  It is well-known that the homology of the classifying space of the unitary
group is isomorphic to the ring of symmetric functions, Symm. We offer the
cohomology of the loop space of the suspension of the infinite complex
projective space as a topological model for the ring of quasisymmetric
functions, QSymm. We exploit standard results from topology to shed light on
some of the algebraic properties of QSymm. In particular, we reprove the
Ditters conjecture. We investigate a product on the loop space that gives rise
to an algebraic structure which generalizes the Witt vector structure in the
cohomology of BU. The canonical Thom spectrum over the loops on the suspension
of BU(1) is highly non-commutative and we study some of its features, including
the homology of its topological Hochschild homology spectrum.
