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Shimura–Teichmüller curves in genus 5
Pseudo-rotations and Steenrod squares
Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada |
In this note we prove that if a closed monotone symplectic manifold $ M $ of dimension $ 2n $, satisfying a homological condition that holds in particular when the minimal Chern number is $ N>n $, admits a Hamiltonian pseudo-rotation, then the quantum Steenrod square of the point class must be deformed. This gives restrictions on the existence of pseudo-rotations. Our methods rest on previous work of the author, Zhao, and Wilkins, going back to the equivariant pair-of-pants product-isomorphism of Seidel.
References:
| [1] |
D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3–36. |
| [2] |
P. Biran and O. Cornea,
Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989.
doi: 10.2140/gt.2009.13.2881. |
| [3] |
B. Bramham,
Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves, Ann. of Math. (2), 181 (2015), 1033-1086.
doi: 10.4007/annals.2015.181.3.4. |
| [4] |
B. Bramham,
Pseudo-rotations with sufficiently Liouvillean rotation number are $C^0$-rigid, Invent. Math., 199 (2015), 561-580.
doi: 10.1007/s00222-014-0525-0. |
| [5] |
I. Charton, Hamiltonian ${S}^1$-spaces with large equivariant pseudo-index, J. Geom. Phys., 147 (2020), 103521, 10 pp.
doi: 10.1016/j.geomphys.2019.103521. |
| [6] |
E. Cineli and V. L. Ginzburg, On the iterated Hamiltonian Floer homology, preprint, arXiv: 1902.06369, 2019. |
| [7] |
E. Cineli, V. L. Ginzburg and B. Z. Gürel, From pseudo-rotations to holomorphic curves via quantum Steenrod squares, preprint, arXiv: 1909.11967, 2019. |
| [8] |
E. Cineli, V. L. Ginzburg and B. Z. Gürel, Pseudo-rotations and holomorphic curves, preprint, arXiv: 1905.07567, 2019. |
| [9] |
M. Entov and L. Polterovich,
Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 30 (2003), 1635-1676.
doi: 10.1155/S1073792803210011. |
| [10] |
B. Fayad and A. Katok,
Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520.
doi: 10.1017/S0143385703000798. |
| [11] |
K. Fukaya and K. Ono,
Arnold conjecture and Gromov-Witten invariant, Topology, 38 (1999), 933-1048.
doi: 10.1016/S0040-9383(98)00042-1. |
| [12] |
V. L. Ginzburg and B. Z. Gürel,
Action and index spectra and periodic orbits in Hamiltonian dynamics, Geom. Topol., 13 (2009), 2745-2805.
doi: 10.2140/gt.2009.13.2745. |
| [13] |
V. L. Ginzburg and B. Z. Gürel,
Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.
doi: 10.4310/JSG.2010.v8.n3.a4. |
| [14] |
V. L. Ginzburg and B. Z. Gürel,
Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms, Duke Math. J., 163 (2014), 565-590.
doi: 10.1215/00127094-2410433. |
| [15] |
V. L. Ginzburg and B. Z. Gürel,
Hamiltonian pseudo-rotations of projective spaces, Invent. Math., 214 (2018), 1081-1130.
doi: 10.1007/s00222-018-0818-9. |
| [16] |
V. L. Ginzburg and B. Z. Gürel,
Conley conjecture revisited, Int. Math. Res. Not. IMRN, 2019 (2019), 761-798.
doi: 10.1093/imrn/rnx137. |
| [17] |
L. Godinho, F. von Heymann and S. Sabatini,
12, 24 and beyond, Adv. Math., 319 (2017), 472-521.
doi: 10.1016/j.aim.2017.08.023. |
| [18] |
D. McDuff,
Hamiltonian $S^1$-manifolds are uniruled, Duke Math. J., 146 (2009), 449-507.
