equivariant $K$- Theory groups $K^{-p}_G(X, A;P)$ in an analogous way.

Let $G$ be a discrete group. A proper, $G$-compact manifold is a smooth, orientable manifold with an orientation preserving, proper and smooth action of the group $G$, in such a way that the quotient space $G/M$ is compact.

Given a proper $G$-compact manifold $M$ of dimension $n$, consider the tangent bundle $T M$. It is a real $G$-vector bundle over $X$ of rank $n$. Let $F(M)$ be the associated ${\operatorname{SO}}(n)$-principal, $G$-equivariant bundle.

A $G$-spin structure on $M$ is a homotopy class of a reduction of the bundle $F(M)$ to a ${\operatorname{Spin}}(c)$-principal, $G$-equivariant bundle. Given a choice of the ${\operatorname{Spin}}(c)$-structure for $M$ we denote by $-M$ the manifold with the opposite ${\operatorname{Spin}}(c)$-structure.

This is the homotopy class of the reduction of the bundle $F(M)$ to the ${\operatorname{Spin}}(c)$-principal, $G$-equivariant bundle, where the right action of ${\operatorname{Spin}}(c)$ is twisted by an automorphism $\epsilon: {\operatorname{Spin}}(c)\to {\operatorname{Spin}}(c)$ induced from the orientation reversing automorphism $- {\rm  id} : {{\mathbb R}}^n\to {{\mathbb R}}^n$.

We recall one of the main results from [@barcenascarrillovelasquez], obtained using a version of the Thom Isomorphism and suitable deformation index maps in Twisted Equivariant $K$-theory. It is stated there (Proposition 5.18) in the 