A second cohomology class of the symplectomorphism group with the discrete topology
type: Item
instance of: scholarly article
title: A second cohomology class of the symplectomorphism group with the discrete topology
zbMATH Open document ID: 1385.57027
author: Ryoji Kasagawa
published in: Algebraic \& Geometric Topology
publication date: 2018-01-22T00:00:00Z
review text: If \((M,\omega)\) is a connected, symplectic manifold of dimension \(2n\) and \(G=\text{Symp}(M,\omega)\) is its symplectomorphism group, then \({\mathcal P}\to M\) is the principal symplectic frame bundle for \((M,\omega)\). If \({\mathcal J}\) is the quotient space \({\mathcal P}/U(n)\), then the projection \(p:{\mathcal J}\to M={\mathcal P}/\text{Sp}(2n,\mathbb R)\) is a fiber bundle over \(M\) with fiber \(\text{Sp}(2n,\mathbb R)/U(n)\) which is identified with the Siegel upper half space and is contractible. In this paper, the author considers a symplectic manifold \((M,\omega)\) of dimension \(2n\) with first Chern class \(c_1(M)\) proportional to the class \([\omega]\) of the symplectic form \(\omega\), that is, the equality \(c_1(M)+\kappa[\omega]=0\) is satisfied in \(H^2(M;\mathbb R)\) for some \(\kappa\in\mathbb R\). A second cohomology class \(\sigma\in H^2(M;\mathbb R)\) in the group cohomology of the symplectomorphism group \(G=\text{Symp}(M,\omega)\) with values in \(\mathbb R\) is defined by means of a fiber bundle \(p:{\mathcal J}\to M\) over \(M\). If \(BG^\delta\) is the classifying space of \(G^\delta\), which is \(G\) with the discrete topology, then the group cohomology \(H^*(G,\mathbb R)\) is identified with the singular cohomology \(H^*(BG^\delta;\mathbb R)\) of \(BG^\delta\). The identity map of \(G\) induces a continuous map \(BG^\delta\to BG\) and a homomorphism \(\imath^*:H^*(BG;\mathbb R)\to H^*(BG^\delta;\mathbb R)\). If \(j\in{\mathcal J}\) and \(J\) is any positive compatible almost complex structure on \((M,\omega)\), then \(G_j\) and \(G_J\) are subgroups of \(G\) consisting of all elements which preserve \(j\) and \(J\), respectively. Among other properties, the author shows that if \(K\) is any subgroup of \(G_j\) for some \(j\) or of \(G_J\) for some \(J\), \(M\) is closed, and \(K\subset G_J\), then the restriction of the class \(\sigma\) to \(K\) vanishes, that is, the equality \(\imath^*\sigma=0\) in \(H^2(K,\mathbb R)\) holds…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 58H10
zbMATH DE Number: 6828003
zbMATH Keywords: symplectomorphism group
zbMATH Keywords: simplicial manifold
zbMATH Keywords: characteristic class
MaRDI profile type: Publication
full work available at URL: agt.2018.18.187
OpenAlex ID: W2785682060
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DOI: 10.2140/AGT.2018.18.187
description: scientific article; zbMATH DE number 6828003