doi: 10.1215/00127094-2009-003. |
| [19] |
M. McLean,
Local Floer homology and infinitely many simple Reeb orbits, Algebr. Geom. Topol., 12 (2012), 1901-1923.
doi: 10.2140/agt.2012.12.1901. |
| [20] |
Y.-G. Oh, Symplectic topology and Floer homology. Vols. 1 and 2, New Mathematical Monographs, vol. 27 and 28, Cambridge University Press, Cambridge, 2015. |
| [21] |
S. Sabatini, On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action, Commun. Contemp. Math., 19 (2017), 1750043, 51pp.
doi: 10.1142/S0219199717500432. |
| [22] |
D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, UT, 1997), IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999,143–229. |
| [23] |
D. Salamon and E. Zehnder,
Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360.
doi: 10.1002/cpa.3160451004. |
| [24] |
P. Seidel,
The equivariant pair-of-pants product in fixed point Floer cohomology, Geom. Funct. Anal., 25 (2015), 942-1007.
doi: 10.1007/s00039-015-0331-x. |
| [25] |
P. Seidel and N. Wilkins,, work in progress. |
| [26] |
E. Shelukhin, On the Hofer-Zehnder conjecture, preprint, arXiv: 1905.04769, 2019. |
| [27] |
E. Shelukhin, Pseudo-rotations and Steenrod squares revisited, arXiv: 1909.12315, 2019. |
| [28] |
E. Shelukhin and J. Zhao, The $ {\mathbb{Z}/(p)}$-equivariant product-isomorphism in fixed point Floer homology, preprint, arXiv: 1905.03666, 2019. |
| [29] |
M. Usher and J. Zhang,
Persistent homology and Floer–Novikov theory, Geom. Topol., 20 (2016), 3333-3430.
doi: 10.2140/gt.2016.20.3333. |
| [30] |
N. Wilkins, A construction of the quantum Steenrod squares and their algebraic relations, Geom. Topol., to appear, arXiv: 1810.02738, 2018. |
| [31] |
N. Wilkins, Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology, preprint, arXiv: 1810.02738, 2018. |
show all references
References:
| [1] |
D. V. Anosov and A. B. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obšč., 23 (1970), 3–36. |
| [2] |
P. Biran and O. Cornea,
Rigidity and uniruling for Lagrangian submanifolds, Geom. Topol., 13 (2009), 2881-2989.
doi: 10.2140/gt.2009.13.2881. |
| [3] |
B. Bramham,
Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves, Ann. of Math. (2), 181 (2015), 1033-1086.
doi: 10.4007/annals.2015.181.3.4. |
| [4] |
B. Bramham,
Pseudo-rotations with sufficiently Liouvillean rotation number are $C^0$-rigid, Invent. Math., 199 (2015), 561-580.
doi: 10.1007/s00222-014-0525-0. |
| [5] |
I. Charton, Hamiltonian ${S}^1$-spaces with large equivariant pseudo-index, J. Geom. Phys., 147 (2020), 103521, 10 pp.
doi: 10.1016/j.geomphys.2019.103521. |
| [6] |
E. Cineli and V. L. Ginzburg, On the iterated Hamiltonian Floer homology, preprint, arXiv: 1902.06369, 2019. |
| [7] |
E. Cineli, V. L. Ginzburg and B. Z. Gürel, From pseudo-rotations to holomorphic curves via quantum Steenrod squares, preprint, arXiv: 1909.11967, 2019. |
| [8] |
E. Cineli, V. L. Ginzburg and B. Z. Gürel, Pseudo-rotations and holomorphic curves, preprint, arXiv: 1905.07567, 2019. |
| [9] |
M. Entov and L. Polterovich,
Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 30 (2003), 1635-1676.
doi: 10.1155/S1073792803210011. |
| [10] |
B. Fayad and A. Katok,
Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520.
doi: 10.1017/S0143385703000798. |
| [11] |
K. Fukaya and K. Ono,
Arnold conjecture and Gromov-Witten invariant, Topology, 38 (1999), 933-1048.
