Some remarks on foliated \(S^ 1\) bundles (Q1823520)

The author investigates flat circle bundles over a surface S, \(genus(S)>1\). These are determined by a homomorphism \(\phi\) : \(\pi_ 1(S)\to Diff^ n_+(S^ 1)\). An important invariant of \(\phi\) is the Euler class eu(\(\phi)\), and Milnor and Wood had previously shown the following theorem: \(| eu(\phi)| \leq | \chi (S)|\). The author shows the theorem: Any two homomorphisms \(\phi_ i: \pi_ 1(S)\to Diff^ n_+(S^ 1)\), \(n\geq 2\), \(i=1,2\), with \(eu(\phi_ 1)=eu(\phi_ 2)=\pm \chi (S)\) are topologically conjugate. He also gives a new, elegant proof of a theorem of Goldman, that any homomorphism \(\phi\) : \(\pi_ 1(S)\to PSL(2,{\mathbb{R}})\) with \(| eu(\phi)| =| \chi (S)|\) is injective onto a discrete subgroup.