Cellular Structures and Stunted Weighted Projective Space

CELLULAR STRUCTURES AND STUNTED WEIGHTED PROJECTIVE SPACE

A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences

2014

Beverley O’Neill School of Mathematics Contents

Abstract 5

Declaration 6

Acknowledgements 8

Dedication 10

1 Introduction 11

2 Cell structures 16 2.1 CW-complexes ...... 16 2.2 Cellular chain complex and homology ...... 19 2.3 ∆-complexes ...... 22 2.3.1 Simplices ...... 22 2.3.2 ∆-complexes ...... 24 2.4 Cellular chain complex of a ∆-complex ...... 29 2.5 Products of ∆-complexes ...... 32 2.5.1 CW-complex on X × Y ...... 32 2.5.2 Cellular chain complex of X × Y ...... 34 2.6 Group actions and G-complexes ...... 36

3 Weighted projective space 41 3.1 Preliminary deﬁnitions and properties ...... 41 3.2 Kawasaki’s results ...... 43

2 3.3 Regarding cellular decompositions of P(χ) ...... 45 3.3.1 Complex projective space revisited ...... 45 3.3.2 Weighted projective space ...... 46 3.4 The way forward ...... 48

n 4 The Cχ-action on CP 52 n 4.1 Properties of the Cχ-action on CP ...... 53 n 4.2 Canonical representatives for points in (CL) /Cχ ...... 56 n n n n 4.3 Canonical representatives for points in T /Cχ, T /Cχ × ∆ and CP /Cχ 62 4.4 Regarding fundamental domains ...... 63

5 CW-complex on P(χ) 67 n n 5.1 A ∆-complex structure on T and T /Cχ ...... 68 n n n n 5.2 A CW-structure on T × ∆c and T /Cχ × ∆c ...... 73 5.3 A CW-structure on CP n ...... 74 5.4 A CW complex structure on P(χ)...... 81

6 Cellular chain complex of P(χ) 86 6.1 Introduction and motivation ...... 86

n n 6.2 Cellular chain complex of TL and TL /Cχ ...... 87 n n n n 6.3 Cellular chain complex of TL × ∆c and TL /Cχ × ∆c ...... 88 n 6.4 Cellular chain complex of CPL ...... 90 n 6.5 Cellular chain complex of CPL /Cχ ...... 97 6.6 Cellular chain complex of P(χ)...... 99

7 Homology calculations for P(χ) 101

8 Stunted weighted projective space and weighted lens space 116 8.1 Weighted lens space ...... 117

2n 2n 8.1.1 A CW-structure on D and D /Cχ ...... 117 0 8.1.2 A CW-structure on L(χ0; χ ) ...... 123 8.2 Stunted weighted projective space ...... 125

8.2.1 A CW-structure on P(χ; I) ...... 125 8.2.2 Homology and cohomology calculations for P(χ; I) ...... 127

3 Bibliography 133

Word count 46736

4 The University of Manchester

Beverley O’Neill Doctor of Philosophy Cellular structures and stunted weighted projective space January 20, 2014

Kawasaki has calculated the integral homology groups H∗(P(χ)) of weighted projective space, and his results imply the existence of a homotopy equivalence between P(χ) and a CW-complex Xχ, with a single cell in each even dimension less than or equal to that of P(χ). When χ satisfies certain divisibility conditions then P(χ) is actually homeomorphic to such an Xχ and such decompositions are well-known in these cases. Otherwise Xχ is not a weighted projective space. Our aim is to give an explicit CW-structure on any P(χ), using an invariant decomposition of CP n with respect to the action of a product Cχ of finite cyclic groups. The result has many cells, in both odd and even dimensions; nevertheless, we identify it with a subdivision of the minimal decomposition whenever χ is divisive. We then extend the decomposition to stunted weighted projective space, defined as the quotient of one weighted projective space by another. Finally, we compute the integral homology groups of stunted weighted projective space, identify generators in terms of cellular cycles, and describe cup products in the corresponding cohomology ring.

5 Declaration

No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualiﬁcation of this or any other university or other institute of learning.

6 Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the “Copyright”) and s/he has given The University of Manchester certain rights to use such Copyright, including for administrative purposes.

ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intellectual property (the “Intellectual Property”) and any reproductions of copyright works in the thesis, for example graphs and tables (“Reproductions”), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The Univer- sity Library’s regulations (see http://www.manchester.ac.uk/library/aboutus/regulations) and in The University’s Policy on Presentation of Theses.

7 Acknowledgements

Firstly I would like to thank my supervisor Nige Ray for his much-needed guidance and support. I have taken great comfort in the knowledge that his door was always open in times of minor or major crises, and I have no doubt taken a great deal of his patience over the years. His concern in ensuring I settled into Manchester is one of the reasons I have quite so many wonderful friends to thank. I have met some fantastic people through studying maths at Manchester, but I have enjoyed some particularly hearty belly laughs (and I bet you could hear them from your office couldn’t you Nige?!) courtesy of Sebastian Law, Phil Haines, Karren Palmer, Philip Bridge, Rob McKemey, Paul Bradley, Simon Baker and Dan Vasey. Cheers to my writing-up buddies Vicki Andrew and Patrick Hurley for the tea and empathy. To Andrew Fenn, Stephen Miller and Alistair Darby, you guys are the best big brothers I could ask for. Particular thanks goes to Dave Naughton for listening to endless maths grumbles, getting me to dust off my fiddle and reminding me to “do a smile” during the last year. Whenever I entered my “maths black holes” and the resultant radio silence that would ensue, the patience of many of my friends, of whom I must mention Gemma Lloyd, Lauren Mondshein, Katie Wray and Carolyne Gregory, has been astounding and humbling. To our Pe, our Kath and our Jane, my personal cheerleaders, you have supported me in your own special ways: Kath with your gentle tellings off keeping me in check; Jane with your rather forceful compliments staving off the inevitable “pity-party”; and Peter with your well-timed messages of Rocky montages to suit any occasion. Finally my parents. Dad, I always find it amazing that any pep talk involves an analogy of how Liverpool must have felt in the dressing room at half time in Istanbul. Mum, thank you for never asking about how maths was going. None of this would

8 have been possible without your love and understanding. You’ve enabled me to get this far, when at times I doubted it dearly. For that reason, this is as much yours as it is mine.

9 Dedication

To me Ma an Pa: Nothing seems to have changed since I was a clumsy toddler-you two are still the ﬁrst people to pick me up whenever I stumble.

10 Chapter 1

Introduction

Weighted projective spaces P(χ) are deﬁned for every positive integral weight vector

χ = (χ0, . . . , χn), and provide the most basic examples of projective toric varieties exhibiting singularities of quotient type. Although they belong to such a restricted class of algebraic varieties, their simplicity has attracted the attention of much study in algebraic geometry, often to test hypotheses and constructions of greater generality. See [10] for a well-known introduction to toric varieties, in which weighted projective spaces feature in §2.2. Outside algebraic geometry, weighted projective spaces and more generally toric varieties arise naturally in many other areas of mathematics. As the quotient of the odd dimensional sphere S2n+1 by the weighted circle action T hχi = {(tχ0 , . . . , tχn ): |t| = 1} < T n+1 , weighted projective space is the underlying space of a representable orbifold [1]. When the weights are such that χ0 = ··· = χn = 1 then T hχi is the diagonal n circle Tδ and P(χ) reduces to the standard complex projective space CP . The case P(1, 2) is the teardrop orbifold studied by Thurston [21] and is the easiest example of a non-global quotient orbifold. The importance of orbifold structures in areas such as theoretical physics has provided the impetus for many recent developments. Further- more, the groupoid representation of an orbifold has become especially inﬂuential, and has encouraged the development of algebraic and topological invariants which reﬂect the structure of the entire orbifold. In fact P(χ) may equally be presented as a global quotient orbifold, by factoring out the coordinate action of the product

Cχ = Cχ0 × · · · × Cχn

11 CHAPTER 1. INTRODUCTION 12 of cyclic groups on CP n. This deﬁnes an alternative orbifold structure on P(χ), however we note that the underlying space is the same as the representable orbifold view- point. For many years algebraic topologists paid little attention to weighted projective spaces. Their integral cohomology and complex K-theory rings were computed in pioneering work of Kawasaki [15] and Al Amrani [2], and rather surprisingly shown to be free of additive torsion. Their homotopy theoretic properties have only recently been investigated [5], and descriptions of their singularities are emerging in terms of iterated Thom spaces [4]. Their Borel equivariant cohomology has also been computed [3]. Most introductory texts in algebraic topology contain an exposition of cellular structures, in particular CW-complexes. Formally introduced by J.H.C. Whitehead [22], CW-complexes are built up inductively by attaching cells by maps of spheres to lower dimensional skeleta and endowing the resulting space with the weak topology. These essentially combinatorial structures make the fundamental group, homology and cohomology calculations much more manageable and often provide invaluable geometrical insight for any student. Although such spaces may be seen as a convenient compromise to studying more general spaces, results such as the Whitehead Theorem and Cellular Approximation Theorem (see [18], for example) illustrate how CW-complexes are suitably nice spaces to work with in homotopy theory. Further- more, the CW-approximation theorem (see [11], for example) shows that every space is weakly homotopy equivalent to a CW-complex.

One of our main aims is to describe an explicit CW-structure on P(χ), by appealing to the theory of invariant triangulations. No such description seems to be available in the current literature. The work of Illman [13] conﬁrms the existence of a Cχ- invariant cell structure on CP n, which therefore induces a cellular decomposition of the underlying orbifold. In order to construct such examples it is convenient to view

CP n as a toric variety, and make implicit use of the associated T n-orbits. The outcome has many cells in every dimension, but may still be reconciled with Kawasaki’s results.

In fact, the calculations of [15] imply that P(χ) is homotopy equivalent to a unique

CW-complex Xχ with a single cell in each dimension (by a theorem of Whitehead [11, Proposition 4C.1]). If the weights obey certain divisibility conditions then P(χ) CHAPTER 1. INTRODUCTION 13

is actually homeomorphic to such an Xχ [4], and we show that our cell-structure is compatible with this fact. Weighted projective spaces have been classiﬁed up to homeomorphism and up to homotopy type, [5]. In particular, two weighted projective spaces P(χ) and P(ρ) are homeomorphic if and only if χ = (χ0, . . . , χn) and ρ = (ρ0, . . . , ρn) are such that

1. χ and ρ diﬀer by reordering of the weight vectors,

2. χ and ρ diﬀer by scaling by an integer d > 1, that is (χ0, . . . , χn) = (dρ0, . . . , dρn).

3. χ and ρ diﬀer by dividing by the greatest common divisor of n of the weights.

For example, if χ and ρ are such that (χ0, . . . , χn) = (ρ0, ρ1/p, . . . , ρn/p) for

gcd(ρ1, . . . , ρn) = p

If χ is not divisive, then P(χ) is homotopy equivalent to Xχ, however it can be shown that Xχ is not homeomorphic to a weighted projective space. Therefore one of the aims of this thesis is to determine CW-structures Yχ such that P(χ) is homeomorphic to Yχ. Nevertheless, the possibility of simplifying our structures remains open. These methods also provide cell decompositions of the stunted weighted projective spaces, which do not yet seem to feature in the literature. For I ⊆ {0, . . . , n}, we follow Kawasaki by writing χI for the restriction of χ to I. So P(χI ) is identiﬁed with the subspace

PI (χ) := {[z]: zj = 0 for j∈ / I} ⊆ P(χ) and we denote the stunted weighted projective space P(χ)/PI (χ) by P(χ; I). The theory of stunted projective spaces is well developed for the standard case.

If I = {0, . . . , k − 1}, then the quotient space P(χ; I), where χ0 = ··· = χn = 1, is n traditionally written as CPk . It is homotopy equivalent to the Thom space T (kζn−k) of a k-fold sum of tautological line bundles over CP n−k [12]. We calculate the integral homology of P(χ; I), via the properties of the cell structures determined for weighted projective space and conclude with calculations of the cohomology ring, which we believe to be new. This thesis is organised as follows. Chapter 2 is more or less a survey of the deﬁnitions and basic properties of the various cellular structures and their associated cellular chain complexes considered throughout this thesis. Although the material is elementary, it is fundamental in forming a solid basis for subsequent work. CHAPTER 1. INTRODUCTION 14

In chapter 3 we give the formal deﬁnitions of weighted projective space and weighted lens space, focussing on the work of Kawasaki and Al Amrani. We provide a discussion of the requirements for prescribing CW-complex structures on P(χ), with a view to calculating the cohomology of stunted weighted projective space. Appealing to the theory of G-complexes established in Chapter 2, we conclude that the global quotient perspective of P(χ) will be the most fruitful. Since CP n is a toric variety over the n-simplex ∆n, it has a decomposition as T n × ∆n/ ∼, [10]. We observe that there is

n n n+1 an associated Cχ-action on T × ∆ , which arises as a subgroup of the T -action on S2n+1. In this chapter we outline the approach we will take in later chapters: via

n n n a Cχ-complex on T × ∆ , we can obtain a Cχ-complex on CP and note that we will see that such an approach is more illuminating for working with the cellular chain complex of P(χ). As an orbit space, there is a choice of representative for each point in P(χ). We n therefore analyse the Cχ-action on CP in Chapter 4, ﬁnding a canonical representative for each orbit, which plays an integral part in deﬁning cells for P(χ). Chapter 5 and Chapter 6 concentrate solely on cell structures and their associated cellular chain complexes respectively. The following commutative diagram,

i q T n / T n × ∆n / CP n (1.0.1)

n ˜i n n q˜ n T /Cχ / T /Cχ × ∆ / CP /Cχ which is introduced in Chapter 3, is key. Cell structures are given for each space in (1.0.1) such that it is a commutative diagram of cellular maps. Therefore once the cellular chain complex is determined for T n, the cellular chain complexes for the remaining spaces quickly follow. We conclude with the CW-complex and chain complex for P(χ) in each chapter. The aim of Chapter 7 is to identify generating cycles of the even dimensional

k integral homology groups of P(χ). We begin by calculating Hk(T /CχI ) and use this to k ∼ n n ﬁnd the generator of H2k(CP /CχI ) = Z. We propose a generator, [G ], of H2k(CP ), where n > k, and prove this is homologous to the generator induced by the inclusion

k n n n n CP ,→ CP . The image of G ∈ C2k(CP ) under the chain map (pχ)∗ : C2k(CP ) → χ n n k C2k(P(χ)) is then shown to be lk Gχ where Gχ is 2k-cycle and lχ is the degree of the n n map (pχ)∗ : H2k(CP ) → H2k(P(χ)) given in [15], therefore H2k(P(χ)) = Z{[Gχ]}. CHAPTER 1. INTRODUCTION 15

In chapter 8 we recall Kawasaki’s identification of P(χ) with the adjunction space of the cone on the weighted lens space, L(χ ; χ ) with (χ ), where for some 0 ≤ i ≤ n i bι P bι we have ι = {0, . . . , n}\ i. We are able to identify a cell structure for CL(χ ; χ ) such b i bι that the attaching map L(χ ; χ ) → (χ ) is cellular, giving back the structure we have i bι P bι for (χ) as well as determining a cell structure for the weighted lens space L(χ ; χ ). P i bι When the weights obey certain divisibility conditions then CL(χ ; χ ) is homeomorphic i bι to D2n and we observe that our cell structure for P(χ) is a subdivision of the CW- complex with a single cell in each even dimension, referred to above. Extending the cofibration sequence of Kawasaki identifies the suspended lens space ΣL(χ ; χ ) with i bι P(χ,bι). The CW-structures we give for P(χ) and PI (χ) are such that (P(χ), PI (χ)) is a CW-pair. Therefore PI (χ) ,→ P(χ) → P(χ, I) is a cofibration sequence, which we use to calculate the integral homology groups H∗(P(χ, I)) and the cohomology ring H∗(P(χ, I)). Furthermore we give a CW-structure for P(χ, I), from which we identify generators of H∗(P(χ, I)). Chapter 2

Cell structures

This chapter is dedicated to introducing the deﬁnitions and concepts of cell structures that are integral and prerequisite to the work contained in this thesis. To avoid detracting from later, sometimes quite combinatorial discussions to establish notation and deﬁne mostly standard terms, this material is presented here for more convenient reference. The main references for this chapter are Algebraic Topology by A. Hatcher [11] and A Basic Course in Algebraic Topology by W. S. Massey [17].

2.1 CW-complexes

In this and subsequent sections we will use the following terminology and notation for any integer n ≥ 1:

n n D = {x ∈ R : |x| ≤ 1} (closed n-dimensional disc or ball)

n−1 n S = {x ∈ R : |x| = 1} ((n − 1)-dimensional sphere)

For n = 0, we let D0 = 0 ∈ R. The following description can be found in the Appendix of [11]. A CW-complex is a space X constructed in the following way:

1. Start with a discrete set X0, the 0-cells of X.

n n−1 n 2. Form the n-skeleton X inductively from X by attaching n-cells eα via maps n−1 n−1 n n−1 F n ϕα : Sα → X . This means, X is the quotient space of X α Dα, under n n the identiﬁcations x ∼ ϕα(x) for x ∈ ∂Dα. The cell eα is the homeomorphic n n image of Dα − ∂Dα under the quotient map.

16 CHAPTER 2. CELL STRUCTURES 17

S n 3. X = n X with the weak topology: a set A ⊂ X is open (or closed) if and only if A ∩ Xn is open (or closed) in Xn for each n.

Note that condition 3 is superﬂuous when X is ﬁnite-dimensional, so that X = Xn

n for some n. Each cell eα has its characteristic map Φα, which is the composition n n−1 F n n n Dα ,→ X α Dα → X ,→ X. Therefore the restriction of Φα to the interior of Dα n is a homeomorphism onto eα. In this thesis we exclusively consider CW-complexes on Hausdorff spaces and find that it is more convenient to use the following definition of CW-complex, which follows from Proposition A.2 [11].

Deﬁnition 2.1.1. A CW-complex is prescribed on a space X (which is always assumed

n to be Hausdorﬀ) by a family of characteristic maps Φα : Dα → X that satisfy the following:

n n 1. Each Φα is injective on the interior of Dα, which we denote by int(Dα); hence n n Φα restricts to a homeomorphism from int(Dα) onto its image, the cell eα ⊂ X, and these cells are all disjoint and their union is X.

n n 2. For each cell eα, then Φα(∂Dα) is contained in the union of a ﬁnite number of cells of dimension less than n.

3. A subset of X is closed if and only if it meets the closure of each cell of X in a closed set.

Remark 2.1.2. The ‘hence’ in condition 1 follows from the fact that as a map from

n the compact set Dα to a Hausdorﬀ space, Φα takes compact sets to compact sets and is therefore a closed map, i.e. it takes closed sets to closed sets. Consequently Φα is n n n a quotient map onto its image Φα(Dα) with int(Dα) as a saturated subset of Dα with n respect to this map. It follows that Φα restricts to a quotient map from int(Dα) onto n n n the cell eα. By deﬁnition of the quotient topology on eα it easily follows that Φα|int(Dα) n n is a closed map and hence a homeomorphism from int(Dα) to eα.

The CW-complexes we define in Chapter 5 are finite, that is, they consist of finitely many cells, and in this case we observe that condition 3 is automatic since the projec- F n tion α Dα → X is a map from a compact space to a Hausdorff space, and therefore CHAPTER 2. CELL STRUCTURES 18 is a quotient map. We note that any space homeomorphic to a disc can be used as the domain of a characteristic map. In Section 2.3 we introduce a particular class of

n CW-complexes in which each cell eα is provided with a distinguished characteristic map from the n-simplex ∆n to the CW-complex X. We also note that the terms “cell complex”, “cell structure” and “CW-structure” are used interchangeably with the term “CW-complex” throughout this thesis.

Example 2.1.3. Complex projective n space CP n is deﬁned as the quotient

n 2n+1 n+1 CP = {(z0, . . . , zn) ∈ S ⊂ C }/ ∼

iθ iθ iθ 1 where (z0, . . . , zn) ∼ (e z0, . . . , e zn) for e ∈ S . Here we are using the notational n+1 2n+2 n convention that (z0, . . . , zn) is a point of C = R . As points in CP are equiva- 2n+1 lence classes of points in S , we will denote by [z0 : ... : zn] the set of all points in 2n+1 iθ iθ S identiﬁed to (z0, . . . , zn) by ∼; that is, [z0 : ... : zn] = [e z0 : ... : e zn]. The 2n+1 n n quotient map ϕn : S → CP invests CP with the quotient topology. There are inclusions

0 1 2 ∗ = CP ⊂ CP ⊂ CP ⊂ · · ·

n−1 n where CP ⊂ CP is given by [z0 : ... : zn−1] 7→ [0 : z0 : ... : zn−1]. An arbitrary point in CP n − CP n−1 can be uniquely represented by

p 2 ( 1 − Σi|zi| , z1, . . . , zn)

2n+1 p 2 in S , where we observe that 1 − Σi|zi| > 0 and is real. We consider the map

2n n Φn : D → CP

p 2 (z1, . . . zn) 7→ [ 1 − Σi|zi| : z1 : ... : zn]

2n for 0 ≤ k ≤ n. From the discussion above we see that the restriction of Φn to int(D ) n n−1 is one-to-one and onto its image CP − CP . By Remark 2.1.2 Φn is therefore a homeomorphism from int(D2n) to CP n − CP n−1. The boundary of D2n, which p 2 n−1 corresponds to the points (z1, . . . zn) where 1 − Σi|zi| = 0, is sent to CP ; that 2n−1 n−1 is, Φn|∂D2n may be identiﬁed with the map ϕn−1 : S → CP . CHAPTER 2. CELL STRUCTURES 19

We consider the maps

2k n Φk : D → CP

p 2 (z1, . . . zk) 7→ [0 : ... : 0 : 1 − Σi|zi| : z1 : ... : zk]

0 n for 1 ≤ k ≤ n. For k = 0, we let Φ0 : D → CP be the map given by Φ0(0) = [0 : ... :

0 : 1]. From the above discussion it is clear that the family of maps Φk for 0 ≤ k ≤ n satisfy the conditions of Deﬁnition 2.1.1 and we obtain a CW-complex for CP n with 2k one cell e := im Φk|int(D2n) in every even dimension such that 0 ≤ 2k ≤ 2n. The (2k + 1)-skeleton is the same as the 2k-skeleton.

A subcomplex of a cell complex X is a closed subspace A ⊂ X that is a union of cells of X. Since A is closed, the characteristic map of each cell in A has its image contained in A, in particular the image of the attaching map of each cell in A is contained in A, so A is a cell complex in its own right. A pair (X,A) consisting of a cell complex X and a subcomplex A ⊆ X will be called a CW-pair.

Example 2.1.4. The image of the inclusion CP k ⊂ CP n given by

[z0 : ... : zk] 7→ [0 : ... : 0 : z0 : ... : zk] is the 2k-skeleton of the CW-complex structure on CP n from Example 2.1.3. Hence CP k ⊂ CP n is a subcomplex of the given cell structure on CP n.

Let X and Y be CW-complexes. A map f : X → Y is called a cellular map if f(Xn) ⊂ Y n for all n.

2.2 Cellular chain complex and homology

To each CW-complex X, we associate a chain complex {(Cn(X), ∂n): n = 0, 1,...} and the n-th homology groups obtained from this chain complex are isomorphic to the singular homology group Hn(X; Z). Throughout this thesis we exclusively consider homology with integer coeﬃcients and so denote Hn(X; Z) simply by Hn(X). Much of the theory behind the material in this section can be found in [17].

n n−1 Given a CW-complex X, then Hn(X ,X ) is a free abelian group with basis in n n−1 one-to-one correspondence with the n-cells of X. We let Cn(X) = Hn(X ,X ) and

∂n : Cn(X) → Cn−1(X), n = 1, 2,... CHAPTER 2. CELL STRUCTURES 20 be deﬁned as the composition of the homomorphisms

n n−1 ∂∗ n−1 jn−1 n−1 n−2 Hn(X ,X ) −→ Hn−1(X ) −−−→ Hn−1(X ,X ), n = 1, 2,...

n n−1 where ∂∗ is the boundary operator in the long exact sequence of the pair (X ,X ) n and jn−1 is the homomorphism induced by the inclusion map. We deﬁne X = ∅ for 0 n < 0 and so C0(X) = H0(X ). n For each n-cell eα, where n > 0, there is a characteristic map

n n n n−1 fα :(Dα, ∂Dα) → (X ,X ).

Passing to homology this induces

n n n n−1 (fα)∗ : Hn(Dα, ∂Dα) → Hn(X ,X ).

n n Since the group Hn(Dα, ∂Dα) is inﬁnite cyclic, there are two ways to choose a generator, which diﬀer by sign. We refer to these choices as ‘choosing orientations for the

n n n domain disc Dα’. Such a choice of generator of the group Hn(Dα, ∂Dα) will be denoted n n n n−1 by [Dα]. The map (fα)∗ takes [Dα] to a generator of the Z summand in Hn(X ,X ) n n corresponding to eα. Let eα denote this generator and if we let An denote the indexing n set of all the n-cells, then we see the set {eα : α ∈ An} is a basis for Cn(X). By setting

n Cn(X) = Z{eα : α ∈ An}

we see that the elements of Cn(X), known as n-chains, can be written unambiguously as ﬁnite formal sums P m en with coeﬃcients m ∈ . α∈An α α α Z For the case n = 0, we observe that the characteristic map for each 0-cell induces the following map in homology

0 0 (fα)∗ : H0(Dα) → H0(X ).

0 For each α, the augmentation homomorphism ∗ : H0(Dα) → Z is an isomorphism. 0 0 0 Hence the choice of generator of H0(Dα) is canonical: we let [Dα] ∈ H0(Dα) be the 0 0 0 0 unique element such that ∗[Dα] = 1 and write (fα)∗[Dα] as eα. The set {eα : α ∈ A0} is a basis for C0(X) , hence

0 C0(X) = Z{eα : α ∈ A0}. CHAPTER 2. CELL STRUCTURES 21

Therefore the homomorphisms ∂n are completely determined by the value of ∂n on n n−1 the basis elements and we express ∂neα as a unique linear combinations of the eβ s. 1 0 0 When n = 1 it is not difficult to see ∂1 : H1(X ,X ) → H0(X ) is the same as the cellular boundary map ∂1 for a ∆-complex, which is defined in Section 2.4. For n > 1, n n−1 the cellular boundary formula, as defined in [11], is given by ∂neα = Σβdαβeβ , where n−1 n−1 n−1 dαβ is the degree of the map Sα → X → Sβ , that is, the composition of the n n−1 n−1 attaching map of eα with the quotient map collapsing X − eβ .

Together the groups Cn(X) for n = 0, 1,... and homomorphisms ∂n for n = 1, 2,... form a chain complex

∂n+1 ∂n ∂1 ∂0 · · · → Cn+1(X) −−−→ Cn(X) −→ Cn−1(X) → · · · → C1(X) −→ C0(X) −→ 0

known as the cellular chain complex of X, which we denote by (C∗(X), ∂∗). We note that we have extended the sequence by 0 at the right end as the boundary of 0-cells is empty, therefore ∂0 = 0. The sequence of groups Cn(X) for n = 0, 1,... are called the cellular chain groups and the sequence of homomorphisms ∂n for n = 0, 1,... are called the cellular boundary homomorphisms of the cellular chain complex (C∗(X), ∂∗).

Let Zn(X) = Ker ∂n be the group of n-cycles of X and Bn(X) = Im ∂n+1 be the group of n-boundaries. The homology groups of this cellular chain complex, namely

{Zn(X)/Bn(X): for n = 0, 1,...}

are called the cellular homology groups of X. Temporarily we will write Zn(X)/Bn(X) CW as Hn (X).

In the following Hn(X) denotes the n-th singular homology group.

CW ∼ Theorem 2.2.1 ([11] Theorem 2.35). Hn (X) = Hn(X).

We may and therefore will drop the “CW” superscript for the cellular homology groups of X.

Example 2.2.2. The cellular chain groups of the CW complex CP n given in Example 2.1.3 are of the form  i n  Z{e } for i = 2k and 0 ≤ k ≤ n Ci(CP ) =  0 otherwise. CHAPTER 2. CELL STRUCTURES 22

n n We observe that each of the boundary homomorphisms ∂i : Ci(CP ) → Ci−1(CP ) must be the zero map. The following result is then immediate:  i n  Z{[e ]} for i = 2k and 0 ≤ k ≤ n Hi(CP ) =  0 otherwise.

i i n where [e ] denotes the homology class of the cycle e ∈ Ci(CP ).

A CW-complex is regular if for each cell en such that n > 0, there exists a char-

n acteristic map Φα : Dα → X which is a homeomorphism onto its image. We refer the reader to [17, Chapter IX, §6, §7] for a dicussion of the properties of regular CW-complexes and the relative ease of determining the cellular boundary formula in comparison with general CW-complexes. A regular CW-complex is homeomorphic to a ∆-complex, which we introduce in the next section.

2.3 ∆-complexes

As previously remarked, any space homeomorphic to Dn can be used as the domain of a characteristic map of a CW-complex. By identifying the unit n-cube In with Dn we exploit the property that the product of two cubes is a cube when calculating the homology of the product of two CW-complexes. In this section we study the standard n-simplex ∆n and ∆-complexes. We will see that a ∆-complex is a special type of CW-complex that utilises the combinatorial structure of the topological boundary of ∆n. Much like regular CW-complexes, this gives ∆-complexes the additional virtue of allowing easier computation of the algebraic boundary.

2.3.1 Simplices

m The n-simplex is the smallest convex set in R containing n + 1 points v0, . . . , vn that do not lie in a hyperplane of dimension less than n; that is, the set of points Pn of the form t0v0 + ··· + tnvn where i=0 ti = 1 and ti ≥ 0 for all i. The points vi are the vertices of the simplex and the simplex itself is denoted [v0, . . . , vn]. If we let n+1 ei = (0,..., 0, 1i, 0,..., 0) ∈ R , where 1i denotes an entry of 1 in the (i + 1)-st n coordinate, then the standard n-simplex ∆ has {e0, . . . , en} as its set of vertices, that CHAPTER 2. CELL STRUCTURES 23 is,

n n+1 X ∆ = {(t0, . . . , tn) ∈ R : ti = 1, ti ≥ 0 for all i}. i The ordering on the set of vertices is very important for homology. Throughout this thesis the term ‘n-simplex’ should be interpreted as the ‘n-simplex with an ordering on its vertices’. As the vertices of ∆n are the unit vectors along the coordinate axes of

Rn+1, they will always be ordered according to the ordering of the coordinate axes. We will see that each n-simplex has a natural orientation based on its ordered vertices and this is why we denote the n-simplex with square brackets. Additionally we observe that ordering the vertices of an n-simplex determines a canonical linear homeomorphism

n ∆ → [v0, . . . , vn] given by

(t0, . . . , tn) 7→ Σitivi that preserves the order of the vertices.

The i-th face of [v0, . . . , vn] is defined as the (n − 1)-simplex obtained by deleting the vertex vi. We order the remaining vertices of a face according to their order in the larger simplex and denote it [v0,..., vî, . . . , vn], where thev î denotes vi has been n n removed from the sequence v0, . . . , vn. The boundary of ∆ , denoted by ∂∆ is the union of all faces of ∆n. The open simplex ∆˚n is ∆n − ∂∆n, the interior of ∆n. Before we define ∆-complexes we pause to fix the homeomorphism between ∆n and Dn. Mindful of Section 2.5, it is favourable to factor this homeomorphism through In.

Let I = [0, 1] ⊂ R and let In = I × I × · · · I ⊂ Rn for n ≥ 1. For n = 0 we let 0 n n I = 0 ∈ R. We deﬁne the map hn : ∆ → I such that   (0,..., 0) if (t0, t1, . . . , tn) = (1, 0,..., 0), hn(t0, t1, . . . , tn) =  λ(t1, . . . , tn) · (t1, . . . , tn) if (t0, t1, . . . , tn) 6= (1, 0,..., 0),

n for (t0, t1, . . . , tn) ∈ ∆ , where λ(t1, . . . , tn) is the scalar function given by

t1 + ··· + tn λ(t1, . . . , tn) = . max{t1, t2, . . . , tn}

n In particular we note that hn is the composite of the homeomorphism from ∆ to n the n-simplex [v0, . . . , vn] in R , where v0 = (0,..., 0) and vi = (0,..., 1i,... 0) for n i ∈ {1, . . . , n}, with the homeomorphism projecting [v0, . . . , vn] onto I . Clearly hn takes ∂∆n to ∂In. CHAPTER 2. CELL STRUCTURES 24

n n For n ≥ 1 we deﬁne the map gn : I → D such that  1 1  (0,..., 0) if (x1, . . . , xn) = ( 2 ,..., 2 ), gn(x1, . . . , xn) = 1 1  µ(x1, . . . , xn) · (x1, . . . , xn) if (x1, . . . , xn) 6= ( 2 ,..., 2 ),

n for (x1, . . . , xn) ∈ I , where µ(x1, . . . , xn) is the scalar function given by

max{|(2x1 − 1|,..., |2xn − 1)|} µ(x1, . . . , xn) = . |(2x1 − 1,..., 2xn − 1)|

1 1 n n Let Ie = [− 2 , 2 ] and Ie = Ie× · · · × Ie ⊂ R . It is clear to see that gn is the composite of the map translating In to Ien with radial projection of Ien onto Dn, mapping ∂In to n ∂D . For n = 0 we let g0(0) = 0. We conclude that both ∆n and In are homeomorphic to Dn. Whenever we use (∆n, ∂∆n) and (In, ∂In) as the domains of characteristic maps of a CW-complex we

n n n n n n will always mean to identify (∆ , ∂∆ ) with (D , ∂D ) via gn ◦ hn and (I , ∂I ) with n n (D , ∂D ) via gn.

2.3.2 ∆-complexes

We now introduce ∆-complexes as a restricted class of CW-complexes obtained by imposing certain restrictions upon the attaching maps of cells.

Deﬁnition 2.3.1. A ∆-complex is prescribed on a space X (which is always assumed

n to be Hausdorﬀ) by a family of characteristic maps σα : ∆ → X that satisfy the following:

˚n 1. Each σα is injective on ∆ ; hence each σα restricts to a homeomorphism from ˚n n ∆ onto its image, the cell ∆α ⊂ X, and these cells are all disjoint and their union is X.

n n−1 2. Each restriction of σα to a face of ∆ is one of the maps σβ : ∆ → X. Here we are identifying the face of ∆n with ∆n−1 by the canonical homeomorphism between them that preserves the ordering of the vertices.

3. A subset of X is closed if and only if it meets the closure of each cell of X in a closed set. CHAPTER 2. CELL STRUCTURES 25

Remark 2.3.2. In light of the comments at the end of Section 2.3.1 this is a slight modification of the definition for ∆-complexes found in [11] to mirror Definition 2.1.1. However from remarks in the aforementioned reference we can easily see that these two definitions are equivalent.

Chapter 5 includes a ∆-complex structure for the n-dimensional torus T n. Its description is combinatorial and to make it more accessible to the reader we provide a couple of low dimensional examples of such structures here. First we set up notation that will be used throughout this thesis.

