(1) − − = 2p ps(1) 2+2g + s(1) − − = 2(p 1) s(1)(p 1)+2g − − − = 2g = px +(s(1) 2)(p 1) ⇒ − −

Thus, by our theorem in the previous section, there exists an o.p. diﬀeomorphism

ψ : S S of order p on any surface S of genus g with exactly s(1) ﬁxed points. → Thus, the Lefshetz sequence of ψ is precisely s.

Conversely, given an o.p. diﬀeomorphism ψ : S S of period p, Dold’s theorem → tells us that it’s Lefshetz sequence s : N Z is a periodic Dold sequence of period → 1 or p. The periodic Dold sequence of period 1 are all constant sequences. Since the Euler characteristic is negative for g 2, the only possibilities are the sequence ≥ of all 2’s and the sequence of all 0’s. These cases have already been considered.

Now, since any diﬀeomorphism of prime power order on a compact smooth manifold cannot have exactly one ﬁxed point, s(i) 2 for all i 0 (mod p). Finally, the ≥ 6≡ existence of ψ implies the existence of a solution in the non-negative integers to

84 2g = xp +(s(1) 2)(p 1). Using the fact that s(p)=2 2g, we must have that: − − −

px = p(2 M (1)) M (p) − s − s and hence the third condition is satisﬁed.

3.4 Review of Fuchsian Groups

Thus far, we have discussed Dold sequences, their relation to mapping class groups, and a solution to the realization problem in the case where the order of the map is an odd prime. The case where m> 1 requires the use of more sophisticated theory and is ultimately independent of this special case. First of all, we must separate the problem into two parts: surfaces of genus 1 and those of genus 2. The first case can ≤ ≥ be solved quite easily as it is unnecessary to deal with anything more complex than a2 2 matrix. This case is discussed in the final section of this chapter. The second × case is more difficult and requires the theory of Fuchsian groups. In this section, we include a review of the basic properties of Fuchsian groups and their actions on the hyperbolic plane. The exposition follows the excellent book by Katok [23]. In the following sections, we discuss the work of Harvey and Machlachlan on surface kernel homomorphisms. With these tools and a few additional lemmas, we can proceed directly to the solution of the general realization problem. Those readers familiar with the topics from Fuchsian groups listed in the table of contents are invited to skip this section and move to the next.

85 3.4.1 Deﬁnition and Properties of Fuchsian Groups

Recall that the hyperbolic plane is the set H = z = x + iy C : y > 0 with { ∈ } Riemannian metric induced by:

dx2 + dy2 ds = p y

The metric will be less useful to us than the hyperbolic area. For A H, the ⊂ hyperbolic area is given by: dxdy µ(A)= y2 ZA if this integral exists. It is well known that the set of orientation preserving isometries of H is given by the projective special linear group i.e.

PSL(2, R) = A M (R) : det(A)=1 / I { ∈ n } {± } az + b = z : a, b, c, d R, ad bc =1 → cz + d ∈ −

It is also known that if T PSL(2, R), A H, and µ(A) exists, then µ(T (A)) = µ(A). ∈ ⊂ For our purposes, we are most concerned with certain subgroups of PSL(2, R) that act nicely on H. By “nicely” we mean “properly discontinuously”. To define this term, we consider only a transformation group (X,G,ϕ). The group G is said to act properly discontinuously on X if each point x X has a neighborhood V such ∈ that g(V ) V = for all but finitely many g G. It is important to note that this ∩ ∅ ∈ definition is different than other definitions of the same pair of words. In the theory of covering spaces, proper discontinuity means that g(V ) V = for all g = 1 in G. ∩ ∅ 6 We will refer to this situation in the future as proper discontinuity where G has no

86 ﬁxed points. For the groups we are working with here, this notion turns out to be equivalent to the deﬁnition from covering space theory.

The set of elements in PSL(2, R) that act properly discontinuously on H have a useful algebraic characterization. First, we deﬁne a topology on PSL(2, R). An element A PSL(2, R) is of the form: ∈

a b A = c d

We consider SL(2, R) as a subspace X of R4:

X = (a, b, c, d) R4 : ad bc =1 { ∈ − }

The map δ : X X deﬁned by δ(x)= x is a homeomorphism of X. We can then → − identify PSL(2, R) as the set X/ , where is the equivalence relation induced by ∼ ∼ δ. We give PSL(2, R) the quotient topology from the projection X X/ . Now, → ∼ a subgroup Γ < PSL(2, R) is said to be discrete if the subspace topology on Γ is discrete. A Fuchsian group is deﬁned to be a discrete subgroup of PSL(2, R), i.e. it is a discrete group of orientation preserving isometries of H. The notions of proper discontinuity and Fuchsian groups are connected by the following theorem.

Theorem 47. Let Γ < PSL(2, R). Then Γ is a Fuchsian group if and only if Γ acts properly discontinuously on H.

87 3.4.2 Parabolic, Hyperbolic, and Elliptic

The elements of P SL(2, R) are characterized by a numerical invariant. Let T (z) =

(az + b)/(cz + d) P SL(2, R). The trace of T is deﬁned as Tr(T )= a + d . ∈ | | If Tr(T ) > 2, then T is said to be hyperbolic. If Tr(T ) = 2, then T is said to be parabolic. Finally, if Tr(T ) < 2, then T is said to be elliptic. Much information about a Fuchsian group can be obtained by analyzing the ﬁxed points of the various kinds of transformations contained in it. The following important lemma can be proved using only the most elementary of computations.

Lemma 48. Let T P SL(2, R) ∈

1. If T is hyperbolic, then T has exactly two ﬁxed points in R . ∪ {∞}

2. If T is parabolic, then T has exactly one ﬁxed point in R . ∪ {∞}

3. If T is elliptic, then T has two complex conjugate ﬁxed points in C, exactly one

of which is in H.

3.4.3 Fundamental Regions

We will denote the coset space of the Γ-action on H, where Γ is a Fuchsian group, by

Γ H. An important object to study is the fundamental region of the action of Γ on \ H. If(X,G,ϕ) is a transformation group, we say that a closed set F is a fundamental region of G if:

88 1. g∈G g(F )= X, and S 2. g(int(F )) int(F )= for all g G. ∩ ∅ ∈

The boundary of a fundamental region is ∂F = F int(F ). A tesselation of X is the − set g(F ) : g G . { ∈ }

If Γ is a Fuchsian group and F1 and F2 are two fundamental regions of Γ, then the hyperbolic area of F1 and F2 are equal as long as one of them has ﬁnite hyperbolic area and it’s boundary has zero hyperbolic area. This fact will be used in the realization theorem. In particular, we will use a special kind of fundamental region called a

Dirichlet region. A Dirichlet region at a point p H is the set: ∈

D (Γ) = z H : ρ(z,p) ρ(T (z),p) for all T Γ p { ∈ ≤ ∈ }

If p is not ﬁxed by some element of Γ, it turns out that Dp(Γ) is a connected fundamental region of Γ.

Recall that the geodesics in H are vertical lines and circles whose centers lie on the real axis. Let Γ be a Fuchsian group, p a point that is not ﬁxed by any element of Γ and F = Dp(Γ) a Dirichlet region of Γ. From the deﬁnition, it can be seen that

∂F consists of segments of hyperbolic geodesics. Any one of the maximal geodesic segments intersecting F is called a side of F . A point of intersection of two sides of

F is called a vertex. Vertices of a Dirichlet region of a Fuchsian group are necessarily isolated. However, it is good to note that an arbitrary Fuchsian group may have inﬁnitely many vertices. We will eventually assume that our Fuchsian groups have a compact Dirichlet region and hence, will have ﬁnitely many vertices.

89 In the coset space Γ H, the sides and vertices of a Dirichlet region are identified \ and the result is a 2-manifold. The way in which this identification occurs forces a certain detectable subgroup structure in Γ. To see this, we define an equivalence relation on the set of vertices of a Dirichlet region. Two points z , z H are said to 1 2 ∈ be congruent if they are in the same Γ orbit, i.e. T (z ) = z for some T Γ. This 1 2 ∈ definition implies that if two points in a Dirichlet region are congruent, then they must lie in the boundary. Now, we apply this relation to the set of vertices of F .

Since elements of PSL(2, R) are angle-preserving and map geodesics to geodesics, the relation on vertices is indeed an equivalence relation. The equivalence classes under this relation are called cycles. Suppose that a vertex u is ﬁxed by an elliptic element

S. Let v = T (u) be some other vertex in the cycle containing u. Then the elliptic transformation T ST −1 fixes v. We see that if a vertex is fixed by an elliptic element, then the vertices of the cycle are fixed by conjugate elliptic elements. For this reason, we call such a cylce an elliptic cycle and the vertices in this class are called elliptic vertices.

In defining the concepts of vertex and side for a Dirichlet region, we neglected to consider fixed points of elliptic involutions. Suppose T Γ is elliptic and T 2 = I. ∈ The transformation T has exactly one fixed point z H and it must lie in ∂S(F ) for ∈ some S Γ. Then w = S−1(z) F , and (S−1TS)(w)= w. Now, S−1TS is also an ∈ ∈ elliptic element of Γ of order 2. Hence, the point w necessarily lies in ∂F and is either a vertex or is interior to one of the geodesic segments bounding F . In the later case,

S−1TS interchanges the parts of this segment that are seperated by w. By a vertex, then, we will either mean a point of intersection of two maximal bounding geodesic

90 segments of F or a fixed point of some elliptic involution that is interior to some maximal bounding geodesic segment of F . For sides, we have the following modified definition:

1. A side is one of the maximal geodesic segments intersecting ∂F whose interior

does not contain the ﬁxed point of an elliptic involution.

2. If l is a maximal geodesic segment intersecting ∂F containing the ﬁxed point z0

of an involution in Γ, each connected component of l z is also called a side. \{ 0}

In the sequel, “elliptic cycle” and “elliptic vertex” will refer to the obvious modiﬁca- tions under these altered deﬁnitions of side and vertex.

3.4.4 Periods, Presentations, and Signatures

The elliptic fixed points of Γ are closely related to the periodic points of a periodic map on a surface. An important part of this connection is the relation between maximal cyclic subgroups of Γ and the elliptic cycles of Γ. Suppose that x H ∈ and that Γ = I . Then Γ contains only elliptic elements, each of which has x as x 6 { } x its fixed point. Now, two elements of PSL(2, R) commute if and only if they have the same fixed point set. Thus, Γx is abelian. A Fuchsian group such that all nonidentity elements have the same fixed point set is necessarily cyclic. Moreover, an elliptic cyclic subgroup of PSL(2, R) is a Fuchsian group if and only if it is finite.

Therefore, Γx is a ﬁnite cyclic subgroup of Γ. If there were any other ﬁnite cyclic

91 subgroup of Γ containing Γx, then all its elements commute with the elements of Γx and therefore are contained in Γx. Thus, we have shown that Γx is a maximal finite cyclic subgroup of Γ. If F is a Dirichlet region of Γ, we now have that every elliptic cycle of F corresponds to a conjugacy class in Γ of maximal finite cyclic subgroups of Γ. This leads to the following very important definition:

Definition 49. A period of a Fuchsian group Γ is the order of some maximal finite cyclic subgroup of Γ. Given a set of conjugacy class representatives of maximal finite cyclic subgroups of Γ, the periods of these subgroups are called the periods of Γ.

It is important to note that if there are several nonconjugate subgroups of the same order, this period will occur in the periods of Γ once for each conjugacy class.

As we shall see, the periods of a Fuchsian group are a useful invariant of that group.

A parabolic element in a Fuchsian group, which has one fixed point in R , ∪ {∞} can be viewed as an elliptic element of infinite order. In this case, we can attempt the same type of characterization of vertices of F . However, the Dirichlet region can never be compact and has vertices “at infinity”. As we are only concerned with

Fuchsian groups with compact Dirichlet regions, our Fuchsian groups will not have parabolic elements and hence we consider this case no further.

To ﬁnd a presentation of Γ, we extend the notion of congruence to the sides of a

Dirichlet region. Two sides s1 and s2 of a Dirichlet region are said to be congruent if there is a T Γ such that T (s )= s . It can be shown that every side of a Dirichlet ∈ 1 2 region is congruent to some other side other that itself and no side is congruent to three or more sides. Thus, the equivalence classes for this relation each contain exactly

92 two sides of the Dirichlet region. If the Dirichlet region is compact, or more generally has a ﬁnite number of sides, we have that the number of sides of a Dirichlet region is even.

If Γ is a Fuchsian group with Dirichlet region F and A is a set of elements of

Γ which “pair” the sides of F , then = Γ. With some extra hypotheses, it is possible to give a presentation of Γ. In general, we will assume that Γ has a fundamental region F of finite hyperbolic area whose boundary has zero hyperbolic area. For our practical purposes, this will be easily verified. In this case, the area of any two fundamental regions is the same. Hence we define µ(Γ H) = µ(F ) and \ we are confident that this does not depend on our choice of F . We will encode all of these hypotheses on Γ by saying that µ(Γ H) < . In this case, the theorem \ ∞ of Siegel says exactly that any Dirichlet region of Γ has finitely many sides. Hence,

F has ﬁnitely many vertices and ﬁnitely many periods. If we also assume that Γ contains no parabolic elements, a presentation of Γ is given by:

r γ Γ= a , b , a , b ,..., a , b , x , x ,..., x xmi =1, x [a , b ]=1 1 1 2 2 γ γ 1 2 r| i i j j * i=1 j=1 + Y Y where [x, y] denotes the commutator and (m 1 i r) is the list of periods. Note i| ≤ ≤ that this means that every ﬁnite cyclic subgroup of Γ is contained in a conjugate of some Γ , 1 i r. xi ≤ ≤ A Fuchsian group with the above presentation has an important invariant called the signature. It is deﬁned to be the following list:

σ(Γ) = (γ; m1, m2,..., mr)

There is also a signature for Fuchsian groups that contain parabolic elements, but it

93 is of no use to us. It is important to note that the order of the above mi is immaterial and a natural number m appears in the list for as many conjugacy classes of maximal

ﬁnite cyclic groups there are in Γ with order m. If a Fuchsian group is said to have signature (γ; m1, m2,..., mr), then it means that Γ satisﬁes all of the hypotheses that allow it to have the above presentation. If we are given a list of integers of the form (γ; m1, m2,..., mr), then we refer to this as an abstract signature.

3.4.5 Uniformization

Now, we are ﬁnally prepared to recognize the orbit space as a bona ﬁde surface.

Here again, we assume that Γ is a Fuchsian group with µ(Γ H) < . Each side of a \ ∞ Dirichlet region gets paired in the quotient to another side and it can be shown that

Γ H is an orientable surface with “marked points” and possibly some “cusps”. A \ marked point is the image of an elliptic cycle. A cusp is either comes from a parabolic cycle or segments of R that bound the closure of F in the closure of H. If Γ has ∪{∞} no parabolic cycles, then of course, Γ H is an orientable surface with marked points. \ If no element of Γ ﬁxes a point of H, then the quotient actually has a complex structure. When this occurs, we have what is called uniformization. It is a special case of the Riemann mapping theorem. In fact, every complex 1-manifold is the quotient of one of the three simply connected Riemann surfaces: S2, C, H. Those that are the quotients of H( i.e. the ones under consideration here) are said to be hyperbolic.

As has been mentioned several times already, we are only concerned about the case in which the Dirichlet region of a Fuchsian group is compact. The important

94 theorem for us states that any Dirichlet region of a Fuchsian group is compact if and only if Γ H is compact if and only if µ(Γ H) < and Γ contains no parabolic \ \ ∞ elements. If a Dirichlet region is compact, it automatically follows that it has a finite number of vertices and hence a finite number of edges. From all this we conclude that a Fuchsian group Γ has µ(Γ H) < and lacks parabolic elements if and only \ ∞ if the orbit space Γ H is a closed orientable surface. \ Suppose Γ is a Fuchsian group that acts properly discontinuously on H without any fixed-points. Then from the theory of covering spaces we have that π : H Γ H, → \ where π is simply projection, is a covering space. The group of covering transformations of Γ H is Γ. Since H is simply connected, π (Γ H) = Γ. The following theorem \ 1 \ ∼ combines the results of the last three paragraphs and identifies exactly which spaces are hyperbolic.

Theorem 50. Every compact Riemann surface (connected complex 1-manifold) of genus g 2 is the orbit space of some Fuchsian group Γ acting on H satisfying the ≥ following properties:

1. Γ acts on H without any ﬁxed-points.

2. Γ has no parabolic elements.

