Infinite symmetric products of rational algebras and spaces

TL;DR

The paper establishes a natural isomorphism between infinite symmetric products of rational graded-commutative algebras and free graded-commutative algebras, providing new proofs and models in rational homotopy theory.

Contribution

It introduces a novel algebraic isomorphism for infinite symmetric products and applies it to simplify proofs and determine models in rational topology.

Findings

Infinite symmetric product is isomorphic to a free graded-commutative algebra.

Provides a quick proof of the Dold-Thom theorem in rational homotopy theory.

Determines minimal Sullivan models for symmetric products of complex projective 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.