I'll assume you're talking about principal G-bundles. These are classified by maps into $BG$, the base of the universal $G$-bundle, so if we have bundles classified by $f:E \to BG$ and $g:F \to BG$, you are looking for a bordism between $f$ and $g$ - whether there exists a $h : W \to BG$ connecting these classifying maps. So there is a bundle cobordism between the two bundles iff the bordism classes of $[f],[g] \f$ and $g$ in \mathfrak{N}n(BG)\mathfrak{N}n(BG)$coincide, and if they do coincide, then the choice of$W$is parametrized by the bordism group$\mathfrak{N}{n+1}(BG)$. \mathfrak{N}_{n+1}(BG)$. I don't know an algorithmic way to obtain the class $[f]$ from $E$, but there is a splitting $\mathfrak{N}_n(BG) = \oplus H_j(BG) \otimes \mathfrak_{n-j}$ mathfrak{N}_{n-j}$which can help identify some bundles' classes. 1 I'll assume you're talking about principal G-bundles. These are classified by maps into$BG$, the base of the universal$G$-bundle, so if we have bundles classified by$f:E \to BG$and$g:F \to BG$, you are looking for a bordism between$f$and$g$- whether there exists a$h : W \to BG$connecting these classifying maps. So there is a bundle cobordism between the two bundles iff the bordism classes$[f],[g] \in \mathfrak{N}n(BG)$coincide, and if they do coincide, then the choice of$W$is parametrized by the bordism group$\mathfrak{N}{n+1}(BG)$. I don't know an algorithmic way to obtain the class$[f]$from$E$, but there is a splitting$\mathfrak{N}_n(BG) = \oplus H_j(BG) \otimes \mathfrak_{n-j}\$ which can help identify some bundles' classes.