{{\mathbb R}})$ over $M$.

The following result, generalization of the “Composition Lemma” on page 81 of [@baumcareywang] in the *non-equivariant* setting, will be useful to study interactions between bordism and vector bundle modification:

\[lemmanormalbordism\] Let $M$ be a proper, $G$-compact, smooth manifold, consider a $G$-equivariant, ${\operatorname{Spin}}(c)$, smooth real vector bundle of even dimensional fibers, $\pi: E\to M$. Let $s:M\to S(E\oplus  {{\mathbb R}})$ be the unit section.

Given any real, even dimensional smooth vector bundle $F$ with a ${\operatorname{Spin}}(c)$-structure over $S(E\oplus  {{\mathbb R}})$, the proper $G$-manifolds $$F\bigstar (E\bigstar M),$$ and $$(s^*(F)\oplus E) \bigstar M$$ are bordant.

Pick up a $G$-invariant, hermitian metric on $E\oplus  {{\mathbb R}}$. This is possible because the action on $M$ is proper. Let $D(E\oplus {{\mathbb R}})$ be the unit Ball bundle therein. Notice that the boundary of $D(E\oplus {{\mathbb R}})$ is $E\bigstar M$.

Consider the map $i: M \to  D(F\oplus  {{\mathbb R}})$ given by the composition of the 1-section $M \to  S(E\oplus {{\mathbb R}})$ and the $0$-section $S(E\oplus {{\mathbb R}})\to D(F\oplus {{\mathbb R}})$.

Let $\nu$ be the normal bundle to this inclusion, and consider the ball bundle inside the normal bundle, denoted by $D_\epsilon( \nu)$ for $\epsilon < \frac{1}{4}$. Then, the sphere bundle of $s^*(F)\oplus E  \oplus  