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Exotic spheres and the topology of symplectomorphism groups. (English) Zbl 1325.53104
The authors of the paper under review show that a certain family of diffeomorphisms \(\phi_s\) of high-dimensional spheres parametrized by \(S^{k}\), gives a non-contractible family of compactly supported symplectomorphisms. They also provide examples where the symplectomorphism group is not simply connected and does not have the homotopy type of a finite CW complex. They also show that the phenomena persist for Dehn twists along the standard matching sphere of \(A_m\)-Milnor fibre. They also find related examples of symplectic mapping classes for \(T^{*}(S^{n} \times S^{1}) \) and of an exotic symplectic structure on \(T^{*}(S^{n} \times S^{1})\) .

MSC:
53D12 Lagrangian submanifolds; Maslov index
53D35 Global theory of symplectic and contact manifolds
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[1] Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres, Ann. of Math. 175 ((2)) pp 71– (2012) · Zbl 1244.53089 · doi:10.4007/annals.2012.175.1.4
[2] M. Abouzaid T. Kragh On the immersion classes of nearby Lagrangians · Zbl 1405.53120
[3] Antonelli, Gromoll groups, DiffSn and bilinear constructions of exotic spheres, Bull. Amer. Math. Soc. 76 pp 772– (1970) · Zbl 0195.53303 · doi:10.1090/S0002-9904-1970-12544-7
[4] V. I. Arnol’d Some remarks on symplectic monodromy of Milnor fibrations The Floer memorial volume Progress in Mathematics 133 Birkhäuser Basel 1995 99 103
[5] M. Audin Cobordismes d’immersions lagrangiennes et legendriennes Travaux en Cours 20 Hermann Paris 1987
[6] Browder, Torsion in H-spaces, Ann. of Math. 74 ((2)) pp 24– (1961) · Zbl 0112.14501 · doi:10.2307/1970305
[7] Burghelea, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 pp 1– (1974) · Zbl 0296.58003 · doi:10.1090/S0002-9947-1974-0356103-2
[8] Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie, Inst. Hautes Études Sci. Publ. Math. 39 pp 5– (1970) · Zbl 0213.25202 · doi:10.1007/BF02684687
[9] Crowley, The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature, Geom. Topol. 17 pp 1773– (2013) · Zbl 1285.57015 · doi:10.2140/gt.2013.17.1773
[10] Dombrowski, On the geometry of the tangent bundle, J. Reine Angew. Math. 210 pp 73– (1962) · Zbl 0105.16002
[11] Ekholm, Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 pp 195– (2014) · Zbl 1287.53068 · doi:10.1007/s00208-013-0958-6
[12] Evans, Symplectic mapping class groups of some Stein and rational surfaces, J. Symplectic Geom. 9 pp 45– (2011) · Zbl 1242.58004 · doi:10.4310/JSG.2011.v9.n1.a4
[13] M. Gromov A topological technique for the construction of solutions of differential equations and inequalities Actes du Congrès International des Mathématiciens Nice 1970, Tome 2 Gauthier-Villars Paris 1971 221 225
[14] Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 pp 307– (1985) · Zbl 0592.53025 · doi:10.1007/BF01388806
[15] Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 pp 47– (1961) · Zbl 0102.38603 · doi:10.1007/BF02566892
[16] Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 pp 242– (1959) · Zbl 0113.17202 · doi:10.1090/S0002-9947-1959-0119214-4
[17] Kervaire, Groups of homotopy spheres. I, Ann. of Math. 77 pp 504– (1963) · Zbl 0115.40505 · doi:10.2307/1970128
[18] Lees, On the classification of Lagrange immersions, Duke Math. J. 43 pp 217– (1976) · Zbl 0329.58006 · doi:10.1215/S0012-7094-76-04319-2
[19] Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. 76 ((2)) pp 531– (1962) · Zbl 0106.37102 · doi:10.2307/1970373
[20] J. Milnor Lectures on the h -cobordism theorem Princeton University Press Princeton, NJ 1965 Notes by L. Siebenmann and J. Sondow
[21] Paechter, The groups \(\pi\)r(Vn,m). I, Quart. J. Math. Oxford Ser. 7 ((2)) pp 249– (1956) · Zbl 0073.18402 · doi:10.1093/qmath/7.1.249
[22] P. Seidel Symplectic automorphisms of T * S 2
[23] Seidel, Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 pp 103– (2000) · Zbl 0992.53059 · doi:10.24033/bsmf.2365
[24] Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 pp 1003– (2003) · Zbl 1032.57035 · doi:10.1016/S0040-9383(02)00028-9
[25] Seidel, Lecture Notes in Mathematics, vol. 1938 pp 231– (2008)
[26] P. Seidel Exotic iterated Dehn twists · Zbl 1333.53126
[27] N. J. A. Sloane The online encyclopedia of integer sequences 2015 http://oeis.org
[28] Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. 69 ((2)) pp 327– (1959) · Zbl 0089.18201 · doi:10.2307/1970186
[29] Wu, Exact Lagrangians in An-surface singularities, Math. Ann. 359 pp 153– (2014) · Zbl 1315.53099 · doi:10.1007/s00208-013-0993-3
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