3. µ(Γ H) < \ ∞

4. Γ = π (Γ H). ∼ 1 \

95 3.5 Surface Groups and Surface Kernel Homomorphisms

Harvey and Machlachlan (see [19] and [29]) refer to a Fuchsian group satisfying these properties as a surface group. One might also say that a surface group has signature (g; ). We will use this terminology in the sequel. Harvey also introduces the − terminology of a surface kernel homomorphism. Let G be a ﬁnite group and Γ a

Fuchsian group with a compact Dirichlet region. A surface kernel homomorphism is a surjective homomorphism Γ G such that Ker(Γ G) is a surface group. The → → following proposition helps to relate the fundamental regions of a Fuchsian group to those of its subgroups.

Proposition 51. Let Γ be a Fuchsian group and Γ′ < Γ a subgroup of ﬁnite index m. Suppose that F is a fundamental region for Γ and T1,...,Tm is a set of coset representatives. Then

′ m ′ 1. F = i=1 Ti(F ) is a fundamental region for Γ , and S 2. if µ(F ) < and µ(∂F )=0, then µ(F ′)= mµ(F ). ∞

From this we conclude that if Γ has compact Dirichlet region, Γ G is a surface → kernel homomorphism, and Ker(Γ G) has ﬁnite index, then Ker(Γ G) has a → → compact Dirichlet region. The study of surface kernel homomorphisms is essential to realizing the periodic Dold sequences because of the following theorem. For references to it’s proof, see [19] or [29].

96 Theorem 52. Let G be a ﬁnite group acting on a compact Riemann surface S of genus g 2 as a group of automorphisms (analytic homeomorphisms whose inverses ≥ are also analytic). Then there is a Fuchsian group Γ and a surface kernel Γ G → such that Ker(Γ G) has signature (g; ). Conversely, if Γ is a Fuchsian group and → − Γ G is a surface kernel homomorphism with Γ′ = Ker(Γ G) then Γ/Γ′ = G → → ∼ deﬁnes a G-action on the Riemann surface Γ′ H. \

Recall that our interest here is not in analytic homeomorphisms on Riemann surfaces but in diffeomorphisms of smooth surfaces. Fortunately, the distinction is not necessary. It is known that if the genus of a surface S is at least 2 and a finite group G of orientation preserving diffeomorphisms acts on S, then there is a hyperbolic metric on S such that G is conjugate to a group of isometries of S under that metric. For our purposes, then, we will not draw a distinction between automorphisms of Riemann surfaces and finite order orientation preserving diffeomorphisms on a smooth surface.

Most of the work that has been done on ﬁnding surface kernel homomorphisms comes down to adding enough hypotheses to the ﬁnite group G so that a particular

Diophantine equation can be solved. The Diophantine equation typically comes from the famous equation of Riemann-Hurwitz. This equation can be derived quite nicely in the context of Fuchsian groups. Let Γ be a Fuchsian group with compact Dirichlet region F . The hyperbolic area of Γ H can be found in terms of the signature of Γ: \ r 1 µ(Γ H)=2π (2γ 2) + 1 \ − − m " i=1 i # X Suppose that Γ has a compact Dirichlet region and Γ Z is a surface kernel. Then → m Γ′ = Ker(Γ Z ) has signature (g; ) and hence µ(Γ′ H)=2π(2g 2). Since → m − \ − 97 µ(Γ′ H)= mµ(Γ H), we have: \ \ 2g 2 r 1 − = (2γ 2) + 1 m − − m i=1 i X This is, of course, the Riemann-Hurwitz equation.

Since Dirichlet regions have interior, their hyperbolic area is positive. A surprising result shows that if an abstract signature (γ; m1,..., mr) corresponds to a positive abstract area, then there is a nice Fuchsian group realizing that abstract signature.

In some sense, it can be thought of as a converse to the Riemann-Hurwitz equation.

Theorem 53 (Poincar´e-Maskit). Let γ 0, r 0. Given a set of r integers m such ≥ ≥ i that m 2, 1 i r and: i ≥ ≤ ≤ r 1 (2γ 2) + 1 0 − − m ≥ i=1 i X then there is a Fuchsian group Γ with compact Dirichlet region whose signature is given by:

(γ; m1, m2,..., mr).

3.6 The Theorems of Harvey

All of the theorems and ideas presented thus far come from the classical theory of

Fuchsian groups. In the past 50 years, a nice theory has been developed to extend the famous Hurwitz 84(g 1) theorem. This theorem states that the largest order − group that can act on an orientable surface of genus g is 84(g 1). A naturally related − problem, then, is to determine the smallest genus g on which a ﬁnite group G can act.

98 The ﬁrst work, and the work that is of most use to us here, seems to have been done by Harvey(see [19]). He determined the minimal genus on which a ﬁnite cyclic group can act. Almost simultaneously, Maclachlan found the minimal genus on which any

finite abelian group can act. While various authors considered different versions of this question, the biggest step forward was taken much later by Kulkarni [26]. Kulka- rni showed that for every finite group G, there is a genus gs and an increment N such that G acts on the surface of genus g g if and only if g = Nk + 1 for some k. The ≥ s minimal genus gs, called the stable genus, and the number N, called the stable genus increment, depend only on the Sylow subgroup structure of G. The problem then becomes one of determining the stable genus increment and genus for various classes of groups. Much detailed work has been done on this problem. Since our interest is only in periodic maps, we can get off this train essentially before it leaves the station.

Below we have the two theorems of Harvey that will be of the most use.

Lemma 54 (Harvey,[19]). Let Γ be a Fuchsian group and G a ﬁnite group. A homomorphism φ :Γ G is a surface-kernel if and only if for every periodic generator T → of period w, φ(T ) also has order w.

Proof. ( ) If x has order w but φ(x)k = 1 for some k

( ) Suppose that φ : Γ G is a homomorphism that preserves periods. We ⇐ → must show that the Ker(φ) contains no elements of finite order. Suppose T Ker(φ) ∈ has finite order. Then T is contained in some maximal finite cyclic subgroup that is

99 conjugate to an isotropy group Γ , where x H is some elliptic ﬁxed point. Without x ∈ loss of generality, we can assume that Γx =< Tx >, where Tx is one of the periodic generators of Γ. Then T = S−1T kS for some S Γ and k Z, k

Theorem 55 (Harvey’s Necessity Theorem,[19]). Let Γ have a presentation as given above and let M be the least common multiple of m ,..., m . Suppose φ : Γ Z 1 r → m is a surface kernel homomorphism. Then either r =0 or the following conditions are satisﬁed.

H1. lcm m ,..., mˆ ,..., m = M for all i, where mˆ denotes the omission of m ; { 1 i r} i i

H2. M divides m and if γ =0, then M = m;

H3. r =1 and if γ =0, then r 3; 6 ≥

Proof. Suppose that φ : Γ Z is a surface kernel homomorphism and that r = 0. → m 6 To show that [H1] is satisﬁed, let 1 j r and deﬁne: ≤ ≤

Mˆ = lcm m ,...,m , m ,...,m { 1 j−1 j+1 r}

Since the image of φ is abelian, the relation x . . . x γ [a , b ] = 1 implies: 1 · · r i=1 i i Q φ(x ) . . . φ(x )=1 1 · · r where the image in Zm is written multiplicatively. More to the point, this equation implies that φ(x )= φ(x−1) . . . φ(x−1). Then Lemma 55 shows that φ preserves the j 1 · · r periods of Γ, and therefore that m Mˆ . Thus, Mˆ = M and [H1] is veriﬁed. j| 100 The fact that M m follows from the fact that the image of φ is a subgroup of Z | m and φ preserves the periods. Since φ is surjective, the possibility that γ = 0 forces the least common multiple of the orders to be m. Thus [H2] follows.

γ −1 Now, suppose that r = 1. Then x1 = i=1[ai, bi] = 1 and we get the contradic- Q tion φ(x1) = 1. The following calculation shows that there is no Fuchsian group with compact Dirichlet region F that has γ = 0 and r = 2. If there was one, its hyperbolic area would be given by:

1 1 µ(F ) = 2π 2(0 1) + 1 + 1 − − m − m 1 2 1 1 = 2π −m − m 1 2 < 0

Thus, the [H3] is true and we’re done.

Theorem 56 (Harvey’s Suﬃciency Theorem, [19]). Let Γ have a presentation as given above and let M be the least common multiple of m1,..., mr. Let m > 1.

Suppose furthermore that either r =0 or the following conditions are satisﬁed.

H1. lcm m ,..., mˆ ,..., m = M for all i, where mˆ denotes the omission of m ; { 1 i r} i i

H2. M divides m and if γ =0, then M = m;

H3. r =1 and if γ =0, then r 3; 6 ≥

H4. If 2 M, the number of periods divisible by the maximum power of 2 dividing M | is even.

Then there is a surface kernel homomorphism φ :Γ Z . → m 101 Proof. We follow the original proof of Harvey. The proof presented here attempts to correct several typographical errors. It also includes some additional cases to aid in the clarity of the exposition. The argument is completed through a series of stages where the number m is built from a single prime power component. First, assume r> 0. Furthermore suppose that γ = 0 (and hence, M = m), and m = pb, p a prime, b 1. The second case will treat the situation in which γ = 0 and M = m is an ≥ arbitrary natural number greater than 1. The general case of γ > 0 is then a simple corollary of the special cases. The proof is ﬁnally completed with the consideration of the r = 0 case.

Case I: γ = 0, M = m = pb: We construct the surface kernel homomorphism φ :

Γ Z explicitly by specifying the images of the generators of Γ. Then it is seen → m that these images satisfy the relations from Γ and preserve the periods. The general m case uses a Chinese Remainder Theorem argument that builds on this base case.

Write each period m = pµi , where µ 1. Since M = m, it must be that i i ≥ some t > 0 of the µi are equal to b. Without loss of generality, we can assume these are mr−t+1, mr−t+2,..., mr−1, mr. Now, we will specify the values of φ(xi) for i = 1,..., r 2 without worrying about whether these satisfy the relations of Γ. −

Then we will set the values of φ(xr−1) and φ(xr) so that these relations are satisﬁed.

In the following, we will always write the operation in Zm additively. Deﬁne:

φ(x )= pb−µi Z , i =1, 2,..., r 2 i ∈ m −

µi It is clear that for these values of i, φ(xi) has order p . Note that the hypothesis that r 3 is used in this step of the proof. ≥

102 To appease Γ, we must have φ(x )+ . . . + φ(x ) 0 (mod m). Substituting, we 1 r ≡ obtain that the values of φ(xr−1) and φ(xr) must satisfy:

r−2 φ(x )+ φ(x ) pb−µi (mod pb) r−1 r ≡ − i=1 Xr−t r−2 = pb−µi 1 (mod pb) − − i=1 i=r−t+1 Xr−t X = pb−µi (t 2) (mod pb) − − − i=1 X Note that since µ < b for 1 i r t, the last summation notation is divisible by i ≤ ≤ − p. Hence we can write the relation from Γ as:

φ(x )+ φ(x ) qp t + 2 (mod pb) r−1 r ≡ −

We deﬁne φ(xr−1) and φ(xr) in two cases.

First suppose that t 1 (mod p). If p = 2, this means that the number of periods ≡ divisible by the maximum power of 2 dividing M = 2b is odd. So in this case, we have that p = 2. Deﬁne: 6

φ(x ) t (mod pb), φ(x ) 2+ qp (mod pb) r−1 ≡ − r ≡

b b b It is clear that φ(xr−1) has order p . Since 2 has order p and qp has order < p , it follows that φ(x ) also has order pb. We also have that r φ(x ) 0 (mod pr), and r i=1 i ≡ hence φ :Γ Z is a surface-kernel homomorphism. P → m Now suppose that t 1 (mod p). Deﬁne: 6≡

φ(x ) 1 (mod pb), φ(x ) 1 t + qp (mod pb) r−1 ≡ r ≡ −

Clearly, φ(x ) has order pb. The fact that 1 t 0 (mod p) and qp has order

m = pµi 1 pµi 2 pµik i 1 2 ··· k where µ 0 for all i and j. Now fix a j, 1 j k. Define m = pµi j . If m = 1, ij ≥ ≤ ≤ ij j ij define ξ = 0. It may occur that there are only two m = 1. If this is the case for ij ij 6 mxj and myj, [H1] implies that mxj = myj. In this case, we define:

m bj m bj ξxj (mod p ), ξyj (mod p ) ≡ mx j j ≡ − my j j

Then ξ + ξ 0 (mod pbj ). If there are at least three m = 1, let ξ Z be an xj yj ≡ j ij 6 ij ∈ m element as in Case I that solves the congruences:

bj pj bj ξij (mod pj ) ≡ mij r ξ 0 (mod pbj ) ij ≡ j i=1 X Perform the above procedure for every j. Now we ﬁx i and use the Chinese Remainder

Theorem to obtain an unique element ξi of Zm such that:

ξ ξ (mod pbj ) i ≡ ij j for all j, 1 j k. Deﬁne φ(x )= ξ . ≤ ≤ i i

In this paragraph, we show that the order of each φ(xi) is mi. Let di be any natural number such that d φ(x ) 0 (mod m). It is clear that d φ(x ) 0 (mod m) if and i i ≡ i i ≡

104 only if d ξ 0 (mod pb1 pbk ) if and only if d ξ 0 (mod pbj ) for all j,1 j k. i i ≡ 1 ··· k i ij ≡ j ≤ ≤ This implies that d pbj /m 0 (mod pbj ) for all j. Then we must have that d is i j ij ≡ j i a multiple of mij for every j. The order of φ(xi) is the smallest such di that works.

k Hence di = j=1 mij = mi.

What weQ have shown so far is that if φ : γ Z is homomorphism, then it is a → m surface kernel homomorphism. To see that our φ is indeed a homomorphism, we need only verify that the relations in Γ are preserved in Z . We have that r ξ 0 m i=1 ij ≡ bj r P (mod pj ) for every j. But this immediately implies that i=1 φ(ξi) is divisible by

bj P pj for all j and hence that:

r φ(x ) 0 (mod m) i ≡ i=1 X Case III: γ > 0, r> 0, m is arbitrary: Now suppose that r > 0 and γ > 0. By hypothesis, M m. Thus, Z has a subgroup isomorphic to Z . If Γ has at least | m M three periods, then map the periodic generators onto Zm as above and map the ai and bi to 1. If Γ has only two periods, then we just do the same trick as in the case

γ = 0, m arbitrary. Since the periods and relations of Γ are preserved, Γ is a surface kernel homomorphism.

Case IV: r = 0: Finally, suppose that r = 0. Then deﬁne φ :Γ Z by φ(a )= → m i

φ(bi) = 1. This is clearly a surface kernel homomorphism, so we are done.

105 3.7 Realization of Periodic Dold Sequences on Surfaces

The requisite background material has now been covered. The goal of this section is to state and necessary and suﬃcient conditions for a periodic Dold sequence to be the

Lefshetz sequence of a periodic orientation preserving diffeomorphism. After stating the theorem, it will be shown that the general case agrees with the prime order case that was discussed in Section 3.3. An overview of the proof for the general case is also presented in this sections. The proofs of the necessary and sufficient parts of the theorem are relegated to subsections. To assist the reader, a schematic diagram of the different actions used in the proof is given in Figure 6. We begin with the following general lemma.

Lemma 57. Let G be a group and H a ﬁnite index normal subgroup of G. Suppose that X is a G set. For x X, let x¯ denote the orbit of x by H and X¯ denote the − ∈ set of orbits. Then G/H acts on X¯ and:

G [G : H] x¯ = H · Gx Gx∩H

Proof. The action of G/H on X¯ is deﬁned by gH x¯ = gx¯. To establish the above · equation, ﬁrst note that the stabilzer ofx ¯ in G/H, [G/H]x¯, is equal to GxH/H. This equation is well-formed since H⊳G. Let gH [G/H] . Then gx¯ =x ¯ and hence ∈ x¯ g x = h x for some h H. Hence, h−1g x = x. Since H is normal in G, h−1g = gh′ · · ∈ · for some h H and thus, gh′ G . This implies that g = gh′(h′)−1 G H and that ∈ ∈ x ∈ x gH G H/H. The other inclusion is clear. ∈ x 106 Now, the theory of G sets provides us with the following G isomorphism, where − − the right hand side merely means “set of cosets”:

G G H x¯ ≈G H · GxH H Since H⊳G, H⊳G H. Then by the second isomorphism theorem, we have G H/H x x ≈ G /(G H). The lemma now follows. x x ∩

Corollary 58. Let Γ be a Fuchsian group and φ : Γ Z a surface kernel homo- → m morphism with surface kernel Γ′. Let x H be an elliptic ﬁxed point of Γ of order k. ∈ Let f : Γ′ H Γ′ H be a generator of the induced Z action on Γ′ H. Then x¯ is \ → \ m \ a periodic point of f with period exactly [Γ:Γ′]/k. Moreover, if y is a periodic point of f, then there is an elliptic cycle of Γ containing an elliptic ﬁxed point x H such ∈ that x¯ = y.

