more general context of Metalinear structures, but, as observed there, the proof readily applies to the ${\operatorname{Spin}}(c)$-case.

\[Pushforward morphism for proper $G$-manifolds\]\[theopushforward\]

Let $X,Y$ be two proper, smooth ${\operatorname{Spin}}(c)$- $G$- manifolds. Let $P$ be a projective unitary stable $G$-bundle over $Y$. Let $f:X\longrightarrow Y$ be a map such the $G$-vector bundle $T X\oplus f^*TY$ is an equivariant ${\operatorname{Spin}}(c)$-vector bundle of even rank. Then, there exists a natural pushforward homomorphism

$$f!: K_{G}^0(X, f^*(P))\to K_{G}^0(Y, P).$$

Geometric Approach to twisted equivariant K-Homology {#sectiongeometricapproach}
====================================================

We describe now a geometric version of Twisted Equivariant $K$-Homology. The results in this section extend and generalize methods by Baum-Higson-Schick and Baum-Carey-Wang [@baumhigsonschickproper], [@wanggeometriccycles], [@baumcareywang].

Recall that a $G$-CW complex structure on the pair $(X,A)$ consists of a filtration of the $G$-space $X=\cup_{-1\leq n } X_{n}$ with $X_{-1}=\emptyset$, $X_{0}=A$ and every space is inductively obtained from the previous one by attaching cells with pushout diagrams $$\xymatrix{\coprod_{\lambda \in I_n} S^{n-1}\times G/H_{\lambda} \ar[r] \ar[d] & X_{n-1} \ar[d] \\ 
  \coprod_{\lambda \in I_n}D^{n}\times G/H_{\lambda} \ar[r]& X_{n}.}$$

The set $I_n$ is called the set of $n$-cells. A $G$-CW complex is finite if it is obtained out of finitely many orbits of cells $S^n\times  G/H$ in a finite number of dimensions $n$.

This 