Notation 2.3.3. Let S ⊆ [n]:={1, . . . , n} and U = [n]\S = {u1, . . . , un−|S|}. We will always write sets of integers in ascending order, that is, given si, sj ∈ S, then si ≤ sj if and only if i ≤ j. Although these sets are ordered and therefore actually sequences we will still use the set notation {· · · } being aware that it is an ordered set. We deﬁne

J = {J1,...,Jk} to be an ordered partition of S into k parts such that

J = {j1, . . . , j1 },J = {j2, . . . , j2 },...,J = {jk, . . . , jk }, 1 1 n1 2 1 n2 k 1 nk

p where jq ∈ [n] denotes the q-th element in the part Jp. Notice that n1 + ··· + nk = |S| and observe that S can be empty. In the case that S = ∅, then J = ∅.

n 2πix1 2πixn n Every point in T can be written as (e , . . . , e ), for some (x1, . . . , xn) ∈ I . k n We denote by ΦJ : ∆ → T the map deﬁned by

2πx1(t0,...,tk) 2πxn(t0,...,tk) (t0, . . . , tk) 7→ (e , . . . , e ),

k where xa : ∆ → I is given as:   tk+1−i + ··· + tk if a ∈ Ji xa(t0, . . . , tk) =  0 if a ∈ U.

When the context is clear we will write xa(t0, . . . , tk) as xa. For convenience we will ¯ k ˚k k denote the image of ΦJ by ∆J and the image of ΦJ |∆ by ∆J .

¯ k n Note 2.3.4. We see that ∆J is exactly the set of points in T of the form CHAPTER 2. CELL STRUCTURES 26

2πix1 2πixn (e , . . . , e ) such that x1, . . . , xn satisfy the following set of inequalities:

0 = xu = ··· = xu ≤ x 1 = x 1 = ··· = xj1 1 n−|S| j1 j2 n1

≤ x 2 = x 2 = ··· = xj2 j1 j2 n2 ≤ · · ·

≤ x k−1 = x k−1 = ··· = x k−1 j1 j2 jnk−1

≤ xjk = xjk = ··· = xjk 1 2 nk ≤ 1. (2.3.5)

k Note that ∆J corresponds to those points given by turning the inequalities in (2.3.5) into strict inequalities. Similarly the image of the restriction of ΦJ to the i-th face of ∆k corresponds to those points given by turning the (k +1−i)-th inequality symbol in

0 n (2.3.5) to an equality. When J = ∅, the image of ΦJ : ∆ → T is the point (1,..., 1).

Remark 2.3.6. Implicit in the deﬁnition of the maps ΦJ is the fact that each ΦJ n factors through the unit n-cube I . The motivation for considering the maps ΦJ is n n the canonical subdivision of I given by observing that all points (x1, . . . , xn) ∈ I lie in a set of the form

n (x1, . . . , xn) ∈ I : 0 ≤ xρ(1) ≤ · · · ≤ xρ(n) ≤ 1 for some permutation ρ ∈ Sym(n). Such a subset of In is homeomorphic to an n- simplex. The n! diﬀerent permutations provide a subdivision of In into n! n-simplices that is a ∆-complex structure on In. This decomposition of In is known as the Coxeter-Freudenthal-Kuhn triangulation, [6],[9],[16]. However for brevity we will call this decomposition the Kuhn triangulation. Given a homeomorphism h: In → X ⊂

Rn, we can obtain a ∆-complex structure for X by composing the characteristic maps of the Kuhn triangulation with h. We call such a ∆-complex on X a Kuhn cube. In Chapter 5 we give a ∆-complex structure on T n that is the image of In subdivided into Ln Kuhn cubes, for some positive integer L, under the quotient map q : In → T n that identiﬁes opposite pairs of faces of the cube together. When L = 1, we will see that the family of maps ΦJ , where J ranges over all ordered partitions of all subsets S ⊆ {1, . . . , n}, are the characteristic maps of such a ∆-complex on T n. Therefore we present the following examples without proof. CHAPTER 2. CELL STRUCTURES 27

2 2 2πix1 2πix2 Example 2.3.7 (T ). In this case we write points in T as (e , e ) for (x1, x2) ∈ 2 k 2 I and note S ⊆ [2] and U = [2] \ S. The maps ΦJ : ∆ → T are the characteristic maps for a ∆-complex structure on T 2. We can summarise this information in the

2 following table, where the third column corresponds to the conditions on (x1, x2) ∈ I

k 2πix1 2πix2 when writing ∆J as a set of points of the form (e , e ). From Note 2.3.4 the k boundaries of each cell ∆J can easily be deduced and are given in the fourth column.

k k J Conditions on (x1, x2) ∂∆J

0 ∅ 0 = x1 = x2 ∅

0 1 {{1}} 0 = x2 < x1 < 1 ∆∅

0 1 {{2}} 0 = x1 < x2 < 1 ∆∅

0 1 {{1, 2}} 0 < x1 = x2 < 1 ∆∅

¯ 1 ¯ 1 ¯ 1 2 {{1}, {2}} 0 < x1 < x2 < 1 ∆{{1}} ∪ ∆{{1,2}} ∪ ∆{{2}}

¯ 1 ¯ 1 ¯ 1 2 {{2}, {1}} 0 < x2 < x1 < 1 ∆{{2}} ∪ ∆{{1,2}} ∪ ∆{{1}}

3 3 2πix1 2πix2 2πix3 Example 2.3.8 (T ). Points in T are written as (e , e , e ) for (x1, x2, x3) ∈ 3 k 3 I with S ⊆ [3] and U = [3] \ S. The maps ΦJ : ∆ → T are the characteristic maps for a ∆-complex structure on T 3. Again we can summarise this information in the following table, with the boundaries in the fourth column following from Note 2.3.4. CHAPTER 2. CELL STRUCTURES 28

k k J Conditions on (x1, x2, x3) ∂∆J

0 ∅ 0 = x1 = x2 = x3 ∅

0 1 {{1}} 0 = x2 = x3 < x1 < 1 ∆∅

0 1 {{2}} 0 = x1 = x3 < x2 < 1 ∆∅

0 1 {{3}} 0 = x1 = x2 < x3 < 1 ∆∅

0 1 {{1, 2}} 0 = x3 < x1 = x2 < 1 ∆∅

0 1 {{1, 3}} 0 = x2 < x1 = x3 < 1 ∆∅

0 1 {{2, 3}} 0 = x1 < x2 = x3 < 1 ∆∅

0 1 {{1, 2, 3}} 0 < x1 = x2 = x3 < 1 ∆∅

¯ 1 ¯ 1 ¯ 1 2 {{1}, {2}} 0 = x3 < x1 < x2 < 1 ∆{{1}} ∪ ∆{{1,2}} ∪ ∆{{2}}

¯ 1 ¯ 1 ¯ 1 2 {{2}, {1}} 0 = x3 < x2 < x1 < 1 ∆{{2}} ∪ ∆{{1,2}} ∪ ∆{{1}}

¯ 1 ¯ 1 ¯ 1 2 {{1}, {3}} 0 = x2 < x1 < x3 < 1 ∆{{1}} ∪ ∆{{1,3}} ∪ ∆{{3}}

¯ 1 ¯ 1 ¯ 1 2 {{3}, {1}} 0 = x2 < x3 < x1 < 1 ∆{{3}} ∪ ∆{{1,3}} ∪ ∆{{1}}

¯ 1 ¯ 1 ¯ 1 2 {{2}, {3}} 0 = x1 < x2 < x3 < 1 ∆{{2}} ∪ ∆{{2,3}} ∪ ∆{{3}}

¯ 1 ¯ 1 ¯ 1 2 {{3}, {2}} 0 = x1 < x3 < x2 < 1 ∆{{3}} ∪ ∆{{2,3}} ∪ ∆{{2}}

¯ 1 ¯ 1 ¯ 1 2 {{1, 2}, {3}} 0 < x1 = x2 < x3 < 1 ∆{{1,2}} ∪ ∆{{1,2,3}} ∪ ∆{{3}}

¯ 1 ¯ 1 ¯ 1 2 {{3}, {1, 2}} 0 < x3 < x1 = x2 < 1 ∆{{3}} ∪ ∆{{1,2,3}} ∪ ∆{{1,2}}

¯ 1 ¯ 1 ¯ 1 2 {{1, 3}, {2}} 0 < x1 = x3 < x2 < 1 ∆{{1,3}} ∪ ∆{{1,2,3}} ∪ ∆{{2}}

¯ 1 ¯ 1 ¯ 1 2 {{2}, {1, 3}} 0 < x2 < x1 = x3 < 1 ∆{{2}} ∪ ∆{{1,2,3}} ∪ ∆{{1,3}}

¯ 1 ¯ 1 ¯ 1 2 {{2, 3}, {1}} 0 < x2 = x3 < x1 < 1 ∆{{2,3}} ∪ ∆{{1,2,3}} ∪ ∆{{1}}

¯ 1 ¯ 1 ¯ 1 2 {{1}, {2, 3}} 0 < x1 < x2 = x3 < 1 ∆{{1}} ∪ ∆{{1,2,3}} ∪ ∆{{2,3}}

¯ 2 ¯ 2 ¯ 2 ¯ 2 3 {{1}, {2}, {3}} 0 < x1 < x2 < x3 < 1 ∆{{1},{2}} ∪ ∆{{1},{2,3}} ∪ ∆{{1,2},{3}} ∪ ∆{{2},{3}}

¯ 2 ¯ 2 ¯ 2 ¯ 2 3 {{1}, {3}, {2}} 0 < x1 < x3 < x2 < 1 ∆{{1},{3}} ∪ ∆{{1},{2,3}} ∪ ∆{{1,3},{2}} ∪ ∆{{3},{2}}

¯ 2 ¯ 2 ¯ 2 ¯ 2 3 {{2}, {1}, {3}} 0 < x2 < x1 < x3 < 1 ∆{{2},{1}} ∪ ∆{{2},{1,3}} ∪ ∆{{1,2},{3}} ∪ ∆{{1},{3}}

¯ 2 ¯ 2 ¯ 2 ¯ 2 3 {{2}, {3}, {1}} 0 < x2 < x3 < x1 < 1 ∆{{2},{3}} ∪ ∆{{2},{1,3}} ∪ ∆{{2,3},{1}} ∪ ∆{{3},{1}}

¯ 2 ¯ 2 ¯ 2 ¯ 2 3 {{3}, {1}, {2}} 0 < x3 < x1 < x2 < 1 ∆{{3},{1}} ∪ ∆{{3},{1,2}} ∪ ∆{{1,3},{2}} ∪ ∆{{1},{2}}

¯ 2 ¯ 2 ¯ 2 ¯ 2 3 {{3}, {2}, {1}} 0 < x3 < x2 < x1 < 1 ∆{{3},{2}} ∪ ∆{{3},{1,2}} ∪ ∆{{2,3},{1}} ∪ ∆{{2},{1}} CHAPTER 2. CELL STRUCTURES 29 2.4 Cellular chain complex of a ∆-complex

As already noted, we will deﬁne a CW-structure on T n that is a ∆-complex. In this section we observe that for ∆-complexes the cellular boundary homomorphism of the associated cellular chain complex takes a much simpler form than for general CW-complexes.

Each characteristic map σα gives the following induced map

n n n n−1 (σα)∗ : Hn(∆ , ∂∆ ) → Hn(X ,X )

n in homology. As described in Section 2.3.1, the vertices, e0, . . . , en, of ∆ are ordered according to the ordering of the standard basis of Rn+1. The motivation for specifying an ordering on the vertices lies in the following lemma.

Lemma 2.4.1 ([8], §4, Example 4.3). The degree of a linear map β : (∆n, ∂∆n) → (∆n, ∂∆n) which permutes the vertices equals the signature of the permutation; that is deg(β) = sign(β|{v0, v1, . . . , vn}).

n n Therefore, the induced endomorphism β∗ of Hn(∆ , ∂∆ ) is given by β∗(x) = sign(β|{v0, v1, . . . , vn}) · x = ±x We conclude that the ordering on the vertices of ∆n ﬁxes a generator [∆n] of

n n Hn(∆ , ∂∆ ), and we deﬁne the cellular chain groups as

n n Cn(X) = Z{∆α = (σα)∗[∆ ]: α ∈ An},

where An denotes the indexing set of the n-cells of X. With these orientation conven- n tions for ∆ the cellular boundary homomorphisms ∂n : Cn(X) → Cn−1(X) are deﬁned by n n X i n−1 ∂n∆α = (−1) (σβi )∗[∆ ], i=0 n−1 where the map σβi : ∆ → X is the restriction σα |[v0,...,vˆi,...,vn] (see [19], for example). In Chapters 7 and 8 we determine generating cycles for the non-trivial integral homology groups of weighted projective space and stunted weighted projective space respectively. We achieve this by making observations regarding generating cycles of the n-th dimensional integral homology groups of T n, and following on from Examples 2.3.7 and 2.3.8 we now provide some low dimensional examples of such observations. CHAPTER 2. CELL STRUCTURES 30

Example 2.4.2. The cellular chain groups of the ∆-complex given for T 2 in Example 2.3.7 are of the form  k 2  Z{∆J } for 0 ≤ k ≤ 2 and all J = {J1,...,Jk} Ck(T ) =  0 otherwise

2 2 and the boundary homomorphism ∂k : Ck(T ) → Ck−1(T ) on the basis elements of 2 Ck(T ) takes the following values:

k J ∂k∆J

∅ 0

{{1}} 0

{{2}} 0

{{1, 2}} 0

1 1 1 {{1}, {2}} +∆{{1}} − ∆{{1,2}} + ∆{{2}}

1 1 1 {{2}, {1}} +∆{{2}} − ∆{{1,2}} + ∆{{1}}

2 2 2 To calculate the homology groups we observe ∆{{1},{2}} − ∆{{2},{1}} ∈ Z2(T ) and as 2 2 there are no 3-simplices then Z2(T ) = H2(T ). Although the lower dimensional homology groups are not used in the considerations of Chapters 7 and 8, with such a simple CW-structure on T 2 we will quickly determine them here. Considering

2 1 1 1 1 1 1 C1(T ) = Z{∆{{1}}, ∆{{1,2}}, ∆{{2}}}, we see that ∆{{1}} − ∆{{1,2}} + ∆{{2}} ∈ Im ∂2, 2 1 1 therefore H1(T ) = Z ⊕ Z with basis the homology classes [∆{{1}}] and [∆{{2}}]. As 2 0 ∂1 = 0 then we see that H0(T ) = Z, with basis the homology class [∆∅].

Example 2.4.3. Considering Example 2.3.8 we see the cellular chain groups are of the form  k 3  Z{∆J } for 0 ≤ k ≤ 3 and all J = {J1,...,Jk} Ck(T ) =  0 otherwise

3 3 and the boundary homomorphism ∂k : Ck(T ) → Ck−1(T ) on the basis elements of 3 Ck(T ) is given in the following table: CHAPTER 2. CELL STRUCTURES 31

k J ∂∆J

∅ 0

{{1}} 0

{{2}} 0

{{3}} 0

{{1, 2}} 0

{{1, 3}} 0

{{2, 3}} 0

{{1, 2, 3}} 0

1 1 1 {{1}, {2}} +∆{{1}} − ∆{{1,2}} + ∆{{2}}

1 1 1 {{2}, {1}} +∆{{2}} − ∆{{1,2}} + ∆{{1}}

1 1 1 {{1}, {3}} +∆{{1}} − ∆{{1,3}} + ∆{{3}}

1 1 1 {{3}, {1}} +∆{{3}} − ∆{{1,3}} + ∆{{1}}

1 1 1 {{2}, {3}} +∆{{2}} − ∆{{2,3}} + ∆{{3}}

1 1 1 {{3}, {2}} +∆{{3}} − ∆{{2,3}} + ∆{{2}}

1 1 1 {{1, 2}, {3}} +∆{{1,2}} − ∆{{1,2,3}} + ∆{{3}}

1 1 1 {{3}, {1, 2}} +∆{{3}} − ∆{{1,2,3}} + ∆{{1,2}}

1 1 1 {{1, 3}, {2}} +∆{{1,3}} − ∆{{1,2,3}} + ∆{{2}}

1 1 1 {{2}, {1, 3}} +∆{{2}} − ∆{{1,2,3}} + ∆{{1,3}}

1 1 1 {{2, 3}, {1}} +∆{{2,3}} − ∆{{1,2,3}} + ∆{{1}}

1 1 1 {{1}, {2, 3}} +∆{{1}} − ∆{{1,2,3}} + ∆{{2,3}}

2 2 2 2 {{1}, {2}, {3}} +∆{{1},{2}} − ∆{{1},{2,3}} + ∆{{1,2},{3}} − ∆{{2},{3}}

2 2 2 2 {{1}, {3}, {2}} +∆{{1},{3}} − ∆{{1},{2,3}} + ∆{{1,3},{2}} − ∆{{3},{2}}

2 2 2 2 {{2}, {1}, {3}} +∆{{2},{1}} − ∆{{2},{1,3}} + ∆{{1,2},{3}} − ∆{{1},{3}}

2 2 2 2 {{2}, {3}, {1}} +∆{{2},{3}} − ∆{{2},{1,3}} + ∆{{2,3},{1}} − ∆{{3},{1}}

2 2 2 2 {{3}, {1}, {2}} +∆{{3},{1}} − ∆{{3},{1,2}} + ∆{{1,3},{2}} − ∆{{1},{2}}

2 2 2 2 {{3}, {2}, {1}} +∆{{3},{2}} − ∆{{3},{1,2}} + ∆{{2,3},{1}} − ∆{{2},{1}} CHAPTER 2. CELL STRUCTURES 32

3 3 As there are no 4-simplices then Z3(T ) = H3(T ). We observe that the parts of J in this case when k = 3 are all singleton sets, therefore J = {{ρ(1)}, {ρ(2)}, {ρ(3)}} for some ρ ∈ Sym(3). If we set J = {{1}, {2}, {3}} then any other ordered partition of the set [3] into three parts can be written in the form ρ(J) = {{ρ(1)}, {ρ(2)}, {ρ(3)}}.

3 By inspection we can see that Z3(T ) is generated by the 3-chain

X 3 sign(ρ)∆ρ(J), ρ hence (" #) 3 X 3 H3(T ) = Z sign(ρ)∆ρ(J) . ρ

2.5 Products of ∆-complexes

In Chapter 5, we consider a cell structure for the product of two ∆-complexes. The product of two simplices is not a simplex, therefore the product of two ∆-complexes will not be a ∆-complex. In this section we describe how to give a CW-structure to the product of two ∆-complexes X and Y , and deﬁne its associated cellular chain complex.

2.5.1 CW-complex on X × Y

The product of two simplices, ∆k and ∆`, is homeomorphic to Dk+`; under any such homeomorphism (∂∆k × ∆`) ∪ (∆k × ∂∆`) corresponds to the boundary Sk+`−1, and ∆˚k × ∆˚` is homeomorphic to int(Dk+`). Suppose X and Y are ∆-complexes, then

k ` X × Y has the structure of a CW-complex with cells the products ∆α × ∆β where k ` ∆α ranges over the cells of X and ∆β ranges over the cells of Y . Note that there are some point-topological subtleties involved in showing that X × Y is a CW-complex in a natural way as one would need to check the product topology on X × Y is the same as the weak topology determined by the closed cells. In Chapter 5 we will prescribe a

n n n n n CW-complex on the product T × ∆c , via CW-complexes on T and ∆c , where ∆c is n n defined below in Notation 2.5.2. The CW-structures we give for T and ∆c are finite, n n therefore the CW-structure we give for T × ∆c is also finite and there is nothing to prove. We use the following theorem (a weaker form of [[11], Theorem A.6]) to determine a CW-structure for the product of two finite ∆-complexes. CHAPTER 2. CELL STRUCTURES 33

Theorem 2.5.1. For ﬁnite ∆-complexes X and Y with characteristic maps Φα and

Ψβ, the product maps Φα ×Ψβ are the characteristic maps for a CW-complex structure on X × Y .

The n-skeleton of the CW-complex X × Y is

[ (X × Y )n = Xk × Y ` k+`=n

S n and X × Y = n(X × Y ) .

n n n Notation 2.5.2. We consider the product T × ∆c , where ∆c denotes the curvilinear n+1 n n+1 n-simplex; that is, the intersection of the positive cone, R+ with S ⊂ R , so

n n+1 2 2 ∆c = {(r0, . . . , rn) ∈ R :(r0) + ... + (rn) = 1, ri ≥ 0, i = 0, 1, . . . , n}.

n n Radial projection shows that ∆c is homeomorphic to ∆ . Let τ = {τ0, τ1, . . . , τ`} ⊆

{0, 1, . . . , n} emphasising that τ is an ordered sequence such that for τi, τj ∈ τ then ` n τi ≤ τj if and only if i ≤ j. We denote by iτ : ∆ → ∆c the map deﬁned by

(t0, . . . , t`) 7→ (ι0(t0, . . . , t`), . . . , ιn(t0, . . . , t`)),

` where ιa : ∆ → I is given by:  √  + tb if a = τb ιa(t0, . . . , t`) =  0 if a∈ / τ.

When the context is clear we will write ιa(t0, . . . , t`) as ιa. It is easy to see that the ` n n set of maps iτ : ∆ → ∆c , for all τ, are the characteristic maps of a ∆-complex on ∆c ˚` and we write ∆τ = iτ (∆ ). We see that {eτi : τi ∈ τ for all i ∈ {0, 1, . . . , `}} is the set of vertices of ∆τ . Therefore it follows that

` [ ¯ ∂∆τ = ∆τ\τi i=0 ¯ where ∆τ\τi denotes the closure of the cell ∆τ\τi . In particular, for the purposes of Chapter 6, which deﬁnes the cellular chain com-

n n n plex of the CW-structure on T × ∆c , we deﬁne the cellular chain group C`(∆c ) as

n ` C`(∆c ) = Z ∆τ = (iτ )∗[∆ ]: τ = {τ0, . . . , τ`} ⊆ {0, . . . , n} CHAPTER 2. CELL STRUCTURES 34

n n and observe that the cellular boundary homomorphism ∂` : C`(∆c ) → C`−1(∆c ) is given by

` X i ∂`∆τ = (−1) ∆τ\τi . i=0

2 2 2 2 Example 2.5.3 (T × ∆c ). In this case we write points in T × ∆c as

2πix1 2πix2 2 (e , e , r0, r1, r2) for (x1, x2) ∈ I and note S ⊆ [2] and U = [2] \ S. The k ` 2 2 maps ΦJ × iτ : ∆ × ∆ → T × ∆c are the characteristic maps for a CW-complex 2 2 k structure on T × ∆c . We denote by ∆J × ∆τ the image of (ΦJ × iτ ) |int(∆k×∆`). The k boundary of the cell ∆J × ∆τ is given by

k k k ∂{∆J × ∆τ } = ∂∆J × ∆τ ∪ ∆J × ∂∆τ

k where ∂(∆J ) is given in Example 2.3.7 and ∂(∆τ ) is given in Notation 2.5.2.

2.5.2 Cellular chain complex of X × Y

n n−1 Given a CW-complex X ×Y , we observe Hn((X ×Y ) , (X ×Y ) ) is the free abelian group with basis in one-to-one correspondence with the n-cells of X × Y . For each

k ` n-cell, ∆α × ∆β such that k + ` = n, there is a characteristic map

k ` k ` n n−1 Φα × Ψβ : (∆ × ∆ , ∂(∆ × ∆ )) → ((X × Y ) , (X × Y ) ) which induces

k ` k ` n n−1 (Φα × Ψβ)∗ : Hn(∆ × ∆ , ∂(∆ × ∆ )) → Hn((X × Y ) , (X × Y ) ) in homology. We recall that an orientation of ∆k was a choice of generator of the inﬁnite cyclic

k k k group Hk(∆ , ∂∆ ) and was denoted by [∆ ]. This was determined by ordering the k k k k vertices of ∆ . Equivalently we could orient ∆ via the homeomorphism hk : ∆ → I given at the end of Section 2.3.1, where Ik is oriented by ordering the basis vec-

k k k k tors of ambient R and we let [∆ ] ∈ Hk(∆ , ∂∆ ) be the unique element such that k k (hk)∗[∆ ] = [I ]. The beneﬁt of this is that there is a canonical choice of ordering of the basis elements of Rk and the orderings for Rk and R` give the ordering for Rk+` under the homeomorphism Rk ×R` = Rk+`. Then the product orientation for ∆k × ∆` k ` k ` k+` is induced by the homeomorphism hk × h` : ∆ × ∆ → I × I = I . The comments CHAPTER 2. CELL STRUCTURES 35 preceding Proposition 3B.1 in [11] show how our previous choice of orientation of ∆k

k k is in fact one such that (hk)∗[∆ ] = [I ]. Therefore we let

k ` k ` n n−1 ∆α × ∆β = (Φα × Ψβ)∗[∆ ] × [∆ ] ∈ Hn((X × Y ) , (X × Y ) ).

Note that

k ` k ` k ` ∆α × ∆β = (Φα × Ψβ)∗[∆ ] × [∆ ] = (Φα × Ψβ)∗(−[∆ ] × −[∆ ]) and

k ` k ` k ` −(∆α × ∆β) = (Φα × Ψβ)∗(−[∆ ] × [∆ ]) = (Φα × Ψβ)∗([∆ ] × −[∆ ]).

We deﬁne

k ` Ck+`(X × Y ) = Z{∆α × ∆β : for all α, β} and by adhering to these standard orientation conventions for product cells consistent with those outlined in [11] we take advantage of the following result.

Proposition 2.5.4 ([11], Proposition 3B.1). The boundary homomorphisms in the cellular chain complex (C∗(X×Y ), ∂∗) are determined by the boundary homomorphisms in the cellular chain complexes (C∗(X), ∂∗) and (C∗(Y ), ∂∗) via the formula

k ` k ` k k ` ∂k+`(∆α × ∆β) = ∂k∆α × ∆β + (−1) ∆α × ∂`∆β.

2 2 2 2 Example 2.5.5 (T × ∆c ). The cellular chain groups for the CW-complex T × ∆c given in Example 2.5.3 are of the form  k ` 2 2  Z{∆J × ∆τ } for n = k + `, all J and all τ. Cn(T × ∆c ) =  0 otherwise.

2 2 It follows from Proposition 2.5.4 that the boundary homomorphism ∂n : Cn(T ×∆c ) → 2 2 Cn−1(T × ∆c ) is determined by

k ` k ` k ` ∂n{∆J × ∆τ } = ∂k∆J × ∆τ + (−1) ∆J × ∂`∆τ

where k + ` = n and ∂k, ∂` are the boundary homomorphisms of the cellular chain 2 2 complexes C∗(T ) and C∗(∆c ) respectively. CHAPTER 2. CELL STRUCTURES 36 2.6 Group actions and G-complexes

Appealing to the theory of G-complexes, we take advantage of the description of weighted projective space P(χ) as the orbit space of a ﬁnite group action on CP n. Therefore this section is dedicated to introducing some basic concepts regarding group actions and CW-complexes that will be illustrative for prescribing a CW-structure on

P(χ).

Deﬁnition 2.6.1. Let G be a group and X a set. By a G-action on X we mean a map ϕ: G × X → X satisfying the following properties:

1. ϕ(e, x) = x for all x ∈ X, where e is the identity of G;

2. ϕ(g2, ϕ(g1, x)) = ϕ(g2g1, x) for all g1, g2 ∈ G and x ∈ X.

Suppose that G is a topological group, X is a topological space and ϕ: G×X → X is a continuous map satisfying the conditions 1 and 2 of Deﬁnition 2.6.1. Then we call X endowed with the action ϕ of G a G-space. If X is a G-space and x ∈ X, then the subspace

G(x) = {ϕ(g, x): g ∈ G} is called the orbit of x (under G). The orbits G(x) and G(y) of two points x, y ∈ X are either equal or disjoint. We let X/G denote the set whose elements are the orbits x∗ = G(x) of G on X; that is x∗ = y∗ if and only if x and y are in the same orbit. Let p: X → X/G denote the natural map taking x to its orbit x∗ = G(x). Then X endowed with the quotient topology is called the orbit space of X (with respect to G).

Lemma 2.6.2 ([14], Lemma 1.29). Let X be a G-space. Given g ∈ G, let ϕg : X → X be the map deﬁned by ϕg(x) = ϕ(g, x). Then ϕg is a homeomorphism of X.

We will now restrict attention to the case that G is a ﬁnite topological group with the discrete topology and X is a Hausdorﬀ. The theory in the remainder of the chapter may be stated in more general terms (see [7] and [14], for example). However this is not necessary for the purposes of this thesis. CHAPTER 2. CELL STRUCTURES 37

As with a G-space, a G-complex is a CW-complex acted upon by a group G in a way that respects the cell structure. The next deﬁnition follows from [[7], §§II Propositions 1.15 and 1.16].

Deﬁnition 2.6.3. Let X be a G-space and a CW-complex. We say that a ﬁnite group G acts cellularly on X if the following hold:

1. ϕg : X → X is cellular for all g ∈ G;

2. ϕg carries a cell to itself if and only if ϕg is the identity on that cell.

The CW-complex X with a cellular G-action is called a G-complex.

Remark 2.6.4. It follows from Lemma 2.6.2 that if X is a G-complex, then ϕg maps each cell homeomorphically onto its image cell; that is, the group action takes n-cells to n-cells.

The following proposition underpins much of our work on CW-structures for weighted projective space, but ﬁrst we require a standard result from the theory of transformation groups.

Lemma 2.6.5. Let G be a ﬁnite group and X a G-space. If X is Hausdorﬀ then so is X/G.

Proof. A proof of the more general case that G is a compact topological group can be found in [[14], Proposition 1.58].

Proposition 2.6.6. Let X be a G-complex, for ﬁnite group G. The orbit space X/G inherits the structure of a CW-complex from X. In particular, the quotient map

n p: X → X/G is a cellular map such that if eα is an n-cell of X, then the restric- n tion p|eα is a homeomorphism onto an n-cell of X/G.

n Proof. As G acts cellularly on X, it follows from Deﬁnition 2.6.3 that each cell eα is n mapped injectively and therefore homeomorphically onto its image p(eα). Note that n the orbit for each cell eα is a disjoint union of n-cells and the set of all orbits of cells in X form a partition of X. Therefore we can choose a representative cell en of the αe n orbit for each cell eα of X and compose its characteristic map with p to obtain a set of characteristic maps for a CW-structure on X/G. CHAPTER 2. CELL STRUCTURES 38

Let X and Y be two G-spaces. A map f : X → Y is called a G-equivariant map if f(ϕ(g, x)) = ϕ(g, f(x)) for all g ∈ G and all x ∈ X. A basic property of a map f : X → Y being G-equivariant is that there exists an induced map f˜: X/G → Y/G deﬁned by f˜((x)∗) = (f(x))∗ such that the following diagram is commutative

f X / Y

f˜ X/G / Y/G.

Suppose that X and Y are G-complexes such that the map f : X → Y is G-equivariant and cellular. We call such a map a map of G-complexes.

Remark 2.6.7. Given a map of G-complexes f : X → Y , it follows from Proposition 2.6.6 that there exist CW-complexes on the orbit spaces X/G and Y/G such that the induced map f˜: X/G → Y/G is cellular.

Note 2.6.8. When ϕ is understood from the context, we will often use g ·x for ϕ(g, x) and g for ϕg.

1 2πi/m Example 2.6.9. For a positive integer m, we set ζm := e and write the cyclic i group Cm of order m as {ζm : i = 0, 1, . . . , m − 1}. Given integers l1, . . . , ln relatively 2n−1 n prime to m, we deﬁne a Cm-action on S ⊂ C by

l1 ln ζm · (z1, . . . , zn) = (ζmz1, . . . , ζm zn).

2n−1 2n−1 The action of Cm on S is free: only the identity element ﬁxes any point of S

2n−1 lj since each point of S has at least one non-zero coordinate, zj say, therefore ζmzj 6= 2n−1 zj. The orbit space S /Cm is denoted L(m, l1, . . . , ln) and is called the lens space. 2n−1 n Given any point (z1, . . . , zn) ∈ S ⊂ C we deﬁne τ = {τ0, . . . , τ`}:={j : zj 6= 0}.

We observe that the point (z1, . . . , zn) can be written in the form

m1 2πix1/m mn 2πixn/m (r1ζm e , . . . , rnζm e ) where

1. r , . . . , r are such that r ≥ 0 for 1 ≤ j ≤ n and P` r2 = 1; 1 n j j=1 τj

n 2. x1, . . . , xn are such that (x1, . . . , xn) ∈ I ; CHAPTER 2. CELL STRUCTURES 39

3. m1, . . . , mn is a sequence of integers such that 0 ≤ mj < m for 1 ≤ j ≤ n and

mj = 0 if j∈ / τ.

2n−1 We see that {(r1, . . . , rn):(r1, . . . , rn) ∈ S } can be identiﬁed with the restriction n n to the n-th face of ∆c ; that is, the set of points in ∆c where r0 = 0. Following this observation and recalling Notation 2.3.3 and Notation 2.5.2,we let τ = {τ0, . . . , τ`} ⊆ {1, . . . , n} and J be an ordered partition of τ into k parts. Furthermore we let m denote the n-tuple (m1, . . . , mn). m k ` 2n−1 We let FJ,τ : ∆ × ∆ → S denote the map deﬁned by

0 0 m1 2πix1/m mn 2πixn/m (t0, . . . , tk, t0, . . . , t`) 7→ (ι1ζm e , . . . , ιnζm e ),

0 0 where xj(t0, . . . , tk) and ιj(t0, . . . , t`) are deﬁned in Notation 2.3.3 and Notation 2.5.2. m Observe that τ determines which coordinates in the image of FJ,τ are non-zero. We m will see in Chapter 8 that the family of maps FJ,τ for all τ, J and m satisfying the properties given in the previous paragraph are a family of characteristic maps for a

2n−1 m m k ` CW-complex on S . We let eJ,τ = im{FJ,τ |int(∆ × ∆ )} denote a (k + `)-cell of this CW-structure.

i Given ζm ∈ Cm, we consider the map

i 2n−1 2n−1 ζm : S → S

m1 2πix1/m mn 2πixn/m i·l1+m1 2πix1/m i·ln+mn 2πixn/m (r1ζm e , . . . , rnζm e ) 7→ (r1ζm e , . . . , rnζm e ).

me 1 me n If we let me = (ζL , . . . , ζL ) such that   i · lj + mj mod m j ∈ τ me j =  0 otherwise and 0 ≤ me j < m for 1 ≤ j ≤ n then we observe that

i m me ζm ◦ FJ,τ = FJ,τ .

i m me i m m In particular ζm · eJ,τ = eJ,τ and whenever ζm · eJ,τ = eJ,τ it is clear that the restriction i m 2n−1 of ζm to eJ,τ is the identity map. Therefore the CW-complex structure S is a

Cm-complex. 2n−1 Denoting the quotient map p: S → L(m, l1, . . . , ln) as the map given by

m1 2πix1/m mn 2πixn/m m1 2πix1/m mn 2πixn/m ∗ p(r1ζm e , . . . , rnζm e ) = (r1ζm e , . . . , rnζm e ) , CHAPTER 2. CELL STRUCTURES 40

m ∗ m m ∗ we let (eJ,τ ) = p(eJ,τ ). It follows from Proposition 2.6.6 that (eJ,τ ) is a (k + `)-cell of a CW-structure on L(m, l1, . . . , ln).

m m 2n−1 me m Let Cm(eJ,τ ) denote the orbit of the cell eJ,τ of S . Then eJ,τ ∈ Cm(eJ,τ ),

me m ∗ m me therefore p(eJ,τ ) = (eJ,τ ) . We see that either p ◦ FJ,τ or p ◦ FJ,τ could be used as a characteristic map for the CW-structure on L(m, l1, . . . , ln). Therefore describing the

CW-structure on L(m, l1, . . . , ln) explicitly requires choosing a representative cell in S2n−1 of each orbit. Such considerations will be the subject of Chapter 4. Chapter 3

Weighted projective space

One of the main aims of this thesis is to give a CW-structure for stunted weighted projective space, which enables us to identify generators of the non-trivial integral homology groups in terms of cycles in the corresponding cellular chain complex. We ﬁnd a solution to this task via the cellular chain complex associated to a CW-structure we prescribe on weighted projective space. In this chapter we recall deﬁnitions and give a brief survey of properties and results regarding weighted projective space, weighted lens space and other associated constructions that are relevant to this problem. The main references for this chapter are [2] and [15].