Proof. Since x is an elliptic ﬁxed point, we know by the theory of Fuchsian groups that

Γ is a maximal ﬁnite cyclic subgroup of Γ with Γ = k. The fact that Γ′ is a surface x | x| kernel means that it contains no subgroups of ﬁnite order. Hence, Γ Γ′ = id . x ∩ { } Then by Lemma 57, we have that

Γ m x¯ = Γ′ · k

Then the stabilizer ofx ¯, [Γ/Γ′] has order k. If ψΓ′ Γ/Γ′ is any generator, then x¯ ∈ ′ m/k ′ ′ [Γ/Γ ]x¯ =< ψ Γ >. Hence,x ¯ is a periodic point of period exactly [Γ : Γ ]/k.

Conversely, suppose that y is a periodic point of f with period m/k and let y =z ¯ for some z H. Then [Γ/Γ′] is nonempty and has order k. Now, Γ Γ′ = id . It ∈ z¯ z ∩ { } follows from the proof of the above Lemma 57 that Γz has order k. This says precisely that z is an elliptic ﬁxed point of period k.

107 Theorem 59. Let s : N Z be a periodic Dold sequence of period m. Deﬁne: →

∆ =2m (m + 1)s(m)+ kM (k) s − s Xk|m Also, deﬁne K = k : M (k) = 0, 1 k m 1 , where k K is counted with s { s 6 ≤ ≤ − } ∈ s multiplicity M (k) (i.e. K is a multi-set). Suppose that s : N Z satisﬁes the s s → following conditions:

1. (MWC1) s(m) 2N and if s(1) = 1, then K 1 = and gcd(K 1 )=1; ∈ − s\{ } 6 ∅ s\{ }

2. (MWC2) M (k) 0 for all k 0 (mod m); s ≥ 6≡

3. (MWC3) ∆ 0 and ∆ 0 (mod 2m); s ≥ s ≡

4. (MWC4) If ∆s =0, then gcd(Ks)=1.

Then there is a periodic orientation preserving diffeomorphism ψ : S S of period → m on any surface of genus g = (2 s(m))/2 2 such that the Lefshetz sequence of ψ − ≥ is s : N Z. Conversely, the Lefshetz sequence of a periodic orientation preserving → diffeomorphism of period m from a surface to itself satisfies (1) (4). −

Before we prove the theorem, we will give an outline for the proofs of necessity and sufficiency. For the proof of sufficiency, we will frequently refer to the diagram in Figure 6. Let s : N Z be a given periodic Dold sequence and M : N Z its → s → Möbius inverse sequence. First we use M : N Z to construct an abstract signature s →

(γ; m1,..., mr). Then [MWC1], [MWC3], and the Poincar´e-Maskit theorem imply that there is a Fuchsian group Γ with signature (γ; m1,..., mr). We then show that

[MWC1]-[MWC4] imply that there is a surface kernel homomorphism φ : Γ Z . → m

108 Define Γ′ = Ker(φ). As shown above, Γ′ is also a “nice” Fuchsian group in that it has signature (γ′; ). The space of orbits Γ′ H is a closed orientable surface of genus − \ γ′ 2 that has H as its universal cover. ≥ The group Γ/Γ′ = Z acts on Γ′ H as a group of orientation preserving diffeo- ∼ m \ morphisms (actually, automorphisms of a Riemann surface). The next task is to show that the Lefshetz sequence of any generator is in fact s : N Z. Using the formula → for the hyperbolic area of a fundamental region, it is shown that γ′ = (2 s(m))/2. − In other words, s(m) is the Euler characteristic of Γ′ H. The remaining terms of \ the sequence that are needed, i.e. s(1), s(2), s(3),..., s(m 1) are determined from − the relationship between conjugacy classes of maximal finite cyclic subgroups of Γ and periodic points of any generator of Γ/Γ′. Finally, we employ Möbius inversion to show that, in fact, the Lefshetz sequence of any generator of Γ/Γ′ is s : N Z. → The converse is much easier. Given a closed orientable surface S and an orientation preserving diffeomorphism ψ : S S of finite period m, Theorem 52 shows that the →

<ψ>∼= Zm-action can be lifted to a Fuchsian group Γ admitting a surface-kernel homomorphism φ : G Z . This nice Fuchsian group has signature (γ, m ,..., m ) → m 1 r and Γ′ = Ker(φ) has signature (g; ). Once again, we connect the periods of Γ to the − periodic points of a generator of Γ/Γ′. Then the necessity part of Harvey’s theorem is used to show that the Lefshetz sequence of ψ satisﬁes [MWC1]-[MWC4].

We now establish that the result for arbitrary m implies the result for the special odd prime case presented in Section 3.3. This is indeed true. Of course, it must be supposed that g 2. Let p be an odd prime and s : N Z be a periodic Dold ≥ → sequence of period p. The proof is broken into 2 cases: s(1) 1 and s(1) = 0. ≥ 109 Figure 6: Diagram for the proof of suﬃciency. Γ H H

Γ ’< Γ

Γ ’

H H Γ ’ Γ ’

Γ S S Γ ’

S S Γ/ Γ ’ Γ/ H Γ ’ H Γ Γ

Case I: Suppose that (1), (2), and (3) of Proposition 46 are satisﬁed. We must show that [MWC1], [MWC2], [MWC3], and [MWC4] are satisﬁed in Theorem 59. By

(1) and (2), [MWC1] is clearly satisﬁed. Since Ms(k) = 0 for all k other that 1 and p, [MWC2] is satisﬁed. Now we compute ∆s.

∆ = 2p (p + 1)s(p)+ kM (k) s − s Xk|p = 2p (p + 1)s(p)+ s(1) + p(s(p) s(1)) − − = 2p s(p)+ s(1) ps(1) − − = p(2 M (1)) M (p) − s − s

By (3), this implies that ∆ 0. The fact that ∆ 0 (mod 2p) follows immediately s ≥ s ≡ from the proof of Proposition 46. Hence, (3) implies [MWC3]. Now suppose that

∆ = 0. Since s(1) 2, it is trivially true that gcd(K )=1. s ≥ s 110 For the reverse direction, suppose that s(1) 1 and that [MWC1]-[MWC4] all ≥ hold. We know that s(k) = s(1) for all k, 1 k p 1. Also, M (k) = 0 for all k ≤ ≤ − s such that 1 < k p 1. By [MWC1], we have that if s(1) = 1, then K 1 = . ≤ − s \{ } 6 ∅ Since K contains only ones, we have the contradiction that s(1) 2. Hence, we s ≥ must have that s(k) 2 for all k such that 1 k p 1. Hence, (1) is satisfied. ≥ ≤ ≤ − It is obvious that (2) is satisfied. An identical argument to the preceding paragraph shows that [MWC3] implies (3). This completes the proof of the first case.

Case II: We will show that the result of Proposition 46 implies the special case of

Theorem 59. Suppose that s(1) = 0 and s(p)=2pk where k 1. It is clear that ≤ −

[MWC1] and [MWC2] are satisﬁed. Computing ∆s gives:

∆ = 2p (p + 1)2pk + p(2pk) s − = 2p 2pk −

Clearly then, ∆ 0 and ∆ 0 (mod 2p). If ∆ = 0, we have k = 1, which is s ≥ s ≡ s impossible.

Now suppose that s(1) = 0 and [MWC1]-[MWC4] all hold. We only need to show that s(m)=2pk where k 1. We have that: ≤ −

2px = ∆ =2p (p + 1)s(p)+ ps(p) s − for some x 0. Thus, s(p)=2p(1 x). Since s(p) 2N, we must have that x 2. ≥ − ∈ − ≥ Let k =1 x 1. This completes the proof of the second case and establishes the − ≤ − overall result.

111 3.7.1 Proof of Suﬃciency

We will begin by establishing the existence of the appropriate Fuchsian group. De-

ﬁne g = (2 s(m))/2 and γ = ∆ /(2m). Since ∆ 0 and ∆ 0 (mod 2m), − s s ≥ s ≡ γ N 0 . Let K be the set as deﬁned above, where k K has multiplicity ∈ ∪{ } s ∈ s M (k). If k K , let m = m/k. Consider the abstract signature (γ; m : k K ), s ∈ s k k ∈ s where of course mk appears in the abstract signature Ms(k) times.

The claim is that there is a Fuchsian group Γ which has the above signature. It will be shown that the above signature satisﬁes the Poincar´e condition (see Theorem

53). Note that m 2 for all k K . Using the fact that for periodic Dold sequences k ≥ ∈ s of period m, Ms(k)=0 for k with gcd(k, m)

1 k m 1 = m 1 − mk − m kX∈Ks kX∈Ks m−1 k = m 1 M (k) − m s k=1 m−X1 = (m k)M (k) − s k=1 Xm−1 m−1 = m M (k) kM (k)+ mM (m) mM (m) s − s s − s Xk=1 Xk=1 = m M (k) kM (k) s − s Xk|m Xk|m = ms(m) kM (k) − s Xk|m

112 This equation implies that:

1 m 2(γ 1) + 1 = ∆s 2m + ms(m) kMs(k) " − − mk # − − kX∈Ks Xk|m = 2m (m + 1)s(m)+ kM (k) 2m − s − Xk|m + ms(m) kM (k) − s Xk|m = s(m) − 2 ≥

Then by Theorem 53, there exists a Fuchsian group Γ with the above signature.

It must now be shown that this signature satisﬁes all of the conditions laid out in

Theorem 56.

If K = 0, then the number of periods in Γ is 0. Then by Harvey’s suﬃciency | s| theorem, there exists a surface kernel homomorphism φ : Γ Z . In the case → m that K > 0, it is necessary to show that conditions [H1]-[H4] are satisﬁed for the | s| Fuchsian group Γ. These cases are broken up explicitly below.

[H1] is satisﬁed: Now suppose that K > 0. Let M = lcm m : k K . | s| { k ∈ s} Suppose k K . Let Mˆ be the least common multiple of m : j K m , where ∈ s { j ∈ s}\{ k} each mj in this set is counted with a mutliplicity of Ms(j) but mk is counted with multiplicity M (k) 1. Now, since k M (k), we have that M (k) 1 k 1. If s − | s s − ≥ − 1 1.

In the second case, M = Mˆ = m. In the first case, we still know that M = m. We have by hypothesis that K 1 = and gcd(K 1 ) = 1. Then Mˆ = lcm( m : s\{ } 6 ∅ s\{ } { k k K m )= m/ gcd(K 1 )= m. This completes the verification of [H1]. ∈ s}\{ } s\{ } 113 [H2] is satisfied: The fact that M m is obvious, since m m for all k. Suppose | k| that γ = 0. Then ∆s = 0. By hypothesis, gcd(Ks) = 1. It is easy to show that

M = m/gcd(Ks)= m. Thus, [H2] is satisﬁed.

[H3] is satisﬁed: The number r in Harvey’s suﬃciency theorem corresponds to the number of periods of Γ. If r = 1, it must be that M (k) = 1 for some k m and s | M (j) = 0 for all j with gcd(j, m) < j, j = m. Since k M (k), we must have k = 1. s 6 | s This is a contradiction of our hypothesis that K 1 = . Hence r = 1. Now suppose s\{ } 6 ∅ 6 that γ = ∆s/(2m) = 0 and that r = 2. This could only occur if (1) Ms(2) = 2 and

M (k) = 0 for all k = 2, m or (2) if M (1) = 2 and M (k) = 0 for all k = 1, m. In s 6 s s 6 these cases, the area of the Dirichlet regions are given by:

1 0 2m(γ 1) + m 2 1 =2mγ 4= 4 ≤ − · − m − − 2

1 0 2m(γ 1) + m 2 1 =2mγ 2= 2 ≤ − · − m − − 1 which are contradictions. Thus, [H3] is satisﬁed.

[H4] is satisﬁed: For the last condition, note that the hypotheses state :

kM (k) (m + 1)s(m) (mod2m) s ≡ Xk|m

Since s(m) is even, we must have that k|m kMs(k) is also even. Now, multiplying a number by an odd number does not changeP the parity of that number. Hence, if k is odd, kM (k) M (k) (mod 2). Let e 1 be the largest exponent of 2 dividing M. s ≡ s ≥

114 If k K is even, then 2e m . But if k K is odd, 2e m . Then we have: ∈ s 6| k ∈ s | k

This shows that the number of periods of Γ divisible by the maximum power of 2 dividing m is even. Hence, [H4] is satisﬁed.

Then by Theorem 56, we have that there is a surface kernel homomorphism Γ → Z with surface kernel Γ′. Then Γ/Γ′ is a Z -action on Γ′ H. We have yet to m m \ show that Γ′ H has genus g and that any generator of Γ/Γ′ has Lefshetz sequence \ s : N Z. → Suppose that Γ′ has signature (γ′; ). Then Γ′ has a compact Dirichlet region − ′ ′ ′ D ′ (Γ ), where p H, and the hyperbolic area of D ′ (Γ ) is given by: p ∈ p

′ ′ ′ µ(Γ H)= µ(D ′ (Γ ))=4π(γ 1) \ p −

′ It was shown above that Γ ⊳Γ is a subgroup of index m. Let F = Dp(Γ) be a Dirichlet region of Γ at some point p H not fixed by an element of Γ. The region F is compact ∈ because it has finite hyperbolic area and Γ contains, by construction, no parabolic elements. Then ∂F is a union of finitely many hyperbolic geodesic segments in H and hence ∂F has 0 hyperbolic area. Let T1,..., Tm be a set of coset representatives of Γ′ in Γ. By Proposition 51, F ′ = m T (F ) is a fundamental region for Γ′. Note ∪i=1 i that ∂F ′ has 0 hyperbolic area and F ′ is compact. Then µ(F ′) = mµ(F ). We have

115 ′ ′ ′ that µ(Dp′ (Γ )) = µ(F ) = mµ(F ). Now we can compute the hyperbolic area of F in terms of the periods of Γ. Yet again using the fact that M (k) =0 for k m, we s 6 | have:

mµ(D (Γ)) 1 p = m 2(γ 1) + 1 2π " − − mk # kX∈Ks 1 = ∆s 2m + m 1 m − − k kX∈Ks = 2m (m + 1)s(m)+ kM (k) 2m + (m k) − s − − Xk|m kX∈Ks = (m + 1) M (k)+ kM (k)+ (m k)M (k) − s s − s Xk|m Xk|m Xk|m = M (k) − s Xk|m = s(m) −

Since g = (2 s(m))/2, s(m)=2g 2 and we have: − − −

′ mµ(D (Γ)) µ(D ′ (Γ )) 2(g 1) = s(m)= p = p = 2(γ′ 1) − − 2π 2π −

Thus γ′ = g. Let ψΓ′ be any generator of Γ/Γ′ and let t : N Z be its Lefshetz → sequence. By Corollary 58, every elliptic ﬁxed point of Γ of period m/k projects to a periodic point of period k and every periodic point of period k is the image of some elliptic cycle of Γ. Since there are Ms(k) elliptic cycles of period k, Mt(k) = Ms(k) for all k, 1 k m 1. Since t(m)=2 2g = s(m), M¨obius inversion gives that ≤ ≤ − − t = s. By Theorem 20, the choice of the generator of Γ/Γ′ is irrelevant. Thus we have shown that every periodic Dold sequence of period m satisfying the above conditions is the Lefshetz sequence of a periodic map of oder m.

116 3.7.2 Proof of Necessity

Let S be a surface of genus g 2 and ψ : S S an orientation preserving dif- ≥ → feomorphism of ﬁnite period m > 1. Let s : N Z be the Lefshetz sequence of ψ. → We will show that s : N Z satisﬁes [MWC1]-[MWC4]. It has already been shown → that the period of s : N Z is m. We complete the proof by lifting the action on S → generated by ψ to some Fuchsian group acting on H.

The group <ψ>∼= Zm deﬁnes a smooth Zm-action on S. Since the genus of S is at least 2, there is a Fuchsian group Γ and a surface kernel homomorphism

φ : Γ Z . Let Γ′ = Ker(φ). The presentations of Γ and Γ′ are determined by → m the following signatures, where the mi are unknown periods. Although the mi are unknown, we must have that they satisfy [H1]-[H4].

