A second cohomology class of the symplectomorphism group with the discrete topology
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, where \(\imath:K\hookrightarrow G\) is the inclusion. Also, it is shown that the homomorphism \(\pi^*:H^2(G,\mathbb R)\to H^2(E^\sigma M)\) is injective and the equality \(\pi^*\sigma=c_1(E^\sigma TM)+\kappa\|\omega\|\) holds in \(H^2(E^\sigma M)\), where \(E^\sigma M=EG^\sigma\times_{G^\sigma}M\to BG^\sigma\) with the universal \(G^\sigma\)-bundle \(EG^\sigma\to BG^\sigma\) over the classifying space \(BG^\sigma\) of \(G^\sigma\).
scientific article; zbMATH DE number 6828003
scientific article