THE NUMBER THEORY OF FINITE CYCLIC ACTIONS ON

SURFACES

A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THEUNIVERSITY OF HAWAI‘I IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

IN

MATHEMATICS

DECEMBER 2006

ByMicah Whitney Chrisman

Dissertation Committee:

Robert D. Little, ChairpersonChris Allday

Karl Heinz DovermannHugh M. HildenGeorge WilkensLynne 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 Möbius

inverse sequence, denoted Ms : N → Z, which is defined as follows:

Ms(n) =∑

d|n

µ(d)s(nd

)

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 fn is compact for all n and s : N → Z

is the Lefshetz sequence of f , then n|Ms(n) for all n. In this thesis, we investigate

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

Ms(n) = 0 for almost all n ∈ N and m = lcm{k ∈ N : Ms(k) 6= 0}. Moreover, it is

shown that a Dold sequence is bounded if and only if it is periodic. This extends a

result of Babenko and Bogaty̆ı.

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 Lefshetz Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.1 Euclidean Neighborhood Retracts . . . . . . . . . . . . . . . . 14

1.2.2 The Lefshetz Index . . . . . . . . . . . . . . . . . . . . . . . . . 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 Dold Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Periodic Dold Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Limit Points of Dold Sequences . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Geometric Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.5 Dold’s Realization Theorem . . . . . . . . . . . . . . . . . . . . . . . . 52

2.6 Periodic Realization of s : N → Z . . . . . . . . . . . . . . . . . . . . . 54

vii

2.6.1 Proof of Lemma 32 . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.6.2 Proof of Theorem 30 . . . . . . . . . . . . . . . . . . . . . . . . 58

Chapter 3: Applications to Maps of Surfaces . . . . . . . . . . . . . . . . . . . . . . 61

3.1 Mapping Class Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.1.1 The Thurston Classification Theorem . . . . . . . . . . . . . . 62

3.1.2 Finite Order Mapping Classes and Their Dold Sequences . . . . 64

3.2 The Similarity Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Realization in the Special Case of Zp . . . . . . . . . . . . . . . . . . . 72

3.4 Review of Fuchsian Groups . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4.1 Definition and Properties of Fuchsian Groups . . . . . . . . . . 85

3.4.2 Parabolic, Hyperbolic, and Elliptic . . . . . . . . . . . . . . . . 88

3.4.3 Fundamental Regions . . . . . . . . . . . . . . . . . . . . . . . 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 The Theorems of Harvey . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.7 Realization of Periodic Dold Sequences on Surfaces . . . . . . . . . . . 106

3.7.1 Proof of Sufficiency . . . . . . . . . . . . . . . . . . . . . . . . 111

3.7.2 Proof of Necessity . . . . . . . . . . . . . . . . . . . . . . . . . 117

3.8 The Sphere and the Torus . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 4: On Zp-actions that extend to the Handlebody . . . . . . . . . . 128

4.1 Slice Types and the G-Signature Theorem . . . . . . . . . . . . . . . . 129

viii

4.2 Rational Linear Independence and the αj/p’s . . . . . . . . . . . . . . 135

4.3 Some Geometry, Topology, and Bordism . . . . . . . . . . . . . . . . . 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-Actions on Surfaces . . . . . . . . . . . . . . . . . 162

Chapter 5: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.1 Chapter 2 Summary and Future . . . . . . . . . . . . . . . . . . . . . 171

5.2 Chapter 3 Summary and Future . . . . . . . . . . . . . . . . . . . . . 172

5.3 Chapter 4 Summary and Future . . . . . . . . . . . . . . . . . . . . . 174

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

ix

List of Figures

1 Illustration of Dold’s simplicial 2-complex. . . . . . . . . . . . . . . . 53

2 Schematic of the basic s(1) = 0 case. . . . . . . . . . . . . . . . . . . 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 Diagram for the proof of sufficiency. . . . . . . . . . . . . . . . . . . . 110

7 Rotation of the torus by π has 4 fixed points. . . . . . . . . . . . . . 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 Bi × I with labeled two-sided collar. . . . . . . . . . . . . . . . . . . 158

13 Pasting V1 × 1 with V2 × 2 along the map ϕ. . . . . . . . . . . . . . . 159

14 Assigning C∞ structure to C. . . . . . . . . . . . . . . . . . . . . . . 161

15 The n-manifold Z is what remains after gluing. . . . . . . . . . . . . 162

16 Construction of Z near two fixed points of ψ. . . . . . . . . . . . . . 169

x

Chapter 1

Introduction

1.1 Overview

The symmetries of planar objects have long fascinated artists and mathematicians.

