to $(M_1,f_1, ,   \sigma_1 )$ if there exists a cycle $(W, f,  \sigma )$, where $W$ is a $G$-compact, orientable and ${\operatorname{Spin}}(c)$-manifold with boundary, $f: W\to X$ is a $G$-equivariant map, and $\sigma:W\to {\ensuremath{{\mathrm{Fred}}}}(- f^*(P))$ is the homotopy class of a $G$-invariant section, with the property that

    $$(\partial  W, f\mid_{\partial W}, \sigma  )= (M_0,f_0,  \sigma_0 )\coprod (-M_1, f_1,  \sigma_1)$$

    Where $(-M_1, f_1, \sigma_1)$ has the reversed orientation with respect to $M_1$.

3.  Vector bundle modification. Given a geometric cycle $(M,   f,  \sigma )$, and an even dimensional real vector bundle with a $G$-equivariant ${\operatorname{Spin}}(c)$-structure $F\to  M$, denote by $\pi:S(F\oplus  {{\mathbb R}})\to  M $ the unit sphere bundle of $F\oplus  {{\mathbb R}}$. We consider the map $S_{M,F}:M\to  S(F\oplus  {{\mathbb R}})$ given by the $1$-section and let $S_{M,F}!:K_G^0(M, -f^*(P)) \longrightarrow  K_G^0(S(F\oplus  {{\mathbb R}}), S_{M,F}^*(-f^*(P)))   $ be the pushforward homomorphism.

    Then, the geometric cycles $(M, f,  \sigma )$ and $S(F\oplus {{\mathbb R}}),\pi\circ  f,   S_{M,F}!(\sigma))$ are equivalent.

    We will introduce the notation $E\bigstar  (M,f, \sigma)$ for the class of the modification of the geometric cycle $(M,f, \sigma)$ with respect to the ${\operatorname{Spin}}(c)$-vector bundle $E\to M$.

    Similarly, we will denote by $E\bigstar M$ the manifold given as the total space of the spherical bundle $S(E\oplus 