Algebra, Geometry and Topology
Infinite symmetric products of rational algebras and spaces
Comptes Rendus. Mathématique, Volume 360 (2022) no. G3, pp. 275-284.

We show that the infinite symmetric product of a connected graded-commutative algebra over 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 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.

Published online:
DOI: 10.5802/crmath.298
Classification: 13A02, 16E45, 55P62
Keywords: Symmetric products, Dold–Thom theorem
Hu, Jiahao 1; Milivojević, Aleksandar 2

1 Stony Brook University, Department of Mathematics, 100 Nicolls Road, 11794 Stony Brook, USA
2 Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
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     title = {Infinite symmetric products of rational algebras and spaces},
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Hu, Jiahao; Milivojević, Aleksandar. Infinite symmetric products of rational algebras and spaces. Comptes Rendus. Mathématique, Volume 360 (2022) no. G3, pp. 275-284. doi : 10.5802/crmath.298.

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