Introduction to the Baum-Connes conjecture by Alain Valette

By Alain Valette

The Baum-Connes conjecture is a part of A. Connes' non-commutative geometry programme. it may be seen as a conjectural generalisation of the Atiyah-Singer index theorem, to the equivariant surroundings (the ambient manifold isn't really compact, yet a few compactness is restored via a formal, co-compact motion of a bunch "gamma" ). just like the Atiyah-Singer theorem, the Baum-Connes conjecture states in simple terms topological item coincides with a merely analytical one. For a given staff "gamma", the topological item is the equivariant K-homology of the classifying area for correct activities of "gamma", whereas the analytical item is the K-theory of the C*-algebra linked to "gamma" in its average illustration. The Baum-Connes conjecture implies a number of different classical conjectures, starting from differential topology to natural algebra. It has additionally powerful connections with geometric workforce idea, because the evidence of the conjecture for a given staff "gamma" often relies seriously on geometric houses of "gamma". This booklet is meant for graduate scholars and researchers in geometry (commutative or not), staff conception, algebraic topology, harmonic research, and operator algebras. It offers, for the 1st time in e-book shape, an advent to the Baum-Connes conjecture. It begins by way of defining rigorously the items in each side of the conjecture, then the meeting map which connects them. Thereafter it illustrates the most instrument to assault the conjecture (Kasparov's theory), and it concludes with a coarse cartoon of V. Lafforgue's facts of the conjecture for co-compact lattices in in Spn1, SL (3R), and SL (3C).

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If X is locally compact, then the action of Γ on X is proper if and only if for all compact subsets K, L ⊂ X, the set {γ ∈ Γ | γ · K ∩ L = ∅} is finite. 5. A proper Γ-space EΓ is said to be universal if it is metrizable with EΓ/Γ paracompact and if for every proper metrizable Γ-space X with X/Γ paracompact there is a Γ-equivariant continuous map X → EΓ, unique up to Γ-equivariant homotopy. The space EΓ is unique up to Γ-homotopy. 6. (1) If Γ is torsion free, every proper action is free, hence gives a covering with group Γ.

But we want a direct construction of βa , so that βa = µΓi ◦ ι ◦ βt is non trivial. This illustrates what the Baum-Connes map does in small homological degree. The case j = 0: Define βt : Z → RK0 (BΓ) by mapping 1 to [i∗ ], the class of the element of RK0 (BΓ) corresponding to the inclusion of the base point. Define βa : Z → K0 (Cr∗ Γ) by mapping 1 to [1], the K-theory class of the unit. 5 shows that µΓ0 ◦βt = βa . The canonical trace gives a map τ∗ : K0 (Cr∗ Γ) → R and τ∗ ([1]) = 1, that is, βa is injective.

Let J be a Jordan curve in S 2 , i. e. a homeomorphic image of S 1 . We must show that S 2 \ J has two connected components. It follows from the second step that, if M is an open, orientable surface, then K 0 (M ) = Zc , where c is the number of connected components of M . Since S 2 \ J is an open, orientable surface, we must show that K 0 (S 2 \ J) = Z2 . Consider for that the short exact sequence / 0 / C0 (S 2 \ J) / C(S 2 ) / C(J) 0 giving, in K-theory, / K 0 (S 2 \ J) O K 1 (J) o / K 0 (S 2 ) K 0 (J)  K 1 (S 2 ) o K 1 (S 2 \ J) Now J is homeomorphic to S 1 , so K 0 (J) = Z, and K 1 (J) = Z.

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