Motivation
In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map
- ,
and proved that is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles
- for any .
It was later shown that the homotopy group is the infinite cyclic group generated by . In 1951, Jean-Pierre Serre proved that the rational homotopy groups
for an odd-dimensional sphere ( odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree . There is an interesting way of seeing this:
Read more about this topic: Hopf Invariant
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