3.1 Preliminary deﬁnitions and properties

We let S1 denote the circle as a topological space, and T < C1 its realisation as the group of unimodular complex numbers with respect to multiplication. The standard action of the (n + 1)-dimensional torus T n+1 on Cn+1 is by coordinate multiplication and restricts to the unit sphere S2n+1 ⊂ Cn+1. The orbit space of the latter is the n n+1 2n+1 n curviliner simplex ∆c ⊂ R+ , and the quotient map r : S → ∆c is given by r(z) = (|z0|,..., |zn|).

A weight vector χ = (χ0, . . . , χn) is a ﬁnite sequence of positive integers. Any weight vector χ determines a subcircle T hχi < T n+1 by

T hχi = {(tχ0 , . . . , tχn ): t ∈ T },

χ0 χn which is a T -space with respect to the action given by t·(s1, . . . , sn) = (t s1, . . . , t sn)

41 CHAPTER 3. WEIGHTED PROJECTIVE SPACE 42

1 for (s1, . . . , sn) ∈ T hχi and t ∈ S . We note that although T hχi and T hdχi for some positive integer d > 1 there is no homeomorphism that respects the T -action, therefore T hχi and T hdχi are diﬀerent as T -spaces.

Deﬁnitions 3.1.1. The weighted projective space P(χ) is the orbit space of the action of T hχi on S2n+1.

2n+1 Notice that the action of T hχi on S is free if and only if χ0 = ··· = χn = 1, in which case T hχi is the diagonal torus Tδ and P(χ) reduces to standard complex projective space CP n. For any positive integer d, T hdχi and T hχi produce homeomorphic orbit spaces. We observe that the action of T hχi on S2n+1 has ﬁnite isotropy groups and therefore P(χ) is an orbifold, which is singular for n > 1, unless χ0 = ··· = χn = d, for some positive integer d. We note that although P(χ) and P(dχ) are homeomorphic, they have diﬀerent orbifold structures.

n+1 n There is a canonical action of T /T hχi on P(χ), with orbit space ∆c . The map r factors through P(χ), that is, r is the composition of the quotient maps

2n+1 qχ pχ n S −−−→ P(χ) −−−→ ∆c . (3.1.2)

n+1 n n As T /T hχi can be identiﬁed with T and ∆c is homeomorphic to the standard n-simplex ∆n, we observe that P(χ) is a toric variety with quotient polytope ∆n. 2n+1 We will denote by [z0 : ... : zn]χ the equivalence class of points in S identiﬁed to (z0, . . . , zn) by the map qχ. In the case χ = (1,..., 1) we drop the subscript χ and follow the notation set up in Example 2.1.3 for points in CP n.

For any k + 1-element subset I = {i0, . . . , ik} ⊆ {0, . . . , n}, we follow Kawasaki by writing χI for the restriction of χ to I, in other words χI = (χi0 , . . . , χik ). So P(χI ) is identiﬁed with the subspace

PI (χ) := {[z0 : ... : zn]χ : zj = 0 for j∈ / I} ⊆ P(χ).

Deﬁnition 3.1.3. For any subset I ⊆ {0, . . . , n}, the stunted weighted projective space

P(χ; I) is the quotient space P(χ)/PI (χ).

If I = {0, . . . , k − 1} where 0 ≤ k ≤ n and χ0 = ··· = χn = 1 the the quotient n space P(χ; I) is traditionally written as CPk .

For any positive integer k we write Zk for the integers modulo k and Ck < T for its realisation as the subgroup generated by the primitive k-th root of unity. CHAPTER 3. WEIGHTED PROJECTIVE SPACE 43

Deﬁnition 3.1.4. For any positive integer k, the weighted lens space L(k; χ) is the

χ0 χn orbit space of the action of the weighted cyclic group Ckhχi := (Ck) × · · · (Ck) < T hχi on S2n+1.

If k is prime to χi for 0 ≤ i ≤ n, then L(k; χ) reduces to a standard lens space deﬁned in Example 2.6.9, which is smooth; otherwise, Ckhχi fails to act freely, and k L(k; χ) is singular. We will use the notation (z0, . . . , zn)χ for points in L(k; χ).

We observe there is a canonical action of the quotient circle T hχi/Ckhχi with orbit n+1 n space P(χ), and of the (n + 1)–torus T /Ck(χ) with orbit space ∆ .

3.2 Kawasaki’s results

In this thesis we build on Kawasaki’s work determining the integral cohomology ring structure of P(χ) in [15] to the case of stunted weighted projective space and so we state his results here.

n χ0 χn We consider the map φ(χ): CP → P(χ) given by φ[z0 : ... : zn] = [z0 : ... : zn ]χ, but before we state the ring structure of H∗(P(χ)) explicitly we recall the following deﬁnition.

χ Deﬁnition 3.2.1. Given a weight vector χ = (χ0, . . . , χn), the numbers lk are associated to χ and given by χ χi0 ··· χik lk = lcm : for all I = {i0, . . . , ik} ⊆ {0, . . . , n} . (χi0 , . . . , χik )

Here, (χi0 , . . . , χik ) denotes gcd{χi0 , . . . , χik }.

χ χ χ0···χn In particular l = lcm{χ0, . . . , χn} and l = . 1 n (χ0,...,χn)

∗ ∼ ∗ n Theorem 3.2.2. Additively, H (P(χ)) = H (CP ). For the multiplicative structure ∗ 2k 2 n of H (P(χ)), there exist generators γk ∈ H (P(χ)) and α ∈ H (CP ) such that ∗ χ k φ (γk) = lk α and ∗ ∼ H (P(χ); Z) = Z[γ1, . . . , γn]/Iχ , (3.2.3) where Iχ is the ideal generated by the elements

χ χ k l1 lk γ1 − χ γk lk+1 for 1 ≤ k ≤ n. CHAPTER 3. WEIGHTED PROJECTIVE SPACE 44

The proof Kawasaki gives for Theorem 3.2.2 shows that the integral homology groups H∗(P(χ)) are given by   if 0 ≤ j = 2k ≤ 2n ∼ Z Hj(P(χ); Z) = (3.2.4) 0 otherwise. and are therefore isomorphic to Hom(H∗(P(χ)); Z) by the Universal Coeﬃcient The- orem. Underpinning Kawasaki’s proof of Theorem 3.2.2 is his observation that there exists a coﬁbre sequence of the form [15, page 245]

f g L(χ ; χ ) −−→χ (χ ) −−→χ (χ) , (3.2.5) i bι P bι P such that for some i ∈ {0, . . . n} we denote by bι the set {0, . . . , n}\ i. Here fχ is the orbit map for the action of T hχ i/C hχ i on L(χ ; χ ) and g is the canonical inclusion. bι χi bι i bι χ Note that this is a generalisation of the cofibre sequence given by Kawasaki, however his proof clearly extends to show (3.2.5) is also a cofibre sequence. In particular, P(χ) is identified with the cofibre (χ ) ∪ CL(χ ; χ ) of f . It follows that g is also a P bι fχ i bι χ χ cofibration, with cofibre (χ ; ι), which is identified with the suspension ΣL(χ ; χ ). P i b i bι

More generally we observe that for any subset I = {i0, . . . , ik} ⊆ {0, . . . , n} the following sequence

gχ PI (χ) −−→ P(χ)−−→P(χ; I) , (3.2.6) is also a cofibre sequence. This sequence provides the basis for our work on stunted weighted projective space. In Chapter 5 we determine a CW-structure for P(χ) such that (P(χ), PI (χ)) is a CW-pair, therefore P(χ; I) has a canonical CW-structure given by collapsing the cells of PI (χ) ⊂ P(χ) to a point. We give this CW-structure in Chapter 8. These cell decompositions are such that (3.2.6) enables us to identify generators of the non-trivial integral homology groups of P(χ; I) via such identifications for P(χ). Following Kawasaki’s lead, we can apply a cofibre sequence of the form (3.2.5) to the augmented weight vector (k, χ) = (k, χ0, . . . , χn) to identify the integral cohomology CHAPTER 3. WEIGHTED PROJECTIVE SPACE 45 ring of L(k; χ) in terms of additive isomorphisms   if j = 2n + 1 Z  Hj(L(k; χ); ) ∼ (3.2.7) Z = Zq if j = 2k for 1 ≤ k ≤ n   0 otherwise,

(k,χ) χ where q = lj /lj .

3.3 Regarding cellular decompositions of P(χ)

Whitehead’s Theorem implies (see [11, Proposition 4C.1] for example) that any simply connected CW complex X with integral homology groups of the form (3.2.4) is homotopy equivalent to a CW-complex with a single cell e2i in each even dimension such that 0 ≤ i ≤ n. We will see in this section that if the weights obey certain divisibility conditions then P(χ) is in fact homeomorphic to such a CW-complex. For more general weights, we refer the reader to [5] for such a homotopy equivalence whose existence is proclaimed by Whitehead.

∗ Let χ and χ be two weight vectors such that χ = (χ0, . . . , χn) and ∗ χ = (χρ(0), . . . , χρ(n)), where ρ is a permutation on a set of (n + 1)-elements. We remark that there is a canonical homeomorphism between P(χ) and P(χ∗) given by permuting the basis vectors of Cn+1 according to ρ. Therefore without loss of generality, we can and will assume in this Section that weight vectors are written in increasing order, that is, χi ≤ χj for i ≤ j. The purpose of this is purely notational: in Sec- tion 3.3.1 we recall the established conventions of Example 2.1.3, which can be easily adapted to the content of Section 3.3.2. We emphasise that in general it will not be necessary to make such an assumption.

3.3.1 Complex projective space revisited

Example 2.1.3 gave a CW-structure for CP n with one cell e2k in every even dimension such that 0 ≤ 2k ≤ 2n. This was determined by showing that the maps Φk exhibit the properties required by Deﬁnition 2.1.1 and are therefore a family of characteristic maps for a CW-structure on CP n. However at the beginning of Chapter 2 an alternative description for CW-structures was given in terms of inductively adjoining cells. CHAPTER 3. WEIGHTED PROJECTIVE SPACE 46

We recall from Example 2.1.3

n 2n+1 n+1 CP = {(z0, . . . , zn) ∈ S ⊂ C }/ ∼ and the map

2n n Φn : D → CP

p 2 (z1, . . . zn) 7→ [ 1 − Σi|zi| : z1 : ... : zn]. is a homeomorphism from the interior of D2n to CP n − CP n−1 and on the boundary, 2n 2n−1 n−1 ∂D , it restricts to the map qn−1 : S → CP . Therefore we have

n n−1 2n CP = CP ∪qn−1 D .

2i By iteration and lettinge ¯ denote the image of Φi, we can write this cell structure

n 0 2 2n CP =e ¯ ∪ e¯ ∪ · · · ∪ e¯ .

3.3.2 Weighted projective space

From Deﬁnition 3.1.1 we see that, given a weight vector χ = (χ0, . . . , χn), we deﬁne weighted projective space as the quotient

2n+1 n+1 P(χ) = {(z0, . . . , zn) ∈ S ⊂ C }/ ≈

iχ0θ iχnθ iθ 1 where (z0, . . . , zn) ≈ (e z0, . . . , e zn) for e ∈ S . We have already noted that 2n+1 [z0 : ... : zn]χ ∈ P(χ) is the equivalence class of points in S identiﬁed to (z0, . . . , zn) 2n+1 by ≈. The quotient map qχ : S → P(χ) equips P(χ) with the quotient topology. 0 0 The inclusion P(χ ) ⊂ P(χ), where χ = (χ1, . . . , χn), is given by [z0 : ... : zn−1]χ0 7→ 0 [0 : z0 : ... : zn−1]χ. An arbitrary point in P(χ) − P(χ ) is represented by a point of 2n+1 the form (z0, . . . , zn) ∈ S for which z0 is non-zero. We consider the map

2n Φχ : D → P(χ)

p 2 (z1, . . . zn) 7→ [ 1 − Σi|zi| : z1 : ... : zn]χ.

2n 2 Assuming (z1, . . . , zn) ∈ int(D ), and so Σi|zi| 6= 1, we observe that (z1, . . . , zn)

χ1 χn and ζ · (z1, . . . , zn) := (ζ z1, . . . , ζ zn) have the same image under Φχ if and only if CHAPTER 3. WEIGHTED PROJECTIVE SPACE 47

χ0 ζ = 1; that is if and only if ζ ∈ Cχ0 . We deduce that Φχ factors through the quotient 2n D / ≈, where (z1, . . . , zn) ≈ ζ · (z1, . . . , zn) for ζ ∈ Cχ0 . The factorisation

2n Fχ : D / ≈ → P(χ) is a homeomorphism from its interior to P(χ) − P(χ0). 2n 2n−1 0 The boundary of D / ≈ is S / ≈ and can be identiﬁed with L(χ0; χ ).

Therefore on the boundary we see that the map Fχ restricts to the map

0 0 fχ : L(χ0; χ ) → P(χ ) of (3.2.5).

2n 0 0 The space D / ≈ is homeomorphic to the cone on L(χ0; χ ), denoted by CL(χ0; χ ), 2n 0 with the point (0,..., 0) ∈ D / ≈ corresponding to the vertex of CL(χ0; χ ). There- fore (as shown in Kawasaki’s proof that (3.2.5) is a coﬁbration sequence) we may write

P(χ) as 0 0 P(χ) = P(χ ) ∪fχ CL(χ0; χ ).

Inductively P(χ) can be given the structure of a q-CW complex, which is a more general notion of CW-complexes deﬁned by Poddar and Sarkar in [20]. In particular, P(χ) is obtained from P(χ0) by attaching the q-cell that is homeomorphic to the interior of the q-disc identiﬁed with D2n/ ≈.

For a certain subset of weights we can give an explicit CW-structure for P(χ) with 0 2n−1 one cell in every even dimension. We observe that L(χ0; χ ) is homeomorphic to S

χi if and only if ζ = 1, i.e. Cχ0 < Cχi for 1 ≤ i ≤ n. This can be equivalently written as the weights χ = (χ0, . . . , χn) having the property that χ0|χi for all i ∈ [n]. In such a situation we write

0 2n P(χ) = P(χ ) ∪fχ D

2n 2n−1 identifying D with CS . In particular, when χi−1|χi for 1 ≤ i ≤ n, then by iteration we obtain the cell structure

0 2 2n P(χ) =e ¯ ∪ e¯ ∪ · · · ∪ e¯ ,

2i wheree ¯ denotes the image of F(χn−i,...,χn). Weights with the property χi−1|χi for

1 ≤ i ≤ n are called divisive. Note that these structures for P(χ), both when χ0 | χi CHAPTER 3. WEIGHTED PROJECTIVE SPACE 48 for 1 ≤ i ≤ n and the divisive case appear in [5, Remark 3.2], albeit from a diﬀerent point of view.

We are concerned with determining CW-structures for P(χ) for general χ; that 0 2n−1 is, when L(χ0; χ ) is not homeomorphic to S . By the comments given in this section we observe that one approach we can take is to build a CW-structure for P(χ) 0 0 inductively from P(χ ) by adjoining a CW-structure for CL(χ0; χ ) chosen such that fχ is a cellular map. An alternative deﬁnition of P(χ) given in the next section presents a seemingly diﬀerent method for prescribing CW-structures on P(χ). However we then show that implicit within this method is the approach suggested above.

3.4 The way forward

Kawasaki observes that the preimage of a point in P(χ) under the map φ is an orbit of n the coordinate action of the product Cχ = Cχ0 ×· · ·×Cχn of cyclic groups on CP and n therefore CP /Cχ is homeomorphic to P(χ) . As Cχ is a finite group, it follows from n n Proposition 2.6.6 that a Cχ-complex on CP will give a CW-structure on CP /Cχ and therefore P(χ). Implicit in Theorem 3.2.2 is the result that the induced map on n χ homology φ∗ : H2k(CP ) → H2k(P(χ)) is multiplication by lk for 0 ≤ k ≤ n. Therefore n we will see that appealing to this description of P(χ) as CP /Cχ is more illuminating when we determine generating cycles for H2k(P(χ)). n n To find a Cχ-complex on CP we inspect the Cχ-action on CP more closely. Observing that it acts on the arguments of the homogenous coordinates of CP n, we see n ∼ n+1 that there is an associated Cχ-action on T = T /Tδ. There are many different ways n+1 ∼ n to define the isomorphism T /Tδ = T and we are required to fix one. Therefore n+1 n −1 −1 we let T /Tδ → T be the isomorphism given by [t0, t1, . . . , tn] 7→ (t0 t1, . . . , t0 tn) that allows us to define a T n-action on CP n that is coordinatewise multiplication on the last n-coordinates of points in CP n, with orbit space ∆n. In the following definitions we recall ζ1 = e2πi/χj and write elements of C as χj χ

a0 a1 an (ζχ0 , ζχ1 , . . . , ζχn ), where a0, . . . , an is a sequence of integers.

n n n Deﬁnition 3.4.1. The Cχ-action on T is the map ρχ : Cχ × T → T deﬁned by

a0 a1 an iθ1 iθn −a0 a1 iθ1 −a0 an iθn ρχ((ζχ0 , ζχ1 , . . . , ζχn ), (e , . . . , e )) = (ζχ0 ζχ1 e , . . . , ζχ0 ζχn e ). CHAPTER 3. WEIGHTED PROJECTIVE SPACE 49

n n n n n n Deﬁnition 3.4.2. The Cχ-action on T ×∆c is the map ρχ : Cχ×(T ×∆c ) → T ×∆c deﬁned by

a0 a1 an iθ1 iθn ρχ((ζχ0 , ζχ1 , . . . , ζχn ),(e , . . . , e , r0, . . . , rn))

−a0 a1 iθ1 −a0 an iθn = (ζχ0 ζχ1 e , . . . , ζχ0 ζχn e , r0, . . . , rn).

n n n Deﬁnition 3.4.3. The Cχ-action on CP is the map ρχ : Cχ × CP → CP deﬁned by

a0 a1 an −a0 a1 −a0 an ρχ((ζχ0 , ζχ1 , . . . , ζχn ), [z0 : ... : zn]) = [z0 : ζχ0 ζχ1 z1 : ... : ζχ0 ζχn zn].

Remark 3.4.4. We deﬁne Dχ as the image of Cχ under the following maps Cχ <

n+1 n+1 ∼ n a0 a1 an −a0 a1 −a0 an T → T /Tδ = T , that is (ζχ0 , ζχ1 , . . . , ζχn ) 7→ (ζχ0 ζχ1 , . . . , ζχ0 ζχn ). We n n ∼ observe that Dχ < T acts eﬀectively on CP , noting Cχ = Dχ if and only if gcd(χ0, . . . , χn) = 1 (that is, precisely when Tδ ∩ Cχ is trivial).

n n n n Notation 3.4.5. We let T /Cχ, T /Cχ × ∆c and CP /Cχ denote the orbit spaces n n n n of the Cχ-actions ρχ on the spaces T , T × ∆c and CP given in Deﬁnitions 3.4.1, n n n n 3.4.2 and 3.4.3. Points in T /Cχ, T /Cχ × ∆c and CP /Cχ will be denoted by

iθ1 iθn ∗ iθ1 iθn ∗ ∗ (e , . . . , e ) ,(e , . . . , e , r0, . . . , rn) and [z0 : ... : zn] respectively.

From these deﬁnitions we see that the inclusion

n n n i: T → T × ∆c

(eiθ1 , . . . , eiθn ) 7→ (eiθ1 , . . . , eiθn , 1, 0,..., 0) and the quotient map

n n n q : T × ∆c → CP

iθ1 iθn iθ1 iθn (e , . . . , e , r0, . . . , rn) 7→ [r0 : r1e : ... : rne ]

are Cχ-equivariant maps. Supposing i and q are maps of Cχ-complexes then, recalling Remark 2.6.7, the following is a commutative diagram of cellular maps:

n i n n q n T / T × ∆c / CP

n ˜i n n q˜ n T /Cχ / T /Cχ × ∆c / CP /Cχ CHAPTER 3. WEIGHTED PROJECTIVE SPACE 50 where ˜i andq ˜ are induced by the maps i and q respectively; the vertical maps are the

n n n n respective orbit maps of the Cχ-action on T , T × ∆c and CP given in Deﬁnitions 3.4.1, 3.4.2 and 3.4.3. As cellular maps induce chain maps, this diagram will be particularly useful in

Chapter 7 where the generators of H2k(P(χ)) for 0 ≤ k ≤ n are determined via n n observations made about the associated cellular chain complexes of T /Cχ × ∆c and CP n. However we ﬁrst return to the remarks given at the end of Section 3.3. Observe that the map q factors through D2n ⊂ Cn: q is the composition of the quotient

n n 2n iθ1 iθn iθ1 iθn map T × ∆c → D given by (e , . . . , e , r0, . . . , rn) 7→ (r1e , . . . , rne ) with the characteristic map Φn.

2n 2n 2n Deﬁnition 3.4.6. The Cχ-action on D is the map Cχ × D → D deﬁned by

a0 a1 an −a0 a1 −a0 an (ζχ0 , ζχ1 , . . . , ζχn )(z1, . . . , zn) = (ζχ0 ζχ1 z1, . . . , ζχ0 ζχn zn).

2n n n 2n It is easy to see that the Cχ-action on D is such that the maps T × ∆c → D 2n n and Φn : D → CP are Cχ-equivariant. 2n n Therefore, provided Φn is map of Cχ-complexes D and CP then the following

Φn D2n / CP n

2n Φen n D /Cχ / CP /Cχ is a commutative diagram of cellular maps.

n ∗ χ0 χn We let hχ : CP /Cχ → P(χ) given by [z0 : ... : zn] 7→ [z0 : ... : zn ] be the homeomorphism described at the beginning of this section. We deﬁne the map

0 2n 0 ∗ χ1 χn k hχ : D /Cχ → CL(χ0; χ ) by (z1, . . . , zn) 7→ (z1 , . . . , zn )χ0 , where the coordinates in the target values are understood to be normalised.

0 2n 0 Theorem 3.4.7. The map hχ : D /Cχ → CL(χ0; χ ) is a homeomorphism.

0 Proof. The map hχ is well-deﬁned and can be easily be seen to be surjective. For ∗ 0 0 0 ∗ 2n injectivity, we let z = (z1, . . . , zn) and z = (z1, . . . , zn) be points in D /Cχ such

0 0 0 a0 0 χ0 χia0 χ0 that hχ(z) = hχ(z ). Then there exists some ζχ0 ∈ Cχ0 such that (zi) = ζχ0 zi χ for all 1 ≤ i ≤ n. Writing (z0)χ0 as ζχia0 ζχiai z 0 for any ζai ∈ C , we see that i χ0 χi i χi χi z0 = ζa0 ζai z . Letting ζa0 = ζ−a˜0 , we conclude that z = z0. As D2n/C is compact and i χ0 χi i χ0 χ0 χ 0 0 CL(χ0; χ ) is Hausdorﬀ, we deduce that hχ is a homeomorphism. CHAPTER 3. WEIGHTED PROJECTIVE SPACE 51

2n n Therefore, given CW-structures on D /Cχ and CP /Cχ such that Φen is cellular, 0 we deduce that we can ﬁnd CW-structures for CL(χ0; χ ) and P(χ) such that

2n Φen n D /Cχ / CP /Cχ

0 hχ hχ

0 Fχ CL(χ0; χ ) / P(χ) is a commutative diagram of cellular maps.

n n n We will see in Chapter 8 that the structures we obtain for T × ∆c and CP 2n in Chapter 5 give rise to a Cχ-complex for D and therefore a CW-structure for 0 0 CL(χ0; χ ) with L(χ0; χ ) as a subcomplex and the restriction to this subcomplex fχ is cellular. In particular such arguments can be easily extended to the augmented weight vector (k; χ) to give a CW-structure on the weighted lens space L(k; χ). Chapter 4

n The Cχ-action on CP

Having settled upon an approach to the problem of ﬁnding a CW-complex structure for P(χ), this chapter deals with the requirements of prescribing a Cχ-complex on CP n. In particular, we recall that the cells of a CW-structure on a space must be n n disjoint. As CP and CP /Cχ are orbit spaces there is a choice of representative for each point. It is easier to ascertain whether a collection of cells is in fact disjoint if such a choice of representative has been made.

Notation 4.0.1. Recall from Example 2.1.3 that we write points of CP n as equivalence 2n+1 classes of points in S . Given a non-negative integer L, to each point [z0 : ... : n zn] ∈ CP we can associate a triple {rτ , x, l} where

1. rτ = {rτ0 , . . . , rτ` } := {rj = |zj|: |zj|= 6 0}

2. x = (x1, . . . , xn) is such that xj ∈ [0, 1) for all j ∈ τ \ τ0, otherwise xj = 0

3. l = (l1, . . . , ln) is such that lj ∈ {0,...,L − 1} for all j ∈ τ \ τ0, otherwise lj = 0 such that

lτ +1 0 2πixτ0+1/L ln 2πixn/L [z0 : ... : zn] = [0 : ... : 0 : rτ0 : rτ0+1ζL e : ... : rnζL e ].

0 The triple {rτ , x, l} is unique to the point [z0 : ... : zn] in that given a point [z0 : ... : 0 n 0 0 0 0 0 zn] ∈ CP with its associated triple {rτ 0 , x , l } then [z0 : ... : zn] = [z0 : ... : zn] if and 0 0 0 only if {rτ , x, l} = {rτ 0 , x , l }.

n Analogously, we wish to ﬁnd a unique way to write all points of CP /Cχ, which will in turn give a unique way to write all points of P(χ).

52 n CHAPTER 4. THE Cχ-ACTION ON CP 53

We note that it will also be beneﬁcial to describe points of T n and T n × ∆n in a similar way to Notation 4.0.1.

Notation 4.0.2. Given a non-negative integer L, to each point (e2πiθ1 , . . . , e2πiθn ) ∈ T n we can associate a pair {x, l} where

1. x = (x1, . . . , xn) such that xj ∈ [0, 1) for all j ∈ [n]

2. l = (l1, . . . , ln) such that lj ∈ {0,...,L − 1} for all j ∈ [n]. such that

2πiθ1 2πiθn l1 2πix1/L ln 2πixn/L (e , . . . , e ) = (ζL e , . . . , ζL e ).

We note that {x, l} is unique to the point (e2πiθ1 , . . . , e2πiθn ) in that given a point

2πiθ0 2πiθ0 n 0 0 (e 1 , . . . , e n ) ∈ T with its associated {x , l } then

2πiθ 2πiθ 2πiθ0 2πiθ0 (e 1 , . . . , e n ) = (e 1 , . . . , e n ) if and only if {x, l} = {x0, l0}.

2πiθ1 2πiθn n n Similarly to each point (e , . . . , e , r0, . . . , rn) ∈ T × ∆c , given a non- negative integer L, we can associate a triple {rτ , x, l} where

1. rτ = {rτ0 , . . . , rτ` } := {rj : rj 6= 0}

2. x = (x1, . . . , xn) such that xj ∈ [0, 1) for all j ∈ [n]

3. l = (l1, . . . , ln) such that lj ∈ {0,...,L − 1} for all j ∈ [n]. such that

2πiθ1 2πiθn l1 2πix1/L ln 2πixn/L (e , . . . , e , r0, . . . , rn) = (ζL e , . . . , ζL e , r0, . . . , rn).

2πiθ1 2πiθn We observe that the triple {rτ , x, l} is unique to the point (e , . . . , e , r0, . . . , rn).

n 4.1 Properties of the Cχ-action on CP

n We recall the Cχ-action on CP given in Deﬁnition 3.4.3 that acts on the arguments of the last n coordinates of points in CP n so that the following sequence of maps

n i n n q n T ,−−→ T × ∆c −−→ CP n CHAPTER 4. THE Cχ-ACTION ON CP 54

n n n are Cχ-equivariant with respect to the Cχ-actions on T and T × ∆c given in Deﬁni- tions 3.4.1 and 3.4.2. From this point forward we let L denote lcm(χ0, . . . , χn). Writing n n points in T in the form given in Notation 4.0.2, we observe that the Cχ-action on T n permutes elements of (CL) . Therefore we deﬁne the following.

n n n Deﬁnition 4.1.1. The Cχ-action on (CL) is the map Cχ × (CL) → (CL) deﬁned by

a0 a1 an l1 ln (ζχ0 , ζχ1 , . . . , ζχn )(ζL , . . . , ζL ) L L L L l1+ a1− a0 ln+ an− a0 χ1 χ0 χn χ0 = ζL , . . . , ζL . (4.1.2)

n n From this deﬁnition it is clear that the inclusion (CL) < T is Cχ-equivariant. We n n let (CL) /Cχ denote the orbit space of the Cχ-action on (CL) , and following the lead

l1 ln ∗ l1 ln of Notation 3.4.5 we let (ζL , . . . , ζL ) be the image of the point (ζL , . . . , ζL ) under n n the orbit map (CL) → (CL) /Cχ. n We will see that understanding this Cχ-action on (CL) is integral to understanding n the orbits of points in CP under the Cχ-action. However before this we are required to set up some notation.

Notation 4.1.3. Given τ ⊆ {0, . . . , n} we let

a0 an (Cχ)τ = {(ζχ0 , . . . , ζχn ) ∈ Cχ : aj = 0 for all j∈ / τ}.

Similarly we let

n b1 bn n (CL)τ\τ0 = {(ζL , . . . , ζL ) ∈ (CL) : bj = 0 for all j∈ / τ \ τ0}.

Deﬁnition 4.1.4. The (C ) -action on (C )n is the map (C ) × (C )n → χ τ L τ\τ0 χ τ L τ\τ0 (C )n deﬁned by L τ\τ0

a0 an b1 bn c1 b1 cn bn (ζχ0 , . . . , ζχn )(ζL , . . . , ζL ) = (ζL ζL , . . . , ζL ζL ) where  L L aj − aτ0 χj χτ0 ζ if j ∈ τ \ τ0 cj  L ζL = (4.1.5)  1 otherwise.

n Given any point [z0 : ... : zn] ∈ CP and therefore τ = {j : zj 6= 0}, the following theorem shows that there is a canonical way in which to write points in the orbit C [z : ... : z ] in terms of orbits of the (C ) -action on (C )n . χ 0 n χ τ L τ\τ0 n CHAPTER 4. THE Cχ-ACTION ON CP 55

0 0 n Theorem 4.1.6. Let [z0 : ... : zn] and [z0 : ... : zn] be two points in CP , with 0 0 0 their respective associated triples {rτ , x, l} and {rτ 0 , x , l }. Then [z0 : ... : zn] and 0 0 n 0 [z0 : ... : zn] are in the same orbit of the Cχ-action on CP if and only if rτ = rτ 0 , 0 0 0 0 l1 ln l1 ln x = x and l and l are such that (ζL , . . . , ζL ) and (ζL , . . . , ζL ) are in the same orbit of the (C ) -action on (C )n . χ τ L τ\τ0

Proof. From Notation 4.0.1 we observe

lτ +1 0 2πixτ0+1/L ln 2πixn/L [z0 : ... : zn] = [0 : ... : 0 : rτ0 : rτ0+1ζL e : ... : rnζL e ].

a0 an Let (ζχ0 , . . . , ζχn ) ∈ Cχ, then

h lτ +1 i a0 a1 an 0 2πixτ0+1/L ln 2πixn/L (ζχ0 ,ζχ1 , . . . , ζχn ) 0 : ... : 0 : rτ0 : rτ0+1ζL e : ... : rnζL e

h aτ aτ +1 lτ +1 i 0 0 0 2πixτ0+1/L an ln 2πixn/L = 0 : ... : 0 : rτ0 ζχτ0 : rτ0+1ζχτ0+1 ζL e : ... : rnζχn ζL e

h −aτ aτ +1 lτ +1 −aτ i 0 0 0 2πixτ0+1/L 0 an ln 2πixn/L = 0 : ... : 0 : rτ0 : rτ0+1ζχτ0 ζχτ0+1 ζL e : ... : rnζχτ0 ζχn ζL e .

. The triple for the point

−aτ aτ +1 lτ +1 −aτ 0 0 0 2πixτ0+1/L 0 an ln 2πixn/L [0 : ... : 0 : rτ0 : rτ0+1ζχτ0 ζχτ0+1 ζL e : ... : rnζχτ0 ζχn ζL e ]

˜ ˜ ˜ ˜ is of the form {rτ , x, l}, where l = (l1,..., ln) = (c1 mod L, . . . , cn mod L) and cj is determined by (4.1.5) for j ∈ {1, . . . , n}. 0 0 0 0 0 ˜ Therefore [z0 : ... : zn] = [z0 : ... : zn] if and only if {rτ 0 , x , l } = {rτ , x, l}, 0 0 0 l1 ln l1 ln which implies l is such that (ζL , . . . , ζL ) and (ζL , . . . , ζL ) are in the same orbit of the (C ) -action on (C )n . χ τ L τ\τ0

Theorem 4.1.6 shows that in order to determine canonical representatives for orbits

n of the Cχ-action on CP , we can reduce the task to determining canonical representatives for orbits of the (C ) -action on (C )n , for all τ ⊆ {0, . . . , n}. χ τ L τ\τ0

Remark 4.1.7. Given τ = {τ , . . . , τ } ⊆ {0, . . . , n} we let C = C × C × · · · × 0 ` χτ χτ0 χτ1 C and see that (C ) ∼= C where the isomorphism is given by χτ` χ τ χτ

a0 an aτ0 aτ` (ζ0 , . . . , ζn ) 7→ (ζτ0 , . . . , ζτ` ).

Similarly we let

b τ\τ0 bτ1 τ` (CL) = {(ζL , . . . , ζL ): bj ∈ {0,...,L − 1} for all j ∈ τ \ τ0} n CHAPTER 4. THE Cχ-ACTION ON CP 56 and see (C )n ∼= (C )τ\τ0 where the isomorphism is given by L τ\τ0 L

b1 bn bτ1 bτ` (ζ0 , . . . , ζn ) 7→ (ζτ1 , . . . , ζτ` ).

Therefore ﬁnding canonical representatives for the orbits of the (C ) -action on (C )n χ τ L τ\τ0

τ\τ0 is equivalent to ﬁnding them for the orbits of the Cχτ -action on (CL) .

In particular, it follows from Remark 4.1.7 that we need only consider ﬁnding

n canonical representatives for the orbits of the Cχ-action on (CL) , as any other case can be deduced from this one.

n 4.2 Canonical representatives for points in (CL) /Cχ

n n In this section we give canonical representatives in (CL) of points in (CL) /Cχ. From 0 0 l1 ln l1 ln n (4.1.2) we see that (ζL , . . . , ζL ) and (ζL , . . . , ζL ) ∈ (CL) are in the same orbit if 0 0 and only if there exists a sequence of integers a0, . . . , an such that l1, . . . , ln, l1, . . . , ln satisfy the following set of linear congruences

0 l1 ≡ l1 + a1 L/χ1 − a0 L/χ0 mod L

0 l2 ≡ l2 + a2 L/χ2 − a0 L/χ0 mod L . .

