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On K 1 of a Waldhausen category 95(R6) For any pair of composable weak equivalences A ! B ! C ,ŒA ! C D ŒB ! CC ŒA ! B:(R7) For any commutative diagram in C as followsA BB=A A 0 B0 B 0 =A 0we haveŒA ! A 0 C ŒB=A ! B 0 =A 0 ŒA D ŒA 0 B 0 B 0 =A 0 C ŒB ! B 0 C ŒA B B=A:(R8) For any commutative diagram consisting of four cofiber sequences in C as followsC=BB=A C=AA B Cwe haveŒB C C=BC ŒA B B=AD ŒA C C=AC ŒB=A C=A C=B ŒA :(R9) For any pair of objects A; B in CHerehŒA; ŒBi D ŒB i 2 A _ B p 1 A C ŒA i 1 A _ B p 2 B:A i 1 A _ Bp 1are the inclusions and projections of a coproduct in C.Remark 1.4. These relations are quite natural, and some illustration of their meaningis given in [4, Figure 2]. They are not however minimal: relation (R3) follows from(R2) and (R5), and (R4) follows from (R6). Also (R5) is equivalent toi 2p 2B
96 F. Muro and A. Tonks(R5 0 ) Œ D 0.This equivalence follows from (R8) on considering the diagrams A 1 A A 1 AA,A1 A A1 A A.We now introduce the basic object of study of this paper.Definition 1.5. A weak cofiber sequence in a Waldhausen category C is a diagramA B C 1Cgiven by a cofiber sequence followed by a weak equivalence in the opposite direction.We associate the elementŒA B C 1C D ŒA B C1 C ŒC ! C 1 ŒA 2 D C (1.1)to any weak cofiber sequence. By (R1) and (R2) we have@ŒA B C 1C D ŒB C ŒC C ŒA:The following fundamental identity for weak cofiber sequences is a slightly lesstrivial consequence of the relations (R1)–(R9).Proposition 1.6. Consider a commutative diagramA0 j AAr AA 000w AA 00j 0 B 0 j BjBr Bj 000 B 000w Bj 00B 00 r 00r 0 D 0 w 0 C 0 rr 000 w 000j D rD D D000 D 00j Cw Cr CC 000 w Dw 00 C 00w C
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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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Page 91 and 92: 76 H. Emerson and R. MeyerExample 4
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Page 97 and 98: 82 H. Emerson and R. MeyerAs a resu
Page 99 and 100: 84 H. Emerson and R. MeyerLet h and
Page 101 and 102: 86 H. Emerson and R. MeyerCorollary
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Page 143 and 144: 128 M. KaroubiThe same method may b
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Page 151 and 152: 136 M. KaroubiTheorem 6.2. 25 The (
Page 153 and 154: 138 M. KaroubiIn order to show this
Page 155 and 156: 140 M. KaroubiThe following theorem
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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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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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