Algebraic & Geometric Topology

Exotic iterated Dehn twists

Paul Seidel

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Abstract

Consider cotangent bundles of exotic spheres with their canonical symplectic structure. They admit automorphisms that preserve the part at infinity of one fiber and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all automorphisms with the same behavior).

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 6 (2014), 3305-3324.

Dates
Received: 22 May 2013
Revised: 15 October 2013
Accepted: 16 March 2014
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513716041

Digital Object Identifier
doi:10.2140/agt.2014.14.3305

Mathematical Reviews number (MathSciNet)
MR3302963

Zentralblatt MATH identifier
1333.53126

Subjects
Primary: 53D40: Floer homology and cohomology, symplectic aspects 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms

Keywords
exotic sphere symplectic automorphism Floer homology

Citation

Seidel, Paul. Exotic iterated Dehn twists. Algebr. Geom. Topol. 14 (2014), no. 6, 3305--3324. doi:10.2140/agt.2014.14.3305. https://projecteuclid.org/euclid.agt/1513716041


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References

  • V I Arnol'd, S M Guseĭn-Zade, A N Varchenko, Singularities of differentiable maps, Vol. II: Monodromy and asymptotics of integrals, Monographs in Mathematics 83, Birkhäuser, Boston (1988)
  • R Avdek, Liouville hypersurfaces and connect sum cobordisms
  • A L Besse, Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb. 93, Springer, Berlin (1978)
  • F Bourgeois, T Ekholm, Y Eliashberg, Effect of Legendrian surgery, Geom. Topol. 16 (2012) 301–389 With an appendix by Sheel Ganatra and Maksim Maydanskiy
  • R Budney, A family of embedding spaces, from: “Groups, homotopy and configuration spaces”, (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13 (2008) 41–83
  • J Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. (1970) 5–173
  • K Cieliebak, Y Eliashberg, From Stein to Weinstein and back: Symplectic geometry of affine complex manifolds, AMS Colloquium Publications 59, Amer. Math. Soc. (2012)
  • J J Duistermaat, On the Morse index in variational calculus, Advances in Math. 21 (1976) 173–195
  • J Johns, The Picard–Lefschetz theory of complexified Morse functions (2009)
  • J Johns, Complexifications of Morse functions and the directed Donaldson–Fukaya category, J. Symplectic Geom. 8 (2010) 403–500
  • L H Kauffman, N A Krylov, Kernel of the variation operator and periodicity of open books, Topology Appl. 148 (2005) 183–200
  • M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504–537
  • A Klein, Symplectic monodromy, Leray residues and quasi-homogeneous polynomials
  • K Lamotke, Die Homologie isolierter Singularitäten, Math. Z. 143 (1975) 27–44
  • L Polterovich, The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1 (1991) 198–210
  • J Robbin, D Salamon, The Maslov index for paths, Topology 32 (1993) 827–844
  • P Seidel, Lagrangian two-spheres can be symplectically knotted, J. Differential Geom. 52 (1999) 145–171
  • P Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000) 103–149
  • P Seidel, Fukaya categories and Picard–Lefschetz theory, Eur. Math. Soc., Zürich (2008)
  • J Stevens, Periodicity of branched cyclic covers of manifolds with open book decomposition, Math. Ann. 273 (1986) 227–239
  • F W Warner, Conjugate loci of constant order, Ann. of Math. 86 (1967) 192–212
  • A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241–251