0 ln ≡ ln + an L/χn − a0 Lχ0 mod L.

Lemma 4.2.1. Given a sequence of integers a0, . . . , an such that L L L a0 ≡ a1 ≡ · · · ≡ an−1 mod L, (4.2.2) χ0 χ1 χn−1 and mn ∈ Z satisfying the linear congruence

mn ≡ an L/χn − a0 L/χ0 mod L (4.2.3) then L(χ0, . . . , χn) mn = · l χn(χ0, . . . , χn−1) for some l ∈ Z and n ≥ 2; and where (χ0, . . . , χn) denotes gcd(χ0, . . . , χn).

Proof. From the sequence of congruences (4.2.2) we see that equation (4.2.3) can be written in the form

mn ≡ an L/χn − ai L/χi mod L (4.2.4) n CHAPTER 4. THE Cχ-ACTION ON CP 57 for 0 ≤ i ≤ n − 1.

The linear congruence (4.2.4) has a solution if and only if gcd(L/χn, L/χi,L) =

L(χi,χn) divides mn for 0 ≤ i ≤ n − 1 and so (4.2.4) must be of the form χiχn L(χi, χn) L L mn = ki ≡ an − ai mod L χiχn χn χi for some ki ∈ Z. That is,

L(χ0, χn) L(χ1, χn) L(χn−1, χn) mn = k0 = k1 = ··· = kn−1 χ0χn χ1χn χn−1χn and so ( ) L(χ0, χn) L(χ1, χn) L(χn−1, χn) mn = lcm k0, k1,..., kn−1 · l χ0χn χ1χn χn−1χn for some l ∈ Z. We will prove L(χ , χ ) L(χ , χ ) L(χ , χ ) L(χ , . . . , χ ) lcm 0 n , 1 n ,..., n−1 n = 0 n . (4.2.5) χ0χn χ1χn χn−1χn χn(χ0, . . . , χn−1)

It is convenient to write χ0, . . . , χn in their prime power decompositions, and let ri denote the index of a given prime p in χi for 0 ≤ i ≤ n. Note that ri ≥ 0 for all i. Observe that for all i ∈ {0, . . . , n − 1} we have L(χ , χ ) L i n = χiχn lcm(χi, χn) and L(χ , . . . , χ ) L 0 n = . χn(χ0, . . . , χn−1) lcm((χ0, . . . , χn−1), χn) Therefore (4.2.5) reduces to give L L L lcm ,..., = . (4.2.6) lcm(χ0, χn) lcm(χn−1, χn) lcm((χ0, . . . , χn−1), χn) We observe that the total index of p in the LHS of (4.2.6) is given by n o max max{r0, . . . , rn} − max{r0, rn} ,..., max{r0, . . . , rn} − max{rn−1, rn} n o = max{r0, . . . , rn} + max − max{r0, rn} ,..., −max{rn−1, rn} n o = max{r0, . . . , rn} − max min{r0, rn} ,..., min{rn−1, rn} n o = max{r0, . . . , rn} − max min{r0, . . . , rn−1} , rn

which is the total index of p in the RHS of (4.2.6). n CHAPTER 4. THE Cχ-ACTION ON CP 58

Theorem 4.2.7. Let a0, . . . , an and m1, . . . , mn be integers such that

m1 ≡ a1 L/χ1 − a0 L/χ0 mod L,

m2 ≡ a2 L/χ2 − a0 L/χ0 mod L, . .

mn ≡ an L/χn − a0 L/χ0 mod L, then the only solution m1, . . . , mn such that

L(χ0, χ1) |m1| < , χ0χ1 L(χ0, χ1, χ2) |m2| < , χ2(χ0, χ1) . .

L(χ0, . . . , χi) |mi| < , χi(χ0, . . . , χi−1) . .

L(χ0, . . . , χn) |mn| < χn(χ0, . . . , χn−1) is m1 = m2 = ··· = mn = 0.

Proof. We prove this by induction. We have already seen in the proof of Lemma 4.2.1 that for there to be a solution to

m1 ≡ a1 L/χ1 − a0 L/χ0 mod L then L(χ0, χ1) m1 = k0 χ0χ1 for some k0 ∈ Z. For there to be a solution within the desired range we conclude then m1 = 0. Suppose that we are now looking for solutions to the congruences

m1 ≡ a1 L/χ1 − a0 L/χ0 mod L (4.2.8)

m2 ≡ a2 L/χ2 − a0 L/χ0 mod L within the range stipulated by the statement of Theorem 4.2.7. As we have just noted m1 = 0 and rearranging (4.2.8) gives

a0 L/χ0 ≡ a1 L/χ1 mod L. n CHAPTER 4. THE Cχ-ACTION ON CP 59

By Lemma 4.2.1 we see L(χ0, χ1, χ2) m2 = k1 χ2(χ0, χ1) for some k1 ∈ Z, therefore m2 = 0. This proves the base case of the induction. We assume that for n = k the statement is true. For the case n = k +1 we see that the inductive hypothesis means that m1 = ··· = mk = 0 and so the linear congruences become

m1 = 0 ≡ a1 L/χ1 − a0 L/χ0 mod L

m2 = 0 ≡ a2 L/χ2 − a0 L/χ0 mod L . .

mk = 0 ≡ ak L/χk − a0 L/χ0 mod L

mk+1 ≡ ak+1 L/χk+1 − a0 L/χ0 mod L.

This gives L L L a0 ≡ a1 ≡ · · · ≡ ak mod L χ0 χ1 χk and

mk+1 ≡ ak+1 L/χk+1 − a0 L/χ0 mod L.

Using Lemma 4.2.1 it follows that

L(χ0, . . . , χk+1) mk+1 = · l χk+1(χ0, . . . , χk)

L(χ0,...,χk+1) for some l ∈ . Therefore if |mk+1| < then mk+1 = 0 as required. Z χk+1(χ0,...,χk)

n Theorem 4.2.9. The orbit of each point in (CL) under the Cχ-action contains a

λ1 λn unique point of the form (ζL , . . . , ζL ) such that

L(χ0, χ1) 0 ≤ λ1 < , χ0χ1 L(χ0, χ1, χ2) 0 ≤ λ2 < , χ2(χ0, χ1) . .

L(χ0, . . . , χi) 0 ≤ λi < , χi(χ0, . . . , χi−1) . .

L(χ0, . . . , χn) 0 ≤ λn < . χn(χ0, . . . , χn−1) n CHAPTER 4. THE Cχ-ACTION ON CP 60

0 0 λ1 λn λ1 λn Proof. Suppose that (ζL , . . . , ζL ) and (ζL , . . . , ζL ) are such that

0 L(χ0, χ1) 0 ≤ λ1, λ1 < , χ0χ1

0 L(χ0, χ1, χ2) 0 ≤ λ2, λ2 < , χ2(χ0, χ1) . .

0 L(χ0, . . . , χi) 0 ≤ λi, λi < , χi(χ0, . . . , χi−1) . .

0 L(χ0, . . . , χn) 0 ≤ λn, λn < χn(χ0, . . . , χn−1)

0 0 are in the same orbit. Then we see that λ1, . . . , λn, λ1, . . . , λn satisfy the linear congruences

0 λ1 ≡ λ1 + a1 L/χ1 − a0 L/χ0 mod L

0 λ2 ≡ λ2 + a2 L/χ2 − a0 L/χ0 mod L . .

0 λn ≡ λn + an L/χn − a0 L/χ0 mod L.

0 Setting mi = λi − λi for 1 ≤ i ≤ n we observe

L(χ0, χ1) |m1| < , χ0χ1 L(χ0, χ1, χ2) |m2| < , χ2(χ0, χ1) . .

L(χ0, . . . , χi) |mi| < , χi(χ0, . . . , χi−1) . .

L(χ0, . . . , χn) |mn| < χn(χ0, . . . , χn−1) and by Theorem 4.2.7 we see m1 = m2 = ··· = mn = 0. Therefore

0 0 λ1 λn λ1 λn (ζL , . . . , ζL ) = (ζL , . . . , ζL ).

n l1 ln n We wish to calculate |(CL) /(Cχ)|. Let ζ = (ζL , . . . , ζL ) ∈ (CL) and {Cχ}ζ denote the isotropy subgroup of ζ. By inspection we can see that

a a {Cχ}ζ = {(ζ(χ0,...,χn), . . . , ζ(χ0,...,χn)) ∈ Cχ : t ∈ {0,..., (χ0, . . . , χn) − 1}}. n CHAPTER 4. THE Cχ-ACTION ON CP 61

n Therefore, by Burnside’s lemma, the number of distinct orbits in (CL) is given by

n n 1 X L (χ0, . . . , χn) |(CL) /(Cχ)| = |{Cχ}ζ | = . |Cχ| n χ0χ1 ··· χn ζ∈(CL)

λ1 λn We observe that the set of points (ζL , . . . , ζL ), such that λ1, . . . , λn are of the form given in the statement of this proof, are representatives of unique orbits and by

n observing that the set of all such points (ζλ1 , . . . , ζλn ) has cardinality L (χ0,...,χn) = L L χ0χ1···χn n |(CL) /(Cχ)|, we see that every orbit is represented.

Recalling Remark 4.1.7 we can deduce the following theorem.

Theorem 4.2.10. Let τ = {τ0, . . . , τ`} ⊆ {0, . . . , n} be given. The orbit of each point in (C )n under the (C ) -action contains a unique point of the form (ζλ1 , . . . , ζλn ) L τ\τ0 χ τ L L such that

L(χτ0 , χτ1 ) 0 ≤ λτ1 < , χτ0 χτ1

L(χτ0 , χτ1 , χτ2 ) 0 ≤ λτ2 < , χτ2 (χτ0 , χτ1 ) . .

L(χτ0 , . . . , χτi ) 0 ≤ λτi < , χτi (χτ0 , . . . , χτi−1 ) . .

L(χτ0 , . . . , χτ` ) 0 ≤ λτ` < χτ` (χτ0 , . . . , χτ`−1) and λj = 0 for all j∈ / τ \ τ0.

λ1 λn The points (ζL , . . . , ζL ) with the properties given in Theorem 4.2.10 will be of particular use throughout the rest of this thesis and the following deﬁnition is introduced to make their description less cumbersome. n CHAPTER 4. THE Cχ-ACTION ON CP 62

Deﬁnition 4.2.11. If (ζλ1 , . . . , ζλn ) ∈ (C )n is such that λ , . . . , λ satisfy L L L τ\τ0 1 n

L(χτ0 , χτ1 ) 0 ≤ λτ1 < , χτ0 χτ1

L(χτ0 , χτ1 , χτ2 ) 0 ≤ λτ2 < , χτ2 (χτ0 , χτ1 ) . .

L(χτ0 , . . . , χτi ) 0 ≤ λτi < , χτi (χτ0 , . . . , χτi−1 ) . .

L(χτ0 , . . . , χτ` ) 0 ≤ λτ` < χτ` (χτ0 , . . . , χτ`−1)

λ1 λn and λj = 0 for all j∈ / τ \ τ0 then we call (ζL , . . . , ζL ) a canonical representative of an orbit in (C )n under the action of (C ) . L τ\τ0 χ τ

n 4.3 Canonical representatives for points in T /Cχ,

n n n T /Cχ × ∆ and CP /Cχ

We are now in a position to give a unique way of writing points for the orbit spaces

n n n n T /Cχ, T /Cχ × ∆c and CP /Cχ.

Theorem 4.3.1. Let (e2πiθ1 , . . . , e2πiθn ) be a point in T n with an associated pair {x, l}.

0 0 2πiθ1 2πiθn 2πiθ 2πiθ The orbit Cχ(e , . . . , e ) contains a unique point (e 1 , . . . , e n ) whose asso-

λ1 λn ciated pair is of the form {x, λ}, where λ = (λ1, . . . , λn) is such that (ζL , . . . , ζL ) is

l1 ln n a canonical representative of the orbit of (ζL , . . . , ζL ) in (CL) under the action of

(Cχ)τ .

Proof. The proof follows directly Theorem 4.2.10. and from inspection of the Cχ-action on T n

2πiθ1 2πiθn n n Theorem 4.3.2. Let (e , . . . , e , r0, . . . , rn) be a point in T ×∆c with an associ-

2πiθ1 2πiθn ated triple {rτ , x, l}. The orbit Cχ(e , . . . , e , r0, . . . , rn) contains a unique point

0 0 2πiθ1 2πiθn 0 0 (e , . . . , e , r0, . . . , rn) whose associated triple is of the form {rτ , x, λ}, where

λ1 λn λ = (λ1, . . . , λn) is such that (ζL , . . . , ζL ) is a canonical representative of the orbit of

l1 ln n (ζL , . . . , ζL ) in (CL) under the action of (Cχ)τ . n CHAPTER 4. THE Cχ-ACTION ON CP 63

Proof. Similarly the proof follows directly Theorem 4.2.10. and from inspection of the

n n Cχ-action on T × ∆

n Theorem 4.3.3. Let [z0 : ... : zn] be a point in CP with an associated triple {rτ , x, l}. 0 0 The orbit Cχ[z0 : ... : zn] contains a unique point [z0 : ... : zn] whose associated triple is

λ1 λn of the form {rτ , x, λ}, where λ = (λ1, . . . , λn) is such that (ζL , . . . , ζL ) is a canonical representative of an orbit in (C )n under the action of (C ) . L τ\τ0 χ τ

Proof. The proof follows from Theorem 4.1.6 and Theorem 4.2.10.

4.4 Regarding fundamental domains

Deﬁnition 4.4.1. Let G be a discrete group acting on a topological space X. Then a fundamental domain for G is a subset F of X satisfying the following properties:

1. Given x ∈ X, there exists g ∈ G such that g · x ∈ F .

2. If g ∈ G, g 6= id on X, and both x, g·x ∈ F , then these points lie on the boundary of F .

n We observe that Theorem 4.2.9 describes a fundamental domain for Cχ on T . We consider some examples of this fundamental domain for low dimensional cases of weights, namely (1, 2, 3) and (2, 3, 5). We then consider how these fundamental domains are related to points in CP 2/Cχ when χ = (1, 2, 3) and χ = (2, 3, 5).

χ0·χ1 Example 4.4.2 (χ = (χ0, χ1)). In this case L = and we observe that every (χ0,χ1) point of T 1 can be written in the form

l1 2πix1/L (ζL e ), where

1. x = (x1) such that x1 ∈ [0, 1)

2. l = (l1) such that l1 ∈ {0,...,L − 1}.

1 Rather trivially the fundamental domain for Cχ on T is the set of points with associated pairs of the form n CHAPTER 4. THE Cχ-ACTION ON CP 64

1. x = (x1) such that x1 ∈ [0, 1)

2. λ = (λ1) such that λ1 = 0.

Example 4.4.3 (χ = (1, 2, 3)). In this case L = 6 and we observe that every point of T 2 can be written in the form

l1 2πix1/6 l2 2πix2/6 (ζ6 e , ζ6 e ), where

1. x = (x1, x2) such that xj ∈ [0, 1) for all j ∈ [1, 2]

2. l = (l1, l2) such that lj ∈ {0,..., 5} for all j ∈ [1, 2].

2 It follows therefore that the fundamental domain for Cχ on T is the set of points with associated pairs of the form

1. x = (x1, x2) such that xj ∈ [0, 1) for all j ∈ [1, 2]

2. λ = (λ1, λ2) such that λ1 ∈ {0, 1, 2} and λ2 ∈ {0, 1}.

The identiﬁcations on the boundary of this fundamental domain can be determined by observing that

3 2 3 2 2 (1, 1) ∼ (ζ6 , 1) ∼ (1, ζ6 ) ∼ (ζ6 , ζ6 ) ∈ (C6) /Cχ.

Remark 4.4.4. It is clear that the fundamental domain given in Example 4.4.3 is homeomorphic to T 2 under the map (eiθ1 , eiθ2 ) 7→ (eiχ1θ1 , eiχ2θ2 ). In fact this is the case for any weight vector such that χ0 = 1 or more generally, such that χ0 divides χ1 and χ2.

Example 4.4.5 (χ = (2, 3, 5)). In this case L = 30 and we observe that every point of T 2 can be written in the form

l1 2πix1/30 l2 2πix2/30 (ζ30e , ζ30e ), where

1. x = (x1, x2) such that xj ∈ [0, 1) for all j ∈ [1, 2]

2. l = (l1, l2) such that lj ∈ {0,..., 29} for all j ∈ [1, 2]. n CHAPTER 4. THE Cχ-ACTION ON CP 65

2 It follows therefore that the fundamental domain for Cχ on T is the set of points with associated pairs of the form

1. x = (x1, x2) such that xj ∈ [0, 1) for all j ∈ [1, 2]

2. λ = (λ1, λ2) such that λ1 ∈ {0, 1, 2, 3, 4} and λ2 ∈ {0, 1, 2, 3, 4, 5}.

The identiﬁcations on the boundary of this fundamental domain can be determined by observing that

5 3 6 2 (1, 1) ∼ (ζ30, ζ30) ∼ (1, ζ30) ∈ (C30) /Cχ. and

3 5 5 6 2 (1, ζ30) ∼ (ζ30, 1) ∼ (ζ30, ζ30) ∈ (C30) /Cχ.

Remark 4.4.6. We emphasise that the fundamental domain given in Example 4.4.5 is not homeomorphic to T 2 under the map (eiθ1 , eiθ2 ) 7→ (eiχ1θ1 , eiχ2θ2 ).

We now consider how such fundamental domains can be used to represent points

2 of CP /Cχ when χ = (1, 2, 3) and χ = (2, 3, 5).

Example 4.4.7 (χ = (1, 2, 3)). It follows from Theorem 4.3.3 that each point in

2 2 CP /Cχ has a unique representative in CP of the form

1. [1 : 0 : 0]

2. [0 : 1 : 0]

3. [0 : 0 : 1]

λ1 2πix1/6 2 2 λ1 2πix1/6 4. [r0 : r1ζ6 e : 0] where r0 + r1 = 1 and r0, r1 > 0; and (ζ6 e ) is an

1 λ1 2πix1/6 element of the fundamental domain of C1×C2 acting on T ; that is (ζ6 e ) ∈ 2πi [0, 2 ).

λ2 2πix2/6 2 2 λ2 2πix2/6 5. [r0 : 0 : r2ζ6 e ] where r0 + r2 = 1 and r0, r2 > 0; and (ζ6 e ) is an

1 λ2 2πix2/6 element of the fundamental domain of C1×C3 acting on T ; that is (ζ6 e ) ∈ 2πi [0, 3 ).

λ2 2πix2/6 2 2 λ2 2πix2/6 6. [0 : r1 : r2ζ6 e ] where r1 + r2 = 1 and r1, r2 > 0; and (ζ6 e ) is an

1 λ2 2πix2/6 element of the fundamental domain of C2×C3 acting on T ; that is (ζ6 e ) ∈ 2πi [0, 6 ). n CHAPTER 4. THE Cχ-ACTION ON CP 66

λ1 2πix1/6 λ2 2πix2/6 2 2 2 7. [r0 : r1ζ6 e : r2ζ6 e ] where r0 + r1 + r2 = 1 and r0, r1, r2 > 0; and

λ1 2πix1/6 λ2 2πix2/6 (ζ6 e , ζ6 e ) is an element of the fundamental domain of C1 × C2 × C3 acting on T 2; that is

h 2πi h 2πi (ζλ1 e2πix1/6, ζλ2 e2πix2/6) ∈ 0, × 0, . 6 6 2 3

Example 4.4.8 (χ = (2, 3, 5)). It follows from Theorem 4.3.3 that each point in

2 2 CP /Cχ has a unique representative in CP of the form

1. [1 : 0 : 0]

2. [0 : 1 : 0]

3. [0 : 0 : 1]

λ1 2πix1/30 2 2 λ1 2πix1/30 4. [r0 : r1ζ30 e : 0] where r0 + r1 = 1 and r0, r1 > 0; and (ζ30 e ) 1 is an element of the fundamental domain of C2 × C3 acting on T ; that is

λ1 2πix1/30 2πi (ζ30 e ) ∈ [0, 6 )

λ2 2πix2/30 2 2 λ2 2πix2/30 5. [r0 : 0 : r2ζ30 e ] where r0 + r2 = 1 and r0, r2 > 0; and (ζ30 e ) 1 is an element of the fundamental domain of C2 × C5 acting on T ; that is

λ2 2πix2/30 2πi (ζ30 e ) ∈ [0, 10 )

λ2 2πix2/30 2 2 λ2 2πix2/6 6. [0 : r1 : r2ζ30 e ] where r1 + r2 = 1 and r1, r2 > 0; and (ζ6 e ) 1 is an element of the fundamental domain of C3 × C5 acting on T ; that is

λ2 2πix2/30 2πi (ζ30 e ) ∈ [0, 15 )

λ1 2πix1/30 λ2 2πix2/30 2 2 2 7. [r0 : r1ζ30 e : r2ζ30 e ] where r0 + r1 + r2 = 1 and r0, r1, r2 > 0; and

λ1 2πix1/30 λ2 2πix2/30 (ζ30 e , ζ30 e ) is an element of the fundamental domain of C2 ×C3 ×C5 acting on T 2; that is

h 2πi h 2πi (ζλ1 e2πix1/30, ζλ2 e2πix2/30) ∈ 0, × 0, . 30 30 6 5 Chapter 5

CW-complex on P(χ)

In Section 3.4 we introduced the approach we will take in determining a CW-structure

n n n n for P(χ): by defining a Cχ-complex on CP via Cχ-complexes on T and T × ∆c . n We begin by defining a Cχ-complex structure on T that is also a ∆-complex and n n n then define Cχ-complexes on T × ∆ and CP , such that the maps

n i n n q n T ,−−→ T × ∆c −−→ CP (5.0.1)

are Cχ-maps. One motivation for this is that the quotient map q is such that any identifications made by q can be pulled back to identifications of cells in T n. Therefore in Chapter 6, when we determine the boundary formulae of the corresponding cellular chain complexes, we have simplified this task to determining how such identifications affect the orderings on the vertices of the cells of the ∆-complex we will prescribe on T n. Furthermore, in Section 5.3 we will see that the ∆-complex structure that is the image of Kuhn subdivision, given in Remark 2.3.6, under the quotient map In → T n, which was introduced in Remark 2.3.6, is sufficient to ensure that q is a cellular map.

n Recalling Deﬁnition 2.6.3 it follows that a Cχ-complex on T is a CW-complex with a cellular Cχ-action; in particular, vertices are mapped to vertices under the Cχ- n n action. In Chapter 4 we saw that Cχ acts on (CL) < T , where L = lcm{χ0, . . . , χn}, n n and so (CL) is a plausible candidate for the set of vertices of the Cχ-complex on T . The image of a subdivision of In into Ln Kuhn cubes under the map In → T n has

n (CL) as its set of vertices and is the ∆-complex structure we propose in Section 5.1. n n n Setting up Cχ-complexes on T , T × ∆c and therefore CW-structures on their respective orbit spaces covers the ﬁrst half of this chapter. The majority of the second

67 CHAPTER 5. CW-COMPLEX ON P(χ) 68

n half of this chapter is dedicated to the Cχ-complex on CP , as an exposition into showing that q is a map of Cχ-complexes is necessary for determining the associated cellular chain complexes for CP n and P(χ) in Chapter 6. We ﬁnish with determining n CW-complexes on CP /Cχ and P(χ).

n n 5.1 A ∆-complex structure on T and T /Cχ

n To prepare for prescribing a Cχ-complex on T we recall and slightly amend Notation 2.3.3.

Notation 5.1.1. Let S, U and J take the same form as they did in Notation 2.3.3.

n l1 2πiφ1/L ln 2πiφn/L n We write points of T in the form (ζL e , . . . , ζL e ), where (φ1, . . . , φn) ∈ I l k n and lj ∈ {0,...,L − 1} for all j ∈ [n]. We consider the map ΦJ : ∆ → T given by

l1 2πix1(t0,...,tk)/L ln 2πixn(t0,...,tk)/L (t0, . . . , tk) 7→ (ζL e , . . . , ζL e ),

k where we recall from Notation 2.3.3 the map xa : ∆ → I is given by:   tk−i+1 + ··· + tk if a ∈ Ji xa(t0, . . . , tk) =  0 if a ∈ U.

l n Following Note 2.3.4, we see the image of ΦJ is exactly the set of points in T of

l1 2πix1/L ln 2πixn/L the form (ζL e , . . . , ζL e ), such that x1, . . . , xn satisfy the following set of inequalities:

0 = xu = ··· = xu ≤ x 1 = x 1 = ··· = xj1 1 n−|S| j1 j2 n1

≤ x 2 = x 2 = ··· = xj2 j1 j2 n2 ≤ · · ·

≤ x k−1 = x k−1 = ··· = x k−1 j1 j2 jnk−1

≤ xjk = xjk = ··· = xjk 1 2 nk ≤ 1. (5.1.2)

l ˚k The image of ΦJ |∆ corresponds to those points given by turning the inequalities in l (5.1.2) into strict inequalities. Similarly the image of the restriction of ΦJ to the i-th face of ∆k corresponds to those points given by turning the (k + 1 − i)-th inequality

l 0 n symbol in (5.1.2) to an equality. When J = ∅, the image of ΦJ : ∆ → T is the point CHAPTER 5. CW-COMPLEX ON P(χ) 69

l1 ln l ¯ l l ˚k (ζL , . . . , ζL ). The image of ΦJ will be denoted by ∆J and the image of ΦJ |∆ will be

l 2πiθ1 2πiθn l denoted by ∆J . In particular, we observe that (e , . . . , e ) ∈ ∆J if and only if its associated pair is of the form {x, l} such that x1, . . . , xn satisfy the following strict inequalities:

0 = xu = ··· = xu < x 1 = x 1 = ··· = xj1 1 n−|S| j1 j2 n1

< x 2 = x 2 = ··· = xj2 j1 j2 n2 < ···

< x k−1 = x k−1 = ··· = x k−1 j1 j2 jnk−1

< xjk = xjk = ··· = xjk 1 2 nk < 1. (5.1.3)

Theorem 5.1.4. The family of maps

l k n ΦJ : ∆ → T for all J and l are the characteristic maps of a ∆-complex structure on T n.

l ˚k Proof. Clearly ΦJ |∆ is an injection and therefore a homeomorphism onto its image

l 2πiθ1 2πiθn n ∆J . Let (e , . . . , e ) ∈ T , with an associated pair {x, l}. We observe that there exists a unique ordered partition J such that x1, . . . , xn satisfy (5.1.3). Therefore the l maps ΦJ satisfy Property 1 of Definition 2.1.1. k Writing the i-th face of ∆ as [e0,..., eî, . . . , ek] where e0, . . . , ek are the vertices of k k−1 ∆ , we identify [e0,..., eî, . . . , ek] with ∆ by the canonical linear homeomorphism k−1 ηi : ∆ → [e0,..., eî, . . . , ek] given by (t0, . . . , tk−1) 7→ (t0, . . . , ti−1, 0, ti, . . . , tk−1). l Therefore to show the maps ΦJ satisfy Property 2 of Definition 2.1.1 we wish to show 0 0 0 0 0 0 l l0 that there exists some l = (l1, . . . , ln) and J = {J1,...,Jk−1} such that ΦJ ◦ ηi = ΦJ0 . l The map ΦJ ◦ ηi is defined by

l1 2πix1(t0,...,ti−1,0,ti,...,tk−1)/L ln 2πixn(t0,...,ti−1,0,ti,...,tk−1)/L (t0, . . . , tk−1) 7→ (ζL e , . . . , ζL e )

l 0 0 j 2πixj (t0,...,ti−1,0,ti,...,tk−1)/L and to determine l and J we calculate ζL e for each j ∈ [n]. This is done by considering the three cases: i ∈ {1, . . . , k − 1}, i = k and i = 0: CHAPTER 5. CW-COMPLEX ON P(χ) 70

1. When i ∈ {1, . . . , k − 1} then   tk−1 if a ∈ J1    tk−2 + tk−1 if a ∈ J2   . .  . .    ti + ··· + tk−1 if a ∈ Jk−i  xa(t0, . . . , ti−1, 0, ti, . . . , tk−1) = 0 + ti + ··· + tk−1 if a ∈ Jk−i+1   t + 0 + t + ··· + t if a ∈ J  i−1 i k−1 k−i+2  . .  . .    t + ··· + t if a ∈ J  1 k−1 k   0 if a ∈ U.

0 0 We observe that if we let li = li and J = {J1,...,Jk−i ∪ Jk−i+1,...,Jk} then

l0 0 l j 2πixj (t0,...tk−1)/L j 2πixj (t0,...,ti−1,0,ti,...,tk−1)/L ζL e = ζL e

for all j ∈ [n].

2. When i = k:

  tk−1 if a ∈ J2    tk−2 + tk−1 if a ∈ J3  . . xa(t0, . . . , ti−1, 0, ti, . . . , tk−1) = . .    t1 + ··· + tk−1 if a ∈ Jk    0 if a ∈ U ∪ J1. 0 0 If we let li = li and J = {J2,...,Jk} then

l0 0 l j 2πixj (t0,...tk−1)/L j 2πixj (t0,...,ti−1,0,ti,...,tk−1)/L ζL e = ζL e

for all j ∈ [n].

3. When i = 0:   t if a ∈ J  k−1 1   t + t if a ∈ J  k−2 k−1 2  . .  . . xa(t0, . . . , ti−1, 0, ti, . . . , tk−1) =  t + ··· + t if a ∈ J  1 k−1 k−1   t + ··· + t = 1 if a ∈ J  0 k−1 k   0 if a ∈ U. CHAPTER 5. CW-COMPLEX ON P(χ) 71

0 If we let li be deﬁned by  li + 1 mod L if i ∈ Jk 0  li = (5.1.5) li otherwise

0 and J = {J2,...,Jk} then

l0 0 l j 2πixj (t0,...tk−1)/L j 2πixj (t0,...,ti−1,0,ti,...,tk−1)/L ζL e = ζL e

for all j ∈ [n].

l l l0 k−1 n We conclude that each face of ∆J restricts ΦJ to a map ΦJ0 : ∆ → T . In particular

 ∆¯ l0 if i = 0  {J1,...,Jk−1}  l l ΦJ (t0, . . . , ti = 0, . . . , tk) = ∆¯ if i = k − j and j 6= 0, k (5.1.6)  {J1,...,Jj ∪Jj+1,...,Jk}  ∆¯ l if i = k.  {J2,...,Jk} Property 3 of Deﬁnition 2.1.1 is automatic as there are only ﬁnitely many maps

l ΦJ .

l n To summarise; each ∆J is a cell of dimension k of a ∆-complex structure on T , n where k = |J|. We will denote this ∆-structure by TL , for a given L. The union of all n (k) cells of dimensions less than or equal to k, which we write as (TL ) , is therefore the n k-skeleton of TL .

n Theorem 5.1.7. The CW-complex TL is a Cχ-complex.

n Proof. Recall the Cχ-action on T given in Deﬁnition 3.4.1. We consider the map

n n ζχ : TL → TL

l1 2πix1/L ln 2πixn/L −a0 a1 l1 2πix1/L −a0 an ln 2πixn/L (ζL e , . . . , ζL e ) 7→ (ζχ0 ζχ1 ζL e , . . . , ζχ0 ζχn ζL e )

n a0 an ˜ induced by the Cχ-action on T for every ζχ = (ζχ0 , . . . , ζχn ) ∈ Cχ. If we let l = ˜ ˜ ˜ L L ˜ (l1,..., ln) such that lj = {lj + aj − a0 } mod L and 0 ≤ lj ≤ L − 1 for 1 ≤ j ≤ n χj χ0 then

l ˜l ζχ ◦ ΦJ = ΦJ .

l ˜l l l In particular ζχ · ∆J = ∆J and so whenever ζχ · ∆J = ∆J we see that the restriction l of ζχ to ∆J is the identity map, as required. CHAPTER 5. CW-COMPLEX ON P(χ) 72

n It follows from Theorem 4.3.1 that each point in T /Cχ can be uniquely represented by a point of the form

λ1 2πix1/L λn 2πixn/L n (ζL e , . . . , ζL e ) ∈ T ,

with its associated pair {x, λ} where x = (x1, . . . , xn) and λ = (λ1, . . . , λn) is such

λ1 λn n that (ζL , . . . , ζL ) is a canonical representative of an orbit in (CL) under Cχ. The

λ1 2πix1/L λn 2πixn/L λ point (ζL e , . . . , ζL e ) is contained in the cell ∆J for some J. λ Therefore we restrict attention to the images of such ∆J under the quotient map n n λ λ p: T → T /Cχ. Letting ΦeJ denote the composition of ΦJ with the quotient map n n ¯ λ ∗ λ ∗ λ λ ˚k p: T → T /Cχ, we denote by (∆J ) and (∆J ) the images of ΦeJ and ΦeJ |∆ respectively.

Theorem 5.1.8. The family of maps

λ k n ΦeJ : ∆ → T /Cχ

λ1 λn n for all J and λ such that (ζL , . . . , ζL ) is a canonical representative of an orbit in (CL) n under Cχ, is a family of characteristic maps of a ∆-complex structure on T /Cχ.

