${K^{\rm geo}_ {G}({X},{P})}$ the abelian group of geometric cycles on $X$, divided by the equivalence relation generated by Isomorphism, Bordism and ${\operatorname{Spin}}(c)$-modification.

Denote by ${K^{\rm geo}_ {G,0}({X},{P})}$ the subgroup generated by those cycles $(M,f, \sigma )$ for which each component of $M$ has even dimension. Similarly, denote by ${K^{\rm geo}_ {G,1}({X},{P})}$ the subgroup generated by cycles $(M,f, \sigma)$ for which each component of $M$ has odd dimension.

We introduce now a version for $G$-CW pairs.

Given a proper $G$-CW pair $(X,A)$ and a $G$-equivariant, projective unitary stable bundle $P$ over $X$, consider the triples $(M, \sigma, f)$ consisting of

-   A $G$-compact, proper manifold $M$ (possibly with boundary).

-   A $G$-equivariant map $f: M\to X$ with $f(\partial M)\subset A)$.

-   A homotopy class of a $G$-invariant section $\sigma:  M\to {\ensuremath{{\mathrm{Fred}}}}(-f^*(P))$ representing an element on the twisted equivariant $K$-theory group $$K_G^0(M, -f^*(P)),$$

Define on the set of geometric cycles for $(X,A)$ and $P$ an equivalence relation generated by

-   Isomorphism.

-   Bordism.

-   ${\operatorname{Spin}}(c)$-vector bundle modification.

Denote by ${K^{\rm geo}_ {G,*}({X,A},{P})}$ the set of such equivalence classes.

As before, disjoint union defines an abelian group structure on ${K^{\rm geo}_ {G,*}({X,A},{P})}$. Notice that the decomposition

$${K^{\rm geo}_ {G,*}({X,A},{P})}= {K^{\rm geo}_ {G,0}({X,A},{P})}\bigoplus {K^{\rm 