By Paul Frank Baum, Guillermo Cortiñas, Ralf Meyer, Rubén Sánchez-García, Marco Schlichting, Bertrand Toën

This quantity is an introductory textbook to K-theory, either algebraic and topological, and to numerous present study subject matters in the box, together with Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparability among algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the idea of dg-categories.

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Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation, (2003) MR1954121 (2003m:20032) 47. , vol. 90. Birkh¨auser, Boston, MA (1996) MR1382659 (97c:19001) 48. : Excision in algebraic K-theory. Ann. Math. (2) 136(1), 51–122 (1992) MR1173926 (93i:19006) 49. : K-theory and C∗ -algebras. A friendly approach. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1993) MR1222415 (95c:46116) 50. : The K-book: An introduction to algebraic K-theory (book in progress), Available at , OPTkey = , OPTmonth = , OPTyear = , OPTannote = .

The groups Ktop ∗ (G, A) are supposed to be computable by topological methods. We present one approach to make this precise that works completely within equivariant Kasparov theory and is a special case of a very general machinery for constructing spectral sequences. We carry over notions from homological algebra like exact chain complexes and projective objects to our category and use them to detop fine derived functors (see [27]). The derived functors of K∗ (G r A) and K∗ (G, A) top agree and form the E 2 -term of a spectral sequence that converges towards K∗ (G, A).

Yu [37]; independently V. Lafforgue [28]). If Γ is a discrete group which is hyperbolic (in Gromov’s sense) then the Baum–Connes conjecture is true for Γ . Theorem 9 (Schick [45]). Let Bn be the braid group on n strands, for any positive integer n. Then the Baum– Connes conjecture is true for Bn . K-Theory for Group C∗ -algebras 21 Theorem 10 (Matthey, Oyono–Oyono, Pitsch [35]). Let M be a connected orientable 3-dimensional manifold (possibly with boundary). Let Γ be the fundamental group of M.