In the form of tesselations, their history can be traced back six thousand years to the

ancient Sumerians. Their use continued with the ancient Greeks and Spanish Moors.

Of course, few modern students are unfamiliar with the works of the Dutch painter

M.C. Escher.

Mathematicians became interested in the symmetries of planar objects as they

provided some of the first examples of finite groups (sets with an associative operation

containing inverses and an identity element). Today, tesselations and symmetries

remain a hotbed of mathematical activity. The questions asked by mathematicians in

the last two hundred years typically focus on much broader notions of symmetry and

geometry. One can naively define a symmetry to be a one-to-one and onto continuous

function from some given object to itself that preserves “distance” and “angle”. One

also needs that the symmetry can be undone. In other words, a symmetry must posses

an inverse symmetry: a one-to-one and onto continuous function from the object to

itself that reverses the action of the original symmetry. Now, every object has an

identity symmetry. It is simply the function that maps every point of the object to

itself. Hence we see that a symmetry can be considered as an element of a group.

1

So instead of speaking intuitively about symmetry, mathematicians ask about how

certain groups can act on certain objects.

What is an object? In this dissertation, we will look at objects that are examples

of topological spaces. One can think of a topological space as a set with an added

neighborhood structure. Any point in the set is contained in a neighborhood. Every

topological space must have at least two neighborhoods: the set itself and the empty

set. Also, the union of any (possibly infinite) number of neighborhoods must be a

neighborhood. Finally, finite intersections of neighborhoods must also remain neigh-

borhoods. Topological spaces are very general and abstract. The easiest concrete

example is that of the real numbers, R. The neighborhood structure is given by the

open intervals, (a, b) = {x ∈ R : a < x < b}. The plane and three-dimensional space

are other examples of topological spaces.

In the most general context, then, the study of symmetries in mathematics is

the study of groups acting on topological spaces. The groups are almost always

assumed to be topological groups: topological spaces with a group structure so that

the operations of multiplication and inversion are continuous. It is probably not clear

why this level of abstraction is necessary. 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 fm 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 alge-

braic 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 them-

selves. 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 pre-

serve the topological notion of equivalence are called algebraic invariants. The three

most commonly used algebraic invariants are the fundamental group functor, the ho-

mology 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) Hq(f) : Hq(X; Q) → Hq(Y ; 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) =

∞∑

q=0

(−1)qTr(Hq(f))

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) 6= 0, then f must have at least one

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

fk(x) = x for some k > 1 but f j(x) 6= x for all j < k. If such a k exists, the smallest

k is called the period of x. Hence, we consider the following sequence, s : N → Z, of

iterates of f :

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 fk is a periodic point of period k.

Something fixed by fk might also be fixed by f d where d|k. To solve this, we can

examine the Möbius inverse sequence of s, denoted Ms : N → Z:

Ms(n) =∑

d|n

µ(d)s(nd

)

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|Ms(n) for all n ∈ N. In this thesis, we consider the number theory of integer

sequences s : N → Z such that n|Ms(n) for all n ∈ N (called a Dold sequence). When

this is applied to the study of finite order symmetries, i.e. maps f : X → X such

that fm = 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 Ms(k) 6= 0. This work is proven in great detail in Section 2.2.

