disjoint union of the Proper $G$-CW complexes $X_\alpha$. Then, the inclusions $X_\alpha\to  X$ induce an isomorphism of abelian groups

    $$\bigoplus_{\alpha \in A }{K^{\rm geo}_ {G,j}({X_\alpha},{P\mid X_\alpha})}\cong  {K^{\rm geo}_ {G,j}({X},{P})}$$

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1.  Associate to the geometric cycle $(M, \sigma, f)$ for the pair $(X,A)$ the cycle $(M, \psi^*(\sigma) , \psi\circ f)$.

2.  It follows from the bordism relation.

3.  We will check exactness of the sequence at ${K^{\rm geo}_ {G,0}({X},{P})}$. Let $(M, f, \sigma )$ be a geometric cycle for the pair $(X,A)$ which maps to $0$ under the map $\delta$. According to Lemma \[lemmanormalbordism\], there exists a real, orientable and ${\operatorname{Spin}}(c)$ $G$-vector bundle $E$ of even dimension over $\partial M$ such that the geometric cycle $E\bigstar (\partial M, f\mid_{\partial M}, \sigma\mid_{\partial M})$ is twisted null bordant. We can assume that $E$ is defined on all of $M$, otherwise using lemma 3.7 of [@lueckoliverbundles], page 599.

    Consider a geometric cycle $(W,F,\Sigma)$, where $W$ is such that $\partial W = E \bigstar \partial M $, $F\mid_{\partial W}= \pi_{E}\circ  f$, $ \Sigma \mid_{\partial W}= \sigma\mid_{\partial M}$.

    Form the manifold $  \hat{W}:=W\cup_{\partial W= \partial M } E\bigstar M $, where the inclusion of $\partial  M$ into the second factor is given as composition 