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.
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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. |
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