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The Geometry of Bundle Gerbes

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The Geometry of Bundle Gerbes L'6 "os The Geometry of Bundle Gerbes Daniel Stevenson Thesis submitted for the degree of Doctor of Philosophy in the Department of Pure Mathematics IJniversity of Adelaide 9 February 2000 Abstract This thesis reviews the theory of bundle gerbes and then examines the higher di mensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced. and it is shown that there is a class in Ha(M;Z) associated to any bundle 2-gerbe. n IV Acknowledgement I would like to thank my supervisor Michael Murray for his patience, generosity and expert guidance in the writing of this thesis. He has greatly enhanced my under- standing and enjoyment of mathematics over the past few years and I would like to say that it has been an honour to work \Mith him during this time. It is my pleasure also to thank AIan Carey for some useful conversations and also Ann Ross for help in ad- ministrative matters. I would also like to gratefully acknowledge the support provided by an Australian Post Graduate Award. v vl Contents 1 Introduction 1 2 Review of some simplicial techniques b 2.7 Simplicial spaces 5 2.2 Geometric realisation 8 2.3 Classifying sPaces 11 2.4 BPC" bundles 11 3 Review of bundle gerbes 15 3.1 Properties of C* bundles 15 3.2 Bundle gerbes and the Dixmier-Douady class ' 18 3.3 Bundle gerbe connections and the extended Mayer-Vietoris sequence 23 3.4 Bundle gerbe morphisms and the 2-category of bundle gerbes . 25 4 Singular theory of bundle gerbes 35 4.1 Construction of a classifying map 35 4.2 The singular theory of C" bundles 37 4.3 The singular theory of bundle gerbes 4I 5 Open covers 51 5.1 Bundle gerbes via oPen covers 51 5.2 Bundle gerbes and Deligne cohomology 52 6 Gerbes and bundle gerbes Ðl 6.1 Torsors 57 6.2 Sheaves of categories and gerbes 58 6.3 Properties of gerbes 61 64 6.4 The gerbe HOM(P,Q . ' 6.5 The gerbe associated to a bundie gerbe 66 6.6 Higher gluing laws 69 7 Definition of a bundle 2-gerbe 77 7.1 Simplicial bundle gerbes and bundle 2-gerbes 77 7.2 Stable bundle 2-gerbes 80 8 Relationship of bundle 2-gerbes with bicategories 83 8.1 Bicategories 83 8.2 Bundle 2-gerbes over a point and bicategories 92 vll I Some examples of bundle 2-gerbes 97 9.1 The lifting bundle 2-gerbe 97 9.2 The tautological bundle 2-gerbe 98 9.3 The bundle 2-gerbe of a principal G bundle 100 10 Bundle 2-gerbe connections and 2-curvings 103 10.1 Definitions and some preliminary lemmas . 103 10.2 Proof of the main result 106 11 Comparison of the Cech, Deligne and de Rham classes associated to a bundle 2-gerbe 111 11.1 A Õech three class 111 1I.2 A Deligne hypercohomology class 115 11.3 The first Pontryagin class IL7 12 Trivial bundle 2-gerbes 119 12.1 Tfivial bundle 2-gerbes 119 L2.2 T)ivial stable bundle 2-gerbes I28 13 The relationship of bundle 2-gerbes with 2-gerbes 131 13.1 Simplicial gerbes and 2-gerbes 131 vnl Chapter 1 Introduction. There has been much recent interest in applying the theory of gerbes to differential geometry l22l and mathematical physics ([Za] and [13]). Gerbes were originally in- troduced by Giraud (see [19]) and were extensively studied in [7]. Loosely speaking a gerbe is a 'sheaf of categories' except that the¡e is an extra level of complication when one comes to discuss sheaf-iike glueing laws for objects. Much of the importance of gerbes in the previously cited applications stems from the fact that equivalence classes of gerbes over a manifold M are in a bijective correspondence wifh H3(M;V'). This is entirely similar to the fact that H'(M;Z) lsin abijective correspondence with isomorphism classes of principal C" bundles on M. One can think of Ct bundles and gerbes as a geometric 'realisation' or 'represen- tation' of the cohomology of the manifold in dimensions two and three respectively. Flom this perspective it is natural to look for such objects in other dimensions. In dimension zero we can consider the set of all maps f , M + Z which are, in fact, in bijective correspondence with Ho(M;Z). In dimension one there are various kinds of objects we can consider which are in bijective correspondence with H'(M;Z). For exampie we could consider (isomorphism classes