On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds

Milivojević, Aleksandar

doi:10.1090/proc/15397

On the realization of symplectic algebras and rational homotopy types by closed symplectic manifolds
HTML articles powered by AMS MathViewer

by Aleksandar Milivojević PDF

Proc. Amer. Math. Soc. 149 (2021), 2257-2263 Request permission

Abstract:

We answer a question of Oprea–Tralle on the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four, along with a question of Lupton–Oprea in all even dimensions. This will also allow us to address, in all even dimensions six and higher, another question of Oprea–Tralle on the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold.

References

Michael Albanese and Aleksandar Milivojević, On the minimal sum of Betti numbers of an almost complex manifold, Differential Geom. Appl. 62 (2019), 101–108. MR 3872943, DOI 10.1016/j.difgeo.2018.10.002
A. K. Bousfield and D. M. Kan, Localization and completion in homotopy theory, Bull. Amer. Math. Soc. 77 (1971), 1006–1010. MR 296935, DOI 10.1090/S0002-9904-1971-12837-9
William Browder, Surgery on simply-connected manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65, Springer-Verlag, New York-Heidelberg, 1972. MR 0358813, DOI 10.1007/978-3-642-50020-6
Urtzi Buijs, Yves Félix, Aniceto Murillo, and Daniel Tanré, Homotopy theory of complete Lie algebras and Lie models of simplicial sets, J. Topol. 11 (2018), no. 3, 799–825. MR 3989431, DOI 10.1112/topo.12073
Yunhyung Cho, Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment maps, Internat. J. Math. 27 (2016), no. 5, 1650043, 14. MR 3498115, DOI 10.1142/S0129167X16500439
Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. MR 1777332, DOI 10.1090/gsm/025
Antonio Gómez-Tato, Stephen Halperin, and Daniel Tanré, Rational homotopy theory for non-simply connected spaces, Trans. Amer. Math. Soc. 352 (2000), no. 4, 1493–1525. MR 1653355, DOI 10.1090/S0002-9947-99-02463-0
Robert E. Gompf, The topology of symplectic manifolds, Turkish J. Math. 25 (2001), no. 1, 43–59. MR 1829078
Bogusław Hajduk and Aleksy Tralle, Exotic smooth structures and symplectic forms on closed manifolds, Geom. Dedicata 132 (2008), 31–42. MR 2396907, DOI 10.1007/s10711-007-9181-5
Friedrich Hirzebruch, Gesammelte Abhandlungen/collected papers. I. 1951–1962, Springer Collected Works in Mathematics, Springer, Heidelberg, 2013 (German). Reprint of the 1987 edition [MR0931775]. MR 3184395
D. Kotschick, The Seiberg-Witten invariants of symplectic four–manifolds, Séminaire Bourbaki, 38, pp.195-220, 1995.
Gregory Lupton and John Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), no. 1-3, 193–207. MR 1255930, DOI 10.1016/0022-4049(94)90142-2
Aleksy Tralle and John Oprea, Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, vol. 1661, Springer-Verlag, Berlin, 1997. MR 1465676, DOI 10.1007/BFb0092608
M. Rivera, F. Wierstra, and M. Zeinalian, Rational homotopy equivalences and singular chains, arXiv preprint arXiv:1906.03655, 2019, Algebr. Geom. Topol., to appear.
Alexander I. Suciu, Iterated spinning and homology spheres, Trans. Amer. Math. Soc. 321 (1990), no. 1, 145–157. MR 987169, DOI 10.1090/S0002-9947-1990-0987169-3
W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. MR 402764, DOI 10.1090/S0002-9939-1976-0402764-6
Aleksy Tralle, Homotopy properties of closed symplectic manifolds, Univ. Iagel. Acta Math. 38 (2000), 105–128. Geometry and topology manifolds (Krynica, 1999). MR 1812105