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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!

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I think this question needs clarification. What do you mean by "bundle"? I assume you mean "(locally trivial) fiber bundle". What do you mean by "boundary"? There are two possibilities: (1) Given a bundle $E$ over a manifold $M$, by boundary of $E$ you mean the bundle $E$ restricted to the boundary $\partial M$. (2) Given a bundle $E$ over a manifold $M$ with fiber $F$, by boundary you mean the corresponding bundle over $M$ with fiber $\partial F$. –  Kevin H. Lin Jan 18 '10 at 22:49
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2 Answers

up vote 4 down vote accepted

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.

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Yeah, this seems to be along the lines of what I was thinking. Although the details of this argument aren't quite clear to me. Can you reference a (preferably) textbook, or a paper that has a little bit of background on where this comes from? –  jeremy Jan 19 '10 at 0:52
For the general argument pairing homotopy classes of maps into $BG$ with principal $G$-bundles, the Wikipedia article for "universal bundle" seems nice enough. For the splitting of bordism groups, my source is Differentiable periodic maps by P. E. Conner and E. E. Floyd, Bull. Amer. Math. Soc. Volume 68, Number 2 (1962), 76-86. ; can be reached via Project Euclid (projecteuclid.org). Is there anything specific that you would like me to address? –  Thorny Jan 19 '10 at 9:46
Well I'm not all that familiar with classifying spaces, or bordism groups. So I have difficulty seeing how $W$ is related to the bordism group! Most of the references I find for this are papers which assume a fairly significant background that I don't have. I was hoping to find a textbook so I could get a reasonably self-contained description. Since these things certainly aren't in my diff. geom. or alg. top. books! The most common claims I seem to be finding are that this is part of "generalized cohomology" which is something I'm completely unfamilliar with. –  jeremy Jan 19 '10 at 13:49
And, further, looking at more references, I am finding difficulty seeing anything concerning $W$. Almost everything I find is about using cobordisms to classify manifolds, and seems to completely ignore the manifold $W$. Perhaps the copious amounts of category theory I run across are making an obvious statement completely opaque to me, but I really am just looking for understanding the allowable (principal) bundle structures on $W$ and anything about $W$ seems very unclear to me from the few references I can find. –  jeremy Jan 20 '10 at 4:09
The bundle structure over $W$ is described up to isomorphism by the homotopy class of the classifying map $W \to BG$, which is why you only need a bordism between the two classifying maps of the boundary in $BG$; this is why $W$ is never mentioned explicitely. If you want a concrete $W$, I can only produce one once the two boundary conditions are in the form compatible with the splitting I referenced, then it is easy. As to the parametrization of the choice of $W$, just notice that any two choices $W_0$ and $W_1$ can be glued along the common boundary, and that will be the identification. –  Thorny Jan 20 '10 at 8:41
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See Daccach and Pergher, Splitting vector bundles up to cobordism, 1985.

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Maybe I don't have enough background here, but this paper doesn't seem especially enlightening. Its introduction claims it wants to find the conditions for a $k m$ dimensional vector bundle to be cobordant to a Whitney sum of $k$ copies of an $m$-dimensional bundle. Which is not quite what I'm thinking of? The paper also seems to have only two references, neither of which appear to be very helpful... But what I am interested in is the bundle structure on $W$ where $\partial W = E \cup F$. –  jeremy Jan 18 '10 at 4:21
In §2 they talk about cobordism of vector bundles (two m-dimensional vector bundles on two manifolds of dimension n are said to be cobordant if there is a vector bundle on an n+1 dimensional manifold which restricts on the boundary to the disjoint union of the former bundles). They say that the resulting cobordism group is isomorphic to the n-th dimensional cobordism group of the classifying space BO(m) -- compare Thorny's answer. This seemed close enough to your question, but maybe I misunderstood? –  Michael Jan 18 '10 at 9:58
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