Exotic iterated Dehn twists
type: Item
instance of: scholarly article
title: Exotic iterated Dehn twists
zbMATH Open document ID: 1333.53126
author: Paul Seidel
published in: Algebraic \& Geometric Topology
publication date: 2015-01-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1305.4615
review text: 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 \(T^*S^n\).
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 57S05
zbMATH DE Number: 6390551
zbMATH Keywords: Dehn twist
zbMATH Keywords: cotangent bundle
zbMATH Keywords: exotic sphere
zbMATH Keywords: homotopy sphere
zbMATH Keywords: Lefschetz fibrations
zbMATH Keywords: monodromy
reviewed by: S. L. Druţă-Romaniuc
MaRDI profile type: Publication
OpenAlex ID: W2092831051
arXiv ID: 1305.4615
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DOI: 10.2140/AGT.2014.14.3305
description: scientific article