Z}})\overset{\cong}{\to} H^2(BH_\sigma , S^1).$$

Denote by $R(\widetilde{H_{P_\sigma}})$ the abelian group of isomorphism classes of complex representations of the group $\widetilde{H_{P_\sigma}}$. Let $R_{S^1}(\widetilde{H_{P_\sigma}})$ be the subgroup generated by the complex representations of $\widetilde{H_{P_\sigma}}$ on which the central subgroup $S^1$ acts by complex multiplication.

\[Coefficients on equivariant contractible covers\]

The restriction of the functors ${K^{\rm geo}_ {G,0}({X},{P})}$ and ${K^{\rm geo}_ {G,1}({X},{P})}$ to the subsets $U_\sigma$ gives covariant functors defined on the category ${\mathcal{N}_{G} \mathcal{U}}$. As abelian groups, the functors ${K^{\rm geo}_ {G,j}({X},{P})}$ satisfy:

$${K^{\rm geo}_ {G,j}({U_\sigma},{P})}\mid_{U_{\sigma}} = \begin{cases} R_{S^{1}}(\widetilde{H_{P_\sigma}})\;  \text{If}\;  j= 0   \\  0 \, \text{If}\;  j= 1    \end{cases}.$$

In degree $0$, a map $ {K^{\rm geo}_ {G,0}({U_\sigma},{P})}\mid_{U_{\sigma}} \to {K^{\rm geo}_ {G,0}({U_\tau},{P})}\mid_{U_{\sigma}}$ is given by induction of representations along group inclusions $\widetilde{H}_\sigma\to  \widetilde{H}_\tau$.

Notice the group isomorphism ${K^{\rm geo}_ {G,j}({U_\sigma},{P})}\mid_{U_{\sigma}}\cong {K^{\rm geo}_ {G,j}({G/H_\sigma},{P})}\mid_{U_{\sigma}}$ due to homotopy invariance, part ii in Theorem \[theoremhomologicproperties\]. The induction structure gives an isomorphism $${K^{\rm geo}_ {G,j}({G/H_\sigma},{P\mid_{U_{\sigma}}})} \cong  {K^{\rm geo}_ {H,j}({{\{\bullet\}}},{\omega_\sigma (P)})}.$$ Consider the $H_\sigma$-trivial Space ${\{\bullet\}}$, and a geometric cycle $$\big (M, f:M\to {\{\bullet\}}, s: M\to {\ensuremath{{\mathrm{Fred}}}}(f^*(\omega_{P_\sigma})) \big ),$$ representing an element in the geometric equivariant $K$-homology group $${K^{\rm geo}_ {H,j}({{\{\bullet\}}},{\omega_\sigma (P)})}.$$ The pushforward map $f_!(s): K_{H_\sigma}^j(M, f^*(\omega_{P_\sigma}))\to K_{H_\sigma}^j({\{\bullet\}}, \omega_{P_\sigma}) $ defined in [@barcenascarrillovelasquez] associated to the 