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 se-

quences. 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 |Ms(k)| periodic points of period k, each with

local Lefshetz number Sign(Ms(k)/|Ms(k)|). The maps constructed in [11] are never

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:

(1) → T (S) α−→ Mod(S) β−→ 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 gs ∈ N and an N ∈ N such the group G acts

on the surface of genus g ≥ gs if and only if g = Nk+1 for some k ∈ N. The number

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

acts without fixed points. By covering space theory, π1(H/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 in-

variant. 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 signa-

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

αj/m =λj + 1

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:

σ(ψ) =m−1∑

j=1

ajαj

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

{α1/m, . . . , α(p−1)/p} which is presented in Section 4.2.

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 eventu-

ally 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 re-

view 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 = 1A. The space A is said to be a neighborhood retract in X if there

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 = Ā ∩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) 6= 0 =⇒ f has a fixed point). The algebraic

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 uK the fundamental class of

V around K. This element is characterized by the property that for all x ∈ K, The

image of uk under the map Hn(V, V \K) → Hn(V, V \ {x}) is the local orientation

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)∗ : Hn(V, V \ F ) → Hn(Rn, Rn \ {0}) ∼= Z

16

Let u0 denote the local orientation of Rn about the origin. The map (ι − f)∗ sends

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)∗(uF ) = If · u0

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

F = ∅), then If = 0.

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

V1, . . . , Vn. Then If =∑n

i=1 If |Vi.

2. (Homotopy Invariance) If g : V → Rn is continuous and f ∼ g, then If = Ig.

3. (Commutativity) Let Uf ⊂ Rnf and Ug ⊂ Rng . Suppose f : Uf → Rng and

g : Ug → Rnf . Then f ◦ g : g−1(Uf) → Rng and g ◦ f : f−1(Ug) → Rng have

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:

Vf

AA

A

A

A

A

A

α

��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 = 1X . This gives the following

sequence of maps:

Vh−→ A i−→ X r−→ A h

−1

−−→ V f−→ 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 If , to be Iαβ. Note that αβ : β−1(V ) → X is a map of Euclidean spaces

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:

Vα1−→ X1 β1−→ Y

Vα2−→ X2 β2−→ Y

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 in-

variance, 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 Hq(f) : Hq(Y,Q) → Hq(Y,Q) is thus a linear

transformation. Recall that the trace function Tr : M(dimHq(Y ; Q),Q) → Q is

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) =∑

q

(−1)qTr(Hq(f))

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 If = Λ(f |K).

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

H0(f) is necessarily the identity on H0(D2; Q) ∼= Q. 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 , If2 , If3 , . . .)

The name for this sequence obviously comes from the case in which Ifk = Λ(fk) 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 θg : V → V to be multiplication

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 θg divides xm − 1 but does not necessarily equal

xm − 1.

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

equals the algebraic multiplicity.

3. Let mj ≥ 0 be the multiplicity of λj as an eigenvalue of θg for 0 ≤ j ≤ m−1. If

λk is an algebraic conjugate of λj (i.e. any other root of the minimal polynomial

of λj), then mj = mk.

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

if mj 6= 0, φj(x) = Φd(x) for some d|m. Define md ≥ 0 to be the common

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

characteristic polynomial of θg is:

charθg(x) =∏

d|m

(Φd(x))md

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

dimZ(A) =∑

d|m

mdϕ(d)

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 Z3 action on Z ⊕ Z:[−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

{e1, . . . , em} be a basis for A. Then {1 ⊗ e1, . . . , 1 ⊗ em} is a basis for V . Note

that:

g(1 ⊗ ei) = 1 ⊗ gei = 1 ⊗ (α1e1 + . . .+ αmem)

= α1(1 ⊗ e1) + . . .+ αm(1 ⊗ em)

where αi ∈ Z for all 1 ≤ i ≤ m. So there is a matrix representation in which all of

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

algebraic number λk 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:

trC(ξ) =∑

(j,m)=1

e2πij/m = µ(m)

where µ is the Möbius function (defined explicitly in Section 2.1).

