Hopf invariant  In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. In 1931 Heinz Hopf used Clifford parallels to construct the "Hopf map"   and proved that formula_2 is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles  is equal to 1, for any formula_4. It was later shown that the homotopy group formula_5 is the infinite cyclic group generated by formula_2. In 1951, Jean-Pierre Serre proved that the rational homotopy groups   for an odd-dimensional sphere (formula_8 odd) are zero unless formula_9 is equal to 0 or "n". However, for an even-dimensional sphere ("n" even), there is one more bit of infinite cyclic homotopy in degree formula_10. Let formula_11 be a continuous map (assume formula_12). Then we can form the cell complex  where formula_14 is a formula_15-dimensional disc attached to formula_16 via formula_17. The cellular chain groups formula_18 are just freely generated on the formula_9-cells in degree formula_9, so they are formula_21 in degree 0, formula_8 and formula_15 and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that formula_12), the cohomology is  Denote the generators of the cohomology groups by  For dimensional reasons, all cup-products between those classes must be trivial apart from formula_28. Thus, as a "ring", the cohomology is  The integer formula_30 is the Hopf invariant of the map formula_17. Theorem: The map formula_32 is a homomorphism. Moreover, if formula_8 is even, formula_34 maps onto formula_35. The Hopf invariant is formula_36 for the "Hopf maps", where formula_37, corresponding to the real division algebras formula_38, respectively, and to the fibration formula_39 sending a direction on the sphere to the subspace it spans. It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.