Exotic spheres and the topology of symplectomorphism groups. (English) Zbl 1325.53104

Rizell, Georgios Dimitroglou; Evans, Jonathan David

J. Topol. 8, No. 2, 586-602 (2015).

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 \(T^{*}(S^{n} \times S^{1})\) .

Reviewer: Saibal Ganguli (Allahabad)

MSC:

53D12	Lagrangian submanifolds; Maslov index
53D35	Global theory of symplectic and contact manifolds

Software:

OEIS

References:

[1]	Abouzaid, Framed bordism and Lagrangian embeddings of exotic spheres, Ann. of Math. 175 ((2)) pp 71– (2012) · Zbl 1244.53089
[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
[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
[7]	Burghelea, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 pp 1– (1974) · Zbl 0296.58003
[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
[9]	Crowley, The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature, Geom. Topol. 17 pp 1773– (2013) · Zbl 1285.57015
[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
[12]	Evans, Symplectic mapping class groups of some Stein and rational surfaces, J. Symplectic Geom. 9 pp 45– (2011) · Zbl 1242.58004
[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
[15]	Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 pp 47– (1961) · Zbl 0102.38603
[16]	Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 pp 242– (1959) · Zbl 0113.17202
[17]	Kervaire, Groups of homotopy spheres. I, Ann. of Math. 77 pp 504– (1963) · Zbl 0115.40505
[18]	Lees, On the classification of Lagrange immersions, Duke Math. J. 43 pp 217– (1976) · Zbl 0329.58006
[19]	Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. 76 ((2)) pp 531– (1962) · Zbl 0106.37102
[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
[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
[24]	Seidel, A long exact sequence for symplectic Floer cohomology, Topology 42 pp 1003– (2003) · Zbl 1032.57035
[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
[29]	Wu, Exact Lagrangians in An-surface singularities, Math. Ann. 359 pp 153– (2014) · Zbl 1315.53099