Timeline for Cobordisms of bundles?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 20, 2010 at 8:41 | comment | added | Thorny | 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. | |
| Jan 20, 2010 at 4:09 | comment | added | jeremy | 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. | |
| Jan 19, 2010 at 13:49 | comment | added | jeremy | 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. | |
| Jan 19, 2010 at 9:46 | comment | added | Thorny | 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? | |
| Jan 19, 2010 at 0:52 | comment | added | jeremy | 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? | |
| Jan 19, 2010 at 0:49 | vote | accept | jeremy | ||
| Jan 18, 2010 at 8:46 | history | edited | Thorny | CC BY-SA 2.5 |
TeX pains
|
| Jan 18, 2010 at 8:41 | history | answered | Thorny | CC BY-SA 2.5 |