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Duality for topological abelian group stacks and T -duality 339Using the construction 2.5.12 we can form a quotient D.P / which fits into the sequenceof maps of Picard stacksBT jB ! D.P / ! D.P /;where H 0 .D.P // Š Z jB and H 1 .D.P // Š Z n jB. The fibre productRD.P / ¹1º Z jBdefines a gerbe R ! B with band Z n jB .6.4.6 By Lemma 6.15 there exists a unique isomorphism class yE ! B of aT n -bundle whose Z n jB -gerbe of Rn -reductions R y EZ n is isomorphic to R. We fix suchan isomorphism and obtain a canonical map of stacks can (see (54)) fitting into thediagramyH op zyH D.P /yE can R y EZ nŠRD.P /(69) B ¹1º Z jB .The gerbes yH op ! yE and z yH ! R are defined such that the squares become twocartesian(we omit to write the two-isomorphisms). In this way the Picard stack P 2Q E defines the second pair . yE; yH/ of the triple ‰.P/ D ..E; H /; . yE; yH/;u/ whoseconstruction has to be completed by providing u.Lemma 6.18. We have the equality Oc.P/ DOc.‰.P// in H 2 .BI Z n jB /.Proof. By the definition in 6.1.12 we have Oc.‰.P// D c. yE/. Furthermore, by (56)Ewe have c. yE/ D d.R y / D d.R/. By Lemma 5.14 we have .D.P// D D..P //,Z n jBwhere is the characteristic class (46), andD W Q E Š Ext 2 Sh Ab S=B .E; T jB / ! Ext 2 Sh Ab S=B .D.T jB /; D.E//is as in Lemma 5.14. The map Oc W Q E! H 2 .BI Z/ by its definition fits into the
340 U. Bunke, T. Schick, M. Spitzweck, and A. ThomdiagramQ ET n jB !EExt 2 Sh Ab S=B .T n jB I T jB /DDExt 2 Sh Ab S=B .D.T jB /; D.E// D.E/!D.T n jB / Ext2Sh Ab S=B .D.T jB /I D.T n jB //aOcŠ Ext 2 Sh Ab S=B .Z jB ; Zn jB /ŠH 2 .BI Z n jB /.By construction the class a.P / 2 Ext 2 Sh Ab S=B .Z jB ; Zn jB/ classifies the Picard stackD.E/. This implies that Oc.P/ classifies the gerbe R ! B, i.e. Oc.P/ D c. yE/.6.4.7 It remains to construct the last entryuW H ˝EB yE y H op ! BT jEB yE(70)of the triple ‰.t/ D ..E; H /; . yE; yH/;u/, where we use Picture 6.1.11. Note thatH 1 .P / Š T jB . Construction 2.5.12 gives rise to a sequence of morphisms of PicardstacksBT jB ! P ! xP(we could write xP D E).We have an evaluation ev W P B D.P / ! BT jB . For a pair .r; s/ 2 Z Z weconsider the following diagram with two-cartesian squares.H r zB H y evsP D.P / BT jBu r;sHzry ˝Er ByR sH sP ˝ xP D.P / D.P /wE r B R s xP D.P /¹r; sº jBZ jB Z jB .
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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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42 A. Bartels, S. Echterhoff, and W
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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 93t
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On K 1 of a Waldhausen category 95(
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On K 1 of a Waldhausen category 97s
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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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228 U. Bunke, T. Schick, M. Spitzwe
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Deformations of gerbes on smooth ma
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Deformations of gerbes on smooth ma
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Torsion classes of finite type and
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Parshin’s conjecture revisitedTho
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Parshin’s conjecture revisited 41
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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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