The Hopf invariant is a topological invariant that arises in the study of mappings between spheres, particularly in the context of homotopy theory and homotopy groups of spheres. Named after Heinz Hopf, the invariant provides a way to classify certain types of mappings and can be used to distinguish between different homotopy classes of maps.
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For , , its mapping cone has integral cohomology generators in degree and in degree . The integer defined by , 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 ; a constant attaching map has invariant zero.