σ(Γ) = (γ; m1,..., mr)

σ(Γ′)=(g; ) −

Each period mi is the common order of the elements of a conjugacy class of a maximal

ﬁnite cyclic subgroup of Γ. A representative of this equivalence class is of the form

Γ where x H. As has already been shown, every conjugacy class of maximal ﬁnite x ∈ cyclic subgroups corresponds to exactly one periodic point of any generator of Γ/Γ′ of period m/mi. Deﬁne ki = m/mi. Now the number of periodic points of period k is M (k ). Hence we know that k M (k ) and Γ has precisely M (m/m ) periods i s i i| s i s i equal to mi.

117 [MWC3] is satisﬁed: From the hyperbolic area of the fundamental regions of Γ and Γ′, we have: 2g 2 r 1 − =2γ 2+ 1 m − − m i=1 i X We now rewrite this in terms of the M¨obius inversion sequence.

r m 2g 2 = m(2γ 2)+ m − − − m i=1 i Xr ( ) s(m) = m(2γ 2)+ (m k ) ⇒ − − − i i=1 Xr ( )2mγ = 2m s(m) (m k ) ⇒ − − − i i=1 X

Using the fact that Ms(k) = 0 when there are no periods of order k or when gcd(k, m)

m−1 2mγ = 2m s(m) (m k)M (k) − − − s k=1 Xm−1 m−1 = 2m s(m) m M (k)+ kM (k)+ mM (m) mM (m) − − s s s − s Xk=1 Xk=1 = 2m s(m) m M (k)+ kM (k) − − s s Xk|m Xk|m = 2m s(m) ms(m)+ kM (k) − − s Xk|m = 2m (m + 1)s(m)+ kM (k) − s Xk|m = ∆s

Since γ 0, ∆ 0 and ∆ 0 (mod 2m). This establishes [MWC3]. ≥ s ≥ s ≡ [MWC2] is satisfied: Note that every power of ψ is also a periodic map and hence has isolated fixed points, each of local Lefshetz number +1. Thus, M (k), 1 k s ≤ ≤ m 1 has the standard interpretation as the number of points fixed by ψk but are − not fixed by ψj for j

m = lcm m 1 i r { i| ≤ ≤ }

However, [H1] implies that:

m = lcm m ,..., m , m ,..., m { 1 a−1 a+1 r} = lcm m ,..., m m { 1 r}\{ }

As each mi is of the form m/ki, where ki is the period of a periodic point and every period appears as many times as there are periodic points of that period, we have:

m m = lcm k K 1 k | ∈ s\{ } n m o = gcd k k K 1 { | ∈ s\{ }} m = gcd(K 1 ) s\{ } Hence, gcd(K 1 )=1. s\{ }

[MWC4] is satisﬁed: Assume that ∆s = 0. Then γ = 0 and Γ has signature:

(0; m1,..., mr)

Then [H2] implies that: m = lcm m ,..., m = lcm m/k k K = m/ gcd(K ). { 1 r} { | ∈ s} s Hence:

gcd(Ks)=1.

This establishes the necessity of the conditions above.

119 3.8 The Sphere and the Torus

Because the universal coverings of S2 and S1 S1 are not H, it is necessary to × take a different tack on these spaces. The case of the sphere is trivial due to the simplicity of its homology. The case of the torus is essentially linear algebraic. It is only necessary to find the elements of finite order in Sp(2, Z). In fact, more is done here. All Dold sequences on the torus can be found in a similar manner. After the proof of this theorem, it is that all periodic Dold sequences on the torus are the

Lefshetz sequence of some periodic map of exactly the same period.

Theorem 60 (Classiﬁcation of Dold sequences on the sphere). Let ψ : S2 S2 be → an orientation preserving diﬀeomorphism of the sphere. Then the Lefshetz sequence of ψ is the constant sequence of all 2’s.

Proof. This is obvious.

Theorem 61 (Classiﬁcation of Dold sequences on the torus). Let T = S1 S1 and × let ψ : T T be an orientation preserving diﬀeomorphism. Let s : N Z be the → → Lefshetz sequence of ψ.

1. If Tr(ψ∗)=2, i.e. Λ(ψ)=0, then

s = (0, 0, 0, 0,...)

2. If Tr(ψ )= 2, i.e. Λ(ψ)=4, then ∗ −

s = (4, 0, 4, 0,...)

120 3. If Tr(ψ∗)=1, i.e. Λ(ψ)=1, then

s = (1, 3, 4, 3, 1, 0, 1, 3, 4, 3, 1, 0,...)

4. If Tr(ψ )= 1, i.e. Λ(ψ)=3, then ∗ −

s = (3, 3 , 0, 3, 3, 0,...)

5. If Tr(ψ∗)=0, i.e. Λ(ψ)=2. then

s = (2, 4, 2, 0, 2, 4, 2, 0,...)

6. If Tr(ψ )= k, k even, k > 2, then s : N Z is the Lefshetz sequence generated ∗ → by the matrix: k k 2 2 +1 k 1 k 2 − 2 Moreover, the Lefshetz sequence has all negative entries and is decreasing.

7. If Tr(ψ )= k, k odd, k > 2, then s : N Z is the Lefshetz sequence generated ∗ → by the matrix:

2 k+1 k −1 1 2 4 − 1 k−1 2 Moreover, the Lefshetz sequence has all negative entries and is decreasing.

8. If Tr(ψ ) = k, k even, k < 2, then s : N Z is the Lefshetz sequence ∗ − → generated by the matrix: k k 2 2 +1 k 1 k 2 − 2 2 Moreover, the entries of the sequence (Tr(ψ∗),Tr(ψ∗),...) alternate in sign and

the sequence ( Tr(ψ ) , Tr(ψ2) ,...) is increasing. | ∗ | | ∗ | 121 9. If Tr(ψ )= k, k odd, k < 2, then s : N Z is the Lefshetz sequence generated ∗ − → by the matrix:

2 k+1 k −1 1 2 4 − 1 k−1 2 2 Moreover, the entries of the sequence (Tr(ψ∗),Tr(ψ∗),...) alternate in sign and

the sequence ( Tr(ψ ) , Tr(ψ2) ,...) is increasing. | ∗ | | ∗ |

Proof. First note that any element A Sp(2, Z) has determinant 1. Moreover, if ∈ A GL(2, Z) and det(A) = 1, then A Sp(2, Z). Thus, Sp(2, Z) = SL(2, Z). Let A ∈ ∈ be the image of ψ∗ in Sp(2, Z).

First, suppose that Tr(ψ∗) = 2. There are two cases to consider: ψ∗ is diagonalizable, and ψ∗ is not diagonalizable. Suppose that ψ∗ is diagonalizable. Then ψ∗ is similar to a matrix of the form:

λ1 0 0 λ 2 where λ1 +λ2 = 2 and λ1λ2 = 1. The characteristic polynomial for this matrix is thus x2 2x +1=(x 1)2. Since ψ is diagonalizable, we must have that ψ = id. It is − − ∗ ∗ clear that the Lefshetz sequence of the identity is the one given above. Now, suppose that ψ∗ is not diagonalizable. Then using the Jordan canonical form, we have that

ψ∗ is similar to a matrix of the form:

λ 1 0 λ where 2λ = 2 and λ2 = 1. Hence λ = 1. An easy inductive argument now shows that

ψ takes the following form and hence Tr(ψk)=2 for all k: ∗ ∗ 1 k Ak ∼ 0 1 122 Now, suppose that Tr(ψ ) = 2. We will use the same notation that was used ∗ − above. If A is diagonalizable, we have that λ λ = 1 and λ + λ = 2. This gives 1 2 1 2 − the characteristic polynomial as x2 +2x +1=(x +1)2. Since A is diagonalizable, we have that A = id and Tr(ψk)=( 1)k 2. If A is not diagonalizable, we have that − ∗ − · 2λ = 2 and λ2 = 1. Then λ = 1, and an easy inductive argument shows that: − −

( 1)k ( 1)k−1k Ak − − ∼ 0 ( 1)k −

Once again, this gives us that Tr(ψk)=( 1)k 2. In either case, s : N Z is as ∗ − · → above.

Suppose that Tr(ψ∗)=1. If A is not diagonalizable, we must have that 2λ = 1 and

λ2 = 1. Since these equations are inconsistent, it must be that A is diagonalizable. In that case, we have that λ1λ2 = 1 and λ1 +λ2 = 1. Then the characteristic polynomial is x2 x + 1 and we have that the eigenvalues are 1/2 i√3/2. Each of these is − ± a primitive sixth root of unity. A calculation then veriﬁes that this is the sequence given above.

If Tr(ψ )= 1, we once again must have that A is diagonalizable. The character- ∗ − istic polynomial is now x2 + x + 1 and hence the eigenvalues are e±i2π/3. The Lefshetz sequence is the one given above.

If Tr(ψ∗) = 0, the fact that A is invertible implies that A is diagonalizable. The characteristic polynomial is x2 + 1 and the eigenvalues are i. It is easy to verify ± that the above sequence is indeed the Lefshetz sequence.

Now suppose that Tr(ψ∗) = k, k > 2. The above matrices have determinant 1 and trace k. Suppose that there is another matrix B SL(2, Z) with trace k. B is ∈

123 diagonalizable and the characteristic polynomial of B, x2 kx + 1, has two distinct − real roots. Clearly, A also has the same characteristic polynomial, and thus the same eigenvalues. It follows that A B and hence, their Lefshetz sequences are equal. To ∼ show that the Lefshetz sequence is negative and decreasing, we only need to show that the sequence (Tr(A),Tr(A2),...) is increasing. It is suﬃcient to show that for all q 2: ≥ 1 1 λq + >λq−1 + λq λq−1 where λ> 1 is an eigenvalue of A. The above relation will hold if an only if the real valued function f(x)= x2k x2k−1 x +1 > 0 for all x> 1. Note that f(1) = 0. We − − have:

f ′(x) = 2kx2k−1 (2k 1)x2k−2 1 − − − = 2kx2k−2(x 1) + x2k−2 1 − − 2k−3 = (x 1) 2kx2k−2 + xj − j=0 ! X But this is clearly greater than 0 for all x> 1. Thus, the sequence is indeed increasing.

Now suppose that Tr(ψ )= k < 2. The same argument as above clearly shows ∗ − that any matrix with trace k will have the same Lefshetz sequence as one of the two cases. We must show that

( Tr(ψ ) , Tr(ψ2) , Tr(ψ3) ,...) | ∗ | | ∗ | | ∗ | is increasing. In this case, note that the eigenvalues of A are both negative. Hence,

λq + λ−q = λq + λ−q for all q 1. This implies that this last case reduces to the | | | | | | ≥ case where k > 2, and we are done. − 124 In the proof of the previous theorem, we used the isomorphism of the Sp(2, Z) with

SL(2, Z). It is well known that the mapping class group of S1 S1 is also isomorphic × to SL(2, Z) (see [34]). Using this isomorphism, we can see that the mapping class group has elements of order 2, 3, 4, and 6. We showed above that there are Dold sequences of each of these orders and that any element of Sp(2, Z) of order 2, 3, 4, or 6 must have the Dold sequence of that period. Here we will explicitly describe periodic diﬀeomorphisms of the appropriate order that realize the Dold sequence of that period.

We begin with the sequence (4, 0, 4, 0,...). The orientation preserving diﬀeomor- phism that realizes this sequence is simply a genus one surface rotating on a spit as shown in Figure 7. Using the Riemann Hurwitz equation, it is possible to show that any map of order 4 or 6 cannot have this sequence as its Lefshetz sequence.

The sequence (3, 3, 0, 3, 3, 0,...) has been described in great detail already. In fact, it is realized by one of the Kosniowski generators. In particular, it is the map

κ : S1 S1 S1 S1 shown in Figure 5. Once again using the Riemann-Hurwitz 1,3 × → × equation, one can show that no map of order 6 can have this as its Lefshetz sequence.

Now, I SL(2, Z) is an element of order 2. It corresponds to a diﬀeomorphism − ∈ which maps σ σ and τ τ, where σ and τ are representatives of the standard → − → − generators of H (S1 S1). This map is indeed orientation preserving. Note that 1 × I is in the center of SL(2, Z). We can thus take the composition of this map with − κ : S1 S1 S1 S1. The resultant diﬀeomorphism has order 6 and Lefshetz 1,3 × → × sequence (1, 3, 4, 3, 1, 0,...).

125 Figure 7: Rotation of the torus by π has 4 ﬁxed points.

For the map of order 4, deﬁne ψ : S1 S1 S1 S1 by: × → ×

ψ(e2πis, e2πit)=(e−2πit, e2πis)

It can be shown that this map is orientation preserving and has exactly two ﬁxed points. One can visualize this map as one that maps a meridian of the solid torus to the longitude and the longitude to meridian in the opposite direction. The induced map in homology is: 0 1 1 0 − One then quickly observes that the Lefshetz sequence is given by (2, 4, 2, 0,...).

In the classiﬁcation of Dold sequences on the torus, it was shown that many of them are decreasing. This raises the question as to whether or not all decreasing Dold sequences are realized by periodic diﬀeomorphisms on the torus. This question can be answered in the negative by the following example. Instead of looking at the Lefshetz sequence, we will look for increasing sequences of the form (Tr(A),Tr(A2),...), where

A is in Sp(4, Z). First note that any matrix of the form:

A 0 0 B where A, B Sp(2, Z) and O is the 2 2 zero matrix, is in Sp(4, Z) if and only if ∈ ×

126 A tB = I. In light of this fact, we consider the following 4 4 matrix C. × 23 0 0 12 0 0 C =  00 2 1  −  0 0 3 2   −    It is clear from the classification above that the sequence (Tr(C),Tr(C2),... ) is the sequence (2Tr(A), 2Tr(A2),...). Since Tr(A) = 4, realization of this sequence by a diffeomorphism on the torus could only occur by the matrix of trace 8 (specified in the statement of Theorem 61): 4 5 B = 3 4 However, a quick calculation gives:

4 5 4 5 31 40 B2 = = 3 4 3 4 24 31 2 3 2 3 7 12 A2 = = 1 2 1 2 4 7 But since 2 Tr(A2)=28 =62= Tr(B2), this cannot be. Thus, not every decreasing · 6 Dold sequence is realized by a diﬀeomorphism on the torus.

127 Chapter 4

On Zp-actions that extend to the handlebody

Every closed orientable surface S of genus g bounds a three-dimensional manifold called a handlebody. Suppose that a group G acts on a handlebody with boundary

S. The restriction of this action to the boundary gives a G-action on the surface S. In the previous chapters, we outlined the entire periodic point structure of ﬁnite cyclic actions on surfaces. This chapter investigates which of those periodic point structures can be the restrictions of maps on handlebodies. In other words, we ask which ﬁnite cyclic actions on surfaces extend to the handlebody bounded by the surface.

An approach to this general problem is presented in the form of a solution to the

Zp-case. As in the previous chapter, the ﬁxed point set F plays the most important role. Here we will need to consider how the group acts near each ﬁxed point. In our case, it will be shown that this local behavior completely determines the global behavior up to equivariant bordism. The proof of this result describes how to get the stronger result about which periodic maps of prime order extend to actions on the handlebody.

The main mathematical tool of this chapter is the g-signature. While the Lef- shetz number provides some information about the local behavior of a map near a

fixed point, it does not provide sufficient differentiation between distinct fixed points for our purposes. A major theorem of Atiyah and Singer shows how the g-signature invariant encodes very precise information about the local behavior of the action at

128 each fixed point. It is the task of the algebraic topologist to decipher this puzzle and return meaningful geometric information. In the first section of this chapter, we review the concepts behind the general theorem of Atiyah and Singer and apply these results to finite cyclic actions on surfaces. The next section investigates the relevant algebraic number theory to the Zp case. The third section discusses the rather complex equivariant constructions that are necessary for a precise proof of the main theorem. The fourth and final section synthesizes the results of this chapter. It gives a necessary and sufficient condition that a Zp-action on a closed orientable surface extends to the handlebody.

4.1 Slice Types and the G-Signature Theorem

In this section, we deﬁne the notion of slice type as it pertains to the g-signature theorem of Atiyah and Singer, where G = Zm. This theorem supplies the algebraic number theory from which we obtain our results concerning extensions of actions on surfaces to handlebodies. The deﬁnitions provided here follow the exposition of

Berand and Katz [5].

