between both constructions in the case of a proper, finite $G$-CW complex. On the way, several results, including versions of $KK$-Theoretical Poincaré Duality [@echterhoffemersonkim], [@echterhoffktheorytwisted] and consequences of the Thom isomorphisms and Pushforward Maps for Twisted Equivariant $K$-Theory of [@barcenascarrillovelasquez] are discussed.

Finally, the relation of the construction depicted below with the $K$-Theory of twisted crossed products [@echterhoffwilliams], [@packerraeburn] is established in section \[Section Crossed Products\].

Acknowledgements
----------------

The author thanks the support of DGAPA-UNAM, and CONACYT research grants.

Parts of this manuscript were written during a visit to Université Toulouse III Paul Sabatier, which was partially founded by the Laboratoire International Solomon Lefschetz.

The author benefited from conversations and joint work with Mario Velásquez and Paulo Carrillo.

Review on Twisted Equivariant $K$-Theory and Pushforward {#sectionreview}
========================================================

Twisted Equivariant K-Theory for proper actions of discrete groups was introduced in [@barcenasespinozajoachimuribe]. We will recall definitions, results and methods from [@barcenasespinozajoachimuribe] and [@barcenascarrillovelasquez].

Let ${\mathcal{H}}$ be a separable Hilbert space and $${\mathcal{U}}({\mathcal{H}}):= \{ U : {\mathcal{H}}\to {\mathcal{H}}\mid U\circ U^*= U^*\circ U = \mbox{Id} \}$$ the group of unitary operators on ${\mathcal{H}}$ with the $*$-strong topology.

Consider the group $P{\mathcal{U}}({\mathcal{H}})$ with the topology determined by the exact sequence $$1\to S^1\to  {\mathcal{U}}({\mathcal{H}})\to P{\mathcal{U}}({\mathcal{H}})\to  1 