of) principalZbundles over M. Or, because the classifying space of. Z is C" we could equally well consider the space of homotopy classes of maps from M into C'. A similar story is true lor H2(M;Z): one could. consider H'(M;Z) to be the set of all homotopy ciasses of maps Í , M - BC', where BCt is the classifying space for the group C', or, we could think of an element of H2(M;Z) corresponding to a homotopy class of maps M -+ BC' as giving rise to a principal Cx bundle on M. In the case of H'(M;Z) we can go one step further and think of a principal C* bundle on M as a principal BZ bundle on M. The crucial fact here, as pointed out in [36], is that BZ, can be chosen to be an abelian topological group since Z is. This last fact is exploited in the paper [1S] by Gajer where the author introduces a notion of a principal BnCx bundle on M and proves that there is a bijection be- tween ¡{n+z(M;Z) and the set of all principai BPC'bundles on M (here BrC' is the pfotd iterated classifying space B...BC" of the group C"). Therefore, when one comes to consider H3(M;Z) there are several possibilities: one option is to think of Ht(M;Z) as the set of all homotopy classes of maps f , M --+ BBC, another is to think of H3(M;Z) as the set of equivalence classes of gerbes on M. Yet another is to follow Gajer and think of. H3(M;Z) as the set of all principal BC" bundles on M or, alternatively, the set of all principal BBZ bundles on M. gerbes give rise In f33] the notion oî. a bundle gerbe was introduced. Bundle also to 1 classes in H3(M;Z). A bundle gerbe on M is a tripie (P,X,M) where ry : X -+ M is a surjection which locally admits sections and where P is a principai CX bundle over thefibre product ylz): X x¡r,tX. P is required tohave aproduct, that is aC" bundle isomorphism which on the fibres of P takes the form P(r",r")Ø Pçqp2¡ - P(rr,rr) for ø1, 12 aîd z3 in the same fibre of X. It can be shown that every bundie gerbe on M gives rise to a gerbe on M. There is a characteristic class associated to bundle gerbes, the Dixmier-Douady class. This is a class in H3(M;Z) which measures the obstruction to the bundle gerbe being 'trivial' in a certain sense. In [7] the notions of a connectiue structure on a gerbe and a curuing for the con- nective structure were introduced. It was shown that one could associate to a gerbe equipped with a connective structure and curving a closed integral three form ø on M - the three curvature of the curving. In [33], it is shown that by considering a connection on P which is compatible with the product above, it is possible to introduce the notion of a bundle gerbe connection and a curving fo¡ a bundle gerbe connection and that, furthermore, a bundle gerbe on M equipped with a bundle gerbe connection and curving gives rise to a closed integrai three form on M. This three form is a repre- sentative of the image of the Dixmier-Douady class of the bundle gerbe in Ë13(M;R.). It can be shown that a bundle gerbe connection and curving on a bundle gerbe induces a connective structure and curving on the gerbe associated to the bundle gerbe. In the series of papers [S] and [9] the authors considered a certain Z-gerbe associated to a principal G bundle P --+ M where G was a compact, simpie, simply connected Lie group. 2-gerbes, introduced in [5], are higher dimensional analogues of gerbes. Broadly speaking, a2-gerbe on M is a'sheaf of bicategories' on M. Bicategories, defined in [1], are higher dimensional analogues of categories. A bicategory consists of a set of objects and for each pair of objects, a category. The objects of this category are then l-arrows of the bicategory and the arrolvs of the category are 2-arrows of the bicategory. As a result, the giueing iaws involved in the definition of a sheaf of bicategories become quite intricate. One can show that if one considers a certain class of 2-gerbes (2-gerbes bound by C") then one can associate an element of. Ha(M;Z) to each 2-gerbe and moreover it can be shown that H (M;Z) is in a bijective correspondence with 'biequivalence' classes of 2-gerbes. In [14] a related object, a bundle 2-gerbe, was introduced. In the latter part of this thesis we will study bundle 2-gerbes using a variation of the definition given in [t4].
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