Let M be a closed orientable smooth manifold of even dimension 2k. Suppose that g : M M is an orientation preserving diffeomorphism of finite order m. Denote by → M g the set of fixed points of g. Note that M g is also the set of stationary points of the

Zm-action on M generated by g. The diﬀerential of g acts on the the normal bundle

g ν of M in M. Hence, ν also admits a Zm-action. To use the G-signature theorem of Atiyah and Singer, we must assume several things. First of all, we assume that ν

129 has a complex structure. Secondly, we assume that the Zm-action on ν preserves this complex structure. Moreover, we need to assume that the orientation on ν is the one determined by the complex structure. The orientation of M g is determined by the complex structure of ν and the orientation of M (see [5]).

For our purposes, these assumptions will not cause a problem. One way to see this is to apply a result of Conner and Floyd[9]. Their theorem (38.3) implies that if M a connected, smooth, closed, and orientable manifold and g : M M is an → orientation preserving diﬀeomorphism of odd prime order p acting on M, then the action of dg : ν ν on the normal bundle has a complex structure that is preserved → by the Zp-action on ν. If the genus is at least 2, one can also use the theory of

Riemann surfaces to show that the assumptions above are satisfied. Every orientation preserving diffeomorphism of finite order on a surface M is conjugate (in the group of diffeomorphisms of M) to an automorphism of a Riemann surface. Moreover, there is exactly one complex structure on M that works for this diffeomorphism. Hence, we can just assume that the surface M has a complex structure and that g : M M → is holomorphic.

Let λ = e2πi/m. In the general case where g is compatible with the complex structure on ν, each connected component of ν can be decomposed into a direct sum of subbundles ν ,1 j m 1, on which the diﬀerential of g acts as multiplication by j ≤ ≤ − j g λ . Deﬁne r = dimC(ν ). Let x M . Since x is in a unique connected component j j ∈ of M g, it is associated to a unique (m 1)-tuple: −

(r1,r2,...,rm−1),

130 where rj corresponds to the dimension of the subbundle νj in the decomposition of that component. This (m 1)-tuple is called the normal slice type of x. Note that this − is different from the (m 1)-tuple called the total slice type (to be defined shortly). − Given an (m 1)-tupler ˆ of integers, define M g to be the set of x M g with normal − rˆ ∈ g g slice typer ˆ. It is possible that Mrˆ is empty. It is also possible that Mrˆ is not connected. Let M g denote the set of connected components of M g. { rˆ } rˆ The following theorem, due to Atiyah and Singer, shows how to compute the g- signature of a periodic orientation preserving diffeomorphism on a closed orientable

C∞-manifold from the normal slice type data of the ﬁxed point set. The original proof can be found in [3]. P.E. Conner [10] discovered a more elementary proof for maps of prime power order.

Theorem 62 (Atiyah-Singer). Let M be a closed orientable C∞-manifold of dimension 2k and g : M M an orientation preserving diﬀeomorphism of ﬁnite order m. → Then the equivariant signature of g is given by:

rj λje2xj,l +1 σ(g)= L(Q), [Q] (4.1) λje2xj,l 1 rˆ Q∈{M g} * j l=1 + X Xrˆ Y Y − where x , 1 j m 1, are the two dimensional Wu classes of ν on Q, L(Q) { j,l} ≤ ≤ − j is the Hirzebruch L-class, and [Q] denotes the preferred fundamental cycle of Q.

It should be noted that in the case of periodic maps, there is another form of the

Atiyah-Singer theorem that separates the algebraic number theory from the algebraic topology. This is the subject and result of a paper by Berend and Katz [5]. This paper should be the starting point of any continuation of the results presented in this chapter. Our focus is on periodic maps of surfaces. In this case, the Atiyah-Singer

131 theorem simpliﬁes dramatically and it is possible to use more elementary methods than those employed in [5].

Let S be a smooth closed orientable surface and let ψ : S S be an orientation → preserving diffeomorphism of order m. The fixed points of ψ are isolated and finite in number. Hence, the normal bundle at each fixed point is identified with the tangent plane to the manifold at that point. The normal slice type of each fixed point is of the form (0,..., 0, 1, 0,..., 0). This means that both of the products in equation 4.1 are over a set containing either one element or no elements. Now, each νj is just a copy of C. Thus, the total Chern class of each νj is just 1. Since the Chern classes are the symmetric polynomials in the Wu classes, it follows that all of the Wu classes are

0. This implies that every exponential in equation 4.1 can be replaced 1. Now, ifr ˆ is

ψ a given normal slice type, then Mrˆ is either empty or a ﬁnite union of points. In this

L g case, the Hirzebruch -class is taken to be 1 on each connected component of Mrˆ .

This gives us the following special case of the Atiyah-Singer g-signature theorem.

Corollary 63 (G-Signature Theorem for Periodic Maps on Surfaces). Let S be a smooth closed orientable surface and let ψ : S S be a orientation preserving diffeo- → morphism of finite order m. Let F be the set of fixed points of ψ. If the action near x F is multiplication by λj, define j = j. Then the equivariant signature is given ∈ x by: λjx +1 σ(ψ)= (4.2) λjx 1 x∈F X − From this we see that algebraic number theory plays an important role in cyclic actions on surfaces. Specifically, we are interested in the theory of algebraic numbers

132 of the form: λj +1 α := j/m λj 1 − where λ = e2πi/m. Let j Z, 1 j m 1. Define a to be the number of fixed ∈ ≤ ≤ − j points of ψ : S S at which the action of the differential on the normal bundle at → that point is λj. We define the total slice type of ψ to be the (m 1)-tuple: −

(a1, a2, ...,am−1)

m−1 Hence, σ(ψ)= j=1 ajαj/p. The goal is to use algebraic number theory to determine P relations amongst the aj. Any such relations will yield topological information about

Zm-actions on surfaces. For the remainder of this section, we show some examples of how the occurrences of certain aj’s in the total slice type appears to force the total slice type to contain other aj’s. Moreover, these examples appear to occur whenever the action extends to the handlebody. The experimental evidence of this section undergoes theoretical development in the remainder of the chapter.

2 3 Let p be an odd prime. Consider S embedded in R in the usual way. A Zp-action on S2 is defined to be generated by a counterclockwise rotation ψ : S2 S2 of size → 2π/p about the z-axis. This action has two fixed points: one at the North pole and one at the South pole. The action of the differential on the tangent plane at the

North pole is rotation by 2π/p. We orient S2 so that with the complex structure on the normal bundle, the action at the North pole is multiplication by λ. It is easy to see that the action at the South pole is just multiplication by λ−1. Hence, we have that the equivariant signature is the expected result of any equivariant boundary:

λ +1 λp−1 +1 σ(ψ)= α + α = + =0 1/p (p−1)/p λ 1 λp−1 1 − − 133 Figure 8: Threefold symmetry of the two-holed torus

Now, let T = S1 S1. We deﬁne a Z -action on T to be generated by the map × p ψ : T T deﬁned below: → ψ(w, z)=(w,λz)

This map has no ﬁxed points, so σ(ψ) = 0. Once again, this action obviously extends to the solid torus.

Let S be the two-holed torus. We embed S into R3 so that it exhibits the threefold symmetry shown in Figure 8. Deﬁne ψ : S S to be rotation by 2π/3. This map has → four ﬁxed points. Orient S so that the action on the normal bundle at the uppermost

fixed point is multiplication by λ. We can think of S as two copies of S2, each missing three 2-disks, along with three copies of S1 I. Each fixed point is on some copy of × S2. Hence, we see that there are two fixed points where the action is multiplication by λ and two fixed points where the action is multiplication is by λ−1. Once again, we have that σ(ψ)=0.

For our final example, we consider the Kosniowski generators of Section 3.3. Let p be an odd prime and j Z, 1 j (p 1)/2. A generator of the action on the ∈ ≤ ≤ − 134 Kosniowski surface K was denoted κ : K K . This map has q + 1 fixed jp jp jp → jp points. There are q fixed points that lie on the disks of the carousel and one fixed point that comes from the disk that is attached to the boundary of the carousel. The action at each of the q fixed points on the carousel is multiplication by λ. A simple combinatorial argument suffices to show that the action near the remaining fixed

j point is multiplication by λ . Hence we have that σ(κjp)= qα1 + αj. It is possible to write σ(κ ) as an element of Z[λ]. From there, it is easy to see that σ(κ ) = 0. This jp jp 6 implies that the action on a Kosniowski generator does not extend to the handlebody.

In our first three examples, we saw that when a map ψ : S S had a fixed → point of type λ and σ(ψ) = 0, there was a corresponding fixed point of type λ−1.

Simultaneously, we saw that when σ(ψ) = 0, the action extends to the handlebody.

Is this generally true? What relationship do these two observations have? In this chapter we will prove the surprising result that for periodic maps ψ : S S of prime → order, σ(ψ) = 0 if and only if the Zp-action generated by ψ extends to the boundary.

The proof extends directly from the fact that if σ(ψ) = 0, then there is a one-to-one correspondence between ﬁxed points of type λj and λ−j.

4.2 Rational Linear Independence and the αj/p’s

From the previous section, we have that in the case of Zp-actions on surfaces, the equivariant signature of a generator is just a linear combination of algebraic numbers

135 of the form: λj +1 α = j/p λj 1 − where λ = e2πi/p. Let µ =(p 1)/2. In this section, we aim to show that the set: −

α , α , α , ...,α (4.3) { 1/p 2/p 3/p µ/p} is a linearly independent set over Q. The proof is very technical and requires us to consider some nontrivial algebraic and analytic number theory. The idea, however, is quite simple. Note that the set:

λ λ−1, λ2 λ−2,...,λµ λ−µ { − − − } is linearly independent over Q because the set λ,..., λp−1 is linearly independent { } over Q. To obtain the linear independence of 4.3, we will ﬁnd a matrix B that solves the following equation:

−1 α1/p b1 1 b1 2 b1 3 ... b1 µ λ λ 2− −2 α2/p b2 1 b2 2 b2 3 ... b2 µ λ λ      3 − −3  α3/p = b3 1 b3 2 b3 3 ... b3 µ λ λ (4.4)  .   . . . . .   −.   .   ......   .       µ −µ   αµ/p   bµ 1 bµ 2 bµ 3 ... bµ µ   λ λ       −       

The determinant of B can be found in terms of odd Dirichlet characters. We then use the functional equation for Dirichlet L-functions to show that this determinant cannot vanish. This implies that the matrix B is just a change of basis matrix and hence, 4.3 is in fact linearly independent over Q. The following lemma shows how to

ﬁnd the entries in the matrix B.

136 Lemma 64. Using the notation of this chapter, we have:

µ λj +1 1 α = = (2k p)(λjk λ−jk) (4.5) j/p λj 1 p − − − Xk=1 Proof. First note that if x C, we have the famous theorem: ∈ µ xµ+1 1 xk = − 1 (4.6) x 1 − Xk=1 − Using this equation, we can ﬁnd a closed form expression of the following sum:

µ µ kxk = x kxk−1 (4.7) Xk=1 Xk=1 = x(1+2x +3x2 + . . . + µxµ−1) (4.8) d = x [x + x2 + . . . + xµ] (4.9) dx d xµ+1 1 = x − 1 (4.10) dx x 1 − − µxµ+1 (µ + 1)xµ = x − (4.11) (x 1)2 − Now, the sum on the right-hand side of 4.5 can be written in the following way:

µ µ µ (2k p)(λjk λ−jk) = 2 kλjk p λjk − − − k=1 k=1 k=1 X Xµ Xµ 2 kλ−jk + p λ−jk − Xk=1 Xk=1 Applying equations 4.11 and 4.6 with x = λj and x = λ−j gives:

µ λj +1 (2k p)(λjk λ−jk)= p − − λj 1 Xk=1 −

We deﬁne the matrix B to be the µ µ matrix over Q from equation 4.4 whose × entries are determined by equation 4.5. Our next task is to compute the determinant of B. First, we need some deﬁnitions.

137 A Dirichlet character modulo m is a group homomorphism f : (Z )× C×. m → × Here (Zm) denotes the multiplicative group of invertible elements modulo m and

C× denotes the multiplicative group of nonzero complex numbers. It is clear that the image of Dirichlet character modulo m is contained in S1. It is also clear that f( 1) = 1. A Dirichlet character is said to be odd if f( 1) = 1 and even if − ± − − f( 1)=1. If f(g)=1 for all g (Z )×, then f is said to be principal. − ∈ m Now, suppose that f is a Dirichlet character modulo n and m is divisible by n.

The natural map (Z )× (Z )× can be used to deﬁne a Dirichlet character modulo m → n m. In fact, we just take the composition:

f (Z )× (Z )× C× m → n −→

Let f be a Dirichlet character modulo m. The smallest n m such that there exists a | homomorphism α : (Z )× (Z )× and a Dirichlet character f ′ : (Z )× C× such m → n n → that the following diagram commutes:

× (Zm) FF FF f α FF FF F# × × (Zn) / C f ′ is called the conductor of f. If a Dirichlet character modulo m does not factor through

× some (Zn) (i.e. the conductor of f is m), then f is said to be primitive. Generally when one speaks about Dirichlet characters modulo m, it is assumed, without loss of generality, that the conductor is m. This convention will be followed here. All

Dirichlet characters will be assumed to be primitive.

It is useful to extend a Dirichlet character f to an arithmetical function χ = χf :

Z C. If k Z, let k¯ Z denote its residue class modulo m. The function χ is → ∈ ∈ m 138 deﬁned to be: f(k¯) if gcd(m, k)=1 χ(k)= χf (k)=    0 ifgcd(m, k) > 1  If f is a Dirichlet character, then χf is completely multiplicative, i.e. χ(ab)= χ(a)χ(b) for all a, b Z. Moreover, χ is periodic with period m. If a function χ : Z C ∈ → is completely multiplicative and periodic with period m, it is also true that χ is the extension of a Dirichlet character. Hence, we will use the term Dirichlet character modulo m to refer to either of these situations.

The following result, due to Ewing [13], computes the determinant of B in terms of the odd Dirichlet characters.

Theorem 65 (Ewing). Using the notation of this section, we have:

p−1 1 det(B)= µ kχ(k) p odd χY Xk=1 Proof. See [13], (3.2).

To prove the independence of 4.3, we need only show that p−1 kχ(k) = 0 for k=1 6 odd characters χ modulo p. This result will eventually followP from the functional equation for Dirichlet L-functions. Before we discuss this, we will brieﬂy review some of the background material necessary to understand the functional equation.

Let χ : Z C be a Dirichlet character modulo m. Write s C as s = σ + it → ∈ where σ, t R. For all s C with σ > 1, we can deﬁne the Dirichlet L function to ∈ ∈ − be: ∞ χ(k) L(s, χ)= (4.12) ks Xk=1

139 This series converges absolutely for s with σ > 1. There are strong connections between this series and the generalized Riemann ζ-function. In fact, there is a formula that explicitly relates the two. Using this formula, it can be shown that if χ is nonprincipal, then L(s, χ) can be analytically continued to an entire function of s. If the reader would like to see the extremely complicated details of the analytic continuation, they are referred to [1]. The following theorem gives some of the properties of

Dirichlet L-functions that will be used in this section.

Theorem 66. Let χ : Z C× be a Dirichlet character modulo m. Suppose that χ is → not the extension of the trivial homomorphism (Z )× C×. Then: m →

1. L(0, χ)= (1/m) m kχ(k). − k=1 P 2. L(1, χ) =0 6

Proof. The ﬁrst fact can be easily proved from the relationship between the the

Dirichlet L-function of χ and the generalized Riemann ζ-function. See [1], (12.13) for more details. The second fact is quite diﬃcult to prove. The best reference seems to be [40], (4.4).

Another important function associated to Dirichlet characters is the Gauss sum.

Let χ be a Dirichlet character modulo m and n Z. Then: ∈ m G(n, χ)= χ(k)e2πikn/m Xk=1 is called the Gauss sum associated to χ.

To use the functional equation for Dirichlet L-functions, we will need to know a certain nonvanishing result about Gauss sums in the case where m is an odd prime.

140 This result follows from a separability condition on G(n, χ). Let m> 1 be an integer and letχ ¯ denote the Dirichlet character modulo m obtained from χ by complex conjugation. The Gauss sum associated to χ is said to be separable at n if:

G(n, χ)=χ ¯(n)G(1, χ)

It can be shown that if χ is a Dirichlet character modulo m and gcd(n, m) = 1, then

G(n, χ) is separable at n(see [1], (8.5)). The following theorem gives a very easy test for separability at every n Z. ∈

Lemma 67. Let χ be a Dirichlet character modulo m. Then the Gauss sum associated to χ is separable at every n Z if and only if G(n, χ)=0 for all n with gcd(n, m) > 1. ∈

Proof. By deﬁnition,χ ¯(n)=0 for all n such that gcd(n, m) > 1. Hence, separability implies that G(n, χ) = 0. Conversely, if gcd(n, m) > 1 and G(n, χ) = 0, it is equal to

χ¯(n)G(1, χ)=0.

