Flat circle bundles, pullbacks, and the circle made discrete

The fact that flat principal circle bundles are characterized by having zero real Euler classes has proved important in recent years in understanding whether total spaces of vector bundles over nonnegatively curved manifolds must support metrics with nonnegative curvature as well (see Theorem 6.3 and the discussion afterwards). Unfortunately, the proofs of this fact of which we are aware use either sheaf-theoretic arguments that require a great deal of background somewhat out of the algebraic topological mainstream (see [4]) or the elaborate differential geometric machinery of Chern-Weil theory (see [7], e.g.). The purpose of this note is simply to observe that an enlightening topological proof exists that only relies on the notions of pullback and classifyingmap, as well the homological algebra surrounding these concepts. Moreover, as our proof demonstrates, the Euler class result is a direct manifestation of the circle’s characterization as the only nontrivial compact connected Lie group with contractible universal cover. Finally, our proof provides a prime example of how homotopy theory and its associated algebra can have beautiful geometric consequences.