---
abstract: 'We define Twisted Equivariant $K$-Homology groups using geometric cycles. We compare them with analytical approaches using Kasparov KK-Theory and (twisted) $C^*$-algebras of groups and groupoids.'
address: |
    Centro de Ciencias Matemáticas. UNAM\
    Ap.Postal 61-3 Xangari. Morelia, Michoacán MÉXICO 58089
author:
- Noé Bárcenas
bibliography:
- 'GEOMETRIC.bib'
title: 'Twisted Geometric K-homology for proper actions of discrete groups. '
---

Introduction
============

In this work, we introduce a geometric notion of twisted equivariant $K$-Homology groups for proper $G$-CW complexes.

Computational evidence [@barcenasvelasquez], [@echterhofflueckphillipswalters], as well as analytical methods related to the Dirac-Dual Dirac method for the Baum-Connes conjecture [@echterhoffchabertgroupextensions], [@echterhoffemersonkim] have suggested the relation of a Twisted Equivariant K-Homology Theory to the topological $K$-Theory of twisted group $C^*$-algebras.

However, the definition of such an invariant has been only given in particular cases [@echterhoffchabertgroupextensions], [@echterhoffktheorytwisted] using Kasparov KK-Theory and rather ad-hoc methods inspired in ideas related the solution of the Connes-Kasparov conjecture by Chabert, Echterhoff and Nest [@echterhoffchabertnest].

We describe twisted equivariant $K$-homology groups using geometric cycles for proper actions of discrete groups on $G$-CW complexes. We extend and generalize results from [@baumhigsonschickproper], [@baumcareywang] to the twisted, respectively equivariant case.

The main methods in this note are a combination of consequences 