Comptes Rendus
Algèbre, Géométrie et Topologie
Infinite symmetric products of rational algebras and spaces
Jiahao Hu1 ; Aleksandar Milivojević2
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
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 275-284.

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

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DOI : 10.5802/crmath.298
Classification : 13A02, 16E45, 55P62
Mots clés : Symmetric products, Dold–Thom theorem
Jiahao Hu 1 ; Aleksandar Milivojević 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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Jiahao Hu; Aleksandar Milivojević. Infinite symmetric products of rational algebras and spaces. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 275-284. doi : 10.5802/crmath.298. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.298/

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