Transversely piecewise linear foliation by planes and cylinders; PL version of a theorem of E. Ghys (Q1185408)

Let \(\Sigma\) be a closed oriented surface of genus \(\geq 2\) and \(p: E\to\Sigma\) an oriented \(S^ 1\)-bundle over \(\Sigma\). Assume that there exists a codimension-one foliation \(\mathcal F\) transverse to each fiber of \(E\). If \(\mathcal F\) is a transversely piecewise linear foliation and has an exceptional minimal set, then \(| \text{eu}(E)| < | \chi(\Sigma)|\) with \(\text{eu}(E)\) being the Euler number of \(E\).