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Quasitoric Manifolds and Cobordism Theory QUASITORIC MANIFOLDS AND COBORDISM THEORY A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2008 Craig Matthew Laughton School of Mathematics Contents Abstract 4 Declaration 5 Copyright Statement 6 Acknowledgements 7 1 Introduction 8 2 Notation and prerequisites 13 2.1 Nomenclature . 13 2.2 Oriented cohomology theories . 14 2.3 Tangential structures . 18 2.3.1 Stably complex structures . 19 2.3.2 Special unitary & stably quaternionic structures . 19 2.3.3 Tangential structures on sphere bundles . 20 2.3.4 Changes of tangential structure . 21 2.4 The Borel construction . 25 2.5 Iterated bundle constructions . 26 3 Quasitoric manifolds 31 3.1 Torus actions and simple polytopes . 31 3.2 Quasitoric manifolds . 34 3.3 Cohomology of quasitoric manifolds . 38 2 3.4 Tangential structures . 40 4 Dobrinskaya towers 44 4.1 Constructing the towers . 44 4.2 Cohomology of Dobrinskaya towers . 50 4.3 Stably complex structure . 51 4.4 Dobrinskaya towers as iterated bundles . 52 4.5 Special cases . 55 5 Quasitoric manifolds with SU-structure 58 5.1 Reducible quasitoric manifolds . 61 5.2 Encoding omniorientations in Λ . 72 5.3 Chern classes . 73 5.4 Characteristic numbers . 75 5.5 Complex projective space . 77 6 Quaternionic towers 85 6.1 Quaternionic line bundles . 86 6.2 Constructing the towers . 90 6.3 Special cases . 92 6.4 Quaternionic quasitoric manifolds . 94 7 Quaternionic towers and cobordism theory 99 7.1 Torsion in MSp∗ ............................. 100 7.2 The manifolds Y 4n+1 ........................... 104 7.3 Tangential structures . 110 7.4 Associated units . 120 7.5 Fundamental classes . 126 7.6 Determination of Ψ(Y 4n+1)........................ 128 Bibliography 135 Word count 37,234 3 The University of Manchester Craig Matthew Laughton Doctor of Philosophy Quasitoric Manifolds and Cobordism Theory July 29, 2008 Our aim is to investigate quasitoric manifolds, and their quaternionic analogues, in the setting of cobordism theory. A quasitoric manifold is said to be reducible if it can be viewed as the total space of an equivariant bundle with quasitoric fibre and quasitoric base space. Buchstaber, Panov and Ray have conjectured that any quasitoric SU-manifold is a boundary in the complex cobordism ring. We prove this conjecture for complex projective space CP n, and for reducible quasitoric manifolds with fibre CP 1. We introduce the notion of a quaternionic tower, as the quaternionic analogue of a certain family of quasitoric manifolds known as Dobrinskaya towers. We com- pute their F -cohomology ring for a quaternionic oriented ring spectrum F , and the properties of various subfamilies are investigated. Approaches to placing the towers within a quaternionic analogue of the theory of toric topology are considered. We undertake a study of one particular subfamily of quaternionic tower in the quaternionic cobordism ring MSp∗. This allows us to construct a new set of man- ifolds, and by studying their possible stably quaternionic structures, we prove that they are simply-connected geometric representatives for an important class of torsion elements in MSp∗. 4 Declaration No portion of the work referred to in this 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. 5 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns any copyright in it (the “Copyright”) and s/he has given The University of Manchester the right to use such Copyright for any administrative, promotional, educational and/or teaching purposes. ii. Copies of this thesis, either in full or in extracts, may be made only in accor- dance with the regulations of the John Rylands University Library of Manch- ester. Details of these regulations may be obtained from the Librarian. This page must form part of any such copies made. iii. The ownership of any patents, designs, trade marks and any and all other intellectual property rights except for the Copyright (the “Intellectual Property Rights”) and any reproductions of copyright works, 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 Rights 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 Rights and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and exploitation of this thesis, the Copyright and any Intellectual Property Rights and/or Reproductions described in it may take place is available from the Head of the School of Mathematics. 