cycle $$(M_1, f\mid_{M_{1}},  \sigma\mid_{M_1})$$ for the pair $(X-U, A-U)$. This construction gives an inverse to the map $l_*$.

5.  This follows easily from manipulations on the components of the manifolds involved in the geometric cocycles.

Given discrete groups $H$ and $G$, a group homomorphism $\alpha: H\to G$, and a proper $H$-CW pair $(X,A)$, the induced space ${\rm ind}_{\alpha}X,$ is defined to be the $G$-CW complex defined as the quotient space obtained from $G\times X $ by the right $H$-action given by $(g,x)\cdot h =( g\alpha(h),h^{-1}x)$. The following result summarizes the behaviour of the geometric model for twisted equivariant $K$-homology with respect to induction and restriction procedures on spaces. Recall the existence of the induction homomorphisms in Borel cohomology ${\rm ind}_{\alpha}:H^3( {\rm ind}_{\alpha} X\times_G EG, {{\mathbb Z}}) \to   H^3(X\times_H EH, {{\mathbb Z}})$ and in Twisted Equivariant $K$-Theory $ {\rm ind}_{\alpha}:  K_{G}^*({\rm ind}_{\alpha} X, {\rm  ind }_{\alpha }P) \to  K_{H}^*(X, P)$ [@barcenasespinozajoachimuribe], (Lemma 5.4 in page 1347).

\[lemmainduction\] There exist natural group homomorphisms $${\rm ind}_{\alpha}:K^{{\rm geo}}_{H, j}(X,A)\longrightarrow K^{{\rm geo}}_{G, j}({\rm ind}_{\alpha}(X,A), {\rm ind}_{\alpha }P)$$ associated with a homomorphism $\alpha:H\to G$ which satisfy the following conditions

1.  [${\operatorname{ind}}_{\alpha}$ is an isomorphism whenever $\ker \alpha$ acts freely on $X$.]{}

2.  [For any $j$, $\partial_{n}^{G}\circ 