Proof. Let Φm(x) be the m−th cyclotomic polynomial. The degree of Φm(x) is ϕ(m)

and the coefficient of xϕ(m)−1 in Φm(x) is −trC(ξ). We will show that this coefficient is

given by −µ(m). First suppose that m = p1p2 . . . ps where pk is a prime for 1 ≤ k ≤ s

and pi 6= pj for i 6= j. If s = 1:

Φm(x) = 1 + x+ . . .+ xm−2 + xm−1

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:

Φp1p2···ps(x) =Φp1···ps−1(x

ps)

Φp1···ps−1(x)

Using long division, one can show:

Φp1p2···ps(x) = x(ps−1)ϕ(p1p2...ps−1) − (−1)s−2x(ps−1)ϕ(p1p2...ps−1)−1 + f(x)

where the degree of f(x) is smaller than ϕ(p1p2 · · · ps−1ps)−1. Thus, tr(ξ) = (−1)s =

µ(m) and the theorem is proved by induction when m is square free. Now, suppose

that m = pa11 · · · pass where the pk are distinct primes and aj > 1 for some j. Also by

[21], we have that:

Φm(x) = Φp1···ps(xp

a1−11 ···p

as−1s )

24

Every power of x on the right hand side of the equation is a multiple of pa1−11 · · · pas−1s .

Now, ϕ(m) ≡ 0 (mod pa1−11 · · · pas−1s ), so xϕ(m)−1 occurs with coefficient 0 on the left

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) =∑

d|m

mdµ(d)

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(θg) = m1 −mp.

2. If m = pq, then Tr(θg) = m1 −mp −mq +mpq.

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 admit-

ting 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 = Zm be the cyclic group of order m. Let A be a finitely

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 θg : V → V by x → gx. In general, we can define a map ρ : Zm →

GL(V ) by g → θg. The eigenvalues of θg can be easily determined. This is due to

the fact θmg = 1. Then the minimal polynomial for θg divides xm − 1 (Note that the

26

minimal polynomial need not be xm− 1. For example, G could act identically on A).

Hence, the eigenvalues of θg are the powers of λ = e2πi/m. The polynomial xm − 1

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

the λj-eigenspace of θg. Then

V =

m⊕

j=1

Vj

Now, ΦC is orthogonal under this decomposition, i.e. ΦC(vj , vk) = 0 if vj ∈ Vj ,

vk ∈ Vk and j 6= k.

ΦC(vj , vk) = ΦC(gvj, gvk)

= ΦC(λjvj , λ

kvk)

= λkλjΦC(vj, vk)

But λkλj = 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 Zm invariant inner

product on V . It can be shown that < ·, · > is orthogonal on the subspaces Vj in a

similar fashion to that used above on ΦC. Define L′ by:

< L′v1, v2 >= ΦC(v1, v2) (1.1)

Recall that V = Vj ⊕ V ⊥j . If w 6∈ Vj, we must have that w ∈ spank 6=j{Vk}. Thus

< L′v, w >= ΦC(v, w) = 0 and L′v ∈ (V ⊥j )⊥ = Vj . This gives us that L′Vj ⊂ Vj .

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< w,w > = µ < w,w >=< 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) = < Lv, u > =< u,Lv >

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 V+j be the

direct sum of the eigenspaces of L that have positive eigenvalues and V −j the direct

sum of the eigenspaces of L that have negative eigenvalues. Since L has only nonzero

eigenvalues, Vj = V+j ⊕ V −j .

If Φ is skew-symmetric, define V +j and V−j via the map L = −iL′. The details of

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

σ(Zm, A,Φ) =

m−1∑

j=0

V +j − V −j

Definition 8. The g-signature is the character of the above representation evaluated

at g.

σ(g, A,Φ) =m−1∑

j=0

tr(gV +j ) − tr(gV −j )

=

m−1∑

j=0

(dimC V+j − dimC V −j )λj

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 = Zm, m ≥ 2. The ideas contained in the proofs of these earlier results

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(V+j ) + dim(V

−j ). Thus,

dim(Vj) ≡ dim(V +j ) − dim(V −j ) (mod 2). Let λ = e2πi/m and let Φd denote, as in

Lemma 3, the d-th cyclotomic polynomial, where d|m. Define Jd = {j ∈ Z : Φd(λj) =