λ Proof. It follows from Theorem 2.6.6 that each ΦeJ restricts to a homeomorphism from ˚k λ ∗ ∆ to its image, (∆J ) . The comments preceding the statement of Theorem 5.1.8 λ ∗ n verify all such (∆J ) are disjoint and their union is T /Cχ. λ We observe from the proof of Theorem 5.1.4 that the restriction of ΦeJ to each face of ∆k is given by   Φλ˜ if i = 0  e{J1,...,Jk−1}  λ λ Φe | [e0,..., eˆi, . . . , ek] = Φ if i = k − j and j 6= 0, k J e{J1,...,Jj ∪Jj+1...Jk}   Φλ if i = k. e{J2,...,Jk}

0 0 λ˜1 λ˜n λ1 λn Here (ζL , . . . , ζL ) is the canonical representative of the orbit of (ζL , . . . , ζL ) under n 0 0 the Cχ-action on (CL) such that (λ1, . . . , λn) is determined by  λi + 1 mod L if i ∈ Jk 0  λi = (5.1.9)  0 λi otherwise. CHAPTER 5. CW-COMPLEX ON P(χ) 73

n λ ∗ n So by analogy with TL , each (∆J ) is a k-cell of a ∆-complex structure on T /Cχ, λ n with ΦeJ as its characteristic map. The ∆-structure itself will be denoted by TL /Cχ n (k) and the k-skeleton will be denoted by (TL /Cχ) . We remark that the set of vertices of

n λ1 λn TL /Cχ are precisely the set of points (ζL , . . . , ζL ) that are canonical representatives n of orbits in (CL) under Cχ.

n n n n 5.2 A CW-structure on T × ∆c and T /Cχ × ∆c

n n n The CW-structure we give for the product T × ∆c is determined from TL and the n ∆-complex ∆c given in Notation 2.5.2. For some τ = {τ0, . . . , τ`} ⊆ {0, . . . , n}, we ` n recall that the map iτ : ∆ → ∆c deﬁned by

(t0, . . . , t`) 7→ (ι0(t0, . . . , t`), . . . , ιn(t0, . . . , t`)),

` where ιa : ∆ → I is given by  √  + tb if a = τb ιa(t0, . . . , t`) =  0 if a∈ / τ,

n n (`) is a characteristic map for an `-cell of ∆c . The `-skeleton, (∆c ) , is therefore the union of all cells ∆τ , such that 1 ≤ |τ| ≤ ` + 1. Consequently we deduce the following from Theorem 2.5.1.

n n Theorem 5.2.1. Given the ∆-complex structure TL , and the ∆-complex structure ∆c , their product ( ) n n [ n (k) n (`) TL × ∆ ; (TL ) × (∆c ) : m = 0, 1,... k+`=m is a CW-complex structure on T n × ∆n.

n n n n We denote the above CW-structure on T × ∆c by TL × ∆c . The m-skeleton, n n (m) l denoted by (TL × ∆c ) , is the set of product cells ∆J × ∆τ , ranging over the cells l n n ∆J of TL and the cells ∆τ of ∆c such that k + ` ≤ m. The characteristic map of the l l product cell ∆J × ∆τ is ΦJ × iτ and its boundary is determined by the restriction of l k ` k ` k ` ΦJ × iτ to ∂(∆ × ∆ ) = (∂∆ × ∆ ) ∪ (∆ × ∂∆ ). Therefore it follows from the proof CHAPTER 5. CW-COMPLEX ON P(χ) 74 of Theorem 5.1.4 and Notation 2.5.2

l l l ∂{∆J × ∆τ } = ∂∆J × ∆τ ∪ ∆J × ∂∆τ (k−1 ) n 0 o [ = ∆¯ l × ∆ ∪ ∆¯ l × ∆ {J1,...,Jk−1} τ {J1,...Ji∪Ji+1,...,Jk} τ i=1 ( ` ) [ ∪ ∆¯ l × ∆ ∪ ∆l × ∆¯ . (5.2.2) {J2,...,Jk} τ J τ\τj j=0

n n We apply similar conventions to the orbit space T /Cχ × ∆c :

n Theorem 5.2.3. Given the ∆-complex structure TL /Cχ, and the ∆-complex structure n ∆c , their product ( ) n n [ n (k) n (`) TL /Cχ × ∆c ; (TL /Cχ) × (∆c ) : m = 0, 1,... k+`=m

n n is a CW-complex structure on T /Cχ × ∆c .

n n n n We denote this CW-structure on T /Cχ × ∆c by TL /Cχ × ∆c . The m-skeleton, n n (m) λ ∗ denoted by (TL /Cχ × ∆c ) , is the set of product cells (∆J ) × ∆τ , ranging over λ ∗ n n the cells (∆J ) of TL /Cχ and the cells ∆τ of ∆c such that k + ` ≤ m. Clearly the λ k ` restriction ΦeJ |∂(∆ × ∆ ) has

λ ∗ ` λ ∗ λ ∗ ∂{(∆J ) × ∆τ } = ∂(∆J ) × ∆τ ∪ (∆J ) × ∂∆τ (k−1 ) n ˜ o [ = (∆¯ λ )∗ × ∆ ∪ (∆¯ λ )∗ × ∆ {J1,...,Jk−1} τ {J1,...Ji∪Ji+1,...,Jk} τ i=1 ( ` ) [ ∪ (∆¯ λ )∗ × ∆ ∪ (∆λ)∗ × ∆¯ (5.2.4) {J2,...,Jk} τ J τ\τj j=0 as its image.

5.3 A CW-structure on CP n

n As stated at the beginning of this chapter we will deﬁne a Cχ-complex on CP such n n n that the quotient map q : TL × ∆c → CP deﬁned by

l1 2πix1/L ln 2πixn/L l1 2πix1/L ln 2πixn/L (ζL e , . . . , ζL e , r0, . . . , rn) 7→ [r0 : r1ζL e : ... : rnζL e ]

n n is a Cχ-map. The following remark describes which points of TL × ∆c are identiﬁed under the map q. CHAPTER 5. CW-COMPLEX ON P(χ) 75

l1 2πix1/L ln 2πixn/L n n Remark 5.3.1. Let y = (ζ e , . . . , ζ e , r0, . . . , rn) ∈ T × ∆c with its associated triple {rτ , x, l} such that x = (x1, . . . , xn), l = (l1, . . . , ln) and rτ =

{rτ0 , . . . , rτ` } be given. Then we observe

l1 2πix1/L ln 2πixn/L q(y) = [r0 : r1ζL e : ... : rnζL e ] l0 0 l0 0 1 2πix1/L n 2πixn/L = [r0 : r1ζL e : ... : rnζL e ]

0 0 0 where x = (x1, . . . , xn) is given by   x − x if τ 6= 0 and i ∈ τ \ τ  i τ0 0 0 0  xi = xi if τ0 = 0 and i ∈ τ \ τ0    0 if i∈ / τ \ τ0

0 0 0 and l = (l1, . . . , ln) is given by   l − l mod L if τ 6= 0 and i ∈ τ \ τ  i τ0 0 0 0  li = li if τ0 = 0 and i ∈ τ \ τ0    0 if i∈ / τ \ τ0.

0 and 0 ≤ li < L for 1 ≤ i ≤ n.

n n Using Remark 5.3.1 we will see that q identiﬁes certain cells of TL × ∆c together.

In the following we let I = {I1,...,Ip} be an ordered partition of a given S ⊆ [n] into p parts such that

1 1 2 2 p p I1 = {i1, . . . , in1 },I2 = {i1, . . . , in2 },...,Ip = {i1, . . . , inp },

α where iβ ∈ [n] denotes the β-th element in the set Iα. As usual τ = {τ0, . . . , τ`} ⊆

{0, 1, ··· n} and l = (l1, . . . , ln) is such that 0 ≤ li < L for all i ∈ [n].

Proposition 5.3.2. There exists an ordered partition, J of a subset σ ⊆ τ \ τ0 and 0 0 0 0 0 some l = (l1, . . . , ln), where 0 ≤ li < L for all i ∈ τ \ τ0 and lj = 0 for all j∈ / τ \ τ0 such that

l l0 im{q ◦ (ΦI × iτ )|int(∆p×∆`)} = im{q ◦ (ΦJ × iτ )|int(∆k×∆`)}.

l Proof. By letting U = [n]\S = {u1, . . . , un−|S|}, we recall that im{ΦI ×iτ |int(∆p×∆`)} =

l l1 2πix1/L ln 2πixn/L ∆I ×∆τ and so corresponds to the set of points y = (ζL e , . . . ζL e , r0, . . . rn) ∈ CHAPTER 5. CW-COMPLEX ON P(χ) 76

T n × ∆n with associated triples of the form {r , x, l}, where Σ` r2 = 1 and r 6= 0 c τ i=0 τj j for all j ∈ τ and where x1, . . . , xn satisfy the following strict inequalities:

0 = xu = ··· = xu < x 1 = x 1 = ··· = xi1 1 n−|S| i1 j2 n1

< x 2 = x 2 = ··· = xi2 i1 i2 n2 < ···

< x p−1 = x p−1 = ··· = x p−1 i1 i2 inp−1

< x p = x p = ··· = x p i1 i2 inp < 1 (5.3.3)

0 0 0 We set σ := S ∩ τ and υ := {υ1, . . . , υ|τ|−|σ0|} = τ \ σ . When σ 6= ∅, we deﬁne

Λ = {λ ∈ [p]: Iλ ∩ τ 6= ∅}. Therefore, there exists some 1 ≤ k ≤ p so that we can write Λ = {λ1, . . . , λk}, where λ1 < ··· < λk. We let J = {J1,...,Jk} be the ordered 0 partition of σ such that Ji = {x: x ∈ Iλi ∩τ}. In other words, J is an ordered partition of σ0, whose ordering is inherited from the ordering on the parts of I.

l 1. τ0 = 0: using Remark 5.3.1 we see that im{q ◦ (∆I × ∆τ )} is the set of points of

l1 2πix1/L ln 2πixn/L 0 0 the form [r0 : r1ζL e : ... : rnζL e ] with associated triples {rτ , x , l }, where Σ` r2 = 1 and r 6= 0 for all j ∈ τ and x0 , . . . , x0 satisfy the following i=0 τj j 1 n inequalities

0 = x0 = ··· = x0 u1 un−|S| 0 0 0 0 0 xυ = ··· = xυ 0 < x 1 = x 1 = ··· = xj1 1 |τ|−|σ | j1 j2 n1 0 0 0 < x 2 = x 2 = ··· = xj2 j1 j2 n2 < ···

0 0 0 < x k−1 = x k−1 = ··· = x k−1 j1 j2 jnk−1 0 0 0 < xjk = xjk = ··· = xjk 1 2 nk < 1. (5.3.4)

l1 2πix1/L ln 2πixn/L The set of all such points [r0 : r1ζL e : ... : rnζL e ] is precisely l0 im{q(∆J × iτ )}, therefore

l l0 im{q ◦ (ΦI × iτ )|int(∆p×∆`)} = im{q ◦ (ΦJ × iτ )|int(∆k×∆`)}. CHAPTER 5. CW-COMPLEX ON P(χ) 77

l 2. τ0 6= 0: from Remark 5.3.1 we see that im{q ◦ (∆I × ∆τ )} is the set of points of l0 0 l0 0 1 2πix1/L n 2πixn/L the form [r0 : r1ζL e : ... : rnζL e ] such that  0  li − lτ0 mod L if i ∈ τ \ τ0 li =  0 if i∈ / τ \ τ0,

0 where 0 ≤ li < L for 1 ≤ i ≤ n and  0  xi − xτ0 if i ∈ τ \ τ0 xi =  0 if i∈ / τ \ τ0.

We split this case up into the cases xτ0 = 0 and xτ0 6= 0.

(a) xτ0 = 0: (that is, τ0 ∈/ S), then  0  xi if i ∈ τ \ τ0 xi =  0 if i∈ / τ \ τ0.

l We see that im{q ◦ (∆I × ∆τ )} is the set of points of the form [r0 : l0 0 l0 0 1 2πix1/L n 2πixn/L 0 0 r1ζL e : ... : rnζL e ] with associated triples {rτ , x , l }, where Σ` r2 = 1 and r 6= 0 for all j ∈ τ and where x0 = (x0 , . . . , x0 ) and i=0 τj j 1 n 0 0 x1, . . . xn are the set of points satisfying the inequalities given in (5.3.4). Therefore we observe

l l0 im{q ◦ (ΦI × iτ )|int(∆p×∆`)} = im{q ◦ (ΦJ × iτ )|int(∆k×∆`)}.

0 (b) xτ 6= 0: that is, τ0 ∈ S and so τ0 ∈ Jα for some α ∈ [k]. We see −1 ≤ xi ≤ 1 for all i ∈ [n]. By deﬁning  x − x τ ∈ J for α < q ≤ k  τp τ0 p q    1 + xτp − xτ0 τp ∈ Jq for 1 ≤ q < α x˜τp =  1 − x τ ∈ υ  τ0 p   0 otherwise

˜ ˜ ˜ and l = (l1,..., ln) such that   l0 mod L i ∈ J for α < q ≤ k  i q   l0 − 1 mod L i ∈ J for 1 ≤ q < α ˜  i q li =  l0 − 1 mod L i ∈ υ  i   0 otherwise CHAPTER 5. CW-COMPLEX ON P(χ) 78

˜ where 0 ≤ li < L for 1 ≤ i ≤ n, we observex ˜i ∈ [0, 1) for all i ∈ [n] and

l0 0 l0 0 ˜ ˜ 1 2πix1/L n 2πixn/L l1 2πix˜1/L ln 2πix˜n/L [r0 : r1ζL e : ... : rnζL e ] = [r0 : r1ζL e : ... : rnζL e ].

l0 0 l0 0 1 2πix1/L n 2πixn/L l Therefore each [r0 : r1ζL e : ... : rnζL e ] ∈ im{q ◦ (∆I × ∆τ )} has associated triples of the form {r , x,˜ ˜l}, where Σ` r2 = 1 and r 6= 0 τ i=0 τj j

for all j ∈ τ. In particularx ˜1,..., x˜n satisfy the following inequalities:

0 =x ˜u1 =x ˜u1 = ··· =x ˜un−|S|

=x ˜jα =x ˜jα = ··· =x ˜jα < x˜ α+1 =x ˜ α+1 = ··· =x ˜ α+1 1 2 nα j1 j2 jnα+1

< x˜ α+2 =x ˜ α+2 = ··· =x ˜ α+2 j1 j2 jnα+2 < ···

< x˜jk =x ˜jk = ··· =x ˜jk 1 2 nk

< x˜υ1 =x ˜υ2 = ... =x ˜υ|τ|\|σ0|

< x˜ 1 =x ˜ 1 = ··· =x ˜j1 j1 j2 n1

< x˜ 2 =x ˜ 2 = ··· =x ˜j2 j1 j2 n2 < ···

< x˜ α−1 =x ˜ α−1 = ··· =x ˜ α−1 j1 j2 jnk < 1.

0 0 0 0 0 Supposing υ 6= ∅ then if we let J = {J1,...,Jk} such that J1 = Jα+1,J2 = 0 0 0 0 0 Jα+2,...,Jk−α = Jk,Jk−α+1 = υ, Jk−α+2 = J1,Jk−α+3 = J2,...,Jk =

Jα−1, then

l ˜l im{q ◦ (ΦI × iτ )|int(∆p×∆`)} = im{q ◦ (ΦJ0 × iτ )|int(∆k×∆`)}.

00 00 00 00 00 If υ = ∅ then we let J = {J1 ,...,Jk−1} such that J1 = Jα+1,J2 = 00 00 00 00 Jα+2,...,J k−α = Jk,Jk−α+1 = J1,Jk−α+2 = J2,...,Jk−1 = Jα−1 and see

l ˜l im{q ◦ (ΦI × iτ )|int(∆p×∆`)} = im{q ◦ (ΦJ00 × iτ )|int(∆k−1×∆`)}.

Remark 5.3.5. It follows from Proposition 5.3.2 that q identiﬁes the (p + `)-cell

l n n l0 ∆I × ∆τ of TL × ∆c with a unique (m + `)-cell ∆J × ∆τ , where m ≤ p. The proof of Proposition 5.3.2 shows how to determine J and l0. CHAPTER 5. CW-COMPLEX ON P(χ) 79

Notation 5.3.6. According to Proposition 5.3.2, in the statement of the following

l theorem we restrict attention to the maps q ◦ (ΦJ × iτ ) such that

1. τ = {τ0, . . . , τ`} ranges over all subsets of {0, . . . , n}

2. J ranges over all ordered partitions of all σ ⊆ τ \ τ0 into k parts

3. l ranges over all n-tuples (l1, . . . , ln) such that 0 ≤ lj < L for all j ∈ τ \ τ0 and

lj = 0 for all j∈ / τ \ τ0.

l l c c c We let eJ,τ :=im{q◦(ΦJ ×iτ ) |int(∆k×∆`)}. In particular, letting σ = {σ1, . . . , σn−|σ|}:= [n] \ σ, we observe that

l1 2πix1/L ln 2πixn/L l [r0 : r1ζL e : ... : rnζL e ] ∈ eJ,τ

if and only if its associated triple {rτ , x, l} is such that x1, . . . , xn satisfy the following strict inequalities:

0 = xσc = ··· = xσc < x 1 = x 1 = ··· = xj1 1 n−|σ| j1 j2 n1

< x 2 = x 2 = ··· = xj2 j1 j2 n2 < ···

< x k−1 = x k−1 = ··· = x k−1 j1 j2 jnk−1

< xjk = xjk = ··· = xjk 1 2 nk < 1. (5.3.7)

Theorem 5.3.8. The family of maps

l k ` n q ◦ (ΦJ × iτ ): ∆ × ∆ → CP is a family of characteristic maps for a CW-structure on CP n.

l Proof. To show the restriction of q ◦ (ΦJ × iτ )|int(∆k×∆`) is an injection onto its image we let

l1 2πix1/L ln 2πixn/L y = (ζL e , . . . ζL e , r0, . . . rn)

l0 0 l0 0 0 1 2πix1/L n 2πixn/L 0 0 y = (ζL e , . . . ζL e , r0, . . . rn) CHAPTER 5. CW-COMPLEX ON P(χ) 80

l 0 0 l be two points in ∆J ×∆τ , such that q(y) = q(y ). We note that as y, y ∈ ∆J ×∆τ , their 0 0 respective associated triples are {rτ , x, l}, {rτ , x , l} and 0 ≤ xi, xi < 1 for 1 ≤ i ≤ n. We see

l1 2πix1/L ln 2πixn/L q(y) = [r0 : r1ζL e : ... : rnζL e ] l0 0 l0 0 1 2πix1/L n 2πixn/L = [r0 : r1ζL e : ... : rnζL e ] = q(y0),

0 and as {rτ , x, l}, {rτ , x , l} are of the form required of associated triples for points of n 0 0 CP , we conclude {rτ , x, l} = {rτ , x , l}. That is, y = y as required. n Let [z0 : ... : zn] ∈ CP , with an associated triple {rτ , x, l}. We observe that there exists a unique ordered partition J such that x1, . . . , xn satisfy (5.3.7). Therefore the l maps q ◦ (ΦJ × iτ ) satisfy Property 1 of Deﬁnition 2.1.1. l The boundary of the cell ∆J × ∆τ is the union of ﬁnitely many cells of dimension less than k + `. It follows from Proposition 5.3.2 and Remark 5.3.5 that the boundary

l of eJ,τ is therefore a union of finitely many cells of dimension less than k +`. Therefore Property 2 of Definition 2.1.1 is satisfied.

l Therefore each eJ,τ such that l, J and τ are of the form given in Notation 5.3.6 is n l a (k + `)-cell of a CW-structure on CP , with q ◦ (ΦJ × iτ ) as its characteristic map. n n (k+`) Let CPL denote this CW-structure and (CPL ) denote its (k + `)-skeleton; that is n (k+`) (CPL ) is the union of cells of dimension less than or equal to k + `.

Corollary 5.3.9. The map q is a cellular map.

l l The boundary of eJ,τ is calculated from the boundary of ∆J × ∆τ given in (5.2.2):

l l ∂eJ,τ = q ∂{∆J × ∆τ } (k−1 ) n 0 o [ = q(∆¯ l × ∆ ) ∪ q(∆¯ l × ∆ ) {J1,...,Jk−1} τ {J1,...Ji∪Ji+1,...,Jk} τ i=1 ( ` ) [ ∪ q(∆¯ l × ∆ ) ∪ q(∆l × ∆¯ ) (5.3.10) {J2,...,Jk} τ J τ\τ` j=0

We will look more closely at the terms in (5.3.10) in Chapter 6.

n Theorem 5.3.11. The CW-complex CPL is a Cχ-complex. CHAPTER 5. CW-COMPLEX ON P(χ) 81

Proof. We consider the map

n n ζχ : CPL → CPL

l1 2πix1/L ln 2πixn/L −a0 a1 l1 2πix1/L −a0 an ln 2πixn/L [r0, r1ζL e , . . . , rnζL e ] 7→ [r0, r1ζχ0 ζχ1 ζL e , . . . , rnζχ0 ζχn ζL e ]

n a0 an induced by the Cχ-action on CP given in Deﬁnition 3.4.3 for every ζχ = (ζχ0 , . . . , ζχn ) ∈ Cχ. ˜ ˜ ˜ l1 ln If we let l = (ζL , . . . , ζL ) such that   l + a L − a L mod L i ∈ τ \ τ i i χi τ0 χτ 0 ˜li = 0  0 otherwise and 0 ≤ ˜li < L for all i ∈ τ \ τ0, then we observe that

l ˜l ζχ ◦ q ◦ (ΦJ × iτ )|int(∆k×∆l) = q ◦ (ΦJ × iτ )|int(∆k×∆l).

l ˜l l l l In particular ζχ · eJ,τ = eJ,τ and whenever ζχ · eJ,τ = eJ,τ we see the restriction of ζχ to eJ,τ is the identity map, as required.

5.4 A CW complex structure on P(χ)

n Recalling Theorem 4.2.9 and Deﬁnition 4.2.11, it follows that each point of CP /Cχ can be uniquely represented by a point of the form

λτ0+1 2πixτ0+1/L λn 2πixn/L n [0 : ... : 0 : rτ0 : rτ0+1ζ e : ... : rnζ e ] ∈ CP with its associated triple {r , x, λ} , where Σ` r2 = 1 and r 6= 0 for all j ∈ τ, τ i=0 τj j

λ1 λn x = (x1, . . . , xn) and λ = (λ1, . . . , λn) is such that (ζL , . . . , ζL ) is a canonical representative of an orbit in (C )n under (C ) . The point L τ\τ0 χ τ

λτ0+1 2πixτ0+1/L λn 2πixn/L [0 : ... : 0 : rτ0 : rτ0+1ζ e : ... : rnζ e ]

λ is contained in the cell eJ,τ for some J. λ Therefore we restrict attention to the images of such eJ,τ under the quotient map n n λ ∗ λ ∗ λ p: CP → CP /Cχ. We denote by (¯eJ,τ ) and (eJ,τ ) the images of p ◦ q ◦ (ΦJ × iτ ) λ n and p ◦ q ◦ (ΦJ × iτ )|int(∆k×∆`) respectively. The map p identiﬁes the cells of CPL in the following way: CHAPTER 5. CW-COMPLEX ON P(χ) 82

l n Remark 5.4.1. Let p ◦ q ◦ (ΦJ × iτ ) be a characteristic map of CPL . There exists

λ1 λn some λ = (λ1, . . . , λn) such that (ζL , . . . , ζL ) is the canonical representative of the orbit of the point (ζl1 , . . . , ζln ) ∈ (C )n under the (C ) -action. We observe L L L τ\τ0 χ τ

l λ p ◦ q ◦ (ΦJ × iτ ) = p ◦ q ◦ (ΦJ × iτ ).

l λ λ ∗ In particular we note that p(eJ,τ ) = p(eJ,τ ) = (eJ,τ ) .

λ n Theorem 5.4.2. Let q ◦ (ΦJ × iτ ) range over the characteristic maps of cells in CPL , where λ is such that (ζλ1 , . . . , ζλn ) is a canonical representative of an orbit in (C )n L L L τ\τ0 under (Cχ)τ . Then the family of maps

λ k ` n p ◦ q ◦ (ΦJ × iτ ): ∆ × ∆ → CP /Cχ

n is a family of characteristic maps for a CW-structure on CP /Cχ.

λ ∗ λ0 ∗ Proof. We observe from the comments preceding Remark 5.4.1 that (eJ,τ ) ∩(eJ0,τ 0 ) 6= 0 0 0 n ∅ if and only if l = l , J = J and rτ = rτ 0 ; and every point of CP /Cχ is contained λ ∗ in some (eJ,τ ) . Therefore it follows from Theorem 5.3.11 and Proposition 2.6.6 that λ k ` n the family of maps p ◦ q ◦ (ΦJ × iτ ): ∆ × ∆ → CP /Cχ given in the statement of the theorem is a family of characteristic maps for a CW-complex on the orbit space

n CP /Cχ.

λ ∗ Therefore, each (eJ,τ ) such that λ, J and τ are of the form required by Theorem n λ 5.4.2 is a (k + `)-cell of a CW-structure on CP /Cχ, with p ◦ q ◦ (ΦJ × iτ ) as its n characteristic map. We will denote this CW-structure by CPL /Cχ. The union of all n (k+`) cells of dimensions less than or equal to k + `, which we write as (CPL /Cχ) , is n λ ∗ therefore the (k + `)-skeleton of CPL /Cχ. Note that the boundary of the cell (eJ,τ ) is determined by

λ ∗ λ ∂(eJ,τ ) = p ∂{eJ,τ }.

n n n As the quotient map q : T × ∆c → CP is Cχ-equivariant, it induces the map

n n n q˜: T /Cχ × ∆c → CP /Cχ

λ1 2πix1/L λn 2πixn/L ∗ λ1 2πix1/L λn 2πixn/L ∗ (ζL e , . . . , ζL e , r0, . . . , rn) 7→ [r0 : r1ζL e : ... : rnζL e ] . CHAPTER 5. CW-COMPLEX ON P(χ) 83

n n n Corollary 5.4.3. The map q˜: TL /Cχ × ∆c → CPL /Cχ is cellular. In particular if λ n p ◦ q ◦ (ΦJ × iτ ) is a characteristic map for CPL /Cχ then

λ λ q˜ ◦ (ΦeJ × iτ ) = p ◦ q ◦ (ΦJ × iτ ),

λ n n where ΦeJ × iτ is a characteristic map for TL /Cχ × ∆c .

Proof. The deﬁnition ofq ˜ is such that the following diagram is commutative:

n n q n TL × ∆c / CP p p

n n q˜ n TL /Cχ × ∆c / CP /Cχ and it follows directly thatq ˜ is cellular.

λ n Let p ◦ q ◦ (ΦJ × iτ ) be a characteristic map for CPL /Cχ. As λ and J are of n λ λ the form required to be a characteristic map of TL /Cχ, we observe ΦeJ = p ◦ ΦJ . n n Furthermore, from the deﬁnition of the characteristic maps for TL /Cχ ×∆c , we observe λ λ ΦeJ × iτ = p ◦ (ΦJ × iτ ). Hence

λ λ p ◦ q ◦ (ΦJ × iτ ) =q ˜ ◦ p ◦ (ΦJ × iτ )

λ =q ˜ ◦ (ΦeJ × iτ ).

Thereforeq ˜ induces a chain mapq ˜∗. Corollary 5.4.3 will be of use in Chapter 7, n as generating cycles of H2k(CPL /Cχ) are identiﬁed via identifying generating cycles of k n n H2k(TL/Cχ × ∆c ) and H2k(CPL ) for 0 ≤ k ≤ n. We recall from Section 3.4 that the map

n h: CP /Cχ → P(χ)

∗ χ0 χn [z0 : ... : zn] 7→ [z0 : ... : zn ]χ is a homeomorphism.

λ n Theorem 5.4.4. Let p ◦ q ◦ (ΦJ × iτ ) range over the characteristic maps of CPL /Cχ. The family of maps

λ k ` h ◦ p ◦ q ◦ (ΦJ × iτ ): ∆ × ∆ → P(χ) is a family of characteristic maps for a CW-structure on P(χ). CHAPTER 5. CW-COMPLEX ON P(χ) 84

λ λ ∗ Therefore, each (eJ,τ )χ := h(eJ,τ ) is a (k + `)-cell of a CW-structure on P(χ), with λ h ◦ p ◦ q ◦ (ΦJ × iτ ) as its characteristic map. Let P(χ)L denote this CW-structure and (k+`) (P(χ)L) denote its (k + `)-skeleton. Following the lead of Examples 4.4.7 and 4.4.8 we will discuss the CW-structures on

2 CPL/Cχ for χ = (1, 2, 3) and χ = (2, 3, 5). Under the homeomorphism h, these CW- structures then determine the CW-structures P(1, 2, 3)6 and P(2, 3, 5)30 respectively.

2 (0,0) ∗ Example 5.4.5 (χ = (1, 2, 3)). There are three 0-cells of CP6 /Cχ, namely (e∅,τ={0}) , (0,0) ∗ (0,0) ∗ ∗ ∗ (e∅,τ={1}) and (e∅,τ={2}) and these 0-cells correspond to the points [1 : 0 : 0] ,[0 : 1 : 0] ∗ 2 and [0 : 0 : 1] in CP /Cχ respectively. The set of points in (4) Example 4.4.7, corresponds to the union of the cells of the

(λ1,0) ∗ form (eJ,τ={0,1}) , where λ1 ∈ {0, 1, 2}. In particular we observe this is made up of (λ1,0) ∗ (λ1,0) ∗ three 1-cells of the form (e∅,τ={0,1}) and three 2-cells of the form (e{1},τ={0,1}) . The set of points in (5) Example 4.4.7, corresponds to the union of the cells of the

(0,λ2) ∗ form (eJ,τ={0,2}) , where λ2 ∈ {0, 1}. In particular we observe this is made up of two (0,λ2) ∗ (0,λ2) ∗ 1-cells of the form (e∅,τ={0,2}) and two 2-cells of the form (e{2},τ={0,2}) . The set of points in (6) Example 4.4.7, corresponds to the union of the cells of the

(0,λ2) ∗ form (eJ,τ={1,2}) , where λ2 ∈ {0}. In particular we observe this is made up of one (0,λ2) ∗ (0,λ2) ∗ 1-cell of the form (e∅,τ={1,2}) and one 2-cell of the form (e{2},τ={1,2}) . The set of points in (7) Example 4.4.7, corresponds to the union of the cells of the

(λ1,λ2) ∗ form (eJ,τ={0,1,2}) , where λ1 ∈ {0, 1, 2} and λ2 ∈ {0, 1}. We observe that this is made (λ1,λ2) ∗ (λ1,λ2) ∗ up of six 2-cells of the form (e∅,τ={0,1,2}) , six 3-cells of the form (e{1},τ={0,1,2}) , six (λ1,λ2) ∗ (λ1,λ2) ∗ 3-cells of the form (e{1,2},τ={0,1,2}) , six 3-cells of the form (e{2},τ={0,1,2}) , six 4-cells of (λ1,λ2) ∗ (λ1,λ2) ∗ the form (e{{1},{2}},τ={0,1,2}) and six 4-cells of the form (e{{2},{1}},τ={0,1,2}) . 2 Therefore we conclude that CP6 /Cχ has a total of three 0-cells, six 1-cells, twelve 2-cells, eighteen 3-cells and twelve 4-cells.

2 (0,0) ∗ Example 5.4.6 (χ = (2, 3, 5)). There are three 0-cells of CP30/Cχ, namely (e∅,τ={0}) , (0,0) ∗ (0,0) ∗ ∗ ∗ (e∅,τ={1}) and (e∅,τ={2}) and these 0-cells correspond to the points [1 : 0 : 0] ,[0 : 1 : 0] ∗ 2 and [0 : 0 : 1] in CP /Cχ respectively. The set of points in (4) Example 4.4.8, corresponds to the union of the cells of the

(λ1,0) ∗ form (eJ,τ={0,1}) , where λ1 ∈ {0, 1, 2, 3, 4}. In particular we observe this is made up (λ1,0) ∗ (λ1,0) ∗ of ﬁve 1-cells of the form (e∅,τ={0,1}) and ﬁve 2-cells of the form (e{1},τ={0,1}) . CHAPTER 5. CW-COMPLEX ON P(χ) 85

The set of points in (5) Example 4.4.8, corresponds to the union of the cells of the

(0,λ2) ∗ form (eJ,τ={0,2}) , where λ2 ∈ {0, 1, 2}. In particular we observe this is made up of (0,λ2) ∗ (0,λ2) ∗ three 1-cells of the form (e∅,τ={0,2}) and three 2-cells of the form (e{2},τ={0,2}) . The set of points in (6) Example 4.4.8, corresponds to the union of the cells of the

(0,λ2) ∗ form (eJ,τ={1,2}) , where λ2 ∈ {0, 1}. In particular we observe this is made up of two (0,λ2) ∗ (0,λ2) ∗ 1-cells of the form (e∅,τ={1,2}) and two 2-cells of the form (e{2},τ={1,2}) . The set of points in (7) Example 4.4.8, corresponds to the union of the cells of the

(λ1,λ2) ∗ form (eJ,τ={0,1,2}) , where λ1 ∈ {0, 1, 2, 3, 4} and λ2 ∈ {0, 1, 2, 3, 4, 5}. We observe that (λ1,λ2) ∗ this is made up of thirty 2-cells of the form (e∅,τ={0,1,2}) , thirty 3-cells of the form (λ1,λ2) ∗ (λ1,λ2) ∗ (e{1},τ={0,1,2}) , thirty 3-cells of the form (e{1,2},τ={0,1,2}) , thirty 3-cells of the form (λ1,λ2) ∗ (λ1,λ2) ∗ (e{2},τ={0,1,2}) , thirty 4-cells of the form (e{{1},{2}},τ={0,1,2}) and thirty 4-cells of the (λ1,λ2) ∗ form (e{{2},{1}},τ={0,1,2}) . 2 Therefore we conclude that CP30/Cχ has a total of three 0-cells, ten 1-cells, forty 2-cells, ninety 3-cells and sixty 4-cells. Chapter 6

Cellular chain complex of P(χ)

6.1 Introduction and motivation

Kawasaki [15] calculates the integral homology of weighted projective space and gives

n the degree of the induced map (pχ)∗ : H2k(CP ) → H2k(P(χ)). In Chapter 5 we determined CW-structures such that the following is a commutative diagram of cellular maps

n n n n TL / TL × ∆ / CPL

n n n n TL /Cχ / TL /Cχ × ∆ / CPL /Cχ and so considering the associated cellular chain complexes of each of these cell structures, the following is a commutative diagram of chain maps

n n n n Ci(TL ) / Ci(TL × ∆ ) / Ci(CPL )

n n n n Ci(TL /Cχ) / Ci(TL /Cχ × ∆ ) / Ci(CPL /Cχ). n Therefore once the cellular chain complex of TL is defined, we can use these chain maps to quickly determine the cellular chain complexes of the remaining CW-structures in the above diagram. One benefit of prescribing a ∆-complex structure on T n is that the cellular chain complex is straightforward to determine. Furthermore, as noted in the introduction to Chapter 5, another benefit we will see is that any identifications

n n n n n made by the quotient maps q : TL ×∆L → CPL and p: CPL → CPL /Cχ can be pulled n back to identiﬁcations of simplices in TL , making the induced maps q∗ and p∗ much more simple to understand.

86 CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 87 n n 6.2 Cellular chain complex of TL and TL /Cχ

n l The k-cells of TL , given in Theorem 5.1.4 are the open cells ∆J , such that J is an ordered partition of S ⊆ [n] into k parts and the n-tuple l is such that 0 ≤ li < L for 1 ≤ i ≤ n.

n We recall from Section 2.2 that the cellular chain group Ck(TL ) is free abelian with n basis the k-cells of TL , but there is a sign ambiguity for the basis element corresponding l n k n (k−1) to ∆J , namely the choice of generator for the Z summand in Hk((TL ) , (TL ) ) l corresponding to ∆J . Only when k = 0 is the choice canonical. For k > 0, we recall l that by ﬁrst orienting the domain simplex, we can use the characteristic map, ΦJ , of l ∆J to make such a choice. We saw in Section 2.4 that for ∆-complexes an orientation for ∆k can be chosen by ordering the vertices of ∆k. We adopt the convention given

k in Section 2.3.1 that the vertices e0, . . . , ek of ∆ are ordered according to the ordering of the standard basis of Rn+1 and we denote this oriented simplex [∆k]. Recalling Section 2.4 and Theorem 5.1.4, in particular (5.1.6) in its proof, we now describe the

n cellular chain complex of TL .

