Exotic iterated Dehn twists
The author considers an exotic \(n\)-sphere \(L\), with \(n\geq 7\), and denotes by \(M\) its cotangent bundle \(T^*L\). This new manifold, endowed with the standard symplectic structure \(\omega_M=d\theta_M,\) is a Liouville manifold, modelled at infinity on the positive part of \(S^*L,\) the symplectization of the contact sphere bundle. The group of symplectic automorphisms \(\phi,\) which are of contact type at infinity is denoted by Symp\((M)\), the cotangent fiber at a fixed point \(q\) is denoted by \(F\), and the corresponding sphere bundle by \(K\). The author provide a Floer-theoretic proof for the fact that if Symp\(^K(M)\) is the subgroup of those symplectic automorphisms \(\phi,\) such that \(F\) and \(\phi(F)\) agree outside a compact subset, including their orientations, then on an exotic \(n\)-sphere with \(n\geq7\), the fundamental group of Symp\((M)\) contains an element of infinite order. The paper is divided in 10 sections, some of them consisting in background material for homotopy spheres, cotangent bundles, topological and symplectic invariants, Dehn twists, Weinstein handles, Lefschetz fibrations, and monodromy. The other two sections are devoted to some constructions of elements of Symp\(^K(M)\), one of them via geodesic flow, and the other one based on Lefschetz fibrations and monodromy. These constructions follow a certain interpretation of the square of a Dehn twist on \(T^*S^n\).
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