Infinite symmetric products of rational algebras and spaces

Abstract

We show that the infinite symmetric product of a connected graded-commutative algebra over the rationals is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over the rationals is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold-Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded commutative algebras are free; this allows us to determine minimal Sullivan models for finite symmetric products of complex projective spaces.