Notation 6.2.1. Let J range over all ordered partitions of S ⊆ [n] into k parts and l range over all n-tuples (l1, . . . , ln) such that 0 ≤ li < L for 1 ≤ i ≤ n. The cellular n chain group Ck(TL ) is given by

n l l k Ck(TL ) = Z ∆J = (ΦJ )∗[∆ ]: all J and l .

n n The cellular boundary homomorphism ∂k : Ck(TL ) → Ck−1(TL ) is deﬁned by

∂ ∆l = ∆l0 k J {J1,...,Jk−1} k−1 X + (−1)i∆l {J1,...,Jk−i∪Jk−i+1,...,Jk} i=1 + (−1)k∆l (6.2.2) {J2,...,Jk} where we let l0 be such that  li + 1 mod L if i ∈ Jk 0  li = (6.2.3) li otherwise

0 and 0 ≤ li < L for 1 ≤ i ≤ n.

n Similarly we recall the ∆-complex structure on TL /Cχ given in Theorem 5.1.8. CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 88

Notation 6.2.4. Let J range over all ordered partitions of S ⊆ [n] into k parts

λ1 λn and let λ range over all n-tuples (λ1, . . . , λn) such that (ζL , . . . , ζL ) is a canonical n n representative of an orbit in (CL) under Cχ. The cellular chain group Ck(TL /Cχ) of n TL /Cχ is n n λ ∗ λ k o Ck(TL /Cχ) = Z (∆J ) = (ΦeJ )∗[∆ ]: all J and λ .

n n The cellular boundary homomorphism ∂k : Ck(TL /Cχ) → Ck−1(TL /Cχ) is deﬁned by

∂ (∆λ)∗ = (∆λ˜ )∗ k J {J1,...,Jk−1} k−1 X + (−1)i(∆λ )∗ {J1,...,Jk−i∪Jk−i+1,...,Jk} i=1 + (−1)k(∆λ )∗ (6.2.5) {J2,...,Jk}

˜ ˜ ˜ ˜ ˜ λ1 λn where λ = (λ1,..., λn) is such that (ζL , . . . , ζL ) is the canonical representative in 0 0 n λ1 λn 0 0 (CL) of the orbit of (ζL , . . . , ζL ) under Cχ, and (λ1, . . . , λn) is determined by  λi + 1 mod L if i ∈ Jk 0  λi = (6.2.6)  0 λi otherwise.

n n n n 6.3 Cellular chain complex of TL ×∆c and TL /Cχ×∆c

n n l We recall that the m-cells of TL ×∆c given in Theorem 5.2.1 are the open cells ∆J ×∆τ , l n n ranging over the k-cells ∆J of TL and the `-cells ∆τ of ∆c such that k + ` = m. The l l k ` n n characteristic map of the product cell ∆J × ∆τ is ΦJ × iτ : ∆ × ∆ → TL × ∆c . We recall the product orientation of ∆k × ∆`. Having deﬁned the cellular chain complexes

n n of ∆c and TL in Notation 2.5.2 and Notation 6.2.1 respectively, we use these to write n n down the cellular chain complex of the product TL × ∆c , following the conventions given in Section 2.5.2.

Notation 6.3.1. Let J range over all ordered partitions of S ⊆ [n] into k parts; l range over all n-tuples (l1, . . . , ln) such that 0 ≤ li < L for 1 ≤ i ≤ n; and τ = {τ0, . . . , τ`} n n n n range over all subsets of {0, . . . , n}. The cellular chain group Ck+`(TL ×∆c ) of TL ×∆c is n n l l k ` Ck+`(TL × ∆c ) = Z ∆J × ∆τ = (ΦJ × iτ )∗[∆ ] × [∆ ]: all J, l and τ . CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 89

n n n n The cellular boundary homomorphism ∂k+` : Ck(TL ×∆c ) → Ck−1(TL ×∆c ) is deﬁned by

l l k l ∂k+`(∆J × ∆τ ) = ∂k∆J × ∆τ + (−1) ∆J × ∂`(∆τ ) = ∆l0 × ∆ {J1,··· ,Jk−1} τ k−1 X + (−1)i∆l × ∆ {J1,··· ,Jk−i∪Jk−i+1,··· ,Jk} τ i=1 + (−1)k∆l × ∆ {J2,··· ,Jk} τ ` k X j l + (−1) (−1) ∆J × ∆τ\τj (6.3.2) j=0 where l0 is given in (6.2.3).

n n Analogously for TL /Cχ × ∆c we have the following.

Notation 6.3.3. Let J range over all ordered partitions of S ⊆ [n] into k parts; λ

λ1 λn range over all n-tuples (λ1, . . . , λn) such that (ζL , . . . , ζL ) is a canonical representative n of an orbit in (CL) under Cχ; and τ = {τ0, . . . , τ`} range over all subsets of {0, . . . , n}. n n n n The cellular chain group Ck+`(TL /Cχ × ∆c ) of TL /Cχ × ∆c is

n n n λ ∗ λ k ` o Ck+`(TL /Cχ × ∆c ) = Z (∆J ) × ∆τ = (ΦeJ × iτ )∗[∆ ] × [∆ ]: all J, λ and τ .

n n n n The cellular boundary homomorphism ∂k+` : Ck(TL /Cχ × ∆c ) → Ck−1(TL /Cχ × ∆c ) is deﬁned by

λ ∗ λ ∗ k λ ∗ ∂k+`((∆J ) × ∆τ ) = ∂k(∆J ) × ∆τ + (−1) (∆J ) × ∂`(∆τ ) = (∆λ˜ )∗ × ∆ {J1,··· ,Jk−1} τ k−1 X + (−1)i(∆λ )∗ × ∆ {J1,··· ,Jk−i∪Jk−i+1,··· ,Jk} τ i=1 + (−1)k(∆λ )∗ × ∆ {J2,··· ,Jk} τ ` k X j λ ∗ + (−1) (−1) (∆J ) × ∆τ\τj , (6.3.4) j=0

˜ ˜ ˜ ˜ ˜ λ1 λn where λ = (λ1,..., λn) is such that (ζL , . . . , ζL ) is the canonical representative in 0 0 n λ1 λn 0 0 (CL) of the orbit of (ζL , . . . , ζL ) under Cχ, and (λ1, . . . , λn) is given in (6.2.6). CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 90 n 6.4 Cellular chain complex of CPL

n We recall the CW-structure CPL given in Theorem 5.3.8 where the characteristic map l of the (k + `)-cell eJ,τ is

l k ` k ` n (k+`) n (k+`−1) q ◦ (ΦJ × iτ ): (∆ × ∆ , ∂(∆ × ∆ )) → ((CPL ) , (CPL ) ).

Letting m = k + `, we can factor through this characteristic map by the pair

n n (m) n n (m−1) ((TL × ∆c ) , (TL × ∆c ) ) as follows:

Φl ×i k ` k ` J τ n n (m) n n (m−1) (∆ × ∆ , ∂(∆ × ∆ )) / ((TL × ∆c ) , (TL × ∆c ) )

q q◦(Φl ×i ) J τ + n (m) n (m−1) ((CPL ) , (CPL ) ) which induces the following commutative diagram in homology

(Φl ×i ) k ` k ` J τ ∗ n n (m) n n (m−1) Hm(∆ × ∆ , ∂(∆ × ∆ )) / Hm((TL × ∆c ) , (TL × ∆c ) )

(q)∗ (q◦(Φl ×i )) J τ ∗ , n (m) n (m−1) Hm((CPL ) , (CPL ) ).

l We observe that the map (q)∗ takes ∆J × ∆τ to the generator of the Z summand in n (m) n (m−1) l Hm((CPL ) , (CPL ) ) corresponding to eJ,τ .

Notation 6.4.1. For all τ, J and l of the form considered in Notation 5.3.6 and such

n n that m = k + `, the cellular chain group Cm(CPL ) of CPL is

n l l Cm(CPL ) = Z eJ,τ = (q)∗(∆J × ∆τ ): all τ, J and l .

Therefore, following from Corollary 5.3.9 and inputting (6.3.2), the cellular boundary

n n homomorphism ∂m : Cm(CPL ) → Cm−1(CPL ) is deﬁned by

l l ∂m(eJ,τ ) = q∗ ∂m(∆J × ∆τ )

n 0 = q ∆l × ∆ ∗ {J1,...,Jk−1} τ k−1 X + (−1)i∆l × ∆ {J1,...,Jk−i∪Jk−i+1,...,Jk} τ i=1 + (−1)k∆l × ∆ {J2,...,Jk} τ ` ) X k+j l + (−1) ∆J × ∆τ\τj ) , (6.4.2) j=0 CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 91 where l0 is given in (6.2.3). We note that with the requirements for l to be of the form considered in Notation 5.3.6, we see that l0 is given by   li + 1 mod L if i ∈ Jk  0  li = li if i ∈ Jα for 1 ≤ α ≤ k − 1 (6.4.3)   0 otherwise In order to make deductions about generating cycles from the cellular chain complex of P(χ)L we are required to deﬁne (6.4.2) explicitly for certain cells. This is given in the following proposition and is achieved by identifying each term in (5.3.10) with

n its corresponding element in Cm−1(CPL ). Before stating the next proposition we are required to set up some notation:

Notation 6.4.4. Let τ = {τ0, . . . , τk} ⊆ {0, . . . , n} and J be an ordered partition of

τ \ τ0 into k parts. Note that the parts of J are singleton sets therefore we can write

J as {{τi1 },..., {τik }}.

Proposition 6.4.5. Let τ, J, and l be given.

∂ el = el0 2k J,τ {J1,...,Jk−1},τ k−1 X + (−1)iel {J1,...,Jk−i∪Jk−i+1,...,Jk},τ i=1 + (−1)kel (6.4.6) {J2,...,Jk},τ where l0 is determined by (6.4.3).

Proof. As q is a chain map, then from Notation 6.4.1 we see

l l ∂2k(eJ,τ ) = q∗ ∂2k(∆J × ∆τ )

n 0 = q ∆l × ∆ ∗ {J1,...,Jk−1} τ k−1 X + (−1)i∆l × ∆ {J1,...,Jk−i∪Jk−i+1,...,Jk} τ i=1 + (−1)k∆l × ∆ {J2,...,Jk} τ ` ) X k+j l + (−1) ∆J × ∆τ\τj ) . (6.4.7) j=0 n n Equation (6.4.7) requires a closer look at the epimorphism q∗ : C2k(TL × ∆c ) → n 0 C2k(CPL ). As l , J and τ are of the form required by Notation 5.3.6 it follows that

q (∆l0 × ∆ ) = el0 . ∗ {J1,...,Jk−1} τ {J1,...,Jk−1},τ CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 92

Similarly

q (∆l × ∆ ) = el ∗ {J1,...,Jk−i∪Jk−i+1,...,Jk} τ {J1,...,Jk−i∪Jk−i+1,...,Jk},τ q (∆l × ∆ ) = el . ∗ {J2,...,Jk} τ {J2,...,Jk},τ

Pk k+j l We now look at q∗ j=0(−1) ∆J × ∆τ\τj in equation (6.4.7). Observe

k k X k+j l X k+j l q∗ (−1) ∆J × ∆τ\τj = (−1) q∗(∆J × ∆τ\τj ). j=0 j=0

l We begin by considering q∗(∆J × ∆τ\τj ) for 1 ≤ j ≤ k. It follows from Proposition 5.3.2 that im{q ◦ (Φl × i )} = im{q ◦ (Φl00 × i )} J τ\τj J\{τj } τ\τj where  li if i ∈ τ \{τ0, τj} for 1 ≤ j ≤ k 00  li = 0 otherwise therefore q(∆l × ∆ ) = el00 . We observe that el00 is a (2k − 2)-cell and J τ\τj J\{τj },τ\τj J\{τj },τ\τj l so on a chain level we see q∗(∆J × ∆τ\τj ) = 0. l For q∗(∆J × ∆τ\τ0 ) it follows from Proposition 5.3.2 that

l l000 im{q ◦ (ΦJ × iτ\τ0 )} = im{q ◦ (ΦJ\{τ1} × iτ\τ0 )} where  li − lτ if i ∈ τ \{τ0, τ1} 000  1 li = 0 otherwise therefore q(∆l ×∆ ) = el000 . We observe that el000 is a (2k −2)-cell and J τ\τ0 J\{τ1},τ\τ0 J\{τ1},τ\τ0 l so on a chain level we see q∗(∆J × ∆τ\τ0 ) = 0. Therefore k X k+j l q∗ (−1) ∆J × ∆τ\τj = 0. j=0

Notation 6.4.8. Let τ = {τ0, . . . , τk+1} ⊆ {0, . . . , n} and J be an ordered partition of τ \{τ0, τs}, for some s ∈ {1, . . . , k + 1} into k parts. Note that the parts of J are singleton sets therefore we can write J as {{τi1 },..., {τik }}. In particular we note there exists some α ∈ [k] such that Jα = {τ1}. CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 93

Proposition 6.4.9. Let τ, J, and l be given. Then

∂ el = el0 2k+1 J,τ {J1,...,Jk−1},τ k−1 X + (−1)iel {J1,...,Jk−i∪Jk−i+1,...,Jk},τ i=1 + (−1)kel {J2,...,Jk},τ k+α(k−α) l000 0 + (−1) eJ ,τ\τ0

k+s l0 + (−1) eJ,τ\τs , (6.4.10) where  li if i ∈ τ \{τ0, τs} 00  li = (6.4.11) 0 otherwise,

0 0 0 0 and J is the ordered partition of τ \{τ0, τ1} given by J1 = Jα+1,J2 = Jα+2,...,Jk−α = 0 0 0 0 000 Jk,Jk−α+1 = {τs},Jk−α+2 = J1,Jk−α+3 = J2,...,Jk = Jα−1 and where l = 000 000 (l1 , . . . , ln ) is such that  l − l mod L i ∈ J 0 for 1 ≤ q < k − α  i τ1 q   0 000  li − lτ1 − 1 mod L i ∈ Jq for k − α < q ≤ k li =  l − l − 1 mod L i = τ  i τ1 s   0 otherwise,

000 where 0 ≤ li < L for 1 ≤ i ≤ n.

Proof. From Notation 6.4.1 we see

l l ∂2k+1(eJ,τ ) = q∗ ∂2k+1(∆J × ∆τ )

n 0 = q ∆l × ∆ ∗ {J1,...,Jk−1} τ k−1 X + (−1)i∆l × ∆ {J1,...,Jk−i∪Jk−i+1,...,Jk} τ i=1 + (−1)k∆l × ∆ {J2,...,Jk} τ k+1 ) X k+j l + (−1) ∆J × ∆τ\τj . (6.4.12) j=0 Anagolously to the proof of Proposition 6.4.5 it follows that

q (∆˜l × ∆ ) = e˜l ∗ {J1,...,Jk−1} τ {J1,...,Jk−1},τ q (∆l × ∆ ) = el ∗ {J1,...,Jk−i∪Jk−i+1,··· ,Jk} τ {J1,...,Jk−i∪Jk−i+1,...,Jk},τ q (∆l × ∆ ) = el . ∗ {J2,...,Jk} τ {J2,...,Jk},τ CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 94

Pk+1 k+j l We now look at q∗ j=0 (−1) ∆J × ∆τ\τj in equation (6.4.12). Observe

k+1 k+1 X k+j l X k+j l q∗ (−1) ∆J × ∆τ\τj = (−1) q∗(∆J × ∆τ\τj ). j=0 j=0

l We begin by considering q∗(∆J × ∆τ\τj ) for 1 ≤ j ≤ k and j 6= s. It follows from Proposition 5.3.2 that

im{q ◦ (∆l × ∆ )} = im{q ◦ (∆l00 × ∆ )} J τ\τj J\{τj } τ\τj where  li if i ∈ τ \{τ0, τj} for 1 ≤ j ≤ k 00  li = (6.4.13) 0 otherwise. Therefore q(∆l × ∆ ) = el00 . We observe that el00 is a (2k − 1)-cell J τ\τj J\{τj },τ\τj J\{τj },τ\τj l and so on a chain level we see q∗(∆J × ∆τ\τj ) = 0. When j = s then it follows from Proposition 5.3.2 that

l l00 im{q ◦ (∆J × ∆τ\τs )} = im{q ◦ (∆J × ∆τ\τs )} where  li if i ∈ τ \{τ0, τs} 00  li = (6.4.14) 0 otherwise. Therefore q(∆l × ∆ ) = el00 and we observe that on a chain level q (∆l × J τ\τs J,τ\τs ∗ J ∆ ) = nel00 for some n ∈ . In fact we can easily observe that the two maps τ\τs J,τ\τs Z l l00 q ◦ (ΦJ × iτ\τs )|int(∆k×∆k) and q ◦ (ΦJ × iτ\τs )|int(∆k×∆k) are equal, hence

l l k k q∗(∆J × ∆τ\τs ) = (q ◦ (ΦJ × iτ\τs ))∗(∆ × ∆ )

l00 k k = (q ◦ (ΦJ × iτ\τs ))∗(∆ × ∆ )

l00 = q∗(∆J × ∆τ\τs )

l00 = +eJ,τ\τs .

l For q∗(∆J × ∆τ\τ0 ) we observe that {τ1} = Jα for some α ∈ [k]. In this case we can l l000 we deduce from Proposition 5.3.2 that im{q ◦ (∆J × ∆τ\τ0 )} = im{q ◦ (∆J0 × ∆τ\τ0 )} 0 0 0 0 0 0 where J1 = Jα+1,J2 = Jα+2,...,Jk−α = Jk,Jk−α+1 = {τs},Jk−α+2 = J1,Jk−α+3 = 0 J2,...,Jk = Jα−1 and where  li − lτ if i ∈ τ \{τ0, τ1} 000  1 li = (6.4.15) 0 otherwise. CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 95

l l000 l Therefore q(∆ × ∆ ) = e 0 and on a chain level we have q (∆ × ∆ ) = J τ\τ0 J ,τ\τ0 ∗ J τ\τ0 l000 n(e 0 ) for some n ∈ . Unlike the other cases in this proof, we need to look more J ,τ\τ0 Z l l000 closely at the maps q ◦ (ΦJ × iτ\τ0 ) and q ◦ (ΦJ0 × iτ\τ0 ). l Recalling the deﬁnition of q ◦ (ΦJ × iτ\τ0 ) we see

l 0 0 q ◦ (ΦJ × iτ\τ0 )(t0, . . . , tk, t0, . . . , tk)

l1 2πx1/L ln 2πxn/L = [ι0, ι1ζL e , . . . , ιnζL e ]

lτ lτ +1 1 2πxτ1 /L 1 2πxτ1+1/L ln 2πxn/L = [0,..., 0, ιτ1 ζL e , ιτ1+1ζL e , . . . , ιnζL e ]

lτ +1−lτ ln−lτ 1 1 2π(xτ1 −xτ1+1)/L 1 2π(xn−xτ1+1)/L = [0,..., 0, ιτ1 , ιτ1+1ζL e , . . . , ιnζL e ]

where xτ1 = tk−α + ··· + tk and

  −tk−α − · · · − tk if a = τs    tk − tk−α − · · · − tk if a ∈ J1    tk−1 + tk − tk−α − · · · − tk if a ∈ J2   . .  . .    tk−α+1 + ··· + tk − tk−α − · · · − tk if a ∈ Jα−1 xa − xτ1 =  tk−α + ··· + tk − tk−α − · · · − tk if a ∈ Jα    tk−α−1 + ··· + tk − tk−α − · · · − tk if a ∈ Jα+1   . .  . .    t1 + ··· + tk − tk−α − · · · − tk if a ∈ Jk    0 if a∈ / τ \ τ0.

By rearranging t0 + ··· + tk = 1 we see that for all a ∈ {τ1, . . . , n} then   t0 + t1 + ... + tk−α−1 − 1 if a = τs    t0 + t1 + ... + tk−α−1 + tk − 1 if a ∈ J1    t0 + t1 + ... + tk−α−1 + tk−1 + tk − 1 if a ∈ J2   . .  . .    t0 + t1 + ... + tk−α−1 + tk−α+1 + ··· + tk − 1 if a ∈ Jα−1 xa − xτ1 =  0 if a ∈ Jα    tk−α−1 if a ∈ Jα+1   . .  . .    t1 + ... + tk−α−1 if a ∈ Jk    0 if a∈ / τ \ τ0. CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 96

l000 We precompose ΦJ0 with the linear transformation

T : (∆k, ∂∆k) → (∆k, ∂∆k)

(t0, t1, . . . , tk) 7→ (tk−α, tk−α+1, . . . , tk−1, tk, t0, t1, . . . , tk−α−2, tk−α−1)

k 0 0 that permutes the vertices of ∆ . Observe that the image of (t0, . . . , tk, t0, . . . , tk) l000 under q ◦ ((ΦJ0 ◦ T ) × iτ\τ0 ) is such that

l000 0 0 q ◦ ((ΦJ0 ◦ T ) × iτ\τ0 )(t0, . . . , tk, t0, . . . , tk)

l000 0 0 = q ◦ (ΦJ0 × iτ\τ0 )((tk−α, tk−α+1, . . . , tk−1, tk, t0, t1, . . . , tk−α−2, tk−α−1), (t0, . . . , tk)) l000 0 l000 0 1 2πx1/L n 2πxn/L = [ι0, ι1ζL e , . . . , ιnζL e ]

000 000 lτ +1 2πx0 /L l 2πx0 /L 1 τ1+1 n n = [0,..., 0, ιτ1 , ιτ1+1ζL e , . . . , ιnζL e ] where for all a ∈ {τ1, . . . , n} we have

0 0 xa = xa(tk−α, tk−α+1, . . . , tk−1, tk, t0, t1, . . . , tk−α−2, tk−α−1).

0 Here xa takes the values  0  tk−α−1 if a ∈ J1 = Jα+1   0  tk−α−2 + tk−α−1 if a ∈ J2 = Jα+2   . .  . .   0  t1 + ··· + tk−α−1 if a ∈ Jk−α = Jk  0 0 xa = t0 + t1 + ··· + tk−α−1 if a ∈ Jk−α+1 = {τs}   t + t + ··· + t + t if a ∈ J 0 = J  0 1 k−α−1 k k−α+2 1  . .  . .    t + t + ··· + t + t + ··· + t if a ∈ J 0 = J  0 1 k−α−1 k−α+1 k k α−1   0 if a∈ / J 0 .

When we take into account the deﬁnitions of the n-tuples l and l000 then it becomes 000 (la−lτ ) l 0 1 2πi(xa−xτ1 )/L a 2πixa/L clear that ιaζL e = ιaζL e for all a ∈ {τ1, . . . , n}. That is, q ◦ l000 0 0 l 0 0 ((ΦJ0 ◦ T ) × iτ\τ0 )(t0, . . . , tk, t0, . . . , tk) = q ◦ (ΦJ × iτ\τ0 )(t0, . . . , tk, t0, . . . , tk) and we deduce

l000 l q ◦ ((ΦJ0 ◦ T ) × iτ\τ0 ) = q ◦ (ΦJ × iτ\τ0 ).

We note that the image of T is the k-simplex

[ek−α, ek−α+1, . . . , ek−1, ek, e0, e1, . . . , ek−α−2, ek−α−1]. CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 97

The two oriented simplices

[e0, . . . , ek] and [ek−α, ek−α+1, . . . , ek−1, ek, e0, e1, . . . , ek−α−2, ek−α−1] have the same orientation if and only if an even permutation transforms one ordering on the vertices into the other; if the permutation is odd, the orientations are said to be opposite. 0 1 2 ··· k−α−1 k−α k−α+1 ··· k The permutation ρ = α α+1 α+2 ··· k−1 k 0 ··· α−1 transforms

[e0, . . . , ek] to [ek−α, ek−α+1, . . . , ek−1, ek, e0, e1, . . . , ek−α−2, ek−α−1].

It is easy to see the sign of this permutation is α(k −α), therefore passing to homology we have

k k k k T∗ : Hk(∆ , ∂∆ ) → Hk(∆ , ∂∆ ) ∆k 7→ (−1)α(k−α)∆k and we see

l l k `−1 q∗(∆J × ∆τ\τ0 ) = (q ◦ (ΦJ × iτ\τ0 ))∗(∆ × ∆ )

l00 k k = (q ◦ ((ΦJ0 ◦ T ) × iτ\τ0 ))∗(∆ × ∆ )

l00 α(k−α) k k = (q ◦ (ΦJ0 × iτ\τ0 ))∗((−1) ∆ × ∆ )

α(k−α) l000 k k = (−1) (q ◦ (ΦJ0 × iτ\τ0 ))∗(∆ × ∆ )

α(k−α) l000 = (−1) q∗(∆J0 × ∆τ\τ0 )

α(k−α) l000 0 = (−1) eJ ,τ\τ0 .

n 6.5 Cellular chain complex of CPL /Cχ

n We recall the CW-structure CPL /Cχ given in Theorem 5.4.2, where the characteristic λ ∗ map of the (k + `)-cell (eJ,τ ) is

λ k ` k ` n (k+`) n (k+`−1) p ◦ q ◦ (ΦJ × iτ ): (∆ × ∆ , ∂(∆ × ∆ )) → ((CPL /Cχ) , (CPL /Cχ) ). CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 98

Letting m = k + `, we can factor through this characteristic map by the pair

n (m) n (m−1) ((CPL ) , (CPL ) ) as follows:

q◦(Φλ×i ) k ` k ` J τ n (m) n (m−1) (∆ × ∆ , ∂(∆ × ∆ )) / ((CPL ) , (CPL ) )

p p◦q◦(Φλ×i ) J τ + n (m) n (m−1) ((CPL /Cχ) , (CPL /Cχ) ) which induces the following commutative diagram in homology

(p◦(Φl ×i )) k ` k ` J τ ∗ n (m) n (m−1) Hm(∆ × ∆ , ∂(∆ × ∆ )) / Hm((CPL ) , (CPL ) )

(p)∗ (p◦q◦(Φl ×i )) J τ ∗ , n (m) n (m) Hm((CPL /Cχ) , (CPL /Cχ) ).

λ We observe that the map (p)∗ takes eJ,τ to the generator of the Z summand in n (m) n (m−1) λ ∗ Hm((CPL ) , (CPL ) ) corresponding to (eJ,τ ) .

Notation 6.5.1. For all τ, J and λ of the form considered in Theorem 5.4.2 and such

n n that m = k + `, the cellular chain group Cm(CPL /Cχ) of CPL /Cχ is

n λ ∗ λ Cm(CPL /Cχ) = Z (eJ,τ ) = (p)∗(eJ,τ ): all τ, J and λ .

Therefore, following from Remark 5.4.1 and inputting (6.4.2) the cellular boundary

n n homomorphism ∂m : Cm(CPL /Cχ) → Cm−1(CPL /Cχ) is deﬁned by

λ ∗ λ ∂m(eJ,τ ) = p∗ ∂m((eJ,τ ) λ = p∗ ◦ q∗ ∂m(∆J × ∆τ ) n = p ◦ q ∆λ˜ × ∆ ∗ ∗ {J1,...,Jk−1} τ k−1 X + (−1)i∆λ × ∆ {J1,...,Jk−i∪Jk−i+1,...,Jk} τ i=1 + (−1)k∆λ × ∆ {J2,...,Jk} τ ` ) X k+j λ + (−1) ∆J × ∆τ\τj ) , (6.5.2) j=0

˜ ˜ ˜ ˜ ˜ λ1 λn where λ = (λ1,..., λn) is such that (ζL , . . . ζL ) is the canonical representative of the λ0 0 orbit of (ζ 1 , . . . ζλn ) ∈ (C )n under (C ) where we see that λ0 is given by L L L τ\τ0 χ τ   λi + 1 mod L if i ∈ Jk  0  λi = λi if i ∈ Jα for 1 ≤ α ≤ k − 1 (6.5.3)   0 otherwise. CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 99

Analogous to Proposition 6.4.5 we can deduce the following:

Proposition 6.5.4. Let τ, J, and λ be given.

∂ (eλ )∗ = (eλ˜ )∗ 2k J,τ {J1,...,Jk−1},τ k−1 X + (−1)i(eλ )∗ {J1,...,Jk−i∪Jk−i+1,...,Jk},τ i=1 + (−1)k(eλ )∗ (6.5.5) {J2,...,Jk},τ

0 0 λ˜1 λ˜n λ1 λn where (ζL , . . . , ζL ) is the canonical representative of the orbit of (ζL , . . . , ζL ) under the (C ) -action on (C )n such that (λ0 , . . . , λ0 ) is determined by (6.5.3) χ τ L τ\τ0 1 n

n n Proposition 6.5.6. The chain map q˜∗ : Cm(TL /Cχ) → Cm(CPL /Cχ) is such that if λ ∗ n λ ∗ λ ∗ (eJ,τ ) ∈ Cm(CPL /Cχ) then (˜q)∗(∆J ) × ∆τ = (eJ,τ ) .

Proof. This is a direct result of Corollary 5.4.3 and Notation 6.2.1

6.6 Cellular chain complex of P(χ)

n We recall the CW-structure CPL /Cχ given in Theorem 5.4.4 where the characteristic λ map of the (k + `)-cell (eJ,τ )χ is

λ k ` k ` (k+`) (k+`−1) h ◦ p ◦ q ◦ (ΦJ × iτ ): (∆ × ∆ , ∂(∆ × ∆ )) → ((P(χ)) , (P(χ)) ).

Analogously to Notation 6.5.1 we describe the cellular chain complex for P(χ)L as follows.

Notation 6.6.1. For all τ, J and λ of the form considered in Theorem 5.4.4 and such that m = k + `, the cellular chain group Cm(P(χ)L) of P(χ)L is

λ λ ∗ Cm(P(χ)L) = Z (eJ,τ )χ = (h)∗(eJ,τ ) : all τ, J and λ . CHAPTER 6. CELLULAR CHAIN COMPLEX OF P(χ) 100

Recalling that h is a homeomorphism and therefore a cellular map the cellular boundary homomorphism ∂m : Cm(P(χ)L) → Cm−1(P(χ)L) is deﬁned by

λ λ ∗ ∂m(eJ,τ )χ = h∗ ∂m(eJ,τ ) λ = h∗ ◦ p∗ ∂m(eJ,τ ) λ = h∗ ◦ p∗ ◦ q∗ ∂m(∆J × ∆τ ) n = h ◦ p ◦ q ∆λ˜ × ∆ ∗ ∗ ∗ {J1,...,Jk−1} τ k−1 X + (−1)i∆λ × ∆ {J1,...,Jk−i∪Jk−i+1,...,Jk} τ i=1 + (−1)k∆λ × ∆ {J2,...,Jk} τ ` ) X k+j λ + (−1) ∆J × ∆τ\τj ) . (6.6.2) j=0

Homology calculations for P(χ)

We use the cellular chain complexes deﬁned in Chapter 6 to determine a cycle of

n n C2k(CPL ) that generates H2k(CPL ). The image of this cycle under the chain map

(h ◦ p)∗ is a multiple of a cycle in C2k(PL(χ)). Observing that this multiple equals the degree of (p)∗, we conclude that such a cycle in C2k(P(χ)L) is indeed the generator of

H2k(P(χ)L).

Remark 7.0.1. The cell structures and associated cellular chain complexes given

n in Chapters 5 and 6 have been determined via the Cχ-action on (CL) where L = lcm{χ0, . . . , χn}. We observe that all work contained up to this point can be extended n more generally in terms of the Cχ-action on (CM ) , where M is a multiple of L as n n (CL) < (CM ) . It is easy to see the CW-structures obtained in this more general case are subdivisions of those given in Chapter 5.

n Having determined (C∗(CPL /Cχ), ∂∗) in Notation 6.5.1, this section contains ob- n servations about the cellular chain complex of CPL /Cχ. First we require the result of k the following lemma regarding the cellular chain complex, (C∗(TL/Cχ), ∂∗) where L is a multiple of the set of weights χ = {χ0, . . . , χk}. k We recall from Notation 6.2.4 that the basis elements of Ck(TL/Cχ) are of the form λ ∗ (∆J ) for any ordered partition J of [k] = {1, . . . , k} and some λ = (λ1, . . . , λk) such

λ1 λk k that (ζL , . . . ζL ) is a canonical representative of the Cχ-action on (CL) . We see that λ ∗ |Ji| = 1 for i ∈ [k] and therefore we can rewrite these basis elements as (∆ρ[k]) , for k some ρ ∈ Sym (k). We represent a chain in Ck(TL/Cχ) as

X X λ λ ∗ nρ[k](∆ρ[k]) λ ρ

101 CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 102

λ for some nρ[k] ∈ Z.

k Lemma 7.0.2. Let m ∈ Ck(TL/Cχ), then ∂km = 0 if and only if

X X λ ∗ m = n sgn(ρ)(∆ρ[k]) λ ρ

λ1 λk for all ρ ∈ Sym(k), for all λ = (λ1, . . . , λk) such that (ζL , . . . ζL ) is a canonical k representative of the Cχ-action on (CL) and for some n ∈ Z.

Proof. We recall from Equation (6.2.5) that the boundary of a general k-chain m ∈

k Ck(T /Cχ) is given by

X X λ λ ∗ X X λ λ ∗ ∂k nρ[k](∆ρ[k]) = nρ[k]∂k(∆ρ[k]) λ ρ λ ρ X X λ λρ ∗ = nρ[k] (∆{ρ(1),...,ρ(k−1)}) λ ρ k−1 X i λ ∗ + (−1) (∆{ρ(1),...,ρ(k−i)∪ρ(k−i+1),...,ρ(k)}) i=1 k λ ∗ +(−1) (∆{ρ(2),...,ρ(k)}) (7.0.3)

˜ ˜ ρ ˜ ˜ λ1 λk where λ = (λ1,..., λn) is such that (ζL , . . . , ζL ) is the canonical representative of the 0 0 λ1 λk k 0 0 orbit of (ζL , . . . , ζL ) in (CL) under the Cχ-action and where (λ1, . . . , λk) is deﬁned by  λi + 1 mod L if i = ρ(k) 0  λi = (7.0.4) λi otherwise. This result is shown by considering the coeﬃcients of the (k − 1)-cells appearing

λ ∗ in Equation (7.0.3). We begin with the coeﬃcient of (∆{ρ(1),...,ρ(k−i)∪ρ(k−i+1),...,ρ(k)}) , where i = 1, . . . , k − 1. If we deﬁne ρ0 ∈ Sym(k) to be the permutation such that   ρ(α) if α ∈ [k] but α 6= k − i, k − i + 1  ρ0(α) = ρ(k − i + 1) if α = k − i   ρ(k − i) if α = k − i + 1

λ ∗ λ ∗ then (∆{ρ(1),...,ρ(k−i)∪ρ(k−i+1),...,ρ(k)}) appears in the boundary of (∆ρ0[k]) :

0 λ ∗ λρ ∗ ∂k(∆ρ0[k]) = (∆{ρ0(1),...,ρ0(k−1)}) k−1 X i λ ∗ + (−1) (∆{ρ0(1),...,ρ0(k−i)∪ρ0(k−i+1),...,ρ0(k)}) i=1 k λ ∗ + (−1) (∆{ρ0(2),...,ρ0(k)}) CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 103

λ ∗ λ ∗ because (∆{ρ(1),...,ρ(k−i)∪ρ(k−i+1),...,ρ(k)}) = (∆{ρ0(1),...,ρ0(k−i)∪ρ0(k−i+1),...,ρ0(k)}) . λ ∗ λ ∗ k It is easy to see that (∆ρ[k]) and (∆ρ0[k]) are the only k-cells in Ck(TL/Cχ) with λ ∗ (∆{ρ(1),...,ρ(k−i)∪ρ(k−i+1),...,ρ(k)}) in their boundary. Therefore, for ∂km = 0, we observe i λ i λ that (−1) nρ[k] + (−1) nρ0[k] = 0. That is,

λ λ nρ0[k] = −nρ[k]. (7.0.5)

The permutations ρ and ρ0 diﬀer by a transposition and so we can rewrite (7.0.5) in the form

λ 0 λ nρ0[k] = sgn(ρ )nρ[k].

