A determinant-line and degree obstruction to foliation transversality
Mostafa Khosravi Farsani
TL;DR
This work identifies two sharp obstructions to keeping a closed embedded submanifold S transverse to a foliation F given by a submersion pi: M^{\ell+n}\to B^n. The determinant-line obstruction provides a canonical, transverse-tangent locus Z with $[Z]=\mathrm{PD}(w_{1}(\mathcal{L}))$ in $H_{n-1}(S;\mathbb{Z}_2)$ for the line bundle \mathcal{L}=\det(TS)^*\otimes\det(\nu\mathcal{F})|_S$, yielding the parity statement when $n=1$. The twisted-degree obstruction uses the orientation local system \mathcal{O}_{B} to show that if $f=\pi|_{S}$ satisfies $f_{*}[S]_{f^{*}\mathcal{O}_{B}}=0$ in $H_n(B;\mathcal{O}_{B})$, then S must be tangent somewhere; this extends the orientable covering-space argument to nonorientable bases via $w_{1}(\mathcal{L})$, and a local smoothing argument ensures these conclusions hold for $C^{1}$ foliations. The paper also applies these obstructions to translation surfaces and billiards, including periodic directions and Zemlyakov–Katok unfoldings, illustrating tangency phenomena in cone singularity settings and rational polygons. Overall, the results provide canonical, computable criteria for when transversality cannot be achieved and highlight the role of the first Stiefel–Whitney class in nonorientable contexts, with potential implications for foliation theory and planar billiard dynamics.
Abstract
Let pi: M^{ell+n} -> B^n be a submersion that presents a regular foliation by its fibers, and let S^n subset M be a closed embedded complementary submanifold, with f = pi|S: S -> B. We give two concise obstructions to keeping S everywhere transverse. (A) Determinant-line obstruction: with L = det(TS)^* tensor f^* det(TB) -> S, a C^1-small perturbation makes the tangency locus Z = {det(df) = 0} subset S a closed (n-1)-dimensional submanifold whose mod 2 fundamental class equals PD(w1(L)) in H{n-1}(S; Z_2). In particular, when n = 1 the set of tangencies is finite and the parity of #Z equals the pairing <w1(L), [S]> mod 2. (B) Twisted homology/degree obstruction: if pi is proper with connected fibers and f_[S]_{f^ O_B} = 0 in H_n(B; O_B) (top homology with the orientation local system), then S must be tangent somewhere. These recover the covering-space argument in the orientable case and extend to nonorientable settings via w1(L). We also give short applications beyond the classical degree test, including the case H_n(B; O_B) = 0 and a nonorientable base with vanishing top homology.