of the pushforward homomorphism constructed in [@barcenascarrillovelasquez], and ideas ultimately originated in geometric pictures of $K$-homology [@baumhigsonschickproper], [@baumcareywang].

We compare the cycle version using an index map to an equivariant Kasparov $KK$-Theory group. We also introduce some computational tools, notably a spectral sequence to compute the equivariant $K$-homology groups described here and we compare the twisted equivariant $K$-homology invariants of the classifying space for proper actions to the topological $K$-theory of twisted versions of the reduced $C^*$ algebra of a discrete group.

This work is organized as follows:

In section \[sectionreview\], we recall definitions, methods, and results related to twisted equivariant $K$-Theory for proper actions [@barcenasespinozajoachimuribe], [@barcenasespinozauribevelasquez], [@barcenascarrillovelasquez].

In section \[sectiongeometricapproach\], geometric cycles for twisted equivariant $K$-homology are introduced, as well as appropriate equivalence relations (bordism, isomorphism and vector bundle modification). It is proved in Theorem \[theoremhomologicproperties\], that the geometric Twisted Equivariant $K$-Homology groups satisfy dual homological properties to the twisted equivariant $K$-Theory groups in [@barcenasespinozajoachimuribe]. In that section computational methods, including a spectral sequence abutting to Twisted Equivariant $K$-Homology are addressed.

In section \[analyticalkhom\], the construction is compared to equivariant Kasparov $KK$-groups via the definition of an index map, which is proved in Theorem \[theoremcoincidencekhomology\] to give an isomorphism 