As every permutation can be written as a sequence of transpositions we deduce that for anyρ ˜ ∈ Sym (k) we have

λ λ nρ˜[k] = nρ[k]sgn (˜ρ) and we see that if ∂km = 0 then

X X λ ∗ m = n sgn(ρ)∂k(∆ρ[k]) λ ρ for some n ∈ Z. In particular (7.0.3) can be rewritten as

X X λρ ∗ k λ ∗ ∂km = n sgn(ρ) (∆{ρ(1),...,ρ(k−1)}) + (−1) (∆{ρ(2),...,ρ(k)}) . (7.0.6) λ ρ

λρ ∗ 00 We now consider the coeﬃcient of (∆{ρ(1),...,ρ(k−1)}) in (7.0.6). If we deﬁne ρ to be the permutation such that   ρ(α − 1) for α = 2, . . . , k ρ00(α) =  ρ(k) for α = 1

λρ ∗ λρ ∗ then it is easy to see that (∆ρ[k]) and (∆ρ00[k]) are the only terms in m that contain λρ ∗ (∆{ρ(1),...,ρ(k−1)}) in their boundaries:

00 λρ ∗ (λρ)ρ ∗ ∂k(∆ρ00[k]) = (∆{ρ00(1),...,ρ00(k−1)}) k−1 X i λρ ∗ + (−1) (∆{ρ00(1),...,ρ00(k−i)∪ρ00(k−i+1),...,ρ00(k)}) i=1 k λρ ∗ + (−1) (∆{ρ00(2),...,ρ00(k)})

λρ ∗ λρ ∗ 00 because (∆{ρ00(2),...,ρ00(k)}) = (∆{ρ(1),...,ρ(k−1)}) . We see ρ = (k k − 1 ··· 2 1) ◦ ρ, hence

sgn(ρ00) = sgn(ρ) · sgn(k k − 1 ··· 2 1) CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 104

λρ ∗ and the coeﬃcient of (∆{ρ(1),...,ρ(k−1)}) in equation (7.0.6) is

n{sgn(ρ) + sgn(ρ00)(−1)k} = n{sgn(ρ) + sgn(ρ)(−1)k+k−1}

= n sgn(ρ){1 + (−1)−1}

= 0.

Ranging over all ρ ∈ Sym(k) and all λ, we observe that ∂km = 0, as required.

k We now turn to the cellular chain complex of CPL/Cχ. The basis elements of k λ ∗ C2k(CPL/Cχ) are of the form (eJ,τ ) for an ordered partition J of [k], a k-tuple λ =

λ1 λk (λ1, . . . , λk) such that (ζL , . . . ζL ) is a canonical representative of the Cχ-action on k λ ∗ λ ∗ (CL) and τ = {0, 1, . . . , k}. With this in mind we write (eρ[k],τ ) for (eJ,τ ) , where k ρ ∈ Sym(k) and see that a chain in C2k(CPL/Cχ) is of the form

X X λ λ ∗ nρ[k](eρ[k],τ ) λ ρ λ for some nρ[k] ∈ Z.

k P P λ ∗ k Proposition 7.0.7. Let Gχ := λ ρ sgn (ρ)(eρ[k],τ ) ∈ C2k(CPL/Cχ), then

k k H2k(CPL/Cχ) = Z{[Gχ]} for all k > 0.

k Proof. As im∂2k+1 = 0 then H2k(CPL/Cχ) = ker∂2k. It follows from Proposition 6.5.4 that

X X λ λ ∗ X X λ λ ∗ ∂2k nρ[k](eρ[k],τ ) = nρ[k]∂2k(eρ[k],τ ) λ ρ λ ρ X X λ λρ ∗ = nρ[k] (e{ρ(1),...,ρ(k−1)},τ ) λ ρ k−1 X i λ ∗ + (−1) (e{ρ(1),...,ρ(k−i)∪ρ(k−i+1),...,ρ(k)},τ ) i=1 k λ ∗ +(−1) (e{ρ(2),...,ρ(k)},τ ) .

From Proposition 6.5.6 we see

X X λ λ ∗ X X λ λ ∗ nρ[k](eρ[k],τ ) =q ˜∗ nρ[k](∆ρ[k]) × ∆τ λ ρ λ ρ ( ) X X λ λ ∗ =q ˜∗ nρ[k](∆ρ[k]) × ∆τ λ ρ CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 105 and observe that ( ( ) ) X X λ λ ∗ X X λ λ ∗ ∂2k nρ[k](eρ[k],τ ) =q ˜∗ ∂k nρ[k](∆ρ[k]) × ∆τ . λ ρ λ ρ

P P λ λ ∗ Therefore, assuming λ ρ nρ[k](eρ[k],τ ) is a 2k-cycle, we have

( ( ) ) X X λ λ ∗ X X λ λ ∗ ∂2k nρ[k](eρ[k],τ ) =q ˜∗ ∂k nρ[k](∆ρ[k]) × ∆τ λ ρ λ ρ = 0.

P P λ λ ∗ We observe ∂k λ ρ nρ[k](∆ρ[k]) = 0 and from Lemma 7.0.2 deduce

X X λ λ ∗ X X λ ∗ nρ[k](∆ρ[k]) = n sgn (ρ)(∆ρ[k]) λ ρ λ ρ for some n ∈ Z. Therefore ( ) X X λ λ ∗ X X λ λ ∗ nρ[k](eρ[k],τ ) =q ˜∗ nρ[k](∆ρ[k]) × ∆τ λ ρ λ ρ (( ) ) X X λ ∗ =q ˜∗ n sgn (ρ)(∆ρ[k]) × ∆τ λ ρ X X λ ∗ = n sgn (ρ)˜q∗ (∆ρ[k]) × ∆τ λ ρ X X λ ∗ = n sgn (ρ)(eρ[k],τ ) λ ρ k = nGχ.

χ Recall from Deﬁnition 3.2.1 the numbers lj for 1 ≤ j ≤ k as

χ n χi0 ··· χij o lj = lcm : 0 ≤ i0 < ··· < ij ≤ k . (χi0 , . . . , χij )

k k From the cellular chain chain complexes for CPL and CPL/Cχ, we are able to recover the following result, implicit in Theorem 3.2.2.

Proposition 7.0.8. The induced map (pχ)∗ on homology is

k k (pχ)∗ : H2k(CPL) / H2k(CPL/Cχ)

χ ·lk Z / Z. CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 106

k k Proof. The chain map (pχ)∗ : C2k(CPL) → C2k(CPL/Cχ) is such that

l λ ∗ eρ[k],τ 7→ (eρ[k],τ )

λ1 λk where l = (l1, . . . , lk) and λ = (λ1, . . . , λk) are such that (ζL , . . . , ζL ) is the canonical

l1 lk k representative of the orbit of (ζL , . . . , ζL ) under the Cχ-action on (CL) . We recall from the proof of Theorem 4.2.9 that

−1 λ ∗ χ0 ··· χk |(pχ)∗ (eρ[k],τ ) | = . (χ0, . . . , χk) Therefore we deduce

X X l χ0 ··· χk X X λ ∗ (pχ)∗ sgn(ρ)eρ[k],τ = sgn(ρ)(eρ[k],τ ) . (χ0, . . . , χk) l ρ λ ρ We let

k X X l k G := sgn(ρ)eρ[k],τ ∈ C2k(CPL) l ρ and

k X X λ ∗ k Gχ = sgn(ρ)(eρ[k],τ ) ∈ C2k(CPL/Cχ). λ ρ k k k k From Proposition 7.0.7 we know H2k(CPL) = Z{[G ]} and H2k(CPL/Cχ) = Z{[Gχ]}. Therefore under the induced map in homology we have

k χ0 ··· χk k (pχ)∗[G ] = [Gχ]. (χ0, . . . , χk)

n n To determine the generator of H2k(CPL /Cχ) we use the map (pχ)∗ : H2k(CPL ) →

H2k(PL(χ)), whose degree we know. Therefore we begin by determining a generator n of H2k(CPL ), but ﬁrst introduce some notation. n We recall from Notation 6.5.1 that the basis elements of Cm+`(CPL /Cχ) are of the λ ∗ form (eJ,τ ) for any τ = {τ0, . . . , τ`} ⊆ {0, . . . , n}; any ordered partition J of σ ⊆ τ \τ0

λ1 λn into m parts and λ = (λ1, . . . , λn) such that (ζL , . . . ζL ) is a canonical representative of an orbit in (C )n under (C ) . L τ\τ0 χ τ

Notation 7.0.9. We slightly amend Notation 2.3.3. Given an ordered partition J 0 =

0 0 {J1,...,Jm} of an ordered set σ ⊆ {1, . . . , n}, we observe that there exists an ordered partition J = {J1,...,Jm} such that

1 1 2 2 k m J1 = {j1 , . . . , jn1 },J2 = {j1 , . . . , jn2 },...,Jk = {j1 , . . . , jnm }, CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 107

1 2 m 0 0 where j1 < j1 < ··· < j1 and {J1,...,Jm} = {Jρ(1),...,Jρ(m)} for some ρ ∈ Sym(m). n Therefore we will write an (m + `)-chain in Cm+`(CPL /Cχ) in the form

X X nλ (eλ )∗ {Jρ(1),...,Jρ(m)},τ {Jρ(1),...,Jρ(m)},τ λ ρ for some ρ ∈ Sym(n) and nλ ∈ , where the ordered partition J = {Jρ(1),...,Jρ(m)},τ Z 1 2 m {J1,...,Jm} is understood to be such that j1 < j1 < ··· < j1 . We will sometimes abbreviate the subscript {Jρ(1),...,Jρ(m)} to ρ(J).

Notation 7.0.10. We emphasise that in what follows L is lcm{χ0, . . . χn}. Let I =

{i0, i1 . . . , ik} ⊆ {0, 1, . . . , n} be such that iα < iβ for all α < β. We let χI =

(χi0 , χi1 , . . . , χik ) and

n iθ1 iθn n (CPL )I /Cχ = [r0, r1e , . . . , rne ] ∈ CPL /Cχ | rj = 0, j∈ / I .

n k Remark 7.0.11. We see that (CPL )I /Cχ is homeomorphic to CPL/CχI and note n n that the closed subspace (CPL )I /Cχ of CPL /Cχ is a subcomplex of the CW-structure n n λ ∗ CPL /Cχ. A typical (m+`)-cell of (CPL )I /Cχ is of the form (eρ(J),τ ) where the set τ ⊆

I is such that τ = {τ0, . . . , τ`}; ρ(J) is an ordered partition of σ ⊆ τ \ τ0 into m parts;

λ1 λn and λ = (λ1, . . . , λn) is such that (ζL , . . . , ζL ) is a canonical representative of an orbit in (C )n of the (C ) -action. We deﬁne the cellular chain group C (( P n) /C ) L τ\τ0 χ τ m+` C L I χ to be the restriction

n λ ∗ n λ ∗ n Cm+`((CPL )I /Cχ) = Z{(eρ(J),τ ) ∈ Cm+`(CPL /Cχ):(eρ(J),τ ) is a cell of (CPL )I /Cχ}.

n ∼ k It is easy to see that the homeomorphism (CPL )I /Cχ = CPL/CχI is cellular, n k taking each cell of (CPL )I /Cχ to a corresponding cell in CPL/CχI . This induces an isomorphism on the cellular chain groups and we can deduce that

n n Ker ∂m+` : Cm+`((CPL )I /Cχ) → Cm+`−1((CPL )I /Cχ) is isomorphic to

k k Ker ∂m+` : Cm+`(CPL/CχI ) → Cm+`−1(CPL/CχI ).

Similarly

n n Im ∂m+`+1 : Cm+`+1((CPL )I /Cχ) → Cm+`((CPL )I /Cχ) CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 108 is isomorphic to

k k Im ∂m+`+1 : Cm+`+1(CPL/CχI ) → Cm+`(CPL/CχI ).

Let

n X X λ ∗ n (GI )χ = sgn (ρ)[(eρ(I\i0),I}) ] ∈ C2k((CPL )I /Cχ), λ ρ then by Proposition 7.0.7 it follows that

n n n H2k((CPL )I /Cχ) = ker∂2k : C2k((CPL )I /Cχ) → C2k−1((CPL )I /Cχ)

n = Z{[(GI )χ]}.

n n In the case when (χ0, . . . , χn) = (1,..., 1), that is (CPL )I /Cχ = (CPL )I , then we n n denote (GI )χ simply by GI . The following Lemma determines a relationship between n n n 0 GI ,GI0 ∈ C2k((CPL ) for particular I,I ⊆ {0, . . . , n}.

Lemma 7.0.12. Given τ = {τ0, τ1, . . . , τk+1} ⊆ {0, 1, . . . , n} and {J1,...,Jk} =

{{τ1},..., {τj−1}, {τj+1},..., {τk+1}}, which is an ordered partition of σ = τ \{τ0, τj}, 0 for some j ∈ {1, . . . , k + 1}, let I = τ \ τ0 and I = τ \ τj, then ( ) n n (k+j+1) X X l GI − GI0 = ∂2k+1 (−1) sgn(ρ)[eρ(J),τ ] l ρ for all ρ ∈ Sym (k) and all l ∈ (C )n . L τ\τ0

Proof. We have (( ) ) X X l X X l sgn(ρ)eρ(J),τ = q∗ sgn(ρ)∆ρ(J) × ∆τ l ρ l ρ CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 109 and so

X X l ∂2k+1 sgn(ρ)eρ(J),τ l ρ (( ) ) X X l = ∂2k+1q∗ sgn(ρ)∆ρ(J) × ∆τ l ρ ( ( ) ) X X l = q∗ ∂k sgn(ρ)∆ρ(J) × ∆τ l ρ ( ( ) ) k X X l +q∗ (−1) sgn(ρ)∆ρ(J) × ∂k+1∆τ l ρ (( ) ) k X X l = (−1) q∗ sgn(ρ)∆ρ(J) × ∂k+1∆τ l ρ ( k+1 ) k X X l X s = (−1) q∗ sgn(ρ)∆ρ(J) × (−1) ∆τ\τs l ρ s=0 k+1 k X X X s l = (−1) q∗ sgn(ρ) (−1) ∆ρ(J) × ∆τ\τs . l ρ s=0 It follows from Proposition 6.4.10 that

k+1 X X X s l q∗ sgn(ρ) (−1) ∆ρ(J) × ∆τ\τs l ρ s=0 X X l j l = q∗ sgn(ρ) ∆ρ(J) × ∆τ\τ0 + (−1) ∆ρ(J) × ∆τ\τj l ρ ( ) X X X X = q sgn(ρ)∆l × ∆ + (−1)j sgn(ρ)el . ∗ ρ(J) τ\τ0 ρ(J),τ\τj l ρ l ρ We conclude that

X X l ∂2k+1 sgn(ρ)eρ(J),τ l ρ k X X l k k+j n 0 = (−1) sgn(ρ)q∗{∆ρ(J) × ∆τ\τ0 } + (−1) GI . l ρ

Suppose τ1 = Jρ(α). Then from Prop 6.4.10

k l α(k−α) l 0 q∗{∆ρ(J) × ∆τ\τ0 } = (−1) eJ ,τ\τ0

0 0 0 0 0 where J1 = Jρ(α+1), Jρ(2) = Jρ(α+2), ... , Jk−α = Jρ(k), Jk−α+1 = {τj}, Jk−α+2 = Jρ(1), 0 0 0 Jk−α+3 = J2, ..., Jk = Jρ(α−1). It is clear that there is no other ρ ∈ Sym(k) such that l k l P k l k q {∆ 0 × ∆ } = ±e 0 . Hence the terms in q (−1) sgn(ρ){∆ × ∆ } ∗ ρ (J) τ\τ0 J ,τ\τ0 ∗ ρ ρ(J) τ\τ0 n are in one-to-one correspondence with the terms in GI . CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 110

By observing ( ) k X X l k k+j k 0 ∂2k (−1) sgn(ρ)q∗{∆ρ(J) × ∆τ\τ0 } + (−1) GI l ρ ( ) k X X l k = ∂2k (−1) sgn(ρ)q∗{∆ρ(J) × ∆τ\τ0 } l ρ = 0

n o and that (−1)k P P sgn(ρ)q {∆l × ∆k } is a chain in C (( P n) /C ) it fol- l ρ ∗ ρ(J) τ\τ0 2k C L I χ lows from Remark 7.0.11 that ( ) k X X l k n (−1) sgn(ρ)q∗{∆ρ(J) × ∆τ\τ0 } = ±GI . l ρ n o To determine the overall sign of (−1)k P P sgn(ρ)q {∆l × ∆k } , we need l ρ ∗ ρ(J) τ\τ0 only consider the sign that (−1)kq {∆l × ∆k } takes in Gn. We see that τ = J in ∗ J τ\τ0 I 1 1 J. That is, α = 1, and from Proposition 6.4.10 we see that

k l k k+(k−1) l 0 (−1) q∗{∆J × ∆τ\τ0 } = (−1) eJ ,τ\τ0

l 0 = −eJ ,τ\τ0 where

0 0 {J1,...,Jk} = {J2,J3,...,Jk, {τj}}

= {{τ2}, {τ3},..., {τj−1}, {τj+1},..., {τk+1}, {τj}}.

n l 0 0 In G , the term −e 0 has coeﬃcient sgn(ρ ) + 1 = k − j + 1 where ρ is the I J ,τ\τ0 permutation (j − 1 j j + 1 ··· k − 1 k). Therefore ( ) k X X l k k−j+1 n (−1) sgn(ρ)q∗{∆ρ(J) × ∆τ\τ0 } = (−1) GI . l ρ

Hence the result follows

(k+j+1) X X l ∂2k+1(−1) sgn(ρ)eρ(J),τ l ρ (k+j+1) k−j+1 n k+j n = (−1) (−1) GI + (−1) GI0

n n = GI − GI0 . CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 111

Remark 7.0.13. Let I and I0 be of the form given in the statement of Lemma 7.0.12.

00 0 0 0 Suppose that we deﬁne I = τ \ τj0 for some j ∈ {1, ··· , k + 1} and let {J1,...,Jk} = 0 {{τ1},..., {τj0−1}, {τj0+1},..., {τk+1}}, which is an ordered partition of σ = τ \{τ0, τj}. Then by Lemma 7.0.12 ( ) n n (k+j0+1) X X l GI − GI00 = ∂2k+1 (−1) sgn(ρ)eρ(J0),τ . l ρ It is easy to see that this implies

n n (k+j0+1) X X l GI0 − GI00 = ∂2k+1{(−1) sgn(ρ)eρ(J0),τ l ρ (k+j+1) X X l − (−1) sgn(ρ)eρ(J),τ }. l ρ Notation 7.0.14. For the remainder of this chapter we emphasise that any subset

S = {s1, . . . , sk} ⊆ {0, . . . , n} will be assumed to be ordered such that sα < sβ for all α < β.

Proposition 7.0.15. Let I,I0 ⊆ {0, 1, . . . n}, such that |I| = |I0| = k + 1. Then

n n n [GI ] ∼ [GI0 ] ∈ H2k(CPL ).

0 0 0 0 0 0 Proof. Writing I = {i0, i1, . . . , ik} and I = {i0, i1, . . . , ik} such that iα < iβ and iα < iβ for all α < β, it follows from Remark 7.0.13 that

n n G − G 0 ∈ im∂2k+1. I {I\i0}∪{i0}

n n n Therefore [G ] ∼ [G 0 ] ∈ H2k( P ). Similarly, I {I\i0}∪{i0} C L

n n G 0 − G 0 0 ∈ im∂2k+1 {I\i0}∪{i0} {I\{i0,i1}}∪{i0,i1} and we see that

n n n [GI ] ∼ [G 0 ] ∼ G 0 0 ∈ H2k( P ). {I\i0}∪{i0} {I\{i0,i1}}∪{i0,i1} C L

By continuing inductuvely, it is easy to see that

n n n n G − G 0 0 0 = G − G 0 ∈ im∂2k+1 I {I\{i0,i1,...,ik}}∪{i0,i1,...,ik} I I

n n n and so [GI ] ∼ [GI0 ] ∈ H2k(CPL ).

Before we state the next Proposition, we require the following result. CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 112

Lemma 7.0.16. The set ( ) lχ k : i ∈ {0, . . . , n} χ\χi lk is a set of coprime integers.

χ χ\χi χ Proof. From its deﬁnition, we observe lk = lcm{lk : i ∈ {0, . . . , n}}. We write lk

χ\χi and lk for i ∈ {0, . . . , n} in their prime power decompositions, and let r denote

χ χ\χi the index of a given prime p in lk and ri for the index of the prime p in lk for i ∈ {0, . . . , n}. χ lk Therefore r = max{r0, . . . , rn} and we see that the total index of p in χ\χi is given lk by max{r0, . . . , rn} − ri = max{r0 − ri, . . . , rn − ri}. χ lk The total index of p in gcd χ\χi : i ∈ {0, . . . , n} is given by lk min{max{r0 − ri, . . . , rn − ri}: i ∈ {0, . . . , n}}. There exists some j ∈ {0, . . . , n} such that ri ≤ rj for all i ∈ {0, . . . , n}. Therefore max{r0 − rj, . . . , rn − rj} = 0 and min{max{r0 −ri, . . . , rn −ri}: i ∈ {0, . . . , n}} = 0. Ranging over all primes we observe that ( ) lχ gcd k : i ∈ {0, . . . , n} = 1 χ\χi lk as required.

n n In the following proposition we let G denote a generating cycle in C2k(CPL ); that n n is, H2k(CPL ) = Z{[G ]}. In particular we recall Remark 7.0.11 and observe that for n+1 n+1 n+1 some I = {i0, . . . , in} ⊂ {0, . . . , n + 1} then H2k((CPL )I ) = Z{[GI ]}, where GI n n ∼ n+1 is the image of G under the homeomorphism CPL = (CPL )I .

Theorem 7.0.17. Let I = {0, 1, . . . , n} and u0, . . . , un be a sequence of integers such that χ χ lk lk u0 + ··· + un = 1. χ\χ0 χ\χn lk ln n n Then H2k(CPL ) = Z{[G ]}, where

χ χ n lk n lk n G = u0G + ··· + unG χ\χ0 I0 χ\χn In lk lk for all n > k. CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 113

Proof. Let I˜ = {0, . . . , k}, then from Remark 7.0.11 we have H (( P n) ) = {[Gn]}. 2k C L I˜ Z I˜ n n From the long exact sequence of the CW-pair (CPL , (CPL )I˜), we know that the in- n n n n clusion iI˜:(CPL )I˜ → (CPL ) induces an isomorphism i∗ : H2k((CPL )I ) → H2k((CPL )) such that H (( P n)) = {[Gn]}. We will show by induction that [Gn] ∼ [Gn] ∈ 2k C L Z I˜ I˜ n H2k(CPL ) for n > k.

We begin with I = {0, . . . , k + 1} and a sequence u0, . . . , uk+1 of integers such that

χ χ lk lk u0 + ··· + uk+1 = 1. χ\χ0 χ\χk+1 lk lk

Note that the existence of the sequence u0, . . . , uk+1 follows from Lemma 7.0.16. We deﬁne Ij to be the ordered subset I \ j for j ∈ I, so we note |Ij| = k + 1. From Proposition 7.0.15 we know Gk+1 − Gk+1 ∈ im∂ : C ( P k+1) → C ( P k+1) for Ij I˜ 2k+1 2k+1 C L 2k C L all j ∈ I and so it follows that

χ χ χ χ lk k+1 lk k+1 lk k+1 lk k+1 u0G + ··· + uk+1G − u0G − · · · − uk+1G χ\χ0 I0 χ\χk+1 Ik+1 χ\χ0 I˜ χ\χk+1 I˜ lk lk lk lk χ χ ( χ χ ) lk k+1 lk k+1 lk lk k+1 = u0G + ··· + uk+1G − u0 + ··· + uk+1 G χ\χ0 I0 χ\χk+1 Ik+1 χ\χ0 χ\χk+1 I˜ lk lk lk lk χ χ lk k+1 lk k+1 k+1 = u0G + ··· + uk+1G − G χ\χ0 I0 χ\χk+1 Ik+1 I˜ lk lk k+1 k+1 ∈ im∂2k+1 : C2k+1(CPL ) → C2k(CPL ).

Therefore [Gk+1] ∼ [Gk+1] ∈ H ( P k+1). I˜ 2k C L We assume that [Gn] ∼ [Gn] ∈ H ( P n) or equivalently I˜ 2k C L

n n n n G − GI˜ ∈ im∂2k+1 : C2k+1(CPL ) → C2k(CPL ).

We now let I = {0, . . . , n + 1}, the subset Ij = {i0, . . . , in} = I \ j for j ∈ I, and deﬁne ˜ Ij = {i0, . . . , ik}. We observe that

n+1 n+1 n+1 n+1 G − G ∈ im∂2k+1 : C2k+1(( P )Ij ) → C2k(( P )Ij ), Ij I˜j C L C L

n+1 n+1 n n n ∼ where G − G is the image of G − G˜ under the homeomorphism P = Ij I˜j I C L n+1 (CPL )Ij . Therefore

n+1 n+1 n+1 n+1 G − G ∈ im∂2k+1 : C2k+1( P ) → C2k( P ), Ij I˜j C L C L for each j ∈ I. CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 114

Using this and Proposition 7.0.15 we observe that

χ χ χ χ lk n+1 lk n+1 lk n+1 lk n+1 u0G + ··· + un+1G − u0G − · · · − un+1G χ\χ0 I0 χ\χn+1 In+1 χ\χ0 I˜0 χ\χn+1 I˜n+1 lk lk lk lk χ χ ( χ χ ) lk n+1 lk n+1 lk lk n+1 = u0G + ··· + un+1G − u0 + ··· + un+1 G χ\χ0 I0 χ\χn+1 In+1 χ\χ0 χ\χn+1 I˜ lk lk lk lk χ χ lk n+1 lk n+1 k+1 = u0G + ··· + un+1G − G χ\χ0 I0 χ\χn+1 Ik+1 I˜ lk lk n+1 n+1 ∈ im∂2k+1 : C2k+1(CPL ) → C2k(CPL ).

Therefore [Gn+1] ∼ [Gn+1] ∈ H ( P n+1). I˜ 2k C L

n n n We let Gχ denote a generating cycle of H2k(CPL /Cχ) and so H2k(CPL /Cχ) = n Z{[Gχ]}. In particular we recall Remark 7.0.11 and observe that for some I = n+1 n+1 n+1 {i0, . . . , in} ⊂ {0, . . . , n + 1} then H2k((CPL )I /Cχ) = Z{[(GI )χ]}, where (GI )χ is the image of Gn under the homeomorphism P n/C ∼ ( P n+1) /C . χI C L χI = C L I χ

Theorem 7.0.18. Let I = {0, 1, . . . , n} and let u0, . . . , un be a sequence of integers such that χ χ lk lk u0 + ··· + un = 1. χ\χ0 χ\χn lk ln ∼ n n Then H2k(P(χ)L) = H2k(CPL /Cχ) = Z{[Gχ]} where

n n n Gχ = u0(GI0 )χ + ··· + un(GIn )χ.

Proof. We prove by induction and begin with the case n = k+1, where I = {0, 1, . . . , k+ 1}. We recall from Remark 7.0.11 that H (( P k+1) /C ) = {[(Gn ) ]} and it fol- 2k C L Ij χ Z Ij χ k+1 k+1 lows from Proposition 7.0.8 that the map pχ :(CPL )Ij → (CPL )Ij /Cχ induces the following map in homology

k+1 k+1 p∗ : H2k((CPL )Ij ) → H2k((CPL )Ij /Cχ) χ ··· χ χ ··· χ [Gk+1] 7→ 0 j−1 j+1 k+1 [Gk+1] = lχ\χj [(Gk+1) ]. Ij χI k Ij χ (χ0, . . . , χj−1, χj+1, . . . , χk+1) j

k+1 k+1 Therefore from Theorem 7.0.17 we see H2k(CPL ) = Z{G } where

χ χ k+1 lk k+1 lk k+1 G = u0G + ··· + uk+1G χ\χ0 I0 χ\χk+1 Ik+1 lk lk CHAPTER 7. HOMOLOGY CALCULATIONS FOR P(χ) 115 and observe that

χ χ ! k+1 lk k+1 lk k+1 χ X k+1 χ k+1 (pχ)∗ u0G + ··· + uk+1G = l uj(G )χ = l G . χ\χ0 I0 χ\χk+1 Ik+1 k Ij k χ lk lk j=0 From Kawasaki [15] we know

k+1 k+1 p∗ : H2k((CPL )) / H2k((CPL )/Cχ)

χ ·lk Z / Z. Thus we can deduce that

k+1 k+1 H2k((CPL )/Cχ) = Z [Gχ ] .

χ 0 0 lk 0 We assume that for I = {0, 1, . . . , n} there exists u0, . . . , un such that χ\χ0 u0 + lk χ lk 0 ··· + χ\χn un = 1 and lk (" n #) X H (( P n)/C ) = u0 (Gk ) . 2k C L χ Z j Ij χ j=0

n n Therefore the map p:(CPL ) → (CPL )/Cχ induces the following map on homology

n n p∗ : H2k((CPL )) → H2k((CPL )/Cχ) " n χ # "n−1 # X lk 0 n χ X 0 n u G 7→ l u (G )χ . χ\χj j Ij k j Ij j=0 lk j=0

χ lk Suppose I = {0, 1, . . . , n + 1}. There exists u0, . . . , un+1 such that χ\χ0 u0 + ··· + lk χ lk χ\χn un+1 = 1. Recall from Theorem 7.0.17 that lk

("n+1 χ #) n+1 X lk n+1 H2k(( P )) = ujG , C L Z χ\χj Ij j=0 lk n n we see that the map (pχ)∗ : H2k((CPL )) → H2k((CPL )/Cχ) is such that

"n+1 χ # " n # " n # X lk n+1 X χ n+1 χ X n+1 (pχ)∗ ujG = l uj(G )χ = l uj(G )χ . χ\χj Ij k Ij k Ij j=0 lk j=0 j=0

n+1 n+1 χ We know that the degree of the map (pχ)∗ : H2k(CPL ) → H2k(CPL /Cχ) is lk and conclude that " n # X H ( P n+1/C ) = u (Gn+1) . 2k C L χ Z j Ij χ j=0 n n n Therefore H2k(CPL /Cχ) = Z{[Gχ]} and it follows H2k(P(χ)L) = Z{[h∗(Gχ)]}. Chapter 8

Stunted weighted projective space and weighted lens space

In this chapter we extend all our work for weighted projective spaces to stunted weighted projective spaces. The motivation for this chapter is the extension of Kawasaki’s coﬁbration sequence

L(χ ; χ ) → (χ ) → (χ) → ΣL(χ ; χ ) , i bι P bι P i bι where for some i ∈ {0, . . . n} we denote by ι the set {0, . . . , n}\i. We observe ΣL(χ ; χ ) b i bι can be identified with the stunted weighted projective space P(χ;bι). Furthermore, we see that implicit within Kawasaki’s cofibration sequence is a cofibration sequence of the form

P(χI ) → P(χ) → P(χ; I), for any I ⊆ {0, . . . , n}. From this sequence we are able to give explicit CW-structures for P(χ; I), calculate the integral homology groups, identifying their generators via the cellular chain complex, and determine the cohomology ring H∗(P(χ; I)). Before we consider stunted weighted projective spaces, we give a little digression into cellular decompositions of weighted lens space, which are determined with relative ease from our corresponding work on weighted projective spaces. In particular, we identify P(χ)L as a subdivision of the minimal decomposition for P(χ) whenever χ is divisive, which was given in Section 3.3.2.

116 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 117 8.1 Weighted lens space

We recall the comments made in Section 3.4 regarding the CW-structure we have for

P(χ) giving a cell structure for weighted lens spaces. We consider this in the following.

2n 2n 8.1.1 A CW-structure on D and D /Cχ

n n We represent points of T × ∆c as we did in Section 5.2. That is

l1 2πix1/L ln 2πixn/L n n (ζL e , . . . , ζL e , r0, . . . , rn) ∈ T × ∆c

lj 2πilj /L where l1, . . . , ln such that lj ∈ {0,...,L − 1} and ζL = e for all j ∈ [n], with n n (x1, . . . , xn) ∈ [0, 1) and (r0, . . . , rn) ∈ ∆c . 2 2 n n 2n Observing r1 + ··· + rn ≤ 1, we deﬁne the quotient map r : T × ∆c → D such that

l1 iθ1 ln l1 2πix1/L ln 2πixn/L (ζL e , . . . , ζL , r0, . . . , rn) 7→ (r1ζL e , . . . , rnζL e ).

n n Using this map we will compose a subset of the characteristic maps for TL ×∆c with r 2n to give a family of characteristic maps for a Cχ-complex on D . Much of what follows n is analogous to the arguments of Section 5.3 regarding the Cχ-complex CPL . n n The following remark describes which points of T × ∆c are identiﬁed under the map r.

l1 2πix1/L ln 2πixn/L n n Remark 8.1.1. Given a point y = (ζL e , . . . , ζL e , r0, . . . , rn) ∈ T × ∆c , 0 0 0 0 then we deﬁne τ := {i ∈ {0, . . . , n}: ri 6= 0} and x1, . . . , xn, l1, . . . , ln such that  xi if i ∈ τ 0  xi = (8.1.2) 0 otherwise  li if i ∈ τ 0  li = (8.1.3) 0 otherwise. Therefore

l1 2πix1/L ln 2πixn/L r(y) = (r1ζL e , . . . , rnζL e ) l0 0 l0 0 1 2πix1/L n 2πixn/L = (r1ζL e , . . . , rnζL e ) l0 0 l0 0 1 2πix1/L n 2πixn/L = r(ζL e , . . . , ζL e , r0, . . . , rn). 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 118

n n Using Remark 8.1.1 we will see that r identiﬁes certain cells of TL × ∆c together.

In the following we let I = {I1,...,Ip} be an ordered partition of a given S ⊆ [n] into p parts such that

1 1 2 2 p p I1 = {i1, . . . , in1 },I2 = {i1, . . . , in2 },...,Ip = {i1, . . . , inp },

α where iβ ∈ [n] denotes the β-th element in the set Iα. As usual τ = {τ0, . . . , τ`} ⊆

{0, 1, ··· n} and l = (l1, . . . , ln) is such that 0 ≤ li < L for all i ∈ [n].

Proposition 8.1.4. There exists an ordered partition, J of a subset σ ⊆ τ and l0 =

0 0 0 0 (l1, . . . , ln), where 0 ≤ li < L for all i ∈ τ and lj = 0 for all j∈ / τ such that

l l0 im{r ◦ (ΦI × iτ )|int(∆p×∆`)} = im{r ◦ (ΦJ × iτ )|int(∆k×∆`)}.

l Proof. By letting U = [n] \ S = {u1, . . . , un−|S|}, we recall that im{ΦI × iτ }|int(∆p×∆`)

l1 2πix1/L ln 2πixn/L n n corresponds to the set of points y = (ζL e , . . . ζL e , r0, . . . rn) ∈ T × ∆c n such that (x1, . . . , xn) are the set of points in [0, 1) satisfying the following inequalities:

0 = xu = ··· = xu < x 1 = x 1 = ··· = xi1 1 n−|S| i1 j2 n1

< x 2 = x 2 = ··· = xi2 i1 i2 n2 < ···

< x p−1 = x p−1 = ··· = x p−1 i1 i2 inp−1

< x p = x p = ··· = x p i1 i2 inp < 1; (8.1.5) and (r , . . . , r ) ∈ ∆n where r2 + ··· + r2 = 1 with r = 0 for all i ∈ {0, 1, . . . , n}\ τ. 0 n c τ0 τ` i

We set σ := S ∩ τ and υ := {υ1, . . . , υ|τ|−|σ|} = τ \ σ. When σ 6= ∅, we deﬁne

Λ = {λ ∈ [p]: Iλ ∩ τ 6= ∅}. Therefore, there exists some 1 ≤ k ≤ p so that we can write Λ = {λ1, . . . , λk}, where λ1 < ··· < λk. We let J = {J1,...,Jk} be the ordered partition of σ such that Ji = {x: x ∈ Iλi ∩τ}. In other words, J is an ordered partition of σ, whose ordering is inherited from the ordering on the parts of I.

l From Remark 8.1.1 we see that im{r ◦ (ΦI × iτ )|int(∆p×∆`)} is the set of points of l0 0 l0 0 1 2πix1/L n 2πixn/L 0 0 the form (r1ζL e , . . . , rnζL e ) such that x1, . . . , xn are determined by (8.1.2) 0 0 0 0 n and l1, . . . , ln by (8.1.3) We observe that (x1, . . . , xn) are the set of points in [0, 1) 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 119 satisfying the following inequalities

0 = x0 = ··· = x0 u1 un−|S| 0 0 0 0 0 xυ = ··· = xυ < x 1 = x 1 = ··· = xj1 1 |τ|−|σ| j1 j2 n1 0 0 0 < x 2 = x 2 = ··· = xj2 j1 j2 n2 < ···

0 0 0 < x k−1 = x k−1 = ··· = x k−1 j1 j2 jnk−1 0 0 0 < xjk = xjk = ··· = xjk 1 2 nk < 1; (8.1.6) therefore

l l0 im{r ◦ (ΦI × iτ )|int(∆p×∆`)} = im{r ◦ (ΦJ × iτ )|int(∆k×∆`)}.

