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Continuous variation of the discrete Godbillion-Vey invariant. (English) Zbl 0786.57011

Let \(PL_ +(S^ 1)\) be the group of orientation preserving piecewise linear homeomorphisms of \(S^ 1\) and \(\pi_ 1(\Sigma_ g)\) the fundamental group of the closed oriented surface of genus \(g\) \((g\geq 1)\). For each homomorphism \(\varphi: \pi_ 1(\Sigma_ g)\to PL_ +(S^ 1)\) two invariants are known. One is the Euler number \(eu(\varphi)\in\mathbb{Z}\), and the other is the discrete Godbillon-Vey invariant \(\overline {GV}(\varphi)\in\mathbb{R}\) which was defined by Ghys and Sergiescu. It is well-known that the Euler number satisfies the so-called Milnor-Wood inequality \(| eu(\varphi)|\leq 2g-2\). When \(g=1\) Ghys showed that the discrete Godbillon-Vey invariant continuously varies. In this paper, the authors consider the case when \(g\geq 2\) and prove that the discrete Godbillon-vey invariant also varies from \(-\infty\) to \(\infty\). More precisely, for any integer \(m\) \((| m|\leq 2g-2)\) they construct a continuous family of representations \(\{\varphi_ t\): \(\pi_ 1(\Sigma_ g)\to PL_ +(S^ 1)\}_{t\in\mathbb{R}}\) such that \(eu(\varphi_ t)=m\) and \(\overline {GV}(\varphi_ t)=t\).
Reviewer: T.Mizutani (Urawa)

MSC:

57R32 Classifying spaces for foliations; Gelfand-Fuks cohomology
58H15 Deformations of general structures on manifolds
57R20 Characteristic classes and numbers in differential topology
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