As mentioned before, a Gauss sum that satisﬁes separability at every n gives a nonvanishing result about Gauss sums. For a proof of the following result, the reader is referred to [1], (8.11).

Theorem 68. If χ is a Dirichlet character modulo m and G(n, χ) is separable at every n, then:

G(1, χ) = √m =0 | | ± 6

Corollary 69. If χ is a nonprincipal Dirichlet character modulo an odd prime p, then

G(1, χ) =0. 6

141 Proof. By Lemma 67, we need only show that G(n, χ) = 0 for all n such that gcd(n, p)= p. Suppose n = qp for some q Z. Then: ∈ p G(n, χ)= G(qp,χ) = χ(k)e2πikqp/p k=1 Xp = χ(k) Xk=1 It is well-known that this last sum is zero. A well-known proof is as follows. Let A be

× the sum and G =(Zp) . Since χ is nonprincipal, there exists a nonprincipal Dirichlet character f : G C×. Hence, there is a g G such that f(g) = 1. Since χ(k)=0 → ∈ 6 for all k such that gcd(k, n) > 1, we have:

p A = χ(k) = f(x) Xk=1 Xx∈G = f(gx) x∈G X = f(g) f(x) x∈G X = f(g)A

Since f(g) = 1, we must have that A = 0. Hence, G(n, χ) is separable at every n and 6 we conclude that G(1, χ) = √p = 0. | | ± 6

Theorem 70 (The Functional Equation for Dirichlet L-functions). Let Γ denote the

Gamma function. If χ is a primitive Dirichlet character modulo m, then

ms−1Γ(s) L(1 s, χ)= e−πis/2 + χ( 1)eπis/2 G(1, χ)L(s, χ¯) − (2π)s − for all s C. ∈

Proof. See [1], (12.11).

142 Finally, we are positioned to prove the main theorem of this section:

Theorem 71. The set α , α , α , ...,α is linearly independent over Q. { 1/p 2/p 3/p µ/p}

Proof. Theorems 65 and 66 imply that showing det(B) = 0 is equivalent to showing 6 that: p−1 p kχ(k)= kχ(k)= pL(0, χ) =0 6 Xk=1 Xk=1 for all odd Dirichlet characters χ modulo p. Note that an odd Dirichlet character modulo p is necessarily nonprincipal and primitive. Using the fact that χ is odd, we have by Theorem 70 that,

Γ(1) L(0, χ)= i G(1, χ)L(1, χ¯) − π

Now, Γ(1) = 1, G(1, χ) = 0 by Theorem 69, and L(1, χ¯) = 0 by Theorem 66. Hence, 6 6 L(0, χ) = 0 and we have that det(B) = 0. 6 6

Corollary 72. If p is an odd prime, then TFAE:

1. α1/p,...,αµ/p is linearly independent over Q. 2. p−1 kχ(k) =0 for odd χ. k=1 6 P 3. L(1, χ) =0 for odd χ. 6

Hence, an elementary proof that 4.3 is linearly independent would give an elementary proof of (2). However, no elementary proof of (2) is known (see [40], pp.

38).

143 4.3 Some Geometry, Topology, and Bordism

We wish to apply the algebraic number theory of the preceding section to the problem of determining when smooth Zp-actions on a surface extend to to the handlebody.

To do this, we must closely investigate how the action near the fixed points (of the action) determines the global action. The major theorem of this section roughly says that if the action near the fixed points of two Zm-manifolds (m odd) are the same, then their disjoint union is cobordant to a Zm-manifold with no fixed points. In some situations, this means that the local action determines the global action up to an equivariant boundary. Before this theorem is proved, we must discuss some rather complicated constructions. First, we look at a method that will allow us to extend the differentiable structure of two smooth Zm-manifolds to a differentiable structure on an identification space of these manifolds so that the Zm-action on the resulting space is smooth. The identification space we will use in the main theorem comes from tubular neighborhoods of the stationary point set. In the second section, we discuss the special kind of tubular neighborhood that we will need for the main theorem.

Finally, the main theorem is stated and proved in the ﬁnal subsection.

4.3.1 Equivariantly Straightening the Angle

Let B be a k-dimensional topological manifold and g : B B a homeomorphism → of M. Let M k−2 be a submanifold of B, invariant under g with the subspace topol-

144 ogy and suppose that M k−2 is known to have a differentiable structure such that g M : M M is a diffeomorphism. Moreover, we will suppose that B M has a | → \ differentiable structure with respect to which g (B M) is a diffeomorphism. Under | \ certain conditions, it can be shown that B has a differentiable structure such that g : B B is a diffeomorphism. The technique is an adaptation of “straightening → the angle”. Although the technique works for general g : B B, we will be most → concerned with the case that g : B B has finite order. In this case, the tech- → nique gives a differentiable structure on B such that G = acts as a group of diffeomorphisms.

Let R+ be the set of nonnegative real numbers with the subspace topology. Now, suppose that there is an open neighborhood U of M in B and a homeomorphism

Φ : U M R R such that the following conditions are satisﬁed: → × + × +

1. If x M, then Φ(x)=(x, 0, 0). ∈

2. Φ: U M (M R R ) (M 0 0) is a diﬀeomorphism. \ → × + × + \ × ×

3. U is g-invariant.

4. If Φ(x)=(y,s,t), then Φ(gx)=(gy,s,t).

A picture of this situation is given in Figure 9. Here we have M S1 and the action ≈ on B nearby M is just rotation by 2π/m about an axis through the center.

Now, let τ : R R R R be any homeomorphism which is a diffeomorphism + × + → × + from R R (0, 0) onto R R (0, 0). This homeomorphism will be fixed and it + × +\ × +\ will partially determine the final differentiable structure on B. Following the example

145 Figure 9: A bent angle to be straightened equivariantly. Rotation M U

R + x R + B

of Conner and Floyd [9], we might choose a polar coordinate system and deﬁne

τ(r, θ)=(r, 2θ) (see Figure 10).

Figure 10: Unbending the angle in R R . + × + 2θ (r, θ ) (r, ) τ

θ 2θ

From this map we can deﬁne a map

τ ′ : M R R M R R × + × + → × × + by τ ′(x, s, t)=(x, τ(s, t)), where the s and t are whatever coordinates you have chosen for R R . Then: + × + τ ′Φ : U M R R → × × + is a homeomorphism. Since M has a differentiable structure and R R has a × + differentiable structure, the image of τ ′Φ has a differentiable structure induced by the product.

The differentiable structure induced by the product also induces, via τ ′Φ a differentiable structure on all of U. More precisely, if (U ,µ ) is a coordinate system for { α α } 146 M R R , then the maximal collection containing ((τ ′Φ)−1(U ),µ τ ′Φ) can be × × + { α α } shown to be a differentiable structure on U. Moreover, τ ′Φ is a diffeomorphism with respect to this differentiable structure.

We have shown that U has a differentiable structure and B M has differentiable \ structure by hypothesis. Now, U (B M) = U M. The differentiable structure ∩ \ \ of U M inherited from U is the same as the differentiable structure inherited from \ B M because τ ′ and Φ are given to be diffeomorphisms from U M and M R \ \ × + × R (M 0 0), respectively. Thus, there is a unique differentiable structure on +\ × × B = U (B M) that induces the differentiable structures on U and (B M). Now we ∪ \ \ have the differentiable structure on B which is said to be obtained by straightening the angle.

The diffeomorphism g M : M M induces a map Γ : M (R R ) M | → × × + → × (R R ) defined by Γ = g id. It is clear that Γ is a diffeomorphism. To show that × + × g is a diffeomorphism, we need only show that g is a diffeomorphism on U, since we already know the story elsewhere. Consider the following diagram:

Γ M R R+ / M R R+ × O × × O × τ ′ τ ′

M R+ R+ M R+ R+ × O × × O × Φ Φ U / U The compositions in the vertical arrows are diﬀeomorphisms. The bottom horizontal arrow is the map (τ ′Φ)−1Γ(τ ′Φ). By property (4) above, this map must be g U. Thus, | g is a diﬀeomorphism.

147 4.3.2 Tubular Neighborhoods of G-Manifolds

In this subsection, we discuss the general properties of tubular neighborhoods in

G-manifolds. The goal is to construct a particular kind of tubular neighborhood that we will use to assign the diﬀerentiable structure to an identiﬁcation space. This application of our construction is covered in the next section. We will not bother to prove many of the results of this section in their full generality. The statements are designed to be as elementary as possible and hence contain little more generality than is necessary for our applications to handlebodies.

Let B be a orientable, closed, connected, smooth manifold of dimension 2k and let G be a finite group of orientation preserving diffeomorphisms of B. Let <,> be some Riemannian metric on B. If g G, then the differential dg : T B T B acts ∈ → differentiably on T B. We will call this action of G via the differentials the G-action on T B. Define a positive definite inner product on <,> : T B T B R by: G × →

1 = < dg(v),dg(w) > G G | | Xg∈G This inner product is invariant under the G-action on T B and is also a Riemannian metric on B. Let F B be a closed submanifold of B of dimension l, with ι : F B ⊂ → the inclusion map. Let p F . The normal bundle ν of F in B is deﬁned to be: ∈

ν (A)= v T B :< v,dι(w) > = 0for all w T F p { ∈ p G ∈ p}

ν(F )= νp(F ) p[∈F A ﬁxed point of a smooth group action on a smooth manifold is a point that is ﬁxed

148 by every element of the group. The following theorem, due to Bochner, can be found in [32], pp. 206.

Theorem 73 (Bochner, 1945). Let G be a compact group of C∞ transformations of a smooth manifold B. Then in a neighborhood of a ﬁxed point, there is an admissible coordinate chart on which all the transformations are linear.

Lemma 74. Let F be the set of ﬁxed points of a ﬁnite group G acting smoothly on a smooth compact manifold B of dimension n. Then F is a compact submanifold of B.

Proof. (Sketch) Since every element of G is continuous, the limit of ﬁxed points is again a ﬁxed point and hence, F is closed. Since B is compact, F is also compact.

Now, let x F . By the above theorem, there is an admissible coordinate chart B ∈ x on which all the transformations are linear. Then we can think of x, locally, as an eigenvector of every g G with associated eigenvalue 1. If V is the λ = 1 eigenspace ∈ g of g in this coordinate chart, U = V is a linear subspace of some dimension n . x ∩g∈G g x For 1 i n, let F = U . It is clear that F F = for i = j. Also, each U ≤ ≤ i ∪nx=i x i ∩ j ∅ 6 x ∞ is seen to be an admissible C -coordinate chart of dimension nx. Then each Fi is an i-dimensional submanifold of F . Note that each Fi need not be connected.

Proposition 75. Let g G and let F be the set of ﬁxed points of G. Let ι : F B ∈ → denote the inclusion map. Then:

1. dg dι = dι ◦

2. For all p F , dg ν (F ) : ν (F ) ν (F ) is an isomorphism on each ﬁber. ∈ | p p → p

3. For all p F , the dimension of the λ =1 eigenspace of dg ν (F ) is 0. ∈ p| p 149 Proof. Let p F . Since F is a smooth submanifold and ι : F B is smooth, ∈ → (g ι) = ι is smooth. Then by the functorial properties of d, we must have that ◦ (dg dι )= dι . This establishes the ﬁrst fact. p ◦ p p Certainly, dg : T B T B is an isomorphism. Let W = dι (T F ). Then p p → p p p ⊥ ⊥ ⊥ W = νp(A) and thus dgp(νp(F ))=(dgp(W )) =(dgp(dιp(T Fp))) = νp(F ).

The last statement is often stated in the literature, but its proof is always omitted.

It seems wise to include it here. We follow Spivak’s[37] overall treatment and notation of geodesics for this proof . Suppose that v is in νp(F ) and that dg(v)= v. Using the properties of geodesics, we know that there is a neighborhood U of p and a number

ǫ > 0 such that if q is in U and w is in T B with w < ǫ, then there is a unique q k kG geodesic γ :( 2, 2) B such that γ (0) = q and: w − → w dγ w = w dt t=0

Now, if dg(v)= v, then any linear multiple of v is also in the 1-eigenspace. Thus, we can assume that v < ǫ. k kG Recall that a geodesic, γ, is deﬁned to be a critical point of the energy function:

1 b dγ dγ , dt 2 dt dt Za G Since the norm is dg-invariant, the chain rule tells us that the energy of γ is the same as the energy of the path g(γ). Moreover, given any variation of γ, g composed with that variation is also a variation with the same energy. Thus, it must be true that g(γ) is also a critical point of the energy function with:

dg(γ) = v dt t=0

150 Since there is a uniqueness property on U, it must be that g(γ(t)) = γ(t) for every t.

Hence, the curve γ(t) is in the ﬁxed point set. Thus, since v is tangent to γ at t = 0, v is in the tangent bundle to the ﬁxed point set. Since v is also in the normal bundle, we must have that v = 0.

We will restrict our attention to the case where G is odd. For the moment, ﬁx a | | g G and let F be the set of ﬁxed points of G. By the above, it is clear that vectors ∈ tangent to F are in the λ = 1 eigenspace of dgp. The following lemma shows that dgp nicely decomposes T Bp into vectors tangent to the stationary point set and vectors that are “rotated” by dgp.

Lemma 76. Let p F . Then there is a basis of ν (F ) such that dg SO(2j). ∈ p p ∈

Moreover, with respect to this basis, there exist real numbers θ1,...,θj such that dgp is: cos θ sin θ cos θ sin θ diag 1 1 ,..., j j sin θ cos θ sin θ cos θ − 1 1 − j j m m Proof. Since g = 1, dgp = 1 and hence det(dgp) = 1. Also, dgp is preserved by the positive deﬁnite inner product <,> . Thus, by deﬁnition of SO(2j), dg G p ∈ SO(2j). It is well known that SO(2j) contains only the matrices which have block diagonal decompositions consisting of 2 2 rotations and blocks of the form ( 1). × ± By Proposition 75 and the fact that m is odd, we cannot have any blocks of the form

( 1). Thus, the existence of θ ,...,θ is necessary. ± 1 j

This tells us that at a ﬁxed point, the group G acts on νp(F ) as a subgroup of

SO(2j). Once we have discussed tubular neighborhoods, we will return to the topic of how G acts on the normal bundle.

151 Let F be any submanifold of B. A tubular neighborhood of F in B is a bundle

τ = (U, F, π), where U is an open neighborhood of F in B such the the 0 section − from F U is the inclusion. The map π is just the projection map. The following → theorem is well-known.

Theorem 77. Let F be any compact submanifold of B. Then F has a tubular neighborhood in B and the exponential map exp : ν τ is an equivalence of bundles. →

Proof. See [37].

In the case where we have a ﬁnite group G acting smoothly on a smooth manifold

B with ﬁxed point set F , the Riemannian metric on B is chosen so as to be G- invariant. In this case, the diﬀerential map commutes with the exponential map in the sense that exp(dg v) = g(exp(v)). To see this, let p F , let v ν (F ), and p ∈ ∈ p γ : [0, 1] B a geodesic with respect to the Riemannian metric <,> such that → G γ(0) = p and dγ/dt (t =0) = v. Since <,> is G invariant, g γ is a geodesic for | G − ◦ any g G. Clearly, g γ(0) = p. By the chain rule: ∈ ◦

dg γ ◦ = dg (v) dt p t=0

Thus, exp(dg (v))=(g γ)(1) = g(γ(1)) = g(exp(v)). This fact will be crucial to the p ◦ main theorem of Section 4.3.3.

Now, for our purposes, we are interested in having a closed tubular neighborhood of the ﬁxed point set. This can be achieved by restricting to the unit disc in the normal bundle and mapping to the tubular neighborhood via the exponential. By

Lemma 76, it is clear that dgp maps the closed unit disk to itself. By the remark of the

152 last paragraph, this means that the exponential map is an equivariant diﬀeomorphism from the unit disk normal bundle to its image.

Since, <,>G is a positive deﬁnite inner product, we can deﬁne a norm on νp(F ) by

v = √ . Let C(r)= v ν (F ) : v = r where r R . If v C(r), k kG G { ∈ p k kG } ∈ + ∈ 2 then < dgpv,dgpv >G=G= r and hence dgp(C(r)) = C(r). Deﬁne:

1 3 T ′ = v ν (F ) : < v < p ∈ p 4 k kG 4 and let M ′ = v ν (F ) : v = 1/2 . By the argument of this paragraph, p { ∈ p k kG } dg (T ′) = T ′ and it follows that T ′ and exp(T ′) have smooth G actions. We will p p p p p − need the following fact in the next section.

