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 ΩΣCP 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 ΩΣCP that gives rise to an algebraic structure which generalizes the Witt vector structure in the cohomology of BU . The canonical Thom spectrum over ΩΣCP is highly non-commutative and we study some of its features, including the homology of its topological Hochschild homology spectrum. Introduction Let us recall some background on the variants of symmetric functions. For a much more detailed account on that see [H03, H05]. The algebra of symmetric functions, Symm, is an integral polynomial algebra Symm = Z[c1, c2, . . .]. The reader is encouraged to think of these ci as Chern classes. This algebra structure can be extended to a Hopf algebra structure by defining the coproduct to be ∆(cn) = ∑