Examples of exceptional homomorphisms which have non-trivial Euler numbers
Let \(G^ r\) denote the group of orientation-preserving \(C^ r\)- diffeomorphisms of the unit circle \(S^ 1\), \(r=0,1,...,\infty\), given the \(C^ r\)-topology. \(S^ 1\) acts on itself by rotations and so includes in \(G^ r\), inducing a map of classifying spaces \(BS^ 1\to BG^ r\) which is a homotopy equivalence. Since \(BS^ 1\) is a K(\({\mathbb{Z}},2)\), \(BG^ r\) has a fundamental cohomology class \(E\in H^ 2(BG^ r).\) Now let \(\Sigma\) be an oriented, closed surface of genus \(\geq 2\), and suppose that \(\pi_ 1\Sigma\) acts on \(S^ 1\), i.e., there is a homomorphism \(\phi: \pi\) \({}_ 1\Sigma \to G^ r\). This induces a map \(B\phi: B\pi\) \({}_ 1\Sigma \to BG^ r\), hence a class \((B\phi)^*(E)\in H^ 2(B\pi_ 1\Sigma)\). Now \(\Sigma\) may be identified with \(B\pi_ 1\Sigma\). Hence, we can evaluate \((B\phi)^*(E)\) on the fundamental class [\(\Sigma\) ], obtaining the Euler number eu(\(\phi\))\(\in {\mathbb{Z}}\). \textit{J. Milnor} [Comment. Math. Helv. 32, 215-223 (1958; Zbl 0196.251)] and \textit{J. Wood} [ibid. 46, 257-273 (1971; Zbl 0217.492)] show that \(| eu(\phi)| \leq | \chi (\Sigma)|.\) Special conditions on \(\phi\) produce sharper results. Henceforth, assume that \(S^ 1\) contains a Cantor set C each of whose points has \(\phi\)- orbit dense in \(S^ 1\). Such a \(\phi\) is said to be of ``type EMS''. \textit{E. Ghys} [Topology 26, 93-105 (1987; Zbl 0607.57020)] has shown that if \(\phi\) has type EMS and \(r\geq 2\), then \(| eu(\phi)| \leq | \chi (\Sigma)| -1.\) In the present paper, the author shows that the Ghys inequality is best possible for surfaces of genus 2 and almost best possible for surfaces of larger genus. More precisely, he constructs homomorphisms \(\phi: \pi\) \({}_ 1\Sigma \to G^{\infty}\) of type EMS and \(eu(\phi)=m\), for any m satisfying (a) \(| m| \leq 1=| \chi (\Sigma)| -1\) if genus \(\Sigma =2\), (2) \(| m| \leq | \chi (\Sigma)| -2\), if genus \(\Sigma >2.\) He applies Milnor's algorithm for computing eu(\(\phi\)) to certain groups of piecewise affine homeomorphisms of \({\mathbb{R}}\). Unfortunately, one important ingredient in the constructions appears in an IHES preprint, which may not be readily accessible.
scientific article; zbMATH DE number 4213025