Quasisymmetric functions from a topological point of view - Baker, Andrew; Richter, Birgit (Mathematica Scandinavica, 2008)
It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions $\mathsf {Symm}$. We offer the cohomology of the space $Ω Σ {\mathsf C} P^{∞}$ as a topological model for the ring of quasisymmetric functions $\mathsf {QSymm}$. We exploit standard results from topology to shed light on some of the algebraic properties of $\mathsf {QSymm}$. In particular, we reprove the Ditters conjecture. We investigate a product on $Ω Σ {\mathsf C} P^{∞}$ that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of $BU$. The canonical Thom spectrum over $Ω Σ {\mathsf C} P^{∞}$ is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum.