  This paper proves three different coherence theorems for symmetric monoidal
bicategories. First, we show that in a free symmetric monoidal bicategory every
diagram of 2-cells commutes. Second, we show that this implies that the free
symmetric monoidal bicategory on one object is equivalent, as a symmetric
monoidal bicategory, to the discrete symmetric monoidal bicategory given by the
disjoint union of the symmetric groups. Third, we show that every symmetric
monoidal bicategory is equivalent to a strict one.
  We give two topological applications of these coherence results. First, we
show that the classifying space of a symmetric monoidal bicategory can be
equipped with an E_{\infty} structure. Second, we show that the fundamental
2-groupoid of an E_n space, n \geq 4, has a symmetric monoidal structure. These
calculations also show that the fundamental 2-groupoid of an E_3 space has a
sylleptic monoidal structure.
