= Hopf invariant
{c}
{title2=$a\smile a=H(\phi)b$}
For $\phi:S^{2q-1}\to S^q$, $q>1$, its mapping cone has integral <cohomology> generators $a$ in degree $q$ and $b$ in degree $2q$. The integer defined by $a\smile a=H(\phi)b$, with fixed cell orientations, is the Hopf invariant. It measures how an attaching map changes multiplication without changing the additive groups. The complex <Hopf fibration> has invariant one because its mapping cone is $\mathbb{CP}^2$; a constant attaching map has invariant zero.
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