6 Acknowledgements I would like to thank my supervisor, Nige Ray, who first encouraged me to pursue my interest in maths when I was an undergraduate at Manchester; his expertise and guidance have since proved invaluable throughout my PhD. Many other mathemati- cians have also imparted plenty of advice and topological wisdom to me over the past four years, in particular, I should thank Tony Bahri, Victor Buchstaber and Reg Wood. I must acknowledge the assistance of my former officemates, Dave, Adrian, Goran and Mark, who all unfailingly came to my rescue whenever I was struggling at the beginning of my PhD. Indeed I am grateful to all my officemates over the years, especially Hadi and Jerry, for their support and friendship. I would also like to express my gratitude to my parents, for all their encouragement throughout my time at university. Finally, my thanks go to Annie, whose determination to overcome much greater difficulties than any I have suffered has been a huge inspiration to me. Without her love and support, I would not have come this far. 7 Chapter 1 Introduction Toric topology encompasses the study of topological spaces that admit well-behaved torus actions. The subject was initiated by the work of Davis and Januskiewicz [16], who introduced the notion of a quasitoric manifold as the topological analogue of a construction in algebraic geometry. In the fifteen years since their pioneering paper, the subject has rapidly flourished, and links between more established areas of topology have begun to be forged. One of the first applications of toric topology was to the theory of complex cobor- dism, the study of which stretches far back into twentieth century mathematics; for instance, the coefficient ring MU∗ of complex cobordism was first determined by Mil- nor [37] and Novikov [40] in the early sixties. Quasitoric manifolds M are amenable to the methods of complex cobordism because once M and certain of its submanifolds are oriented in a particular manner, M is furnished with a canonical stably complex structure. This fact allowed Buchstaber and Ray [9] to fabricate an alternative basis for MU∗ using quasitoric manifolds. Their work led to a solution to the topological analogue of a long standing problem in algebraic geometry, first posed by Hirzebruch in 1958 (see e.g. [5, Section 5.3]). Our thesis continues the study of quasitoric manifolds in complex cobordism the- ory. Buchstaber, Panov and Ray [7] have conjectured that every quasitoric manifold with an SU-structure is a boundary in the complex cobordism ring, and it is this problem which we investigate for complex projective space CP n, and for the class of 8 CHAPTER 1. INTRODUCTION 9 reducible quasitoric manifolds introduced by Dobrinskaya [17]. A quasitoric manifold M 2n is said to be reducible if it can be viewed as the total space of an equivariant bundle, with quasitoric fibres and quasitoric base space. In the second half of our thesis, we consider the effect of replacing the complex numbers by the quaternions in some of the constructions of toric topology. In par- ticular, we study a quaternionic analogue of a family of quasitoric manifolds known as Dobrinskaya towers. The seemingly innocuous fact that the quaternions do not commute throws up many obstacles, but the quaternionic viewpoint does allow us to construct a collection of quaternionic towers with interesting applications in quater- nionic cobordism theory. As in the case of complex cobordism, the coefficient rings of the other clas- sical cobordism theories MG∗(−), which arise from the Thom spectra MG, for G = O, SO, SU, Spin, have long been understood (see e.g. [54]). In contrast, the coefficient ring MSp∗ of quaternionic cobordism, remains something of an enigma. In the past thirty years there have been several attempts to compute this ring. Gargantuan calculations with spectral sequences that converge to MSp∗ were un- dertaken by Kochman [30] and Vershinin [58], while more recent efforts utilised the transfer map [4], and the theory of cobordism with singularities [59]. Though these approaches offered many new insights into the structure of MSp∗, a full description of the ring is still lacking. However, it is clear that a collection of elements ϕm ∈ MSp8m−3 defined by Ray in [42], play a crucial role in MSp∗. The ϕm are multiplicatively indecomposable elements of order 2, and they have been the starting point for all of the investigations described above. Finding geometrical representatives for elements in MSp∗ is a key problem. Once we have obtained such representatives, we would hope that we can gain further infor- mation about the cobordism ring, which would have remained hidden, or been harder to obtain by using purely algebraic methods. Ray went on to realise the torsion elements ϕm geometrically in [47], with the aid of a special subfamily of quaternionic tower. He conjectured that it would be CHAPTER 1. INTRODUCTION 10 possible to create another collection of geometrical representatives by constructing (4n + 1)-dimensional manifolds Y 4n+1 that are closely related to the original family.
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