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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.10799 |
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| _version_ | 1866912686340571136 |
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| author | Farsani, Mostafa Khosravi |
| author_facet | Farsani, Mostafa Khosravi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10799 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A determinant-line and degree obstruction to foliation transversality Farsani, Mostafa Khosravi Geometric Topology 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. |
| title | A determinant-line and degree obstruction to foliation transversality |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2509.10799 |