Algebra, Geometry and Topology
Infinite symmetric products of rational algebras and spaces
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.

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.298
Classification: 13A02,  16E45,  55P62
Keywords: 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
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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/

[1] A. K. Bousfield; V. K. Gugenheim On PL De Rham theory and rational homotopy type, Memoirs of the American Mathematical Society, 179, American Mathematical Society, 1976 | Zbl: 0338.55008

[2] Aldo Conca; Christian Krattenthaler; Junzo Watanabe Regular sequences of symmetric polynomials, Rend. Semin. Mat. Univ. Padova, Volume 121 (2009), pp. 179-199 | Article | Numdam | MR: 2542141 | Zbl: 1167.05051

[3] John Dalbec Multisymmetric functions, Beitr. Algebra Geom., Volume 40 (1999) no. 1, pp. 27-51 | MR: 1678567 | Zbl: 0953.05077

[4] Albrecht Dold; René Thom Quasifaserungen und unendliche symmetrische Produkte, Ann. Math., Volume 67 (1958), pp. 239-281 | Article | MR: 97062 | Zbl: 0091.37102

[5] Yves Félix; Daniel Tanré Rational homotopy of symmetric products and spaces of finite subsets, Homotopy theory of function spaces and related topics (Contemporary Mathematics), Volume 519, American Mathematical Society, 2010, pp. 77-92 | Article | MR: 2648705 | Zbl: 1215.55004

[6] Kathryn Hess Rational homotopy theory: a brief introduction, Interactions between homotopy theory and algebra. Summer school, University of Chicago, IL, USA, July 26–August 6 (Contemporary Mathematics), Volume 436, American Mathematical Society, 2007, pp. 175-202 | Article | MR: 2355774 | Zbl: 1128.55010

[7] Philip S. Hirschhorn Notes on homotopy colimits and homotopy limits (2014) (http://www-math.mit.edu/~psh/notes/hocolim.pdf)

[8] Ian G. Macdonald The Poincaré polynomial of a symmetric product, Proc. Camb. Philos. Soc., Volume 58 (1962), pp. 563-568 | Article | Zbl: 0121.39601

[9] Ian G. Macdonald Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Oxford University Press, 1979

[10] Hans Scheerer; Manfred Stelzer Fibrewise infinite symmetric products and M-category, Bull. Korean Math. Soc., Volume 36 (1999) no. 4, pp. 671-682 | MR: 1736612 | Zbl: 0941.55005

[11] Dennis Sullivan Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977) no. 1, pp. 269-331 | Article | Numdam | Zbl: 0374.57002

[12] Francesco Vaccarino The ring of multisymmetric functions, Ann. Inst. Fourier, Volume 55 (2005) no. 3, pp. 717-731 | Article | Numdam | MR: 2149400 | Zbl: 1062.05143

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