proper actions of a discrete group.

\[theoremhomologicproperties\] The groups ${K^{\rm geo}_ {G,j}({X,A},{P})}$ satisfy the following properties.

1.  (Functoriality). A $G$-equivariant map of pairs $\psi:(X,A)\to (Y,B)$ induces a group homomorphism $ \psi^*:  {K^{\rm geo}_ {G,j}({X,A},{\psi^* P})} \to {K^{\rm geo}_ {G,j}({Y,B},{P})} $

2.  (Homotopy invariance). Two $G$-equivariantly homotopic maps $\psi_0:(X,A)\to (Y,B)$ and $\psi_1:(X,A)\to (Y,B)$ induce the same group homomorphism $\psi:=\psi_0= \psi_1: {K^{\rm geo}_ {G,j}({X,A},{P})}\to {K^{\rm geo}_ {G,j}({Y,B},{\psi^*(P)})}$.

3.  (Six term exact sequence). For a proper $G$-CW pair $(X,A)$, the following sequence is exact:

    $$\xymatrix{ {K^{\rm geo}_ {G,0}({X,A},{P})} \ar[r]_-{\delta} & {K^{\rm geo}_ {G,1}({A},{P})} \ar[d]_-{i^ *} \\ {K^{\rm geo}_ {G,0}({X},{P})}  \ar[u]_-{j_*} & {K^{\rm geo}_ {G,1}({X},{P})} \ar[d]_-{j_*} \\ {K^{\rm geo}_ {G,0}({A},{i^*P})} \ar[u]_-{i_*} &  {K^{\rm geo}_ {G,1}({X,A},{P})} \ar[l]_-{\delta}  },$$ where the maps $i_*$, $j_*$ are induced by the maps of proper $G$-CW pairs $i: A\to X$ and $j: X:= (X,\emptyset)\to (X,A)$ .

4.  (Excision). Given a $G$-CW pair $(X,A)$ and a $G$-invariant, open set $U$ with the property that the closure of $U$ is contained in the interior of $A$, the inclusion $l:(X-U, A-U)\to (X,A)$ induces an isomorphism of abelian groups $${K^{\rm geo}_ {G,j}({X,A},{P})} \to{K^{\rm geo}_ {G,j}({X-U,A-U},{l^{*}(P)})}$$

5.  (Disjoint union Axiom). Let $X=\underset{\alpha\in A} {\coprod} X_\alpha$ be a proper $G$-CW complex which is obtained as the 