′ Lemma 78. With the notations of the preceding paragraph, Tp is equivariantly diffeomorphic to M ′ R, where the smooth G-action on M ′ R acts trivially on R. p × p ×

Proof. Define f : T ′ M ′ (1/4, 3/4) by: p → p × v f(v)= , v 2 v k kG k kG This is clearly well-defined, continuous, and C∞. Moreover, f −1 : M ′ (1/4, 3/4) p × → T ′ defined by f −1(v,r)=2rv is truly the inverse of f. Thus, T ′ M ′ (1/4, 3/4) in p p ≈ p × the C∞ category. Note that G acts on M ′ (1/4, 3/4) by g (v, t)=(dg v, t). From p × • p this we see that the G action on T ′ is equivariantly diffeomorphic to the G action − p − on M ′ (1/4, 3/4). Now, define h : M ′ (1/4, 3/4) M ′ R by: p × p × → p × t 1/2 h(v, t)= v, − −(t 3/4)(t 1/4) − − Clearly, h is well-defined, continuous, C∞ and a diffeomorphism. A G-action on

153 M R is simply g (v, t)=(dg v, t). Then T ′, and exp(T ′) are both equivariantly × • p p p diﬀeomorphic to M ′ R. p ×

Now we are ready to construct the special kind of tubular neighborhood that we will use in the next section. We will call it a closed tubular neighborhood with two- sided collar. Let F be the ﬁxed point set of a ﬁnite odd order group G acting on a smooth manifold B. We know that F has a tubular neighborhood τ =(U, F, π) that is equivalent via the exponential map to the normal bundle. This equivalence gives rise to a unit normal bundle consisting of all the vectors in ν(F ) of length less than or equal to 1. The image of this is a closed neighborhood of F whose interior is a tubular neighborhood of F .

Define M ′ = v ν (F ) : v = 1/2 , M = exp(M ′ ), M ′ = M ′ and p { ∈ p k kG } p p ∪p∈F p ′ ∞ M = exp(M ). Since the G-invariant positive definite inner product <,>G is C and exp is a diffeomorphism, M is a compact submanifold of B. By the above remark,

M has tubular neighborhood T which is equivariantly diﬀeomorphic to M ′ R and × hence, M R. The space T is called a two-sided collar of M. × Now, let V ′ = v ν (F ) : v 1/2 , V = exp(V ′), and V = V . Since p { ∈ p k k ≤ } p p ∪p∈F p ∞ exp is an equivariant diﬀeomorphism and <,>G is a C Riemannian metric, V is also a closed, G invariant, tubular neighborhood of F with boundary M. Given V − and T for F , we call the pair a closed tubular neighborhood V of F with two-sided collar T . The following lemma summarizes the above. A schematic of a case where

B is a 3-manifold and F is a 1-manifold are given in Figure 11.

154 Figure 11: A tubular neighborhood with two-sided collar. F

V= T=

Lemma 79. Let G be a cyclic group of odd order acting smoothly on a compact smooth manifold B. Let F be the set of ﬁxed points of G. Then there exists a closed tubular neighborhood V of F with two sided collar T satisfying the following:

1. V and T are G invariant, −

2. There is a diﬀeomorphism ω : T (∂V )n−1 R such that ω(∂V )= ∂V 0. → × ×

3. The G-action on T is equivariantly diﬀeomorphic to the G-action on ∂V R, × where the G-action on the latter is given by:

g (x, r)=((g ∂V )(x),r) • |

4.3.3 An Equivariant Bordism Theorem

In this section, we prove the fundamental geometric result for the main theorem.

Suppose the B1 and B2 are two G-manifolds. The geometric result roughly states that if the actions on the normal bundles to the fixed point sets of B1 and B2 are the equivalent, then the B ( B ) is bordant to some manifold V admitting a fixed 1 ∪ − 2 155 point free action Let G be a topological group. Recall that a G-vector bundle is a vector bundle (E, X, π) such that the following three conditions are satisfied:

1. E and X are both G-spaces.

2. π is a G-map.

3. For every x X and g G, the linear transformation g : π−1(x) π−1(gx) ∈ ∈ → deﬁned by g(e)= g e is an isomorphism •

If B is a smooth n-manifold and a topological group G acts on B as a group of self-diffeomorphisms, the group of differentials of G forms a G-action on the tangent bundle. All three of the above conditions are satisfied, and hence, the tangent bundle to B is a G-vector bundle. From the work in the previous section, we also see that if F is the set of fixed points of the G-action on B, then the normal bundle to F is also a G-vector bundle. This is the G-vector bundle which will be considered in the theorem of this section.

Let (E ,X , π ) and (E ,X , π ) be G-vector bundles. Suppose that H : E E 1 1 1 2 2 2 1 → 2 and h : X X are G-maps such that the following diagram commutes: 1 → 2

H E1 / E2

π1 π2 X / X 1 h 2

If such a pair exists and the restriction of H to each ﬁber is linear, it is called a

G-vector bundle homomorphism. We will say that the G-vector bundles (E1,X1, π1) and (E ,X , π ) are equivariantly equivalent if there are bijective maps H : E E 2 2 2 1 → 2 and h : X X such that the pair form a G-vector bundle homomorphism. If the 1 → 2 156 G-vector bundles are in the category of smooth G manifolds, then we will addition- − ally require the pair of maps to be C∞ diffeomorphisms. If all of the hypotheses are satisfied in this case, we will say that the G-vector bundles are equivariantly diffeomorphic.

The theorem of this subsection follows this paragraph. It is a partial generalization of a theorem found in the tour de force of Conner and Floyd [9]. We show here that with a stronger hypothesis, their bordism result extends to an equivariant bordism result. In the following section, we show how this cooperates with the G- signature theorem to obtain new results on which G-actions on surfaces extend to the handlebody it bounds.

Theorem 80. Let G be a ﬁnite cyclic group of odd order that acts smoothly on two

n n closed smooth orientable manifolds B1 and B2 . Let <,>1 and <,>2 be G-invariant

Riemannian metrics of B1 and B2, respectively. Suppose that the nonempty sets F1 and F2 of fixed points of G on B1 and B2 have equivariantly diffeomorphic normal bundles defined under these metrics. Then there exists a orientable smooth compact manifold Cn+1 and a closed orientable smooth manifold Zn such that:

1. G acts smoothly on Cn+1,

2. G acts smoothly on Zn and has no ﬁxed points,

3. The boundary of Cn+1 is Bn Bn Zn, and 1 ∪ 2 ∪

n+1 4. The restriction of the G-action on C to each of B1 and B2 is the given

G-action.

157 Figure 12: B I with labeled two-sided collar. i × T=i M i V= i Bi 1 Fi

Bi 0

Proof. Let H : ν(F ) ν(F ), f : F F be the given equivariant equivalence. 1 → 2 1 → 2 From the construction above we have:

(M ′ ) = v ν(F ) : v =1/2 1 p { ∈ 1 k k1 }

(V ′) = v ν(F ) : v 1/2 1 p { ∈ 1 k k1 ≤ }

∞ ′ ′ The C map H sends (M1)p and (V1 )p to the ﬁber (T B2)p monomorphically. Let

M ′ = (M ′ ) , V ′ = (V ′) , M ′ = H(M ′ ), and V ′ = H(V ′). The idea is 1 ∪p∈F1 1 p 1 ∪p∈F1 1 p 2 1 2 1 to create a closed tubular neighborhood of F2 in B2 that behaves identically to the

′ ′ closed tubular neighborhood of F1 in B1 under the action of G. Since M1 and V1 are

G-invariant, M ′ and V ′ must also be G invariant. Also, since H is an isomorphism 2 2 − ′ on each fiber and F1 gets mapped diffeomorphically onto F2, it must be that exp(V2 ) is a closed G invariant tubular neighborhood of F . Then if exp and exp are the − 2 1 2 ′ ′ exponential maps on ν(F1) and ν(F2), we have that V1 = exp1(V1 ) and V2 = exp2(V2 ) are equivariantly diffeomorphic closed tubular neighborhoods of F1 and F2. This diffeomorphism is precisely given by ϕ = exp H exp−1. 2 ◦ ◦ 1 It is clear that the same thing can be done to construct a closed tubular neighborhood V1 of F1 with two-sided collar T1 and it’s equivariantly diffeomorphic copy V2

158 with two-sided collar T of F B . Their preimages via the respective exponential 2 2 ⊂ 2 maps will be denoted each time with the embellishment ′. Note that the “slice” of the normal bundle at a point p gets mapped to a equivariantly diﬀeomorphic “slice” in the normal bundle at h(p). It is this condition that allows us to verify the third criterion when we equivariantly “straighten the angle”. Also note that for i = 1, 2,

M has dimension n 1 and both V and T have dimension n. A diagram of the case i − i i where n = 2 is given in Figure 12.

Figure 13: Pasting V 1 with V 2 along the map ϕ. 1 × 2 ×

T= i ϕ V=i

Deﬁne C = B [0, 1] and C = ( B ) [0, 1], where B simply means B 1 1 × 2 − 2 × − 2 2 with the opposite orientation. We deﬁne an (n + 1)-manifold C by pasting C1 and

C2 along V1 and V2 as follows. An equivalence relation on the disjoint union of C1 and C is given by: (x, 1) V 1 is related to (ϕ(x), 0) V 0 (see Figure 13). 2 ∈ 1 × ∈ 2 × With our reversed orientation, the identiﬁcation space is now an orientable topological

(n + 1) manifold with boundary. We need to show that C is smooth and admits a − smooth G-action.

159 First we will describe the G-action on C. Since T1 and T2 are equivariantly diﬀeomorphic, we have the following data:

1. A group G of diﬀeomorphisms of T and an isomorphism ψ : G G, for i i i i → i =1, 2.

2. The map ψ : G G deﬁned by ψ = ψ−1ψ is an isomorphism. 1 → 2 2 1

In G G , the set X of elements of the form (g , ψ(g )) is a subgroup. Now, the 1 × 2 1 1 map α : G X deﬁned by sending g (ψ−1(g), ψ(ψ−1(g))) is an isomorphism. We → → 1 1 deﬁne the G action on the disjoint union of C and C by: − 1 2

−1 (ψ1 (g)(x),s) , if x C1  ∈  g(x, s)=    (ψ(ψ−1(g))(x),s) , if x C  1 ∈ 2   Then by the remarks of this paragraph, we have a continuous G-action on the disjoint union of C1 and C2. By deﬁnition of ϕ and the fact that the exponential maps

“commute” with the action (see Section 4.3.2), we have that the action on the disjoint union projects to an action on the quotient space C.

Now we show that C has a differentiable structure. For i = 1, 2, we have T i ≈ ∂V R. In C = B [0, 1], ∂V has dimension n 1=(n+1) 2 and a neighborhood i × 1 1 × 1 − − equivariantly diffeomorphic to ∂V R (1/2, 1] ∂V R R . Similarly, in 1 × × ≈ 1 × × + C = B [0, 1], ∂V has dimension n 1=(n+1) 2 and a neighborhood equivariantly 2 2× 2 − − diffeomorphic to ∂V R [0, 1/2) ∂V R R . 2 × × ≈ 2 × × + It is our goal to use these diffeomorphisms to define a differentiable structure on the identification space. First note that through the diffeomorphisms, we must

160 Figure 14: Assigning C∞ structure to C.

C2 deform M 2 ϕ ϕ part identifiedby M1 C 1 straigthen angle here

have that if x ∂V and s R 0 , (x, s, 1) is identified with (ϕ(x), s, 0) ∈ 1 ∈ − ∪{ } ∈ ∂V (R 0 ) 0. For positive values of s, a point is only identified with itself. 2 × − ∪{ } × The differentiable structure on C comes from equivariantly straightening the angle.

First note that G acts ﬁberwise and only on the ﬁrst component of ∂V R R . Let i × × +

M be the equivalence class of points in ∂V1 and ∂V2. We can convince ourselves (see

Figure 14) that, ∂V ∂V in C has a neighborhood equivariantly diffeomorphic to 1 ∼ 2 ∂V R R such that G only acts on the first component. Let g be a generator of 1 × + × + G. Our construction satisfies all 4 conditions (from Section 4.3.1) necessary to assign a differentiable structure on C such that g : C C is a diffeomorphism. Then G → acts as a group of diffeomorphisms on C.

Now, the boundary of C is:

∂C =(B 0) (B 1) ((B int(V )) 1 (B int(V )) 0) 1 × ∪ 2 × ∪ 1 − 1 × ∪ϕ 2 − 2 ×

Since the set of ﬁxed points of G in B and B have been removed, the n manifold 1 2 −

Z = ((B int(V )) 1 (B int(V )) 0) 1 − 1 × ∪ϕ 2 − 2 × has no ﬁxed points. A schematic diagram of this is given in Figure 15.

161 Figure 15: The n-manifold Z is what remains after gluing. B2

ZZZZ VVVi i i

4.4 Applications to Zp-Actions on Surfaces

The main theorem of this section states that a Zp-action on a surface extends to the handlebody if and only if the g-signature vanishes. The crux of the proof is the rational linear independence of the αj/ps. We can use the same argument of the main theorem to get other bordism results about cyclic actions of prime order on surfaces. Also, we can use the independence result to show that that there are no relations amongst the Kosniowski generators in the Zp-bordism ring. First we need the following easily proved lemma.

Theorem 81. Let S be a surface and ψ : S S an orientation preserving dif- → feomorphism of odd prime order p. Suppose that the total slice type of ψ is given by:

(a1, a2,..., a(p−1)/2, a(p+1)/2,..., ap−1)

If σ(ψ)=0, then a = a for all j, 1 j p 1. Moreover, we have that Λ(ψ) 0 j p−j ≤ ≤ − ≡ (mod 2).

162 Proof. Let α = α . Note that for 1 j p 1, α = α . Since σ(ψ) = 0, the i i/p ≤ ≤ − p−j − j G-signature theorem says that:

p−1

0 = ajαj j=1 X (p−1)/2 (p−1)/2

= ajαj + ap−jαp−j j=1 j=1 X X (p−1)/2 = (a a )α j − p−j j j=1 X

Since the set α ,...,α is linearly independent over Q, we must have that { 1 (p−1)/2} a = a for all j, 1 j (p 1)/2. Thus, Λ(ψ)= p a is even. j p−j ≤ ≤ − i=1 i P Before considering the main theorems of this section, we need to delve into the language of bordism and equivariant bordism. Let B1 and B2 be two closed oriented smooth k manifolds. B and B are said to be bordant if there is a smooth (k + 1)- − 1 2 manifold with boundary B ( B ), where B simply means B with orientation 1 ∪ − 2 − 2 2 opposite that of the given one. This bordism relation can be shown to be an equivalence relation. The set of equivalence classes under the operation of disjoint union forms an abelian group denoted by Ωk. The additive identity of this abelian group is the equivalence class of closed oriented k-manifolds that are boundaries of smooth

∞ oriented (k + 1)-manifolds. Deﬁne Ω∗ = k=0 Ωk. This can be given the structure of

P n n a graded ring where the product of the equivalence classes [B1 ] and [B2 ] is given by the topological product [B B ]. This product is commutative in the graded sense. 1 × 2

Consider Ω2. Since every closed oriented 2-manifold bounds a handlebody, it is clear that Ω2 ∼= 0. This fact will be important in the theorems of this section.

163 Let G be a finite group. Recall that a G-manifold of dimension k is a smooth closed orientable k-manifold B such that the the group G acts on B as a group of orientation preserving diffeomorphisms. A principal oriented G-manifold of dimension k is a closed oriented G-manifold of dimension k such that every non-identity element of G has no fixed points. A principal oriented G-manifold of dimension k is said to be

G-bordant if there is an compact oriented G-manifold of dimension k + 1 such that the restriction of the G-action to ∂B is the original G-manifold B. Two principal oriented G-manifolds B1 and B2 of dimension k are said to be bordant if and only if the principal oriented G-manifold B ( B ) is bordant. This relation can also be 1 ∪ − 2 shown to be an equivalence relation. The principal G-bordism group of dimension k, denoted Ωk(G), is the set of equivalence classes of principle oriented G-manifolds of dimension k under the relation of G-bordism. The operation on this set is deﬁned to be the equivalence class of the disjoint union.

