Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold
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| Format: | Preprint |
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2026
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| _version_ | 1866918428233695232 |
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| author | Chow, Bennett Freedman, Michael |
| author_facet | Chow, Bennett Freedman, Michael |
| contents | Let \(M\) be a closed connected oriented topological \(4\)-manifold. We prove that if \(F_1,\dots,F_r\subset M\) are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera \(g_i\), same-sign twisted normal Euler numbers \(e_i\), and \( [F_1]+\cdots+[F_r]=0\in H_2(M;\F_2), \) then the normal-Euler excess \( \sum_{i=1}^r \bigl(\abs{e_i}-2g_i\bigr) \) is bounded above by a constant depending only on \(M\). Thus same-sign mod-\(2\)-null families of disjoint nonorientable surfaces in a fixed ambient \(4\)-manifold have uniformly bounded total excess over Massey's \(S^4\) bound.
The proof combines a tubing construction with the signature and Euler-characteristic formulas for \(2\)-fold branched covers. As corollaries, every closed oriented topological \(4\)-manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of \(\RP^2\) with \(\abs{e}>2\), and only finitely many pairwise disjoint tubular neighborhoods modeled on real \(2\)-plane bundles over \(\RP^2\) whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than \(2\). When \(M\) is a homology \(4\)-sphere, the ambient error term vanishes, and the theorem recovers Massey's sharp inequality \(\abs{e(F)}\le 2g(F)\) for nonorientable surfaces in \(S^4\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_03812 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold Chow, Bennett Freedman, Michael Geometric Topology Differential Geometry Let \(M\) be a closed connected oriented topological \(4\)-manifold. We prove that if \(F_1,\dots,F_r\subset M\) are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera \(g_i\), same-sign twisted normal Euler numbers \(e_i\), and \( [F_1]+\cdots+[F_r]=0\in H_2(M;\F_2), \) then the normal-Euler excess \( \sum_{i=1}^r \bigl(\abs{e_i}-2g_i\bigr) \) is bounded above by a constant depending only on \(M\). Thus same-sign mod-\(2\)-null families of disjoint nonorientable surfaces in a fixed ambient \(4\)-manifold have uniformly bounded total excess over Massey's \(S^4\) bound. The proof combines a tubing construction with the signature and Euler-characteristic formulas for \(2\)-fold branched covers. As corollaries, every closed oriented topological \(4\)-manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of \(\RP^2\) with \(\abs{e}>2\), and only finitely many pairwise disjoint tubular neighborhoods modeled on real \(2\)-plane bundles over \(\RP^2\) whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than \(2\). When \(M\) is a homology \(4\)-sphere, the ambient error term vanishes, and the theorem recovers Massey's sharp inequality \(\abs{e(F)}\le 2g(F)\) for nonorientable surfaces in \(S^4\). |
| title | Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2604.03812 |