Non-commutativity of the exponential spectrum

In a Banach algebra, the spectrum satisfies σ(ab)∖{0}=σ(ba)∖{0} for each pair of elements a,b. We show that this is no longer true for the exponential spectrum, thereby solving a problem open since 1992. Our proof depends on the fact that the homotopy group π4(GL2(C)) is non-trivial.