Is there a notion of a cobordism which is compatible with bundle structure?


That is, if I have bundles $E$ and $F$, is it the case that the manifold $W$ with $E$ and $F$ as boundary components, can be made into a bundle whose bundle structure, when restricted to $E$ or $F$, is the bundle structure of $E$ or $F$.


And, particularly, when can I connect $E$ and $F$ this way (not just when they're cobordant, but when this cobordism is compatable with this structure)? And what can I say about the bundle structure of $W$, knowing what $E$ and $F$ look like? (e.g., if $E$ and $F$ are G-bundles what can I say about the group action on $W$?)


Also, can anyone point me to any particular references which discuss this? I spent a few hours in our (fairly small) math library looking for something like this, but haven't been able to find anything that seems to discuss this. But I may just not know the right catch phrases to search for!



Cobordisms of bundles?
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 $f$ and $g$ 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 $\mathfrak{N}\_n(BG) = \oplus H\_j(BG) \otimes \mathfrak{N}\_{n-j}$ which can help identify some bundles' classes.


