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86 H. Emerson and R. MeyerCorollary 11. Let D 2 KK G .P; C/ be a Dirac morphism for G. Then the followingdiagram commutes:K topC1G; credŠ G .jGj/ @ D K .C 0 .EG;P/ Ì G/ K .C 0 .EG/ Ì G/‰ ŠKK G .C; P/ pEG RKK G D .EGI C; P/ Š RKKG .EGI C; C/D pEG.C; C/,KK G where ‰ is as in Theorem 10 and is the G-equivariant coarse co-assembly mapfor jGj, and the indicated maps are isomorphisms.Proof. This follows from Theorem 10 and the general properties of the Dirac morphismdiscussed in §2.1.Definition 12. We call a 2 RKK G .EGI C; C/(a) a boundary class if it lies in the range of W K topC1 G; cred .jGj/ ! RKK G .EGI C; C/I(b) properly factorisable if a D p EG .b ˝A c/ for some proper G-C -algebra A andsome b 2 KK G .C;A/, c 2 KKG .A; C/;(c) proper Lipschitz if a D p EG .b ˝C 0 .X/ c/, where b 2 KK G C;C 0.X// is constructedas in §3.1.1 and §3.1.2 and c 2 KK G .C 0.X/; C/ is arbitrary.Proposition 13. Let G be a totally disconnected group with G-compact EG.(a) A class a 2 RKK G .EGI C; C/ is properly factorisable if and only if it is a boundaryclass.(b) Proper Lipschitz classes are boundary classes.(c) The boundary classes form an ideal in the ring RKK G .EGI C; C/.(d) The class 1 EG is a boundary class if and only if G has a dual-Dirac morphism; inthis case, the G-equivariant coarse co-assembly map is an isomorphism.(e) Any boundary class lies in the range ofp EG W KKG .C; C/ ! RKKG .EGI C; C/and hence yields homotopy invariants for manifolds.ŠŠ