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I am trying to finish an exercise on Hatcher's Algebraic Topology, it is at page 428, stated as follows:

Show that if $S^k\to S^m\to S^n$ is a fiber bundle, then $m = 2n-1, k =n-1$ and if n > 1 the map $S^m\to S^n$ has Hopf Invariant $\pm 1$.

I have finished part of this, that is, $m = 2n-1,k = n-1$, by a spectral sequence argument. However, I have no idea calculating the Hopf Invariant. The hint says that I may apply poincare duality but I do not know how.

Thanks for any help!

北方火柴
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  • Perhaps I am stating the obvious, but Hatcher seems to be implying that if you have such a bundle then the cofiber is a manifold. This happens to be true in the case of all the Hopf fibrations that actually exist. – Connor Malin Jul 01 '22 at 17:28

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