In Norman Steenrod's book "The Topology of Fibre Bundles", on page 37, one can find the following conjecture: if $n$ is a power of two then the fibre bundle with the projection $SO(n)\to SO(n)/SO(n-1)=\mathbb S^{n-1}$ is equivalent to the product bundle. Is this conjecture true?
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asked Mar 16, 2017 at 13:22
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6$\begingroup$ Famously, the answer is 'no'. It is now known that this bundle is trivial if and only if $n=1,2,4,8$. $\endgroup$– Robert BryantCommented Mar 16, 2017 at 14:37
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$\begingroup$ Thank you. Could you please provide a reference? $\endgroup$– William of BaskervilleCommented Mar 16, 2017 at 14:52
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4$\begingroup$ There are two papers by J. F. Adams in 1962, both titled "Vector fields on spheres", and they contain the proof of the above statement. They are: "Vector fields on spheres". Topology 1 (1962) 63–65 and "Vector fields on spheres" Ann. of Math. (2) 75 (1962) 603–632. I believe that the latter contains the full details and, in fact, proves much more, determining the maximum number of linearly independent vector fields on the $n$-sphere for all $n$. The former doesn't give quite as complete a solution, but it does contain enough to solve the parallelizability problem on spheres. $\endgroup$– Robert BryantCommented Mar 16, 2017 at 15:56
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