Rationality of piecewise linear foliations and homology of the group of piecewise linear homeomorphisms (Q1803627)

The author studies the discrete Godbillon-Vey class \(GV({\mathcal F}) \in H^ 3(M;\mathbb{R})\), where \({\mathcal F}\) is a transversely piecewise linear foliation of codimension one on a \(3k\)-manifold. This invariant was introduced by \textit{E. Ghys} and \textit{V. Sergiescu} [Comment. Math. Helv. 62, 185-239 (1987; Zbl 0647.58009)]. This class leads to the discontinuous invariant of \textit{S. Morita} [Foliations, Proc. Symp., Tokyo 1983, Adv. Stud. Pure Math. 5, 169-193 (1985; Zbl 0678.57011)], an element \(GV_ k({\mathcal F})\in \mathbb{R} \wedge_ \mathbb{Q} \cdots \wedge_ \mathbb{Q} \mathbb{R}\) (the \(k\)-fold wedge over \(\mathbb{Q})\). For \(k \geq 2\), the author proves that \(GV_ k({\mathcal F})=0\), implying a certain ``rationality'' for \(GV({\mathcal F})\). In particular, if \(M=S^ 3 \times S^ 3\), \[ GV({\mathcal F})=(a,b) \in H^ 3(S^ 3 \times S^ 3;\mathbb{R})=\mathbb{R} \oplus \mathbb{R} \] satisfies \(a/b \in \mathbb{Q} \cup \{\infty\}\). The rest of the paper is a study of the homology of the group \(PL_ c(\mathbb{R})\) of compactly supported, piecewise linear homeomorphisms of \(\mathbb{R}\). It is shown that the *-product is not graded commutative, which also insures the rationality.