0, 0 ≤ j ≤ m − 1} for all d|m. By Lemma 3, |Jd| = ϕ(d) and dim(Vj) = md for all

j ∈ J . This gives:m−1∑

k=0

dim(Vk)λk =

∑

d|m

md∑

j∈Jd

λj

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 αj : λ1,d → λj,d, 1 ≤ j ≤ ϕ(d). Thus,∑

j∈Jdλj = tr(λ1,d). But since Q is separable, tr(λ1,d) equals the negative of the

coefficient of x in Φd(x), which is an integer. Finally,

σ(g, A,Φ) =

n−1∑

j=0

(dim(V +j ) − dim(V −j ))λj (1.2)

≡n−1∑

j=0

dim(Vj)λj (mod 2Z[λ]) (1.3)

≡ −∑

d|n

mdtr(λ1,d) (mod 2Z[λ]) (1.4)

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(V+j ) for all j or dim(Vj) = dim(V

−j ). 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 → GLn(A) be the regular representation of G. Moreover, suppose

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 equivari-

antly 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 +∑

d|n

mdϕ(d)

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

intersects each Vλj and hence that each Vλj is split. Applying Definition 8 then shows

that σ(g, A,Φ) = 0.

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

Then Nλj ⊂ Vλj . By hypothesis, we have that dim(N) = k/2. Using Lemma 3 again,

we have the following formula for N :

k

2= m′1 +

∑

d|n

m′dϕ(d)

Subtracting the two equations gives:

0 = (m1 − 2m′1) +∑

d|n

(md − 2m′d)ϕ(d) (1.5)

We claim that md ≥ 2m′d for all d|n. Suppose d|n, 1 ≤ d ≤ n. Since Nλj ⊂ N ,

we must have that Nλj ⊂ N⊥λj . Thus, the largest Nλj could be is N⊥λj . In this case,

md = 2m′d. Hence, md ≥ 2m′d for all d|n. This implies that each of the terms in the

sum in equation 1.5 are all 0. Therefore, md = 2m′d for all d|n and we conclude that

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 Poincaré duality isomorphism. More specifically, let M be a

smooth closed orientable manifold of dimension 2k. Let G be a group of orientation

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

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

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

Φ(x, y) =< x,Dy >

where D : Hk(M ; Z) → Hk(M ; Z) is the Poincaré duality isomorphism and:

< ·, · >: Hk(M ; Z) ×Hk(M ; Z) → Z

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∗k(M) is an isomorphism. Since D is

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) = < g∗x ∪ g∗y, [M ] >

= < g∗(x ∪ y), [M ] >

= < 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:

σ(g) = σ(g∗, Hk(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 corre-

sponding 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 = M1 ∪ (−M2) and the restriction of the action to

Mi is the original G-action on Mi. Here, −M2 just means M2 with the opposite ori-

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

boundary. Then (G,Hk(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 M̃ be the G-

manifold guaranteed by the hypotheses, ∂M̃ = M . The result will be established in

the case that H∗(M ; Z) and H∗(M̃ ; Z) are torsion free. The general result can be

established by performing a similar procedure on the appropriate quotients. We have

the following long exact sequence of cohomology:

. . .→ Hk(M̃,M ; Z) → Hk(M̃ ; Z) j∗

−→ Hk(M ; Z) δ∗

−→ Hk+1(M̃,M ; Z) → . . .

Define W = image(j∗ : Hk(M̃ ; Z) → Hk(M ; Z)). We will show that W = W⊥ and

hence that A is split.

Let g ∈ G and suppose g̃|M = g. Clearly, j ◦ g = g̃ ◦ j. Hence if w = j∗(u) ∈ W ,

we have g∗(w) = g∗j∗(w) = j∗(g̃∗(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 ] >

= < u ∪ u′, j∗([M ]) >

The orientation on M is induced by the orientation on M̃ . In other words, if [M̃,M ]

denotes the fundamental homology class in H2k+1(M̃,M ; Z), then ∂∗[M̃,M ] = [M ]

(by [36], 6.3.10, pg. 304). Here, ∂∗ : H2k+1(M̃,M ; Z) → H2k(M ; Z) is the connecting

homomorphism. Since the homology long exact 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(M̃ ; Z), we have:

0 = Φ(v, j∗u) = < v ∪ j∗u, [M ] >

= (−1)k2 < j∗u, v ∩ [M ] >

= (−1)k2 < u, j∗(v ∩ [M ]) >

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

Now consider the following commutative diagram:

Hk(M ; Z)δ∗ //

∩[M ]

��

Hk+1(M̃,M ; Z)

∼=��

Hk(M) j∗// Hk(M̃ ; Z)

This implies that δ∗(v) = 0 and hence that v ∈ im(j∗) = W . The fact that σ(g) = 0

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 Ms : N → Z

which is defined by the following equation:

Ms(n) =∑

d|n

µ(d)s(n

d)

where µ is the Möbius function:

µ(m) =

1 , m = 1(−1)s , m = p1p2 . . . ps, pk prime for 1 ≤ k ≤ s, pj 6= pk

0 , else

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

Ms(n) =∑

τ⊂P (n)

(−1)|τ |s(n : τ)

where P (n) is the set of primes dividing n and if τ ⊂ P (n), n : τ = n∏p∈τ p−1. In

1983, Dold [12] proved the following theorem:

Theorem 12 (Dold). Let Y by an ENR and f : Y → Y a continuous map. If Fix(fn)

is compact for some n > 1 and s : N → Z is the Lefshetz sequence of f , then n|Ms(n).

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(fk) = |Fix(fk)| for all k.

Now, define

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

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

|Fixn(f)| =∑

τ⊂P (n)

(−1)|τ |I(fn:τ) = Ms(n)

If x ∈ Fixn(f), then x, f(x), f 2(x), . . . , fn−1(x) are also in Fixn(f). Thus, n|Ms(n).

We will say that any sequence s : N → Z is a Dold sequence if it satisfies n|Ms(n)

for all n ∈ N.

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 Ms(k) = 0 for all but

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:

Ms

(k

pr(p′)r

)= Ms(k)

Proof. Let k′ = kpr

(p′)r. Then |P (k)| = |P (k′)|. Suppose that τ ⊂ P (k) and p ∈ τ .

Define τ ′ ⊂ P (k′) by τ ′ = (τ\{p}) ∪ {p′}. Then:

k′ : τ ′ =k

pr(p′)r

∏

q∈τ ′

q−1

=k

pr(p′)r−1

∏

q∈τ ′

q 6=p′

q−1

≡ kp

∏

q∈τ ′

q 6=p′

q−1 (mod m)

≡ k∏

q∈τ

q−1 (mod m)

≡ k : τ (mod m)

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

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 Ms(k) = 0.

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 pi ≡ p (mod m). By Lemma 1,

39

we have:

Ms

(k

prpri

)= Ms(k)

But this implies that pi|Ms(k) for all i. Hence, Ms(k) = 0.

While Ms : N → Z is defined in terms of the function s : N → Z, we can also

write s in terms of Ms. This process is known as Möbius inversion[39].

Theorem 15 (Möbius Inversion). Suppose that f, g : N → Z are functions satisfying

f(n) =∑

d|n µ(d)g(n/d). Then g(n) =∑

d|n f(d) for all n ∈ N.

Lemma 16. Let s : N → Z be a periodic Dold sequence with period m. If {ki} is a

finite set of n natural numbers and gcd(ki, kj) = 1 for i 6= j, then,

s

(n∏

i=1

ki

)=∑

d1|k1

∑

d2|k2

. . .∑

dn|kn

Ms(d1d2 · · · dn)

Proof. Let k =∏n

i=1 ki. Using Möbius inversion, we have s(k) =∑

d|kMs(d). If

d|k, then d can be written uniquely as a product d1d2 · · · dn where di|ki for 1 ≤

i ≤ n. Thus, the above formula is merely a rearrangement of the Möbius inversion

formula.

Theorem 17. Let s : N → Z be a periodic Dold sequence with period m. Suppose

that k ∈ N and gcd(k,m) < k. Then Ms(k) = 0.

