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It was an important problem of topology to determine for which dimensions the Hopf invariant was one. There are several clear expositions giving the definition of the Hopf invariant including the Wikipedia article in the link of this post. However none give any insight into why the Hopf invariant is useful.

What information about the spheres $S^n, S^m$ is contained in knowing that the Hopf invariant of $\phi : S^n \to S^m$ is zero or one?

snailspace
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    It proves, for example, that none of the spheres except $S^1, S^3, S^5,$ and $S^7$ can be Lie groups, or parallelizable, or H-spaces. (The first two were known previously by other means, but this is a much stronger fact). There is also a many-step program called the 'Adams spectral sequence' which computes the homotopy groups of spheres, and the resolution to the Hopf invariant one problem tells you something about lots of the steps... (There must be a better way to explain that last bit. If you know what a spectral sequence is, I'm trying to say that this provides several permanent cycles – Dylan Wilson Jun 13 '13 at 14:12
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    and implies the existence of infinitely many differentials.) – Dylan Wilson Jun 13 '13 at 14:13

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Helpfully in the original Hopf invariant 1 paper, Adams provides this nice diagram of implications.

Adams Image

Drew
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