geo}_ {G,1}({X,A},{P})}$$ holds, where the summands $0$ and $1$ correspond to the subgroup generated by geometric cycles for $(X,A)$ and $P$ given by manifolds of even, respectively odd dimensions.

There exists a boundary map

$$\delta: {K^{\rm geo}_ {G,j}({X,A},{P})}\longrightarrow  {K^{\rm geo}_ {G,j-1}({A},{P})},$$ which is defined as follows.

Given a geometric cycle $(M, \sigma, f)$ for $(X,A)$ and $P$, the cycle $\delta ( M,\sigma, f)$ has as underlying manifold $\partial M$. The map $\partial M \to A$ is given as the composition $\partial M\overset{j}{\to} M \overset{f}{\to} X$, where $j$ denotes the inclusion of $\partial  M$ into $M$. The section $\partial M \to {\ensuremath{{\mathrm{Fred}}}}(- f^*(P))$ is given as the composition of the maps $j:\partial  M \overset{j}{\to} M \overset{f}{\to} {\ensuremath{{\mathrm{Fred}}}}(-f ^*(P))\overset{\mid j}{\to} {\ensuremath{{\mathrm{Fred}}}}(-f^*(P))$, where the map $\mid j$ restricts a bundle with fiber ${\ensuremath{{\mathrm{Fred}}}}(-f^*(P))$ over $M$ to a bundle over $\partial M$.

We introduce an equivalence relation on geometric cycles, which is more technically convenient. This is a generalization of several constructions appearing in the literature [@emersonmeyertriviality], [@baumcareywang] under the name of normal bordism.

\[defnormalvectorbundle\] Let $M$ be a proper, $G$- compact Manifold together with a $G$-equivariant map $f: M \to X$. Let $TM$ be the tangent bundle A normal bundle for $TM$ with 