{{\mathbb R}}$ is $G$-equivariantly diffeomorphic to the boundary of the Ball bundle in $\nu$. Consider the manifold with boundary $W= D(E\oplus {{\mathbb R}})- D_\epsilon(\nu)$. This is a proper, ${\operatorname{Spin}}(c)$-manifold with boundary, which is a bordism between $F\bigstar (E\bigstar M)$ and $(s^*(F)\oplus  E)\bigstar M$.

Given a geometric cycle $(M, f, \sigma)$ and real orientable, and ${\operatorname{Spin}}(c)$-vector bundles $ \pi: E\to  M$ and $F\to  S(E\oplus   {{\mathbb R}}) $,

Denote by $\rho_E:E\bigstar M \to M$ and $\rho_F: F\bigstar (E\bigstar M)\to  E\bigstar  M$ the projection maps. Let $S_{M, E}: M\to E\bigstar M$ and $S_{M, S^*(F)\oplus  {{\mathbb R}}}:M \to S^*F\oplus E \bigstar M $ be the $1$-sections into the spherical bundles.

Then, the geometric cycles $$\big (  F\bigstar(E\bigstar M), f\circ \rho_E \circ \rho_F,  S_{M,E}!\circ S_{E\bigstar M, F}! (\sigma)\big )$$ and $$\big ( (S^*(F) \oplus E )\bigstar M, f\circ \rho_{S^*(F)\oplus  F},  S_{M, S^*(F)\oplus {{\mathbb R}}} !(\sigma) \big )$$ are equivalent.

Addition on the set of geometric cycles is defined as disjoint union: $$( M, \sigma, f)+ (M^{'} \sigma^{'}, f^{'})=  ( M, \sigma, f)\coprod (M^{'} \sigma^{'}, f^{'}),$$ and turns the equivalence classes into an abelian group.

Let $X$ be a proper $G$-CW complex and a projective unitary stable $G$-equivariant bundle $P$ over $X$. We will denote by 