Quasisymmetric functions from a topological point of view

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 space Ω Σ C P∞ 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 Ω Σ C P∞ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of $BU$. The canonical Thom spectrum over Ω Σ C P∞ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.