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 Copyright Statement 7 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 pro- jective 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 ac- tually 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 pro- jective 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 intel- lectual 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 rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) 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 x2:2. Outside algebraic geometry, weighted projective spaces and more generally toric va- rieties 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 = f(tχ0 ; : : : ; tχn ): jtj = 1g < T n+1 , weighted projective space is the underlying space of a representable orb- ifold [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.