respect to $f$ is a real, $G$-equivariant vector bundle $ NM\to M$ such that $TM\oplus NM$ is isomorphic to a pullback $f^*(F)$, where $F$ is a $G$-equivariant, ${\operatorname{Spin}}(c)$-vector bundle over $X$.

The following result concerns the existence of normal bundles:

\[lemmaexistencenormalbundles\] Let $X$ be a finite, proper $G$-CW complex. Let $M$ be a proper, $G$-compact manifold with a $G$-equivariant map $f:M\to X$ . Then, there exists a normal bundle for $TM$ with respect to $f
$.

This follows from Lemma 3.7 in page 599 of [@lueckoliverbundles], after noticing that the manifold $M$ has a proper $G$-CW structure, which consists of finitely many cells.

\[defgeneralizedbordism\]\[Normal Bordism for cycles\] Two cycles $$(M,f,  \sigma),$$ and $$(M^{'}, f^{'},  \sigma{'})$$ for ${K^{\rm geo}_ {G,*}({X,A},{ P})}$ are said to be normally bordant if there exist finite dimensional, real orientable vector bundles for $TM$ with respect to $f$, denoted by $E\to M$, and a normal bundle for $T M^{'}$ with respect to $f^ {'}$ denoted by $E^{'}\to M^{'}$,

such that the cycles

$$\big (E\bigstar M,\pi_{E}\circ  f,  S_{M, E}!(\sigma)\big )$$

and

$$\big (E^{'}\bigstar M^{'},\pi_{E^{'}}\circ  f^{'},   S_{M^{'},E^{'}}!(\sigma^{'}) \big )$$

are bordant.

Two cycles are normal bordant if and only if they are equivalent.

The implication from normal bordism to 