Infinite symmetric products of rational algebras and spaces
- Resource Type
- article
- Authors
- Hu, Jiahao; Milivojević, Aleksandar
- Source
- Comptes Rendus. Mathématique, Vol 360, Iss G3, Pp 275-284 (2022)
- Subject
- Symmetric products
Dold–Thom theorem
Mathematics
QA1-939
- Language
- English
French
- ISSN
- 1778-3569
We show that the infinite symmetric product of a connected graded-commutative algebra over $\mathbb{Q}$ 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 $\mathbb{Q}$ 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.