Applications
The most famous early applications of the Steenrod algebra to outstanding topological problems were the solutions by J. Frank Adams of the Hopf invariant one problem and the vector fields on spheres problem. Independently Milnor and Bott, as well as Kervaire, gave a second solution of the Hopf invariant one problem, using operations in K-theory; these are the Adams operations. One application of the mod 2 Steenrod algebra that is fairly elementary is the following theorem.
Theorem. If there is a map S2n - 1 → Sn of Hopf invariant one, then n is a power of 2.
The proof uses the fact that each Sqk is decomposable for k which is not a power of 2; that is, such an element is a product of squares of strictly smaller degree.
Read more about this topic: Steenrod Algebra