By Ivan Kolář, Prof. RNDr.; Peter W Michor; Jan Slovák
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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.