A determinant-line and degree obstruction to foliation transversality

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.