Proof. Let m = pa11 · · · pann for some primes p1, . . . , pn having exponents ai ≥ 1 for

1 ≤ i ≤ n. Let k = qb11 · · · qbrr for some primes q1, . . . , qn having exponents bj ≥ 1

for 1 ≤ j ≤ r. If qj 6∈ {pi} for some j, then we know by the previous lemma that

Ms(k) = 0. Now, suppose that {qj} ⊂ {pj}. For all primes in {pi}\{qj}, insert them

40

in the prime factorization of k with exponent 0. We relabel the primes qj so that

pi = qi for 1 ≤ i ≤ n. The only remaining case is the case when the exponent bi > ai

for some prime pi (for otherwise, k|m). Without loss of generality, we assume that

this prime is pn.

Let m′ = m/pann and k′ = k/pbnn . Then gcd(m

′, pn) = 1. By Dirichlet’s theorem

again, there exist infinitely many primes αz ≡ pn (mod m′). Multiplying through the

modular equation by k′pbn−1n gives the following congruence modulo m:

k′αzpbn−1n ≡ k′pbnn ≡ k (mod m)

Thus we have the equation s(k′pbn−1n 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:

s(k′pbn−1n αz) =∑

d1|k′

∑

d2|pbn−1n

∑

d3|αz

Ms(d1d2d3)

=∑

d1|k′

∑

d2|pbn−1n

(Ms(d1d2) +Ms(αzd1d2))

=∑

d1|k′

∑

d2|pbn−1n

Ms(d1d2)

Also, we have that:

s(k) = s(k′pbnn ) =∑

d1|k′

∑

d2|pbnn

Ms(d1d2)

These equations can be simplified by noting that:

∑

d2|pbnn

Ms(d1d2) = Ms(pbnn d1) +

∑

d2|pbn−1n

Ms(d1d2)

41

Then from the above we have:

0 = s(k) − s(k′pbn−1n αz) =∑

d1|k′

Ms(pbnn d1)

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

k′ = 1 and the above equation reduces to Ms(k) = Ms(pbnn ) = 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:

∑

d1|k′

Ms(pbnn d1) = 0

For all d1|k′ with d1 6= k, the number of divisors of d1 is less than or equal to

ν. Hence Ms(pbnn d1) = 0 for all d1|k, d1 6= k. But by the relation above, we have

Ms(k) = Ms(pbnn 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 6 |m, we use Möbius inversion to obtain:

s(k) =∑

j|k

Ms(j)

If j 6 |m, then gcd(j,m) < j and by Lemma 17, Ms(j) = 0. If j|m and d|j, then

jd−1|m and hence s(j/d) is known. Each such Ms(j) is then determined, and hence

so is s(k).

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 6= k, then s(k) = s(d)

Proof. Suppose gcd(k,m) = 1. Then s(k) =∑

j|kMs(j) = Ms(1) = s(1). If

gcd(k,m) = d < k, then:

s(k) =∑

j|k

Ms(j) =∑

j|k,

j 6 |m

Ms(j) +∑

j|k,

j|m

Ms(j)

=∑

j|d

Ms(j)

= s(d)

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

(a, a, . . . , a︸ ︷︷ ︸(p−1) times

, a+ pα, a, a, . . . , a︸ ︷︷ ︸(p−1) times

, a+ pα, . . .)

All periodic Dold sequences of period 4 must 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, fk = g for some k, 1 < k < |G| = m. Moreover, gcd(k,m) =

1. Let mt be the period of t. Now, t(k) = s(1). Since mt|m, gcd(mt, k) = 1. Then by

the above, s(1) = t(k) = t(1). Now, let j > 1. Since gj = fkj we have s(j) = t(kj).

Then:

s(j) = t(kj) =∑

l|kj

Mt(l) =∑

l|kj

l|mt

Mt(l) +∑

l|kj

l 6 |mt

Mt(l)

=∑

l|j

Mt(l)

= t(j)

Thus, s = t.

Theorem 21. Let s : N → Z be a Dold sequence such that Ms(k) = 0 for all but

finitely many k. Let m = lcm{k : Ms(k) 6= 0}. Then s : N → Z is a periodic Dold

sequence with period m.

