process

82 H. Emerson and R. MeyerAs a result, any map ˛ W X ! R N that satisfies the displacement condition aboveinducesŒ˛ WD ˛Œ@R N 2 KK G N C;C 0.X/ :The commutative diagram (13) computes p EG Œ˛ 2 RKKG NEGI C;C 0.X/ in purelytopological terms.3.1.2 Principal bundles over coarse spaces. As in [3], we may replace a fixed mapX ! R N by a section of a vector bundle over X. But we need this bundle tohave a G-equivariant spin structure. To encode this, we consider a G-equivariantSpin.N /-principal bundle W E ! B together with actions of G on E and B suchthat is G-equivariant and the action on E commutes with the action of H WD Spin.N /.Let T WD E Spin.N / R N be the associated vector bundle over B. It carries a G-invariantEuclidean metric and spin structure. As is well-known, sections ˛ W B ! Tcorrespond bijectively to Spin.N /-equivariant maps ˛0 W E ! R N ; here a section ˛corresponds to the map ˛0 W E ! R N that sends y 2 E to the coordinates of ˛.y/ inthe orthogonal frame described by y. Since the group Spin.N / is compact, the map ˛0is proper if and only if b 7! k˛.b/k is a proper function on B.As in §3.1.1, a Spin.N /-equivariant proper continuous map ˛0 W E ! Y for a coarsespace Y allows us to pull back the coarse structure of Y to E; then Spin.N / acts byisometries. The group G acts by translations if and only if ˛0 satisfies the displacementcondition from §3.1.1. If Y D R N , we can rewrite this in terms of ˛ W B ! T :weneedsupfkg˛.g 1 b/ ˛.b/kjb 2 B; g 2 Kgto be bounded for all compact subsets K G.If the displacement condition holds, then we are in the situation of Lemma 7 withH D Spin.N / and X D E. Since H acts freely on E, C 0 .E/ Ì H is G-equivariantlyMorita–Rieffel equivalent to C 0 .B/. We obtain canonical mapsK Spin.N /C1c redSpin.N / .RN / .˛0/ ! K Spin.N /C1c redSpin.N / .E/! KK G C;C 0.E/ Ì Spin.N / Š KK G C;C 0.B/ :The Spin.N /-equivariant coarse co-assembly map for R N is an isomorphism by [7]because the group R N Ì Spin.N / has a dual-Dirac morphism. Using also the uniformcontractibility of R N and Spin.N /-equivariant Bott periodicity, we getK Spin.N /C1c redSpin.N / .RN / Š KX Spin.N / .RN / Š K Spin.N / .RN / Š K CNSpin.N / .point/:The class of the trivial representation in Rep.SpinN/ Š K Spin.N /0.C/ is mapped to theusual dual-Dirac morphism Œ@R N 2 K Spin.N /1 Nc redSpin.N / .RN / for R N . As a result, anyproper section ˛ W B ! T satisfying the displacement condition inducesŒ˛ WD ˛Œ@R N 2 KK G N C;C 0.B/ :