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I'm interested in examples of sphere bundles which do not arise from vector bundles.

I'm not quite clear about the following. So please let me know if anything is false.

I believe that a $(k-1)$-sphere bundle arises from a vector bundle iff its structure group can be reduced to $O(k)$. I think that for $k\geq 4$ it is not known if $O(k)$ is homotopy equivalent to $\operatorname{Diff}(S^{k-1})$, and in general it is false. So there are sphere bundles that do not arise from vector bundles.

I'm also interested in more details/clarification of this argument.

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    Your belief is correct: Reducing structure group to $O(k)$ is equivalent to picking a bundle metric, and once you have a bundle metric, you can identify the sphere bundle. For sphere bundles that are not (a priori) constructed from a vector bundle by reducing the structure group, try the Hopf fibrations.2012-12-05
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    I think the interest is in sphere bundles which *actually* don't arise from vector bundles. I don't know about the higher Hopf maps, but I'm pretty sure the first one just comes from the tautological bundle over $\mathbb{C}P^1$.2012-12-05
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    Certainly if there is an example then the universal case will do.2012-12-06
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    Yeah but... $\mbox{Diff}(S^k)$ is pretty intractable, so I'd imagine its classifying space is pretty messy too. But yes, I suppose really what we should be asking about are the properties of the map $BO(k+1) \to B \mbox{Diff}(S^k)$ (e.g. relative cohomology groups, since we should be thinking contravariantly -- this is why characteristic classes are cohomology classes, of course).2012-12-07
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    @Aaron I suppose you're right. But it's going to be pretty hard to analyze the map $BO(k+1)\to B\operatorname{Diff}(S^k)$ since we don't really know much about $B\operatorname{Diff}(S^k)$ in general (as you pointed out)... But if I'm not mistaken, for certain values of $k$, it is known that $\operatorname{Diff}(S^k)$ and $O(k+1)$ are not homotopy equivalent (unfortunately i can't find a reference). This implies the existence of sphere bundles not coming from vector bundles, right?2012-12-07
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    Right. Given any space $X$ with a multiplication that is associative up to all possible higher homotopies (in short, $X$ is an $\mathcal{A}_\infty$-space) we can form the classifying space $BX$. Then, $\Omega BX$ will be the "group-completion" of $X$. So if you start with two inequivalent topological groups, they must have inequivalent classifying spaces. In particular, the universal $S^k$-bundle doesn't come from an $O(k+1)$-bundle, or else there would be a classifying map $B\mbox{Diff}(S^k) \to BO(k)$, which would induce a homotopy equivalence for categorical reasons.2012-12-09
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    Oops, I meant $B\mbox{Diff}(S^k) \to BO(k+1)$ of course. @DaveHartman2012-12-09
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    @AaronMazel-Gee : Thank you for your explanation! I'm a bit surprised that no one has posted an example yet. I assumed that many people had thought about my question and that there would be some "well known" examples of such sphere bundles... Might it be possible that no explicit examples are known?2012-12-09
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    Haha. Well, besides the universal example.... I don't know any others, but I'd imagine that plenty are known. You could ask on MO? I'd suggest maybe asking specifically for interesting / enlightening examples, to avoid someone just responding with this. I'd imagine there could be some nice differential geometry here, perhaps some examples will have to do with exotic spheres. (Oh, incidentally I guess an exotic sphere is trivially a sphere bundle over the one-point space, and I'd imagine this may not come from a vector bundle either. But there should be better examples than this.)2012-12-09
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    @AaronMazel-Gee : Right, but I don't really consider the universal example to be "explicit" :). Thanks for the suggestions, I will most likely ask on MO soon if nothing further turns up here. (I'm not quite sure about what follows, and possibly I'm missing something, but here is what I think: An exotic sphere bundle over the one-point space is not really a _smooth_ sphere bundle. But as a topological sphere bundle it is trivial, hence its structure group can be reduced to $\{id\}\subset\operatorname{Homeo}(S^k)$ and thus it comes from a vector bundle.)2012-12-10
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    Yes, I realized that last night. $\mbox{Diff}(S^k_{exotic})$ shouldn't be the same as for a standard sphere, so it's really a different question. (And of course, over the 1-point space there are no transition functions anyways, so you could consider this to be a principal bundle for *any* structure group.) And as for explicitude -- it's all a matter of perspective! I think it's really beautiful to be able to reduce such questions to universal examples, which is what drew me to algebraic topology in the first place. But I can equally well understand wanting to see an *actual* example...2012-12-10
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    @AaronMazel-Gee : I completely agree, it is very beautiful to reduce such questions to the universal examples. Though in this particular cases I wasn't aiming for that. Thanks again for all your comments. (By the way: I have asked on MO just now...)2012-12-10
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    Cool. You should comment in both places that you've cross-posted -- people tend to be very picky about that, so it's worth preempting.2012-12-10
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    @AaronMazel-Gee : Oh, I'll do that. Thanks for the warning.2012-12-10

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