The Hopf invariant one problem

An expository talk given for zygotop in Spring 2024, a learning seminar for young graduate students interested in homotopy theory and related areas. I gave an introduction to the Hopf invariant one problem and its solution, including the classical results relating real division algebra structures on \(\mathbb{R}^n\), parallelizability of \(S^{n-1}\), H-space structures on \(S^{n-1}\), and elements of Hopf invariant one in \(\pi_{2n-1}(S^n)\). I defined the Hopf invariant in terms of complex topological K-theory, and gave Adams-Atiyah’s proof (using the splitting principle and Adams operations on topological K-theory) that elements of Hopf invariant \(\pm 1\) exist only for \(n = 1, 2, 4, 8\).