Remark 8.1.7. It follows from Proposition 8.1.4 that r identiﬁes the (p + `)-cell

l n n l0 ∆I × ∆τ of TL × ∆c with the (k + `)-cell ∆J × ∆τ , where k ≤ p. The proof of Proposition 8.1.4 shows how to determine J and l0.

Notation 8.1.8. According to Proposition 8.1.4 in the statement of the following

l theorem we restrict attention to the maps r ◦ (ΦJ × iτ ) such that

1. τ = {τ0, . . . , τ`} ranges over all subsets of {0, . . . , n},

2. J ranges over all ordered partitions of all σ ⊆ τ into k parts,

3. l ranges over all n-tuples (l1, . . . , ln) such that 0 ≤ li < 0 for all i ∈ τ and lj = 0 for all j∈ / τ.

Theorem 8.1.9. The family of maps

l k ` 2n r ◦ (ΦJ × iτ ): ∆ × ∆ → D is a family of characteristic maps for a CW-structure on D2n.

l Proof. To show the restriction of r ◦ (ΦJ × iτ )|int(∆k×∆`) is an injection onto its image we let

l1 2πix1/L ln 2πixn/L y = (ζL e , . . . ζL e , r0, . . . rn) 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 120

l0 0 l0 0 0 1 2πix1/L n 2πixn/L 0 0 y = (ζL e , . . . ζL e , r0, . . . rn)

l ` 0 0 l ` be two points in ∆J × ∆τ such that r(y) = r(y ). We note that as y, y ∈ ∆J × ∆τ , 0 0 then li = li and 0 ≤ xi, xi < 1 for 1 ≤ i ≤ n. We can see that

l1 2πix1/L ln 2πixn/L r(y) = (r1ζL e , . . . , rnζL e ) l0 0 l0 0 1 2πix1/L n 2πixn/L = (r1ζL e , . . . , rnζL e ) = r(y0)

0 q n 2 0 0 and deduce rj = rj for 1 ≤ j ≤ n. This implies r0 = 1 − Σj=0rj = r0 and xi = xi 0 l l for 1 ≤ i ≤ n. Therefore y = y as required. Let dJ,τ := im{q ◦ (ΦJ × iτ ) |int(∆k×∆`)} l l0 and suppose (z1, . . . , zn) ∈ dJ,τ ∩ dJ0,τ 0 . Therefore

l1 2πix1/L ln 2πixn/L (z1, . . . , zn) = (r1ζL e , . . . , rnζL e ) l0 0 l0 0 0 1 2πix1/L 0 n 2πixn/L = (r1ζL e , . . . , rnζL e ).

0 0 We can see ri = ri for 1 ≤ i ≤ n and as 0 ≤ xi, xi < 1 for 1 ≤ i ≤ n, then it follows 0 0 0 xi = xi for 0 ≤ i ≤ n. It is evident that not only do we have l = l and τ = τ , the 0 l partitions J and J are equal. Therefore every dJ,τ is either disjoint or equal to any l0 n n other dJ0,τ 0 . Since r is surjective and Proposition 8.1.4 shows that any cell of TL × ∆c l is identified with a cell ∆J × ∆τ such that l, J and τ satisfy the properties given in 2n l Notation 8.1.8, we deduce that each point of D is contained in some dJ,τ . Therefore Property 1 of Definition 2.1.1 is satisfied.

l The boundary of the cell ∆J × ∆τ is the union of ﬁnitely many cells of dimension less than k + `. It follows from Proposition 8.1.4 and Remark 8.1.7 that the boundary

l of dJ,τ is therefore a union of finitely many cells of dimension less than k +`. Therefore Property 2 of Definition 2.1.1 is satisfied.

l Therefore each dJ,τ such that l, J and τ are of the form given in Notation 8.1.8 is 2n l a (k + `)-cell of a CW-structure on D , with r ◦ (ΦJ × iτ ) as its characteristic map. 2n 2n (k+`) Let DL denote this CW-structure and (DL ) denote its (k + `)-skeleton; that is 2n (k+`) (DL ) is the union of cells of dimension less than or equal to k + `.

n n 2n 2n n Corollary 8.1.10. The maps r : TL × ∆c → DL and Φn : DL → CPL are cellular maps. 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 121

2n Proof. Given the CW-complex structure DL determined in Theorem 8.1.9, it follows from Proposition 8.1.4 that r is a cellular map. However r satisﬁes a stronger property

n n than cellularity. Recalling Remark 8.1.7 we observe that r maps cells of TL × ∆c to 2n cells of DL . n Similarly, q is not only cellular. Given the CW-structure CPL determined in The- n n orem 5.3.8, it follows from Remark 5.3.5 that q maps cells of TL × ∆c to cells of n CPL .

As q = Φn ◦ r, where Φn is the characteristic map of the 2n-cell for the CW- n 2n structure on CP given in Example 2.1.3, then we conclude that Φn maps cells of DL n to cells of CPL and therefore Φn is cellular.

2n Theorem 8.1.11. The CW-complex DL is a Cχ-complex.

Proof. We consider the map

2n 2n ζχ : DL → DL

l1 2πix1/L ln 2πixn/L −a0 a1 l1 2πix1/L −a0 an ln 2πixn/L (r1ζL e , . . . , rnζL e ) 7→ (r1ζχ0 ζχ1 ζL e , . . . , rnζχ0 ζχn ζL e )

2n a0 an induced by the Cχ-action on D given in Deﬁnition 3.4.6 for every ζχ = (ζχ0 , . . . , ζχn ) ∈ Cχ. ˜ ˜ ˜ l1 ln If we let l = (ζL , . . . , ζL ) such that   l + a L − a L mod L if i ∈ τ i i χi 0 χ0 ˜li =  0 otherwise and 0 ≤ ˜li < L for all i ∈ τ \ τ0, then we observe that

l ˜l ζχ ◦ r ◦ (ΦJ × iτ )|int(∆k×∆l) = r ◦ (ΦJ × iτ )|int(∆k×∆l).

l ˜l l l l In particular ζχ · dJ,τ = dJ,τ and whenever ζχ · dJ,τ = dJ,τ we see the restriction of ζχ to dJ,τ is the identity map, as required.

We deﬁne

0 2n 2n pχ : D → D /Cχ

∗ (z1, . . . , zn) 7→ (z1, . . . , zn)

2n as the orbit map of the Cχ-action on D given in Deﬁnition 3.4.6. ∗ 2n We can prove analogously to Theorem 4.1.6 that each point (z1, . . . , zn) ∈ D /Cχ can be uniquely represented by the point

λ1 2πix1/L λn 2πixn/L 2n (r1ζL e , . . . , rnζL e ) ∈ D 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 122

2 where we deﬁneτ ˜ = {j ∈ {0, 1, . . . , n}: rj 6= 0}; the moduli r1, . . . , rn satisfy r1 + 2 ··· + rn ≤ 1; the arguments x1, . . . , xn are such that xi ∈ [0, 1) for all i ∈ τ˜ and

λ1 λn xi = 0 otherwise; and λ1, . . . , λn are such that (ζL , . . . , ζL ) is a canonical represen- n tative of an orbit in (CL)τ˜ under the action of (Cχ)τ˜∪{0}. We observe that the point

λ1 2πix1/L λn 2πixn/L λ (r1ζL e , . . . , rnζL e ) is contained in the cell dJ,τ where we deﬁne τ by   τ˜ ∪ {0} if r2 + ··· + r2 < 1 τ = 1 n 2 2  τ˜ if r1 + ··· + rn = 1.

Notation 8.1.12. According to these comments, in the statement of the following

0 λ theorem we restrict attention to the maps pχ ◦ r ◦ (ΦJ × iτ ) such that

1. τ = {τ0, . . . , τ`} ranges over all subsets of {0, . . . , n},

2. J ranges over all ordered partitions of all σ ⊆ τ into k parts,

λ1 λn 3. λ ranges over all n-tuples (λ1, . . . , λn) such that (ζL , . . . , ζL ) is a canonical n representative of an orbit in (CL)τ\{0} under the action of (Cχ)τ∪{0}.

λ ∗ 0 λ We denote by (dJ,τ ) the image of pχ ◦ r ◦ (ΦJ × iτ )|int(∆k×∆`).

λ 2n Theorem 8.1.13. Let r ◦ (ΦJ × iτ ) range over the characteristic maps of cells in DL ,

λ1 λn n where λ is such that (ζL , . . . , ζL ) is a canonical representative of an orbit in (CL)τ\{0} under the action of (Cχ)τ∪{0}. Then the family of maps

0 λ k ` 2n pχ ◦ r ◦ (ΦJ × iτ ): ∆ × ∆ → D /Cχ

2n is a family of characteristic maps for a CW-structure on D /Cχ.

λ ∗ λ0 ∗ Proof. We observe from the comments preceding Notation 8.1.12 that (dJ,τ ) ∩(dJ0,τ 0 ) 6= 0 0 0 2n ∅ if and only if λ = λ , J = J and τ = τ ; and every point of D /Cχ is contained λ ∗ in some (dJ,τ ) . Therefore it follows from Theorem 8.1.11 and Proposition 2.6.6 that 0 λ k ` 2n the family of maps pχ ◦ r ◦ (ΦJ × iτ ): ∆ × ∆ → D /Cχ given in the statement of the theorem is a family of characteristic maps for a CW-complex on the orbit space

2n D /Cχ.

λ ∗ Therefore, each (dJ,τ ) such that λ, J and τ are of the form required by Notation 2n 0 λ 8.1.12 is a (k + `)-cell of a CW-structure on D /Cχ, with pχ ◦ r ◦ (ΦJ × iτ ) as its 2n characteristic map. We will denote this CW-structure by DL /Cχ. The union of all 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 123

2n (k+`) cells of dimensions less than or equal to k + `, which we write as (DL /Cχ) , is 2n therefore the (k + `)-skeleton of DL /Cχ.

Corollary 8.1.14. The maps

n n 2n r˜: TL /Cχ × ∆c → DL /Cχ

λ1 2πix1/L λn 2πixn/L ∗ λ1 2πix1/L λn 2πixn/L ∗ (ζL e , . . . , rnζL e , r0, . . . , rn) 7→ (r1ζL e , . . . , rnζL e ) and

˜ 2n n Φn : DL /Cχ → CPL /Cχ q λ1 2πix1/L λn 2πixn/L ∗ n 2 λ1 2πix1/L λn 2πixn/L ∗ (r1ζL e , . . . , rnζL e ) 7→ [+ 1 − Σi=1ri : r1ζL e : ... : rnζL e ] are cellular maps.

n n 2n 2n n Proof. This follows by observing that as r : TL × ∆c → DL and Φn : DL → CPL are ˜ maps of Cχ-complexes that map cells to cells then the induced mapsr ˜ and Φn take ˜ cells to cells. Thereforer ˜ and Φn are cellular.

0 2n 2n Recalling the homeomorphism hχ : D /Cχ → D / ≈ from Section 3.4 given by

∗ χ1 χn ∗ (z1, . . . zn) 7→ (z1 , . . . , zn ) , then the next Theorem follows directly from Theorem 8.1.13.

0 λ Theorem 8.1.15. Let pχ ◦ r ◦ (ΦJ × iτ ) range over the characteristic maps of cells in 2n DL /Cχ, then the family of maps

0 0 λ k ` 2n hχ ◦ pχ ◦ r ◦ (ΦJ × iτ ): ∆ × ∆ → D / ≈ is a family of characteristic maps for a CW-structure on D2n/ ≈.

˜λ 0 0 λ We denote by dJ,τ the image of hχ ◦ pχ ◦ r ◦ (ΦJ × iτ )|int(∆k×∆`) and see that each ˜λ dJ,τ such that λ, J and τ are of the form required by Notation 8.1.12 is a (k + `)-cell 2n 0 0 λ of a CW-structure on D / ≈, with hχ ◦ pχ ◦ r ◦ (ΦJ × iτ ) as its characteristic map. 2n We will denote this CW-structure by DL / ≈.

0 8.1.2 A CW-structure on L(χ0; χ )

0 We now turn our attention to a CW-structure for the weighted lens space L(χ0; χ ). 2n 0 2n−1 We recall that D / ≈ is homeomorphic to CL(χ0; χ ) and S / ≈ is homeomorphic 0 to L(χ0; χ ). 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 124

0 n ¯ 2n−1 We observe that the map r restricts to the map r : T × ∆{1,...,n} → S , where ¯ n n ∆{1,...,n} is the 0-th face of ∆c ; that is, the closure of the (n − 1)-cell ∆{1,...,n} ⊂ ∆c , 2 2 which corresponds to the set of points {(0, r1, . . . , rn): r1 + ··· + rn = 1}. Clearly n ¯ n n TL × ∆{1,...,n} is a subcomplex of TL × ∆c , with a typical cell being of the form l ∆J × ∆τ , such that τ ⊆ {1, . . . , n}.

l 2n Theorem 8.1.16. Let r ◦ (ΦJ × iτ ) range over the characteristic maps of DL such that τ ⊆ {1, . . . , n}, then the family of maps

l k ` 2n−1 r ◦ (ΦJ × iτ ): ∆ × ∆ → S is a family of characteristic maps for a CW-structure on S2n−1.

l Therefore each dJ,τ such that l, J and τ are of the form given in Notation 8.1.8 2n l and τ ⊆ {1, . . . , n} is a (k + `)-cell of a CW-structure on D , with r ◦ (ΦJ × iτ ) as its 2n−1 characteristic map. We let SL denote this CW-structure.

0 n ¯ 2n−1 2n n Corollary 8.1.17. The maps r : TL × ∆{1,...,n} → SL and Φn|S2n−1 : DL → CPL are cellular maps.

Proof. This proof follows that of Corollary 8.1.10. By restricting attention to τ ⊆ {1, . . . , n} in Proposition 8.1.4 and Remark 8.1.7 we observe r0 maps cells of the n ¯ 2n−1 2n 0 subcomplex TL × ∆{1,...,n} to cells of the subcomplex SL ⊂ DL and we deduce r is cellular. Similarly for Proposition 5.3.2 and Remark 5.3.5, we see that q|S2n−1 maps 2n−1 n 0 cells of SL to cells of CPL and as q|S2n−1 = Φn|S2n−1 ◦ r then the result follows.

0 λ Theorem 8.1.18. Let pχ ◦ r ◦ (ΦJ × iτ ) range over the characteristic maps of cells in 2n DL /Cχ such that τ ⊆ {1, . . . , n}, then the family of maps

0 λ k ` 2n−1 pχ ◦ r ◦ (ΦJ × iτ ): ∆ × ∆ → S /Cχ

2n−1 is a family of characteristic maps for a CW-structure on S /Cχ.

2n−1 Proof. Restricting attention to the case τ ⊆ {1, . . . , n} in Theorem 8.1.11 shows SL is a Cχ-complex, then the result clearly follows.

0 0 λ Theorem 8.1.19. Let hχ ◦pχ ◦r ◦(ΦJ ×iτ ) range over the characteristic maps of cells 2n in DL / ≈ such that τ ⊆ {1, . . . , n}, then the family of maps

0 0 λ k ` 0 hχ ◦ pχ ◦ r ◦ (ΦJ × iτ ): ∆ × ∆ → L(χ0; χ )

0 is a family of characteristic maps for a CW-structure on L(χ0; χ ). 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 125

˜λ Therefore each dJ,τ such that λ, J and τ are of the form given in Notation 8.1.12 and 0 0 0 λ τ ⊆ {1, . . . , n} is a (k+`)-cell of a CW-structure on L(χ0; χ ), with hχ ◦pχ ◦r◦(ΦJ ×iτ ) 0 as its characteristic map. We let L(χ0; χ )L denote this CW-structure. 0 0 We recall the map fχ : L(χ0; χ ) → P(χ ) given in Section 3.2 where for a given 0 χ = (χ0, . . . , χn) then χ = (χ1, . . . , χn).

0 0 Corollary 8.1.20. The map fχ : L(χ0; χ )L → P(χ )L is cellular.

Proof. From Corollary 8.1.14 and Corollary 8.1.17 we see that the map

˜ 2n−1 n Φn|S2n−1 : SL /Cχ → (CP{1,...,n}/Cχ)L

2n−1 ∼ 0 n ∼ 0 is cellular. We recall S /Cχ = L(χ0; χ ) and CP{1,...,n}/Cχ = P(χ )L, and noting 0 0 that the CW-structures L(χ0; χ )L and P(χ )L have been deﬁned so that these home- omorphisms are cellular, then the result follows.

0 0 Remark 8.1.21. Adjoining CL(χ0; χ )L to P(χ )L via the attaching map fχ gives the

CW-structure P(χ)L determined in Theorem 5.4.4. In particular, when χ0 divides the 0 0 2n weights χ , then recall from Section 3.3.2 that CL(χ0; χ ) is homeomorphic to D . We 0 2n observe that P(χ)L is a subdivision of the CW-structure P(χ) = P(χ ) ∪ e . Similarly 0 2 2n P(χ)L is a subdivision of the CW-structure P(χ) = e ∪ e ∪ · · · ∪ e when the weights χ are divisive.

8.2 Stunted weighted projective space

We noted in Remark 7.0.1 that for some I = {i0, . . . , ik} ⊂ {0, . . . , n} then PI (χ)L is a subcomplex of P(χ)L. Therefore we are able to use the homology and cohomology long exact sequences of the pair (P(χ)L, PI (χ)L) to calculate the integral homology and cohomology groups of P(χ; I) := P(χ)/PI (χ). Moreover we are able the collapse the cells of PI (χ)L to a point to obtain a cell structure for the stunted weighted projective space P(χ; I) and input the results of Chapter 7 to identify generating cycles in the associated cellular chain complex of the integral homology groups.

8.2.1 A CW-structure on P(χ; I)

Following on from Proposition 5.4.4, we describe a CW-complex structure for P(χ; I). 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 126

Recalling Remark 7.0.1, it follows that a typical (m+`)-cell of PI (χ)L is of the form λ (eρ(J),τ )χ where the set τ ⊆ I is such that τ = {τ0, . . . , τ`}; ρ(J) is an ordered partition

λ1 λn of σ ⊆ τ \τ0 into m parts; and λ = (λ1, . . . , λn) is such that (ζL , . . . , ζL ) is a canonical representative of an orbit in (C )n of the (C ) -action. Therefore all these cells in L τ\τ0 χ τ

P(χ)L are identiﬁed to a point by the quotient map sI : P(χ)L → P(χ)L/PI (χ)L. We will represent s ( (χ) ) by the image of the 0-cell, (e(0,...,0)) , which corresponds to I PI L ∅,{i0} χ the point [0 : ... : 0 : 1i0 : 0 : ..., 0]χ, under sI .

λ Theorem 8.2.1. Let h ◦ p ◦ q ◦ (ΦJ × iτ ) range over the characteristic maps of P(χ)L, such that τ * I. The family of maps

λ k ` sI ◦ h ◦ p ◦ q ◦ (ΦJ × iτ ): ∆ × ∆ → P(χ; I) and the map

(0,...,0) 0 0 sI ◦ h ◦ p ◦ q ◦ (Φ∅ × i{i0}): ∆ × ∆ → P(χ; I) is a family of characteristic maps for a CW-structure on P(χ; I).

λ I λ λ Therefore we let (eJ,τ )χ := sI (eJ,τ )χ, where (eJ,τ )χ is a cell of P(χ)L such that τ * I and let (e(0,...,0))I = s (e(0,...,0)) . We denote the CW-structure given for (χ; I) in ∅,{i0} χ I ∅,{i0} χ P

Theorem 8.2.1 by P(χ; I)L.

We use the cellular chain complex of P(χ)L given in Notation 6.5.1 to deﬁne the cellular chain complex for P(χ; I).

Notation 8.2.2. For all τ, J and λ of the form considered in Theorem 8.2.1 and such that m = k + `, the cellular chain group Cm(P(χ; I)L) of P(χ; I)L is

λ I λ Cm(P(χ; I)L) = Z (eJ,τ )χ = (sI )∗(eJ,τ )χ : all τ, J and λ .

As sI is a cellular map the cellular boundary homomorphism ∂m : Cm(P(χ; I)L) →

Cm−1(P(χ; I)L) is deﬁned by

λ I λ ∂m(eJ,τ )χ = (sI )∗ ∂m(eJ,τ )χ .

We note that as (sI )∗ is a chain map, given a cycle c ∈ Cm(P(χ)L), it is true that

(sI )∗(c) is a cycle in Cm(P(χ; I)L). 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 127

8.2.2 Homology and cohomology calculations for P(χ; I)

Since PI (χ)L is a subcomplex of P(χ)L, we can use the long exact sequence of homology groups for the pair (P(χ)L, PI (χ)L), identifying Hk(P(χ)L, PI (χ)L) with Hek(P(χ)L/PI (χ)L) for k > 0 and then deduce the homology groups of P(χ)L/PI (χ)L. Similarly the long exact sequence of cohomology groups for the pair will yield the cohomology groups of

P(χ)L/PI (χ)L. Moreover, we are able to identify generators of the non-trivial homology groups of P(χ)L/PI (χ)L via Theorem 7.0.18.

Theorem 8.2.3. The integral homology groups of the stunted weighted projective space, P(χ; I)L, are given by  j  Z{[(sI )∗ ◦ (hχ)∗(Gχ)]} if i = 0 and i = 2j for k + 1 ≤ j ≤ n  j χ χI Hi(P(χ; I)L) = Zq{[(sI ◦ hχ)∗(Gχ)]} q = lj /lj , if i = 2j 1 ≤ j ≤ k   0 otherwise.

Proof. In [15], Kawasaki gives the following commutative diagram of spaces

n iI n CPI / CP

pχI pχ

(iχ)I PI (χ) / P(χ) and shows that when passing to homology for 0 ≤ j ≤ k this becomes

n (iI )∗ n = H2j(CPI ) / H2j(CP ) Z / Z χ χ pχ p I I χ ·lj ·lj

((iχ)I )∗ ·a(χ,I) H2j(PI (χ)) / H2j(P(χ)) Z / Z for some a(χ,I) ∈ Z. The commutivity of the above diagram means that a(χ,I) =

χ χI lj /lj := q. We observe that the long exact sequence of the homology groups for the 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 128

CW-pair (P(χ)L, PI (χ)L) can be written as

···

/ H2j(PI (χ)L) / H2j(P(χ)L) / H2j(P(χ; I)L)

/ H2j−1(PI (χ)L) / H2j−1(P(χ)L) / H2j−1(P(χ; I)L)

/ H2j−2(PI (χ)L) / H2j−2(P(χ)L) / H2j−2(P(χ; I)L)

/ ······

/ H0(PI (χ)L) / H0(P(χ)L) / He0(P(χ; I)L)

/ 0.

As the integral homology groups of PI (χ)L and P(χ)L are trivial for dimensions greater than 2n, we know that the integral homology groups for P(χ; I) are therefore trivial in these dimensions. As P(χ)L has trivial odd homology and H2j(PI (χ)L) ,→ H2j(P(χ)L) for 1 ≤ j ≤ n, it follows that P(χ; I)L also has trivial odd homology. For the remaining cases we observe the above long exact sequence therefore breaks down to give the following exact sequences:

·q 0 → Z −→ Z → H2j(P(χ)L/PI (χ)L) → 0 when 1 ≤ j ≤ k

0 → Z → H2j(P(χ)L/PI (χ)L) → 0 when k + 1 ≤ j ≤ n

= 0 → Z −→ Z → He0(P(χ)L/PI (χ)L) → 0 when 1 ≤ j ≤ k.

Thus we are able to deduce   Z if i = 0 and i = 2j for k + 1 ≤ j ≤ n  χ χI Hi(P(χ; I)) = Zq q = lj /lj , if i = 2j 1 ≤ j ≤ k   0 otherwise.

From the above exact sequences we can see that the induced map in homology

(sI )∗ : H2j(P(χ)L) → H2j(P(χ; I)L) is an epimorphism. Therefore recalling Theorem n 7.0.18 which proves H2j(P(χ)L) = Z{(hχ)∗[Gχ]} for 1 ≤ j ≤ n, we deduce (sI )∗ ◦ n (hχ)∗[Gχ] generates H2j(P(χ; I)L) as required. 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 129

2j 2j Lemma 8.2.4. The induced map (sI )∗ : H (P(χ)) → H (PI (χ)) is multiplication by

χ χI q := lj /lj on Z for 1 ≤ j ≤ k.

Proof. We refer the reader to [11, p.201] to see that sI : PI (χ) → P(χ) induces the following commutative diagram

2j f 0 / Ext(H2j−1(P(χ))) / H (P(χ)) / Hom(H2j(P(χ))) / 0

∗ ∗ ∗ ((sI )∗) (sI ) ((sI )∗) 2j f 0 / Ext(H2j−1(PI (χ)) / H (PI (χ))) / Hom(H2j(PI (χ))) / 0,

∗ ∗ where (sI ) is the induced map on cohomology and ((sI )∗) is precomposing with (sI )∗, the induced map on homology.

By recalling that H2j−1(P(χ)) = 0 = H2j−1(PI (χ)) and therefore Ext(H2j−1(P(χ))) =

0 = Ext(H2j−1(PI (χ))), we obtain the following diagram, noting that the horizontal maps are isomorphisms

2j f H (P(χ)) / Hom(H2j(P(χ)))

∗ ∗ (sI ) ((sI )∗) 2j f H (PI (χ)) / Hom(H2j(PI (χ))).

∗ The map ((sI )∗) takes a function α: Z → Z (which we can identify with α(1) ∈ Z) to the function α ◦ (·q) given by

α ◦ (·q)(1) = α ◦ (q · 1) = α(q) = qα(1).

∗ −1 Therefore ((sI )∗) is multiplication by q on Z. Hence the map (sI )∗ is f ◦ (·q) ◦ h and we see f −1 ◦ (·q) ◦ f(x) = f −1(qf(x) = qf −1(f(x)) = qx, hence multiplication by q.

Theorem 8.2.5. The integral cohomology for the stunted weighted projective space

P(χ; I) is given by   Z if i = 0 and i = 2j for k + 1 ≤ j ≤ n  i χ χI H (P(χ; I)) = Zq q = lj /lj , if i = 2j + 1 1 ≤ j ≤ k   0 otherwise.

∗ ∗ ∗ The induced map (sI ) : H (P(χ; I)L) → H (P(χ)L) determines the ring structure of ∗ H (P(χ; I)L) completely. 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 130

Proof. The cohomology long exact sequence of the CW-pair (P(χ)L, PI (χ)L) can be written as follows

··· o

2j 2j 2j H (PI (χ)L) o H (P(χ)L) o H (P(χ; I)L) o

2j−1 2j−1 H (PI (χ)L) o H2j−1(P(χ)L) o H (P(χ; I)L) o

2j−2 2j−2 2j−2 H (PI (χ)L) o H (P(χ)L) o H (P(χ; I)L) o

··· ··· o

0 0 0 H (PI (χ)L) o H (P(χ)L) o He (P(χ; I)L) o

0. As in the proof of Theorem 8.2.3 we see that for dimensions greater than 2n, where the integral cohomology groups of PI (χ)L and PI (χ)L are trivial, the integral cohomology groups for P(χ; I)L are also trivial. From Lemma 8.2.4 we deduce that the above long exact sequence reduces to give the following exact sequences

2j ·q 2j+1 0 → H (P(χ; I)L) → Z −→ Z → H (P(χ; I)L) → 0 when 1 ≤ j ≤ k

2j 0 → H (P(χ; I)L) → Z → 0 when k + 1 ≤ j ≤ n hence   Z if i = 0 and i = 2j for k + 1 ≤ j ≤ n  i χ χI H (P(χ; I)) = Zq q = lj /lj , if i = 2j + 1 1 ≤ j ≤ k   0 otherwise.

For the cup product structure we observe that the quotient map sI : P(χ) → P(χ; I) ∗ ∗ ∗ induces a ring homomorphism (sI ) : H (P(χ; I)) → H (P(χ)). For dimensional reasons there are only two cases we need to consider:

2i+1 2j+1 1. u2i+1 ∈ H (P(χ; I)) for 1 ≤ i ≤ k and v2j+1 ∈ H (P(χ; I)) for 1 ≤ j ≤ k 2(i+j+1) such that 2k < 2(i + j + 1) ≤ 2n: then u2i+1v2j+1 ∈ H (P(χ; I)) = Z, but 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 131

∗ ∗ ∗ (sI ) (u2i+1v2j+1) = (sI ) (u2i+1)(sI ) (v2j+1) = 0 and as

∗ 2(i+j+1) 2(i+j+1) (sI ) : H (P(χ; I)) → H (P(χ))

is an isomorphism, we deduce that u2i+1v2j+1 = 0.

2i 2j 2. u2i ∈ H (P(χ; I)) for k+1 ≤ i ≤ n and v2j ∈ H (P(χ; I)) for k+1 ≤ j ≤ n such

that 2(i+j) ≤ 2n and u2i, v2j are both generators of their respective cohomology 2(i+j) groups: then u2iv2j ∈ H (P(χ; I)) = Z. Therefore there must exist some 2(i+j) 2(i+j) α2(i+j) ∈ Z and some w2(i+j) ∈ H (P(χ; I)) that generates H (P(χ; I))

such that u2iv2j = α2(i+j)w2(i+j). As

∗ 2` 2` (sI ) : H (P(χ; I)) → H (P(χ))

2i is an isomorphism for all k + 1 ≤ ` ≤ n, if we let γ2i be a generator of H (P(χ)) then by inputting the cup product structure of H∗(P(χ)) as calculated in [15] we see

∗ ∗ ∗ (sI ) (u2iv2j) = (sI ) (u2i)q (v2j)

= γ2iγ2j χ χ li lj = χ γ2(i+j) li+j and

∗ ∗ (sI ) (u2iv2j) = (sI ) (α2(i+j)w2(i+j))

= α2(i+j)γ2(i+j).

χ χ li lj We conclude that u2iv2j = χ w2(i+j). li+j

Remark 8.2.6. We recall the generalisation of Kawasaki’s coﬁbre sequence given in Section 3.2: f g L(χ ; χ ) −−→χ (χ ) −−→χ (χ) , (8.2.7) i bι P bι P where for some 0 ≤ i ≤ n we have ˆι = {0, . . . , n}\ i. Extending (8.2.7) to the right by the quotient map sˆι identiﬁes P(χ; ˆι) with ΣL(χι, χˆι), which clearly has trivial cup product structure. In fact we observe from Theorem 8.2.5 that H∗(P(χ; I)) has trivial cup product if I = {i0, . . . , ik} ⊂ {0, . . . , n} is such that n ≤ 2(k + 1). 0 CHAPTER 8. P(χ; I) AND L(χ0; χ ) 132

An interesting problem would be to consider if the CW-structure P(χ; ˆι)L determined in Theorem 8.2.1 desuspends to give the CW-structure L(χi; χˆι)L in Theorem 8.1.19. However we have not been able to ﬁnd a solution to this question. Bibliography

[1] Alejandro Adem, Johann Leida, and Yongbin Ruan. Orbifolds and Stringy Topol- ogy. Cambridge Tracts in Mathematics 171. Cambridge University Press (2007).

[2] Abdallah Al Amrani. Cohomological study of weighted projective spaces. S Sert¨oz (editor), Algebraic geometry, Proceedings of the Bilkent Summer School (Ankara 1995). Lecture Notes in Pure and Applied Mathematics 193:1–52, Dekker (1997).

[3] Anthony Bahri, Matthias Franz, and Nigel Ray. The equivariant cohomology ring of weighted projective space. Mathematical Proceedings of the Cambridge Philo- sophical Society 146(02):395405, (2009). Society. 146(02):395–405 (2009).

[4] Anthony Bahri, Matthias Franz, and Nigel Ray. Weighted projective spaces and iterated Thom spaces. Available at arxiv.org/abs/1109.2359; to appear Osaka Jour- nal of Mathematics, (2013).

[5] Anthony Bahri, Matthias Franz, Dietrich Notbohm, and Nigel Ray. The classiﬁ- cation of weighted projective space. Fundamenta Mathematica 220:217226, (2013)

[6] H. S. M. Coxeter. Discrete groups generated by reﬂections. Annals of Mathematics 6:13-29 (1934).

[7] Tammo tom Dieck Transformation Groups de Gruyter Studies in Mathematics (1987).

[8] Albrecht Dold. Lectures On Algebraic Topology. Classics in mathematics. Springer- Verlag (1995).

[9] H. Freudenthal. Simplizialzerlegungen von beschr¨ankter Flachheit. Annals of Mathematics 43:580-582 (1942).

133 BIBLIOGRAPHY 134

[10] William Fulton. Introduction to toric varieties. Annals of Mathematics Studies 131, Princeton University Press (1993).

[11] Allen Hatcher. Algebraic Topology (seventh edition). Cambridge University Press (2006).

[12] Dale Husemoller. Fiber Bundles (third edition). Graduate Texts in Mathematics 20. Springer-Verlag (1994).

[13] S¨orenIllman. Smooth equivariant triangulations of G-manifolds, for G a ﬁnite group. Mathematische Annalen 233:199–220 (1978).

[14] Katsuo Kawakubo. The theory of transformation groups. Oxford University Press (1991).

[15] Tetsuro Kawasaki. Cohomology of twisted projective spaces and lens complexes. Mathematische Annalen 206:243–248 (1973).

[16] H. W. Kuhn. Some combinatorial lemmas in topology. IBM Journal of Research and Development 45:518-524 (1960).

[17] William S. Massey. A Basic Course in Algebraic Topology. Graduate Texts in Mathematics 127. Springer-Verlag (1991).

[18] J Peter May. A Concise Course in Algebraic Topology. Chicago lectures in mathematics series. (1999).

[19] James R. Munkres Elements of Algebraic Topology(seventh edition). Westview Press (1984).

[20] M. Poddar and S. Sarkar. On quasitoric orbifolds. Osaka J. Math. 47: 1055–1076 (2010).

[21] W.P. Thurston. Three-dimensional geometry and topology Edited by Silvio Levy. Princeton Mathematical Series, 35, Princeton University Press, Princeton, NJ, (1997).

[22] J.H.C. Whitehead. Combinatorial homotopy I. Bulletin of the American Mathe- matical Society 55:213–245 (1949).