An important group for our work is the reduced G-bordism group. Deﬁne ǫk :

Ω (G) Ω by [B] [B/G], where B/G denotes the orbit space. It can be shown k → k → that this is a well deﬁned group homomorphism. We deﬁne the reduced G-bordism group, denoted Ω˜ k(G), by Ω˜ k(G) = Ker(ǫk). The following proposition, found in [9], will be useful in the future.

Proposition 82. Let G be a ﬁnite group. With the notations deﬁned above, we have the following short exact sequence of abelian groups:

ǫ 0 Ω˜ (G) Ω (G) k Ω 0 → k → k −→ k →

Moreover, the sequence is split-exact. Hence, we have that Ω (G) = Ω Ω˜ (G). k k ⊕ k 164 Proof. The only controversial part of this theorem is that ǫk is surjective. Let B be a k-manifold. Take G copies of B and label them by the elements of the group. | | Let A = B , where the union is disjoint. We define a group action on A as ∪g∈G g follows. Let g, h G. If x B , we denote it by x . Define g x to be x i.e. the ∈ ∈ h h • h gh corresponding point x in Bgh. It is clear that G acts on A as a group of orientation preserving diffeomorphisms. We will show that A/G B. Suppose that x B. Let ≈ ∈ g ,g G be distinct. Note that: i j ∈

(g g−1) x = x j i • gi gj

Hence, the set of x ’s projects to x B and we have that A/G B. Thus, every k- gi ∈ ≈ manifold is the quotient of a G-action on some (not necessarily connected) G-manifold of dimension k. Hence, ǫk is onto.

The map f : Ω Ω (G) deﬁned by [B] [A] (using the notation of the previous k → k → paragraph) can be shown to be a well-deﬁned Z-module homomorphism. Hence, the sequence above is split-exact.

˜ Proposition 83. If k is even, Ωk(Zp) ∼= 0.

Proof. See [9], (34.2).

We are now ready to begin the main theorems of this section. They show to what extent slice types and the equivariant signature determine the Zp-bordism class of a smooth surface admitting a Zp-action. Note that the ﬁrst theorem does not use the independence of the αj/ps. In fact, the g-signature is not mentioned at all.

165 Theorem 84. Let S be a closed connected orientable smooth surface of genus g.

Suppose that S has two smooth Zp-actions generated by diﬀeomorphisms ψ1 and ψ2.

If the total slice type of ψ1 is the same as the total slice type of ψ2, then the actions are equivariantly bordant.

Proof. The sets of fixed points of the two Zp-actions are just the set of fixed points of ψ1 and ψ2. In this case, the normal bundle of each of these sets coincides with the disjoint union of the tangent spaces at each fixed point. If the total slice type contains only zeros, both actions extend to the handlebody and hence are equivariantly bordant

(their diﬀerence bounds a Zp-manifold of dimension 3). Since the slice types are the same and the action is compatible with the complex structure, we can form an equivariant equivalence of the normal bundles simply by sending a point with slice type λk to a point with slice type λk. By the Theorem 80, there exists a smooth orientable 3 manifold C with a smooth Z -action whose boundary is S ( S) Z. − p ∪ − ∪ Since Z is a closed smooth orientable surface with no ﬁxed points, it is in the principle

Z bordism group Ω (Z ). By Propositions 82 and 83, we have that Ω (Z ) = p− 2 p 2 p ∼ Ω Ω˜ (Z ) = 0. Thus, Z is an equivariant boundary. 2 ⊕ 2 p ∼

Let Y be the Zp-manifold of dimension 3 that Z bounds. By the equivariant collaring lemma ([9],(21.2)), there is a neighborhood of Z in Y that is equivariantly diffeomorphic to Z [0, 1], where Z acts only on the first coordinate. There is also × p a neighborhood of Z in C which is equivariantly diffeomorphic to Z [0, 1], where ×

Zp acts only on the ﬁrst coordinate. We glue Y to C via an orientation reversing diﬀeomorphism of Z that commutes with the Zp-action on Z. Give the resulting 3-

166 manifold with boundary the obvious diﬀerentiable structure. The resulting 3-manifold has boundary S ( S) and the restriction of the Z -action to each of the bound- ∪ − p ary components gives the two original actions on S. Hence, they are equivariantly bordant.

Theorem 85. Let S be a surface of genus g and ψ : S S an orientation preserving → diﬀeomorphism of odd prime order p. Then σ(ψ)=0 if and only if the Zp-action on S is an equivariant boundary. In other words, σ(ψ)=0 if and only if there is a Zp-action on the genus g handlebody whose restriction to the boundary gives the

Zp-action generated by ψ.

Proof. By Theorem 11, we know that if the action is an equivariant boundary, then

σ(ψ) = 0. Now, suppose that σ(ψ) = 0. Suppose that the total slice-type of ψ is given by:

(a1, a2,..., a(p−1)/2, a(p+1)/2,..., ap−1)

Then by Theorem 81, we must have that a = a for all j, 1 j p 1. If a =0 j p−j ≤ ≤ − j for all j, then we know by Propositions 82 and 83 that the Zp-action extends to handlebody. Suppose that a = 0 for some j. Then a = 0. To prove the theorem j 6 p−j 6 in this case, we construct a disjoint union of spheres so that a generator of the defined action has the same total slice type as ψ. We then finish off the proof with Theorem

80 in the same way we proved Theorem 84.

Let Q = (p−1)/2 a . We take Q copies of S2 and label them A ,1 j (p 1)/2 j=1 j jk ≤ ≤ − and for eachPj the index k satisfies 1 k a . Let A be the disjoint union of all ≤ ≤ j the Ajk. We define a Zp-action on A as follows. Define a Zp-action on Ajk to be a

167 rotation of size 2πj/p about the z-axis. It is easy to see that the total slice type of this action on A has a 1 in the j-th and (p j)-th positions and zeros elsewhere. jk −

The action on A is simply deﬁned so that the restriction to each component Ajk of

A is just the given action on Ajk. Thus, the total slice type of this action on A is

(a1,...,ap−1)..

In the case of Zp-actions on surfaces, this implies that the normal bundles of the

ﬁxed point sets of A and S are equivariantly diﬀeomorphic. By Theorem 80, there is a smooth Zp-manifold C of dimension 3 with boundary:

S ( A) Z ∪ − ∪

where Z is a smooth Zp-manifold of dimension 2 such that the action has no ﬁxed points. Then by Proposition 82, Proposition 83, and the fact that Ω2 ∼= 0, we have that there is Zp-manifold D1 of dimension 3 with boundary Z such that the restriction of the action to Z is the smooth action given by Theorem 80.

The proof will be completed by plugging in the “holes” of the 3-manifold C.

To visualize this, we will look at the way Z was constructed in Theorem 80. The equivariant diﬀeomorphism between the normal bundles of the ﬁxed point sets of

S and A maps a point with slice type λj to a point with slice type λj. Thus, the diﬀeomorphism can be chosen so that it maps two points with slice type λj and λ−j to the the ﬁxed points of the Zp-action on some sphere Ajk.

We will focus on how this sphere is attached to S in the construction of Z in

Theorem 80. Let z+ be the ﬁxed point on S that gets mapped to the North pole of

Ajk and z− be the ﬁxed point that gets mapped to the South pole. The manifold

168 Figure 16: Construction of Z near two ﬁxed points of ψ.

surface near fixed point

z+ N

S z

surface near fixed point

Z is essentially formed by first deleting Zp-invariant tubular neighborhoods of z+, z−, and the fixed points on Ajk. These spaces are then glued together along the equivariant diffeomorphism of the tubular neighborhoods. It is important to recall that the orientation on Ajk has been reversed and hence the diffeomorphism is now orientation reversing. The identification space is a topological manifold that inherits its differentiable structure from C. See Figure 16 for a picture of Z near z+ and z−.

The surface Z is obtained by performing the procedure described in this paragraph for every Ajk. Note that this argument can also be used to show that Z is connected.

We proceed in the manner of Theorem 84. Using the equivariant collaring lemma

([9],(21.2)), we can paste D1 to C to get a smooth Zp-manifold C1 of dimension 3 with boundary S ( A). On each A , the rotation obviously extends to the 3-ball ∪ − jk 3 B . Let D2 be this three manifold that A equivariantly bounds. Pasting D2 to C1 gives a smooth Zp-manifold of dimension 3 with boundary S. This completes the proof. Lastly, note that this result diﬀers from the last because two actions with the same nonzero signature need not have the same total slice type.

169 As a ﬁnal consequence of our independence result, we show that there are no relations amongst the Kosniowski generators (see Section 3.3) in the Zp-bordism group.

This result appears to be previously unknown.

Theorem 86. Let p be an odd prime. In the oriented Zp-bordism group, there are no relations amongst the Kosniowski generators.

Proof. In Section 3.3, we denoted the Kosniowski surfaces by K ,1 j (p 1)/2. j,p ≤ ≤ −

The speciﬁed generator of the Zp-action was denoted by κj,p. For each j, we will also denote the corresponding element of the Zp-bordism group by κj,p. Suppose there are integers r1, r2,...,rµ such that:

0= r1κ1,p + r2κ2,p + . . . + rµκµ,p

Now, recall that σ(κj,p) = qjα1 + αj, where qj is the unique integer between 1 and p 1 satisfying jq 1 (mod p). Since the equivariant signature of an equivariant − j ≡ − boundary is zero, we can apply the equivariant signature to the above equation to obtain:

0 = r1(q1α1/p + α1/p)+ r2(q2α1/p + α2/p)+ ...rµ(qµα1/p + αµ/p)

= [r1(q1 +1)+ r2q2 + ...rµqµ]α1/p + r2α2/p + . . . + rµαµ/p

Then by Theorem 71, we have that ri = 0 for all i> 1 and r1(q1 +1)+r2q2 +...rµqµ =

0. This implies that r1 = 0 and hence completes the proof.

170 Chapter 5 Summary

5.1 Chapter 2 Summary and Future

The main results of this chapter dealt with periodic Dold sequences. It was shown that a Dold sequence s : N Z is periodic with period m if and only if M (k)=0for → s all but ﬁnitely many k and m = lcm k N : M (k) =0 . This structure of periodic { ∈ s 6 } Dold sequences is important because the only bounded Dold sequences are the periodic ones (see Theorem 22). A possible future project would be to attempt a purely number theoretic characterization of the sequences which are unbounded. Would such a characterization run parallel to some geometrically meaningful invariant? Perhaps a characterization would open up new geometry.

Chapter 2 also discussed a geometric approach to the theory of Dold sequences.

It was shown that all “surface-like” periodic Dold sequences are realized by periodic homeomorphisms from a compact connected two-dimensional simplicial complex.

These results need to be extended to the case in which the M¨obius inversion sequence is negative at some values. Of course, if a number theoretic characterization of Dold sequences is achieved, the geometric approach should also be extended along these lines as well.

As has been already mentioned, the theorems presented here may shed some light on a continuous version of the Shub-Sullivan theorem (see Theorem 28). We showed

171 that the Shub-Sullivan theorem will be valid on ENRs X having the property that for every map f : X X, if M (k) = 0, then f has a periodic point of period k. Here, as → s 6 always, s : N Z is the Lefshetz sequence of f. Fagella and Llibre have shown that → if X is a compact complex manifold and f : X X a holomorphic map, then there → exists an N > 0 such that for all p > N, M (p) = 0 implies that f has a periodic s 6 point of period p. This proves a weaker version of the Shub-Sullivan theorem. It would be interesting to see other kinds of spaces and maps for which this is true.

5.2 Chapter 3 Summary and Future

In Chapter 3, we tackled the problem of determining all periodic Dold sequences that are realized by periodic orientation preserving diﬀeomorphisms on a surface.

The result of this analysis is contained in Theorem 59. Fuchsian groups were used to prove the theorem in the case that the genus of the surface is at least 2. For spheres and tori, it was only necessary to use elementary linear algebra. An alternate approach to this problem was given in the case that the order of the map is an odd prime.

The solution to this realization problem was shown to have applications to the theory of mapping class groups. In particular, we investigated the map Mod(S) → Sp(2g, Z) for a closed orientable surface S of genus g. This map is surjective and the kernel of this map does not contain any elements of finite order. However, there are still some difficulties in dealing with maps of finite order. For example, we showed

172 that there are maps of finite order in Sp(2g, Z) that are not the image of a map of finite order in Mod(S). The problem of differentiating between these fake finite order maps and the real ones can be partially accomplished by determining all of the realized periodic Dold sequences. It was shown that via an integral matrix BG, which is only dependent on the group G, that a periodic Dold sequence can be converted to the multiplicities of the eigenvalues of the induced map. Improvements on this defect detection could be made by including more information about the integral symplectic group. Much is known about these groups and a fruitful investigation is almost certain.

The most obvious generalization of this chapter lies in the Thurston classiﬁcation of mapping classes. What number theoretic properties of Dold sequences, if any, correspond to pseudo-Anosov or reducible maps? And of those, which are realized by pseudo-Anosov or reducible, resp., mapping classes. In this case, it may be ad- vantageous to look at the Nielson number ﬁrst. There are interesting bounds for

Lefshetz numbers in terms of the Nielson number (see [22]) that may help enlighten the remaining cases.

How does one solve the realization problem for closed orientable three-dimensional manifolds? In this case, the ﬁxed point sets are all links. A good place to start would be to consider the paper by Przytycki and Sokolov [33]. They show that all 3 manifolds exhibiting p-fold symmetry (p an odd prime) that have exactly one ﬁxed S1, can be obtained from S3 by integer surgery along a periodic link. A generalization of this result to maps of arbitrary period is due to Sakuma [35]. A disadvantage to looking at Lefshetz numbers of periodic maps on 3 manifolds is that the Euler characteristic is

173 always 0. This suggests that the resulting number theory might not be as interesting.

If this proves to be the case, the generalization to 4-manifolds would certainly be interesting. In this case, a periodic map could have isolated ﬁxed points and linked

ﬁxed surfaces.

Of course, there is no reason not to investigate the Lefshetz sequences of diﬀeo- morphisms on nonorientable surfaces. Also, it might prove interesting to investigate orientation reversing homeomorphisms of surfaces. In this case, the ﬁxed point sets might contain multiple copies of S1.

It would also be useful to establish the invertibility of the matrix BG(see Section

3.2). This appears to be the case in all computational examples that I have considered, but I have been unable to prove it generally.

5.3 Chapter 4 Summary and Future

In Chapter 4, we looked at some number theory associated with the equivariant signature theorem. The Atiyah-Singer g-signature theorem states that for ﬁnite cyclic actions on surfaces, the g-signature is a linear combination of algebraic numbers of the form: λj +1 α = j/m λj 1 − where m is the order of the group and λ = e2πi/m. In the case that m = p, an odd

174 prime, it was shown that the set:

α , α ,...,α { 1/p 2/p µ/p} where µ = (p 1)/2, is linearly independant over Q. This result is equivalent to − nonvanishing of the ﬁrst moment of any odd Dirichlet character modulo p. If an elementary proof of the independence of the αs were known, there would be a corresponding elementary proof of the the nonvanishing result. However, there is no known elementary proof of this fact. Attempting to ﬁnd such a proof would be a potential area of further research.

The last section of Chapter 4 provided some applications of this independence result to the theory of Z -actions on surfaces. It was shown that if ψ Z , ψ = 0 and p ∈ p 6 σ(ψ) = 0, then ψ has an even number of ﬁxed points. This result, along with some general bordism results presented in Section 4.3.3, was used to show that σ(ψ)=0if and only if the Zp-action of ψ extends to the handlebody.

One would like to determine a maximally linearly independent subset of the αs for each m. There would be a corresponding theorem for surfaces as the only permissible slice types are of the form λj where gcd(j, m) = 1. The next step is to determine the theorem for m = pn.

Unfortunately, independence results do not help that much for general semi- free actions on higher dimensional manifolds. A quick look at the Atiyah-Singer g-signature formula (Theorem 62) suﬃces to convince the reader that such actions would typically involve products of αs and not merely sums. This mandates an en- tirely diﬀerent set of resources from algebraic number theory.

175 Another possible direction would be to determine all of the possible slice types for Zp-actions on surfaces. Is it possible to say that a surface can have a certain number of ﬁxed points of one type but not a certain number of another type? For example, the independence of the αs shows that equivariant boundaries are required to have very balanced total slice types. It seems like this problem can be considered from a number theory perspective without having to consider algebraic independence directly.

In general, the more algebraic relations that can be determined about the αs, the more useful this invariant will be. For more on the some of the various tricks that can be used, the reader is referred to Hirzebruch and Zagier’s book [20]. For practical applications to complex projective spaces, see Little’s papers [27] and [28].

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181