Historical note on fiber bundles

C. N. Yang, in his clear review of Maxwell’s equations and gauge theory (Physics Today, November 2014, page 45 ), reports that his colleague mathematician James Simons exclaimed, “[Paul] Dirac had discovered trivial and nontrivial bundles before mathematicians.” Remarkably, however, in 1931, the same year that Dirac discovered his monopole, Heinz Hopf discovered its fiber-bundle equivalent, now known as the Hopf fibration of the 3-sphere. ¹

Although Eli Lubkin pointed out the bundle structure of the Dirac monopole ² in 1963 and Tai Tsun Wu and Yang provided a widely read description, ³ Andrzej Trautman apparently first noted its identification with the Hopf fibration ⁴ in 1977. Trautman’s 1967 lectures at King’s College London introduced some physicists to the mathematical equivalence of gauge theories and fiber-bundle theory, but not until 1970 were those lectures published. ⁵ Yang notes that the equivalence came as a shock to both physicists and mathematicians in the 1970s.