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Hopf invariant
Let \(f: S^{2n-1} \to S^n\), then \(\operatorname{cone}(f)\) has a CW structure \(e^0 \cup e^n \cup e^{2n}\). Let \(H^n(\operatorname{cone}(f)) = \alpha{\mathbf{Z}}\) and \(H^{2n}(\operatorname{cone}(f)) = \beta {\mathbf{Z}}\) where \(\alpha = [e^n] {}^{ \vee }\) and \(\beta = [e^{2n}] {}^{ \vee }\). Then \(\alpha^2 \in H^{2n}(\operatorname{cone}(f))\) so \(\alpha^2 = n\beta\) for some \(n\in {\mathbf{Z}}\); this \(h(f) \coloneqq n\) is the Hopf invariant of \(f\).
Alternatively, take \(f\) smooth and \(\operatorname{vol}\in \Omega^n(S^n)\) a volume form. Since \(\int_{S^n} \operatorname{vol}= 1\), the pullback \(f^*\operatorname{vol}\) is exact since \(H^n(S^{2n-1}) = 0\), so \(f^*\operatorname{vol}= d\alpha\) for some \(\alpha \in \Omega^{n-1}(S^{2n-1})\). Then \begin{align*} h(f) = \int_{S^{2n-1}} \alpha\wedge \,d\alpha \end{align*}
Hopf invariant one
Motivation
Hopf famously discovered an essential (i.e. non-nullhomotopic) map \(S^3\to S^2\), which you can think of as the being defined via \(S^3\to {\mathbf{CP}}^1\) and then choosing an identification of \({\mathbf{CP}}^1\) with the sphere. The mapping cone of this map is a CW complex and asking whether or not the map \(S^3\to S^2\) is non-nullhomotopic is equivalent to asking whether or not this mapping cone is homotopy equivalent to a wedge \(S^4\vee S^2\).
We can see that it is not since the mapping cone is actually just \({\mathbf{CP}}^2\) and the cup square of the generator in \(H^2\) is the generator in \(H^4\). More generally, given a map \(S^{2n−1}\to S^n\) we can form the 2-cell complex given by gluing \({\mathbb{D}}^{2n}\) along this map and ask if the cup square of the generator in degree \(H^n\) squares to the generator in degree \(2n\).
If it does we say this map has Hopf invariant 1, otherwise it has Hopf invariant 0. It’s natural to ask how many maps of Hopf invariant one can we build? The answer is: not many. They only exist when \(n=1,2,4,8\). While nowadays we usually learn the proof of this fact using K-theory, which is very short.