Proof. We must show that s(k+m) = s(k) for all k. If j 6 |m, then j 6∈ {k : Ms(k) 6= 0}

and hence Ms(j) = 0. First suppose that k|m. Then:

s(k +m) =∑

j|(k+m)

Ms(j) =∑

j|k

Ms(j) +∑

j|(k+m)

j 6 |k, j 6 |m

Ms(j) =∑

j|k

Ms(j) = s(k)

Suppose that k 6 |m. Then:

s(k +m) =∑

j|m, j|k

Ms(j) +∑

j|(k+m)

j 6 |k, j 6 |m

Ms(j)

=∑

j|m, j|k

Ms(j) +∑

j 6 |m, j|k

Ms(j)

=∑

j|k

Ms(j)

= 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 Ms(k) 6= 0. Then by

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

Ms(k) = 0 for all but finitely many k ∈ N and m = lcm{k ∈ N : Ms(k) 6= 0}.

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(fk) 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 Ms(k) = 0 for all but finitely many k and s : N → Z is

periodic.

Proof. Applying Möbius inversion to s : N → Z, we get Ms(n) =∑

k|n µ(k)s(n/k).

Then:

|Ms(n)| ≤∑

k|n

|µ(k)|∣∣∣s(nk

)∣∣∣

≤∑

k|n

∣∣∣s(nk

)∣∣∣

≤ M∑

k|n

1

= Mσ0(n)

Dividing by n on both sides gives:

|Ms(n)|n

≤ Mσ0(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, σ0(m) ≤ 2

√n and:

0 ≤ σ0(n)n

≤ 2√n

n=

2√n

Then by the squeeze theorem, limn→∞ σ0(n)/n = 0. This implies that there exists

an N such that Ms(n) = 0 for all n > N . By the previous theorem, s : N → Z is

periodic.

Corollary 24. Let X be an ENR and f : X → X a continuous map. Suppose that

either X is compact or Fix(fk) 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

Möbius inverse sequence, Ms : N → Z is contained in the set {−∞, 0,∞}.

Proof. Suppose that Ms has a finite limit point a. It is clear that a is an integer and

hence that Ms(k) = a for infinitely many k. Since k|Ms(k) = a for each of these k,

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 Möbius 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(Ms), then that one is in L(s).

2. If L(Ms) = {∞}, then L(s) = {∞}.

3. If L(Ms) = {−∞}, then L(s) = {−∞}.

Proof. It suffices to establish the verity of the statements only when ∞ ∈ L(Ms) and

when {∞} = L(Ms). For the first, we divide the problem into two cases: all the

values taken by Ms are positive, and a finite number of the Ms values are negative.

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

A = {k ∈ N : Ms(k) > 0}

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

and:

s(k) =∑

j|k

Ms(j) ≥ Ms(k) ≥ k > M

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(Ms) and a finite number of the Ms are negative. Define:

J =∑

k s.t. Ms(k)

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) =∑

j|k

Ms(j)

≥ Ms(k) + J

≥ k + J

> M

The statement follows as in the first case.

Now suppose that L(Ms) = {∞} and let M > 0 be arbitrary. The proof of this

statement is very similar to the last argument. Suppose first that Ms(k) ≥ 0 for all k.

Then s(k) ≥ Ms(k). There are only finitely many k such that Ms(k) = 0, so choose

K such that Ms(k) 6= 0 for all k > K. Choose N = max{K,M}. Then if k > N :

s(k) ≥Ms(k) ≥ k > M

The result now follows. If Ms(k) ≤ 0 for finitely many k, define J as above. Let K

be such that Ms(k) > 0 for all k > K. Set N = max{K,M − J}. Then if k > N :

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

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 Ms : N → Z, i.e. Ms is unbounded.

Proof. If not, then 0 would be the only limit point ofMs : N → Z and hence s : N → Z

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, Ms(f)(k) 6= 0 implies f has a periodic point of period k. Then

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 , Ms(f)(p) 6= 0 implies f has a periodic point

of period p. Of course, with our work, this is strong enough to prove a much weaker