doi: 10.1016/S0040-9383(98)00042-1. |
| [12] |
V. L. Ginzburg and B. Z. Gürel,
Action and index spectra and periodic orbits in Hamiltonian dynamics, Geom. Topol., 13 (2009), 2745-2805.
doi: 10.2140/gt.2009.13.2745. |
| [13] |
V. L. Ginzburg and B. Z. Gürel,
Local Floer homology and the action gap, J. Symplectic Geom., 8 (2010), 323-357.
doi: 10.4310/JSG.2010.v8.n3.a4. |
| [14] |
V. L. Ginzburg and B. Z. Gürel,
Hyperbolic fixed points and periodic orbits of Hamiltonian diffeomorphisms, Duke Math. J., 163 (2014), 565-590.
doi: 10.1215/00127094-2410433. |
| [15] |
V. L. Ginzburg and B. Z. Gürel,
Hamiltonian pseudo-rotations of projective spaces, Invent. Math., 214 (2018), 1081-1130.
doi: 10.1007/s00222-018-0818-9. |
| [16] |
V. L. Ginzburg and B. Z. Gürel,
Conley conjecture revisited, Int. Math. Res. Not. IMRN, 2019 (2019), 761-798.
doi: 10.1093/imrn/rnx137. |
| [17] |
L. Godinho, F. von Heymann and S. Sabatini,
12, 24 and beyond, Adv. Math., 319 (2017), 472-521.
doi: 10.1016/j.aim.2017.08.023. |
| [18] |
D. McDuff,
Hamiltonian $S^1$-manifolds are uniruled, Duke Math. J., 146 (2009), 449-507.
doi: 10.1215/00127094-2009-003. |
| [19] |
M. McLean,
Local Floer homology and infinitely many simple Reeb orbits, Algebr. Geom. Topol., 12 (2012), 1901-1923.
doi: 10.2140/agt.2012.12.1901. |
| [20] |
Y.-G. Oh, Symplectic topology and Floer homology. Vols. 1 and 2, New Mathematical Monographs, vol. 27 and 28, Cambridge University Press, Cambridge, 2015. |
| [21] |
S. Sabatini, On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action, Commun. Contemp. Math., 19 (2017), 1750043, 51pp.
doi: 10.1142/S0219199717500432. |
| [22] |
D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, UT, 1997), IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999,143–229. |
| [23] |
D. Salamon and E. Zehnder,
Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math., 45 (1992), 1303-1360.
doi: 10.1002/cpa.3160451004. |
| [24] |
P. Seidel,
The equivariant pair-of-pants product in fixed point Floer cohomology, Geom. Funct. Anal., 25 (2015), 942-1007.
doi: 10.1007/s00039-015-0331-x. |
| [25] |
P. Seidel and N. Wilkins,, work in progress. |
| [26] |
E. Shelukhin, On the Hofer-Zehnder conjecture, preprint, arXiv: 1905.04769, 2019. |
| [27] |
E. Shelukhin, Pseudo-rotations and Steenrod squares revisited, arXiv: 1909.12315, 2019. |
| [28] |
E. Shelukhin and J. Zhao, The $ {\mathbb{Z}/(p)}$-equivariant product-isomorphism in fixed point Floer homology, preprint, arXiv: 1905.03666, 2019. |
| [29] |
M. Usher and J. Zhang,
Persistent homology and Floer–Novikov theory, Geom. Topol., 20 (2016), 3333-3430.
doi: 10.2140/gt.2016.20.3333. |
| [30] |
N. Wilkins, A construction of the quantum Steenrod squares and their algebraic relations, Geom. Topol., to appear, arXiv: 1810.02738, 2018. |
| [31] |
N. Wilkins, Quantum Steenrod squares and the equivariant pair-of-pants in symplectic cohomology, preprint, arXiv: 1810.02738, 2018. |
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