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arXiv · 2108.09794

Infinite symmetric products of rational algebras and spaces

Jiahao Hu·Aleksandar Milivojević
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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.

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Jiahao Hu, Aleksandar Milivojević. 2021-11-08. Infinite symmetric products of rational algebras and spaces. https://arxiv.org/abs/2108.09794

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