process

Coarse and equivariant co-assembly maps 81If, in addition, D D C and the action of H on X is free, then we can further simplifythis toK topC1H; credH .X/ @KXH .X/KK G C;C 0.X=H / p EG RKK G EGI C;C 0.X=H / .We may also specialise the space X to jGj, with G acting by multiplication on theleft, and with H G a compact subgroup acting on jGj by right multiplication. Thisis the special case of (13) that is used in [7]. The following applications will requireother choices of X.3.1 Applications to Lipschitz classes. Now we use Lemma 7 to construct interestingelements in KK G C;C 0.X/ for a G-space X. This is related to the method of Lipschitzmaps developed by Connes, Gromov and Moscovici in [3].3.1.1 Pulled-back coarse structures. Let X be a G-space, let Y be a coarse spaceand let ˛ W X ! Y be a proper continuous map. We pull back the coarse structureon Y to a coarse structure on X, letting E X X be an entourage if and only if˛.E/ Y Y is one. Since ˛ is proper and continuous, this coarse structure iscompatible with the topology on X. For this coarse structure, G acts by translationsif and only if ˛ satisfies the following displacement condition used in [3]: for anycompact subset K G, the set˚˛.gx/; ˛.x/2 Y Y j x 2 X; g 2 K(15)is an entourage of Y . The map ˛ becomes a coarse map. Hence we obtain a commutingdiagramK C1 c red .Y / ˛ K C1 c red .X/ C;C 0.X/ KK G @ YKX .Y /˛@ X KX .X/p EG RKK G EGI C;C 0.X/ :with and as in Lemma 7.The constructions of [3, §I.10] only use Y D R N with the Euclidean coarse structure.The coarse co-assembly map is an isomorphism for R N because R N is scalable.Moreover, R N is uniformly contractible and has bounded geometry. Hence we obtaincanonical isomorphismsK C1 c red .R N / Š KX .R N / Š K .R N /:In particular, K C1 c red .R N / Š Z with generator Œ@R N in K 1 N c red .R N / . Thisclass is nothing but the usual dual-Dirac morphism for the locally compact group R N .