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Duality for topological abelian group stacks and T -duality 3236 T -duality of twisted torus bundles6.1 Pairs and topological T -duality6.1.1 The goal of this subsection is to introduce the main objects of topological T -dualityand review the structure of the theory. Let B be a topological space.Definition 6.1. A pair .E; H / over B consists of a locally trivial principal T n -bundleE ! B and a gerbe H ! E with band T jE . An isomorphism of pairs is a diagramH H0EB E0Bconsisting of an isomorphism of T n -principal bundles and an isomorphism ofT-banded gerbes . By P.B/ we denote the set of isomorphism classes of pairsover B. Iff W B ! B 0 is a continuous map, then pull-back induces a functorial mapP.f/W P.B 0 / ! P.B/.6.1.2 The functorP W TOP op ! Setshas been introduced in [BS05] for n D 1 and in [BRS] for n 1 in connection withthe study of topological T -duality.The main result of [BS05] in the case n D 1 is the construction of an involutivenatural isomorphism T W P ! P , the T -duality isomorphism, which associates to eachisomorphism class of pairs t 2 P.B/ the class of the T -dual pair Ot WD T.t/2 P.B/.In the higher-dimensional case n 2 the situation is more complicated. First ofall, not every pair t 2 P.B/ admits a T -dual. Furthermore, a T -dual, if it exists, maynot be unique. The results of [BRS] are based on the notion of a T -duality triple. Inthe following we recall the definition of a T -duality triple.6.1.3 As a preparation we recall the following two facts. Let W E ! B be aT n -principal bundle. Associated to the decomposition of the global section functor.E;:::/ as composition .E;:::/ D .B;:::/ı W Sh Ab S=E ! Ab there is adecreasing filtrationF n H .EI Z/ F n 1 H .EI Z/ F 0 H .EI Z/of the cohomology groups H .EI Z/ Š R .EI Z/ and a Serre spectral sequencewith second term E i;j2Š H i .BI R j Z/ which converges to GrH .EI Z/.
324 U. Bunke, T. Schick, M. Spitzweck, and A. Thom6.1.4 The isomorphism classes of gerbes over X with band T jX are classified byH 2 .XI T/ Š H 3 .XI Z/. The class associated to such a gerbe H ! X is calledDixmier–Douady class d.X/ 2 H 3 .XI Z/.If H ! X is a gerbe with band T jX over some space X, then the automorphismsof H (as a gerbe with band T jX ) are classified by H 1 .XI T/ Š H 2 .XI Z/.6.1.5 In the definition of a T -duality triple we furthermore need the following notation.We let y 2 H 1 .TI Z/ be the canonical generator. If pr i W T n ! T is the projectiononto the ith factor, then we set y i WD pr i y 2 H 1 .T n I Z/. Let E ! B be a T n -principal bundle and b 2 B. We consider its fibre E b over b. Choosing a base pointe 2 E b we use the T n -action in order to fix a homeomorphism a e W E ! T n such thata e .et/ D t for all t 2 T n . The classes x i WD a e .y i/ 2 H 1 .E b I Z/ are independentof the choice of the base point. Applying this definition to the bundle yE ! B belowgives the classes Ox i 2 H 1 . yE b I Z/ used in Definition 6.2.Definition 6.2. A T -duality triple t WD ..E; H /; . yE; yH/;u/ over B consists of twopairs .E; H /; . yE; yH/ over B and an isomorphism uW Op yH ! p H of gerbes withband T jEBdefined by the diagramyEp H uOp yH H E ByE p E O B,Op yEyH(49)where all squares are two-cartesian. The following conditions are required:1. The Dixmier–Douady classes of the gerbes satisfy d.H/ 2 F 2 H 3 .EI Z/ andd. yH/ 2 F 2 H 3 . yEI Z/.2. The isomorphism of gerbes u satisfies the condition [BRS, (2.7)] which says thefollowing. If we restrict the diagram (49) to a point b 2 B, then we can trivializethe restrictions of gerbes H jEb , yH j yE bto the fibres E b ; yE b of the T n -bundlesover b such that the induced isomorphism of trivial gerbes u b over E b yE b isclassified by P niD1 pr E bx i [ pr yE Ox i 2 H 2 .E b yE b I Z/.6.1.6 There is a natural notion of an isomorphism of T -duality triples. For a mapf W B 0 ! B and a T -duality triple over B there is a natural construction of a pull-backtriple over B 0 .
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EMS Series of Congress ReportsEMS S
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Editors:Guillermo CortiñasDepartam
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ContentsPreface....................
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IntroductionSince its inception 50
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Introductionxiwith D. Blecher, he c
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Program list of speakers and topics
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Categorical aspects of bivariant K-
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72 H. Emerson and R. MeyerIn this s
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74 H. Emerson and R. MeyerK-theoret
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76 H. Emerson and R. MeyerExample 4
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78 H. Emerson and R. Meyerand exact
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80 H. Emerson and R. Meyercondition
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82 H. Emerson and R. MeyerAs a resu
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84 H. Emerson and R. MeyerLet h and
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86 H. Emerson and R. MeyerCorollary
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88 H. Emerson and R. Meyer5 Dual-Di
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On K 1 of a Waldhausen categoryFern
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On K 1 of a Waldhausen category 992
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118 M. Karoubi[35], M. F. Atiyah an
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120 M. Karoubihere by GrK n .A/, ar
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122 M. Karoubi(resp. 1) in A. In th
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124 M. Karoubithe class ˛ of A in
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126 M. Karoubiested in the complex
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128 M. KaroubiThe same method may b
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130 M. KaroubiBy the usual excision
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132 M. KaroubiLet A be a graded twi
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134 M. Karoubithe same ideas as in
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136 M. KaroubiTheorem 6.2. 25 The (
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138 M. KaroubiIn order to show this
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140 M. KaroubiThe following theorem
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142 M. KaroubiWe would like to poin
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144 M. KaroubiZ=n identified with t
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146 M. Karoubiwhere n is the ring
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148 M. Karoubi[6] M. F. Atiyah, R.
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Equivariant cyclic homology for qua
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174 C. Voigtand using ˇ.x; y/ D .S
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176 C. Voigtalgebra A. The space Cn
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178 C. VoigtKK-theory [16]. Hence,
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A Schwartz type algebra for the tan
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202 J. CuntzWe denote by N the set
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204 J. CuntzConsider the action of
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206 J. Cuntzprime numbers p and q,
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208 J. Cuntz6 Representations as cr
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210 J. CuntzProof. The spectral pro
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212 J. CuntzThe operator s 1 plays
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214 J. CuntzProposition 7.4. The K-
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On a class of Hilbert C*-manifoldsW
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On a class of Hilbert C*-manifolds
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Deformations of gerbes on smooth ma
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Parshin’s conjecture revisitedTho
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428 C. WeibelGiven (0.4) and axiom
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430 C. WeibelAs pointed out in the
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432 C. WeibelConsider the cohomolog
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434 C. WeibelWe need to see that th
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List of contributorsArthur Bartels,
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List of participantsNatalia Abad Sa
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