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I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and structure group $G$ is a

topological sphere bundle if $G=\mathrm{Homeo}^+(S^n)$,

smooth sphere bundle if $G=\mathrm{Diffeo}^+(S^n)$,

linear sphere bundle if $G=\mathrm{Gl}^+(n,\mathbb{R})$.

It is known that there exist smooth sphere bundles that are not equivalent (as smooth sphere bundles) to linear sphere bundles (see e.g. the following posts here on MO):

Examples of sphere bundles

Is it true that all sphere bundles are boundaries of disk bundles?

For example, in an answer to the second question linked above R. Budney shows that there are smooth sphere bundles over $S^2$ which are not linear. Here comes my first question:

  1. Is it possible to construct a smooth sphere bundle over $S^1$ which is not smoothly equivalent to a linear one? One could take an exotic diffeomorphism $f\colon S^n\to S^n$, and consider the mapping cone of $f$, which is a smooth sphere bundle. Is this bundle linear?

The second question is about topological bundles:

  1. Do there exist a topological sphere bundle which is not topologically isomorphic to a linear one (or even to a smooth one)?
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Concerning your second question (topological $\ne$ linear), isn't it what $J$-groups are about? –  Alex Degtyarev Feb 19 at 18:31

1 Answer 1

up vote 6 down vote accepted

Recall that linear, smooth, and topological $S^k$-bundles over a finite complex $X$ are classified by the sets of homotopy classes $[X, BO_{k+1}]$, $[X, B\mathrm{Diff}(S^k)]$ and $[X, B\mathrm{Homeo}(S^k)]$. There are obvious maps between the sets, and we are interested in their cokernels.

For example, if $X=S^1$, then the smooth nonlinear bundles are elements of the cokernel of $\pi_1(BO_{k+1})\to \pi_1(B\mathrm{Diff}(S^k))$. By the exact homotopy sequence the cokernel is isomorphic to $\pi_0(\mathrm{Diff}(S^k)/O_{k+1})$. The space $\mathrm{Diff}(S^k)/O_{k+1}$ is simply $\mathrm{Diff}(D^k, \partial)$, the diffeomorphisms of the disk that are the identity on the boundary. The latter space is classically studied, and by results of Smale and Cerf the group $\pi_0(\mathrm{Diff}(D^k, \partial))$ is isomorphic to the group of homotopy $(k+1)$-spheres, and hence it is often nonzero. Thus there are lots of smooth non-linear bundles over a circle.

EDIT: the crossed out text below is nonsense, and I retract it. What one actually needs is that $\mathrm{Diff}(S^k)\to \mathrm{Homeo}(S^k)$ is not $\pi_i$-surjective for some $i$, and at the moment I do not know any $k, i$ for which this is true.

Non-smoothable sphere bundles over spheres also exist because the space $\mathrm{Homeo}(S^k)/\mathrm{Diff}(S^k)$ is not contractible, for otherwise the inclusion $\mathrm{Diff}(S^k)\to \mathrm{Homeo}(S^k)$ would be a $\pi_i$-isomorphism for all $i$, which it is not for some $k$ (this is due to Alexander's trick, see page 3 of the reference 2 below).

Some references to online sources which you may find useful, and which contains references to the above claims):

  1. A. Hatcher's survey: http://www.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf

  2. P.L. Antonelli, D. Burghelea, P.J. Kahn, http://www.sciencedirect.com/science/article/pii/0040938372900213

  3. D. Crowley, T. Schick: http://front.math.ucdavis.edu/1204.6474

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